Maths new edition

["5.6 Chapter 5 Case 1:\u2002 When the sum and the product of the roots of a quadratic equation are both positive, then each root is positive. For example, if the sum of the roots is 5 and the product of the roots is 6, then the roots are 2 and 3. Case 2:\u2002 When the sum of the roots is positive and the product of the roots is negative, the root having the greater magnitude is positive and the other root is negative. For example, the sum of the roots is 6 and the product of the roots is \u221216, the roots are +8 and \u22122. Because 8 \u00d7 (\u22122) = \u221216 and +8 \u2212 2 = 6. The root with the greater magnitude, i.e., 8, has a positive sign and the root with the lesser magnitude, i.e., 2 has the negative sign. Case 3:\u2002 When the sum of the roots is negative and the product of the roots is positive, then both the roots are negative. For example, the sum of the roots is \u221210 and the product of the roots is 24, the roots are \u22126 and \u22124. \u22126 \u22124 = \u221210 and (\u22126)(\u22124) = 24. In this case, both the roots, i.e., \u22126 and \u22124 bear the negative sign. Case 4:\u2002 When the sum of the roots is negative and the product of the roots is negative, the root having the greater magnitude has the negative sign and the other root has the positive sign. For example, the sum of the roots is \u22124 and the product of the roots is \u221221, then the roots are \u22127 and +3. (\u22127) \u00d7 3 = \u221221 and \u22127 + 3 = \u22124. The root with the greater magnitude, i.e., 7, has the negative sign and the root with the lesser magnitude, i.e., 3, has the positive sign. The above information can be tabulated as follows: Sign of the Sum Sign of the Sign of the Roots of the Roots Product of the +ve Both roots are positive +ve Roots +ve One root is positive, the other is negative. \u2212ve \u2212ve Numerically larger root is positive. \u2212ve \u2212ve One root is positive and the other is negative. Numerically larger root is negative. +ve Both roots are negative.","Quadratic Expressions and Equations 5.7 Constructing the Quadratic Equation when its Roots are Given Let us say that a and b are the roots of a quadratic equation. The quadratic equation can be written as (x \u2212 a)(x \u2212 b) = 0. \u21d2 x2 \u2212 (a + b)x + ab = 0 That is, x2 \u2212 (sum of the roots) x + (product of the roots) = 0 \u21d2 x2 \u2212 \uf8eb \u2212b \uf8f6 x + c = 0. \uf8ec\uf8ed a \uf8f7\uf8f8 a Example 5.3 Solve the equation x2 \u2212 11x + 30 = 0 by using the formula. Solution Given equation is x2 \u2212 11x + 30 = 0. The roots of ax2 + bx + c = 0, are \u2212b \u00b1 b2 \u2212 4ac . 2a Here, a = 1, b = \u221211 and c = 30. That is, x = \u2212(\u221211) \u00b1 (\u221211)2 \u2212 4(1)(30) = 11 \u00b1 121 \u2212 120 2\u00d71 2 x = 11 \u00b1 1 2 x = 11 + 1 and 11 \u2212 1 \u21d2 6 and 5. 2 2 \u2234 The roots are 5 and 6. Example 5.4 Find the nature of the roots of the equations given below: (a)\t x2 \u2212 13x + 11 = 0 (b)\t 18x2 \u2212 14x + 17 = 0 (c)\t 9x2 \u2212 36x + 36 = 0 (d)\t 3x2 \u2212 5x \u2212 8 = 0 Solution (a)\t Given x2 \u2212 13x + 11 = 0. \t Comparing it with ax2 + bx + c = 0, we have a = 1, b = \u221213 and c = 11. \t Now, b2 \u2212 4ac = (\u221213)2 \u2212 4(1)(11) = 169 \u2212 44 = 125 > 0.","5.8 Chapter 5 \t \u21d2 b2 \u2212 4ac > 0 and is not a perfect square. \t \u2234 The roots are distinct and irrational. (b)\t Given 18x2 \u2212 14x + 17 = 0. \t Comparing it with ax2 + bx + c = 0, a = 18, b = \u221214 and c = 17. \t Now, b2 \u2212 4ac = (\u221214)2 \u2212 4(18)(17) = 196 \u2212 1224 = \u22121028 < 0 \u21d2 b2 \u2212 4ac < 0. \t \u2234 The roots are imaginary. (c)\t Given 9x2 \u2212 36x + 36 = 0. \u21d2 9(x2 \u2212 4x + 4) = 0 \u21d2 x2 \u2212 4x + 4 = 0 \t Comparing the above equation, with ax2 + bx + c = 0, a = 1, b = \u22124 and c = 4. \t Now, b2 \u2212 4ac = (\u22124)2 \u2212 4(1)(4) = 16 \u2212 16 = 0 \u21d2 b2 \u2212 4ac = 0 \t \u2234 The roots are real and equal. (d)\t Given 3x2 \u2212 5x \u2212 8 = 0. \t \u0007Comparing the above equation with ax2 + bx + c = 0, we get, a = 3, b = \u22125 and c = \u22128. \t Now, b2 \u2212 4ac = (\u22125)2 \u2212 4(3)(\u22128) = 25 + 96 = 121 > 0 \t \u21d2 b2 \u2212 4ac > 0 and is a perfect square. \u2234 The roots are rational and distinct. Example 5.5 1 1 \u03b12 \u03b22 If a, b are the roots of the equation x2 \u2212 lx + m = 0, then find the value of + in terms of l and m. Solution Given a, b are the roots of x2 \u2212 lx + m = 0. \u21d2 Sum of the roots =\u03b1 + \u03b2 = \u2212(\u2212l ) = l \b(1) 1 \u21d2 Product of the roots = ab = m = m\b(2) 1","Quadratic Expressions and Equations 5.9 Now, 1 + 1 = \u03b12 + \u03b22 \u03b12 \u03b22 \u03b1 2\u03b2 2 = (\u03b1 + \u03b2 )2 \u2212 2(\u03b1\u03b2 ) (\u03b1\u03b2 )2 Substituting the values of a + b and ab in the above equation, we get, 1 + 1 = l2 \u2212 2m \u03b12 \u03b22 m2 The value of 1 1 l 2 \u2212 2m . \u03b12 \u03b22 m2 \u2234 + = Example 5.6 Write a quadratic equation whose roots are 5 and 8 . 2 3 Solution Sum of the roots = 5 + 8 = 31 . 2 3 6 Product of the roots = 5 \uf8eb 8 \uf8f6 = 20 . 2 \uf8ed\uf8ec 3 \uf8f7\uf8f8 3 The required quadratic equation is, x2 \u2212 (sum of the roots)x + (product of the roots) = 0. \u21d2 x2 \u2212 \uf8eb 31\uf8f6 x + 20 = 0 \uf8ed\uf8ec 6 \uf8f8\uf8f7 3 \u21d2 6x2 \u2212 31x + 40 = 0 \u2234 A quadratic equation with roots 5 and 8 is 6x2 \u2212 31x + 40 = 0. 2 3 Example 5.7 If one root of the equation, x2 \u2212 11x + (p \u2212 3) = 0 is 3, then find the value of p and also its other root. Solution Given that 3 is one of the roots of the equation x2 \u2212 11x + p \u2212 3 = 0. \u21d2 x = 3 satisfies the given equation. \u21d2 (3)2 \u2212 11(3) + p \u2212 3 = 0 \u21d2 p = 33 + 3 \u2212 9 \u21d2 p = 27 \u2234 The value of p is 27. Since the sum of the roots of the equation is 11 and one of the roots is 3, the other root of the equation is 8.","5.10 Chapter 5 Equations which can be Reduced to Quadratic Form Example 5.8 Solve (x2 \u2212 2x)2 \u2212 23(x2 \u2212 2x) + 120 = 0. Solution Let us assume that x2 \u2212 2x = y \u21d2 The given equation reduced to a quadratic equation in y That is, y2 \u2212 23y + 120 = 0 \u21d2 y2 \u2212 15y \u2212 8y + 120 = 0 \u21d2 y(y \u2212 15) \u2212 8(y \u2212 15) = 0 \u21d2 (y \u2212 8)(y \u2212 15) = 0 \u21d2 y \u2212 8 = 0 (or) y \u2212 15 = 0 \u21d2 y = 8 (or) y = 15 But x2 \u2212 2x = y When y = 8, x2 \u2212 2x = 8 \u21d2 x2 \u2212 2x \u2212 8 = 0 \u21d2 x2 \u2212 4x + 2x \u2212 8 = 0 \u21d2 x(x \u2212 4) + 2(x \u2212 4) = 0 \u21d2 (x + 2)(x \u2212 4) = 0 \u21d2 x + 2 = 0 (or) x \u2212 4 = 0 \u21d2 x = \u22122 (or) x = 4 When y = 15, x2 \u2212 2x = 15 \u21d2 x2 \u2212 2x \u2212 15 = 0 \u21d2 x2 \u2212 5x + 3x \u2212 15 = 0 \u21d2 x(x \u2212 5) + 3(x \u2212 5) = 0 \u21d2 (x \u2212 5)(x + 3) = 0 \u21d2 x \u2212 5 = 0 (or) x + 3 = 0 \u21d2 x = 5 (or) x = \u22123 \u2234 x = \u22122, \u22123, 4 and 5 are the required solutions of the given equation. Example 5.9 Solve x + 5 + 5 \u2212 x = 4. Solution Squaring the terms on both the sides, we get","Quadratic Expressions and Equations 5.11 ( x + 5 + 5 \u2212 x )2 = 42 \u21d2 x + 5 + 5 \u2212 x + 2 (x + 5)(5 \u2212 x) = 16 \u21d2 10 + 2 25 \u2212 x2 =16 \u21d2 25 \u2212 x2 = 3 Squaring the terms on both the sides again, we get 25 \u2212 x2 = 32 \u21d2 x2 = 25 \u2212 9 \u21d2 x2 = 16 \u21d2 x = \u00b14. \u2234 -4 and 4 are the required solutions of the given equation. Reciprocal Equation Any equation of the form ax4 + bx3 + cx2 + bx + a = 0, in which the coefficients of terms equidistant from first and last are equal in magnitude, is called a reciprocal equation. This is one of the case of a reciprocal equation. This can be reduced to quadratic form by dividing by x2 on both sides and with a proper substitution. Example 5.10 Solve 3x4 \u2212 8x3 \u2212 6x2 + 8x + 3 = 0. Solution The above equation is a reciprocal equation. Dividing the equation by x2, we get 3x4 \u2212 8x3 \u2212 6x2 + 8x + 3 = 0 x2 \u21d2 3 \uf8eb x2 + 1 \uf8f6 \u2212 8 \uf8eb x \u2212 1 \uf8f6 \u2212 6 = 0 \b(1) \uf8ec\uf8ed x2 \uf8f7\uf8f8 \uf8ed\uf8ec x \uf8f7\uf8f8 Now put x \u2212\u2009 1 =y x \u2234 y2 = \uf8eb x \u2212 1 \uf8f62 \uf8ec\uf8ed x \uf8f8\uf8f7 \u21d2 y2 = x2 + 1 \u22122 x2 \u21d2 x2 + 1 = y2 + 2 x2","5.12 Chapter 5 Substituting x \u2212 1 and x2 + 1 in terms of y in the Eq. (1) we get, x x2 When y = 0, x \u2212 1 = 0 3(y2 + 2) \u2212 8(y) \u2212 6 = 0. x \u21d2 3y2 + 6 \u2212 8y \u2212 6 = 0 \u21d2 3y2 \u2212 8y = 0 \u21d2 y(3y \u2212 8) = 0 \u21d2 x2 = 1 \u21d2 x = \u00b11 When y = 8 , x\u2212 1 = 8 3 x 3 \u21d2 3x2 \u2212 3 = 8x \u21d2 3x2 \u2212 8x \u2212 3 = 0 \u21d2 3x2 \u2212 9x + x \u2212 3 = 0 \u21d2 3x(x \u2212 3) + 1(x \u2212 3) = 0 \u21d2 (3x + 1)(x \u2212 3) = 0 \u21d2x= \u2212 1 (or) x = 3 3 \u2234 x = \u00b11, \u22121 and 3 are the required solutions of the given equation. 3 Constructing a New Quadratic Equation by Changing the Roots of a Given Quadratic Equation If we are given a quadratic equation, we can build a new quadratic equation by changing the roots of the given equation as directed. For example, consider the quadratic equation ax2 + bx + c = 0, whose roots are a and b. The new equations can be constructed in the following manner: 1.\t A quadratic equation whose roots are the reciprocals of the roots of the equation ax2 + bx + c = 0, can be formed by substituting 1 for x. The new equation is a \uf8eb 1 \uf8f62 + b \uf8eb 1 \uf8f6 + c = 0, x \uf8ed\uf8ec x \uf8f7\uf8f8 \uf8ec\uf8ed x \uf8f7\uf8f8 i.e., cx2 + bx + a = 0. 2.\t A quadratic equation, whose roots are k more than the roots of the equation ax2 + bx + c = 0, is obtained by substituting (x \u2212 k) for x in the given equation. 3.\t A quadratic equation whose roots are k less than the roots of the equation ax2 + bx + c = 0 can be obtained by substituting (x + k) for x in the given equation.","Quadratic Expressions and Equations 5.13 4.\t A quadratic equation whose roots are k times the roots of the equation ax2 + bx + c = 0 can be obtained by substituting x for x in the given equation. k 5.\t \u0007A quadratic equation whose roots are 1 times the roots of the equation ax2 + bx + c = 0 k can be obtained by substituting kx for x in the given equation. Example 5.11 The roots of x2 \u2212 (a + 1)x + b2 = 0 are equal. Then choose the correct value of a, b from the following option: (a)\u2002 5, 2\t (b)\u2002 3, 4\t (c)\u20025, \u22123\t (d)\u2002 5, 4 Solution The roots of x2 \u2212 (a + 1)x + b2 = 0 are equal \u21d2 (a + 1)2 \u2212 4b2 = 0 \u21d2 a + 1 = \u00b12b From the options a = 5, b = \u22123 satisfies the above relation. Maximum or Minimum Value of a Quadratic Expression The quadratic expression ax2 + bx + c takes different values, as x takes different values. As x varies from \u2212\u221e to +\u221e (i.e., when x is real), the quadratic expression ax2 + bx + c 1.\t has the minimum value, when a > 0. 2.\t has the maximum value, when a < 0. The minimum or the maximum value of the quadratic expression ax2 + bx + c occurs at x = \u2212b and is equal to 4ac \u2212 b2 . 2a 4a When x has an imaginary value, ax2 + bx + c may have a real value or an imaginary value. For some imaginary values of x, ax2 + bx + c will be real and it may have minimum or maximum value. But such cases will be dealt in the higher stages of learning. Example 5.12 Find the value of x, to get the maximum value of \u22123x2 + 6x + 5. (a)\u20021\t (b)\u2002 5 \t (c)\u2002 \u22125 \t (d)\u20026 3 6 Solution Maximum value of a quadratic expression occurs at x = \u2212b . 2a \u21d2 For \u22123x2 + 6x + 5, maximum value occurs at x = \u22126 , i.e., 1. 2(\u22123)","5.14 Chapter 5 Example 5.13 Choose the minimum value of 2x2 + 12x \u2212 3 from the following options: 1 + 18x \u2212 3x2 (a)\u2002 \u221215 \t (b)\u2002 15 \t (c)\u2002 \u221215 \t (d)\u2002 None of these 29 28 28 Solution For the minimum value of 2x2 \u221212x + 3 , 2x2 \u2212 12x + 3 is minimum and 1 + 18x \u2212 3x2 is 1+18x \u2212 3x2 maximum. The minimum value of 2x2 \u2212 12x + 3 occurs at x = \u2212b = \u2212(\u221212) = 3. 2a 2\u00d72 The maximum value of 1 + 18x \u2212 3x2 occurs at x = \u2212b = \u221218 = 3. 2a 2 \u00d7 \u22123 Minimum value of given expression is 2(3)2 \u2212 12(3) + 3 = 18 \u2212 36 + 3 = \u221215 . 1 + 18(3) \u2212 3(3)2 55 \u2212 27 28","Quadratic Expressions and Equations 5.15 TEST YOUR CONCEPTS Very Short Answer Type Questions \t1.\t For the expression ax2 + 7x + 2 to be quadratic, the 1\t 0.\t The roots of the equation x2 + ax + b = 0 possible values of a are _____. are _____. \t2.\t The equation, ax2 + bx + c = 0 can be expressed \t11.\t x = 2 is a root of the equation x2 \u2212 5x + 6 = 0. Is as aa2 + ba + c = 0, only when \u2018a\u2019 is _____ of the the given statement true? equation. 1\t 2.\t If the equation 3x2 \u2212 2x \u2212 3 = 0 has roots a and b, \t3.\t If \u22123 and 4 are the roots of the equation (x + k) then a \u22c5 b = _____. (x \u2212 4) = 0, then the value of k is ______. 1\t 3.\t If the discriminant of the equation ax2 + bx + c \t4.\t The polynomial, 3x2 + 2x + 1 is a _____ = 0 is greater than zero, then the roots are _____. expression. \t14.\t If the roots of a quadratic equation ax2 + bx + c are \t5.\t For the equation 2x2 \u2212 3x + 5 = 0, sum of the roots complex, then b2 < _____. is ______. \t15.\t The roots of a quadratic equation ax2 + bx + c = 0 \t6.\t The quadratic equation having roots \u2212a, \u2212b c is ______. are 1 and a , then a + b + c = _____. \t7.\t A quadratic equation whose roots are 2 more than \t16.\t If the roots of a quadratic equation are equal, then the roots of the quadratic equation 2x2 + 3x + 5\u00a0=\u00a00 can be obtained by substituting _____ for x. the discriminant of the equation is ______. [(x \u2212 2)\/(x + 2)] \t17.\t For what values of b, the roots of x2 + bx \t8.\t For the expression 7x2 + bx + 4 to be quadratic, the + 9 = 0 are equal? possible values of b are _____. \t18.\t If the sum of the roots of a quadratic equation \t9.\t If (x \u2212 2)(x + 3) = 0, then the values of x is positive and product of the roots is negative, are ______. the numerically greater root has ______ sign. [positive\/negative] 1\t 9.\t If x = 1 is a solution of the quadratic equation PRACTICE QUESTIONS ax2\u00a0\u2212 bx + c = 0, then b is equal to _____. Short Answer Type Questions \t20.\t Factorize the following quadratic expressions: 2\t 4.\t For what value of m does the equation, mx2 + \t\t(a) x2 + 5x + 6 (3x\u00a0 \u2212 1)m + 2x + 5 = 0 have equal roots but of \t\t(b) x2 \u2212 5x \u2212 36 opposite sign? \t\t(c) 2x2 + 5x \u2212 18 \t25.\t Find the value of m for which the quadratic equa- 2\t 1.\t Determine the nature of the roots of the following tion, 3x2 \u2212 10x + (m \u2212 3) = 0 has roots which are equations: reciprocal to each other. \t\t(a) x2 + 2x + 4 = 0 2\t 6.\t If a, b are the roots of the equation x2 \u2212 px \t\t(b) 3x2 \u2212 10x + 3 = 0 \t\t(c) x2 \u2212 24x + 144 = 0 + q = 0, then find the equation which has a and b as its roots. b a Solve the following quadratic equations: \t22.\t If f\u2009(x) = x2 \u2212 5x \u2212 36 and g(x) = x2 + 9x + 20, then 2\t 7.\t If one root of the equation x2 \u2212 mx + n = 0 is twice the other root, then show that 2m2 = 9n. for what values of x is 2f(x) = 3g(x)? 2\t 8.\t The square of one-sixth of the number of s\u00adtudents \t23.\t Solve: 16x4 \u2212 28x2 \u2212 8 = 0 in a class are studying in the library and the","5.16 Chapter 5 \u00adremaining eight students are playing in the ground. 3\t 6.\t The roots of the equation 2x2 + 3x + c = 0 (where What is the total number of students of the class? c < 0) could be ______. \t29.\t If 2a and 3b are the roots of the equation x2 + ax \t37.\t The roots of the equation 30x2 \u2212 7 3x + 1 = 0 are + b = 0, then find the equation whose roots are a, b. \t38.\t If a and b are the roots of the quadratic equation x2 + 3x \u2212 4 = 0, then a\u22121 + b\u22121 = _____. 3\t 0.\t If a, b are the roots of the quadratic equation lx2 + mx + n = 0, then evaluate the following expressions. \t39.\t The roots of the equation x \u2212 3 + 2x =1 , where x \u2212 2 x+3 \t\t(a) a2 + b\u20092 x \u2260 2, \u22123, are \t\t(b) \u03b1 + \u03b2 \t40.\t If the roots of the quadratic equation 4x2 \u2212 16x + \u03b2\u03b1 p = 0 are real and unequal, then find the value\/s of p. \t\t(c) 1 + 1 \u03b13 \u03b23 4\t 1.\t If one root of the quadratic equation ax2 + bx + c = 0 is15 + 2 56 and a, b and c are rational, \t31.\t If the price of sugar is reduced by `1 per kg, then\u00a0find the quadratic equation. 5\u00a0 \u00adkilograms more can be purchased for `1200. What was the original price of sugar per kilogram? \t42.\t If the roots of the equation ax2 + bx + 4c = 0 are in the ratio of 3 : 4, then find the relation between a, 3\t 2.\t The zeroes of the quadratic polynomial x2 \u2212 24x + b and c. 143 are 3\t 3.\t Find the quadratic equation in x whose roots are \t43.\t For which value of p among the following, does the quadratic equation 3x2 + px + 1 = 0 have real \u22127 and 8 . roots? 2 3 3\t 4.\t If k1 and k2 are the roots of x2 \u2212 5x \u2212 24 = 0, then \t44.\t If the product of the roots of ax2 + bx + 2 = 0 find the quadratic equation whose roots are \u2212k1 is\u00a0 equal to the product of the roots of px2 + qx and \u2212k2. \u22121\u00a0= 0, then a + 2p = _____. \t35.\t The product of the roots of the equations \t45.\t Find the roots of quadratic equation ax2 + (a \u2212 b +\u00a0c) x \u2212 b + c = 0. PRACTICE QUESTIONS 1 + 1 = 1 is _____. x +1 x \u2212 2 x + 2 \t46.\t If (2x \u2212 9) is a factor of 2x2 + px \u2212 9, then p = _____. Essay Type Questions \t49.\t x \u2212 3 + 3x + 4 = 5. Solve the given equations: 5\t 0.\t 3x4 \u2212 10x3 \u2212 3x2 + 10x + 3 = 0. \t47.\t (x + 3) (x + 4) (x + 6) (x + 7) = 1120. \t48.\t (x2 + 3x)2 \u2212 16 (x2 + 3x) \u2212 36 = 0. CONCEPT APPLICATION Level 1 \t2.\t The discriminant of the equation x2 \u2212 7x + 2 = 0 is Direction for questions 1 to 20: Select the correct alternative from the given choices. \t\t(a) 47\t\t (b) 40 \t1.\t The solution of the equation x2 + x + 1 = 1 is are \t\t(c) 41\t\t (d) \u221241 \t\t(a) x = 0 \t\t(b) x = \u22121 \t3.\t Find the maximum value of the quadratic \t\t(c) Both (a) and (b) expression \u22123x2 + 7x + 4. \t\t(d) Cannot be determined","Quadratic Expressions and Equations 5.17 \t\t(a) 8 1 \t\t (b) 8 1 \t\t(a) 9x2 \u2212 2x + 7 = 0 6 12 \t\t(b) 9x2 \u2212 2x \u2212 7 = 0 \t\t(c) 9x2 + 2x \u2212 7 = 0 \t\t(c) 8 1 \t\t (d) 12 \t\t(d) 9x2 + 2x + 7 = 0 4 \t4.\t If a and b are the roots of the equation x2 + 3x \u2212\u00a02\u00a0= 0, then a2b + ab\u20092 = ? \t12.\t The number of real roots of the quadratic equation (x \u2212 4)2 + (x \u2212 5)2 + (x \u2212 6)2 = 0 is \t\t(a) \u22126 \t\t (b) \u22123 \t\t(a) 1\t\t (b) 2 \t\t(c) 6\t\t (d) 3 \t\t(c) 3\t\t (d) None of these \t5.\t If one of the roots of an equation, x2 \u2212 2x + c = 0 is thrice the other, then c = ? 1\t 3.\t The number of distinct real solutions of x 2 \u2212 5 x + 6 = 0 is \t\t(a) 1 \t\t (b) 4 23 \t\t(a) 4\t\t (b) 3 \t\t(c) \u2212 1 \t\t (d) 3 \t\t(c) 2\t\t (d) 1 2 4 1\t 4.\t The number of real solutions of x 2 \u2212 5 x + 6 \t6.\t The number of real roots of the quadratic equa- = 0 is tion 3x2 + 4 = 0 is \t\t(a) 1\t\t (b) 2 \t\t(a) 0\t\t (b) 1 \t\t(c) 3\t\t (d) 4 \t\t(c) 2\t\t (d) 4 \t15.\t In writing a quadratic equation of the form x2 +\u00a0px + q = 0, a student makes a mistake in writing \t7.\t If a and b are the roots of the equation x2 + px the coefficient of x and gets the roots as 8 and 12. +\u00a0q\u00a0= 0 then (a - b)2 = _____. Another student makes a mistake in writing the constant term and gets the roots as 7 and 3. Find \t\t(a) q2 \u2212 4p\t\t (b) 4q2 \u2212 p the correct quadratic equation. \t\t(c) p2 \u2212 4q\t\t (d) p2 + 4q \t8.\t Which of the following equations does not have \t\t(a) x2 \u2212 10x + 96 = 0 PRACTICE QUESTIONS real roots? \t\t(b) x2 \u2212 20x + 21 = 0 \t\t(c) x2 \u2212 21x + 20 = 0 \t\t(a) x2 + 4x + 4 = 0\t\t (b) x2 + 9x + 16 = 0 \t\t(d) x2 \u2212 96x + 10 = 0 \t\t(c) x2 + x + 1 = 0\t\t (d) x2 + 3x + 1 = 0 \t9.\t The sum of the roots of the equation, ax2 + bx \t16.\t The roots of the equation 6x2 \u2212 8 2x + 4 = 0 are + c = 0 where a, b and c are rational and whose one of the roots is 4 \u2212 5, is \t\t(a) 1 , 2 \t\t (b) 2 , 1 3 3 \t\t(a) 8\t\t (b) \u22122 5 \t\t(c) 2 5 \t\t (d) 11 \t\t(c) 2 , 2 \t(d) 3, 2 3 2 \t10.\t For the quadratic equation x2 + 3x \u2212 4 = 0 which of the following is a solution? 1\t 7.\t Which of the following equations has roots as a, b and c? \t\t(A) x = \u22124\t\t (B) x = 3 \t\t(a) x3 + x2(a + b + c) + x(ab + bc + ca) + abc = 0 \t\t(C) x = 1 \t\t(b) x3 + x2(a + b + c) + x(ab + bc + ca) \u2212 abc = 0 \t\t(a) A and B\t\t (b) B and C \t\t(c) x3 \u2212 x2(a + b + c) + x(ab + bc + ca) \u2212 abc = 0 \t\t(c) A and C\t\t (d) Only A \t\t(d) x3 \u2212 x2(a + b + c) \u2212 x(ab + bc + ca) \u2212 abc = 0 \t11.\t Find the quadratic equation whose roots are \t18.\t If the roots of the equation 2ax2 + (3b \u2212 9) x + 1 = reciprocals of the roots of the equation 7x2 \u2212 2x + 0 are \u22122 and 3, then the values of a and b respec- 9\u00a0=\u00a00. tively are","5.18 Chapter 5 \t\t(a) 1 , 5 \t\t (b) \u22121 , \u221253 \t25.\t The age of a father is 25 years more than his son\u2019s 12 18 12 18 age. The product of their ages is 84 in years. What will be son\u2019s age in years, after 10 years? \t\t(c) \u22121 , \u22125 \t\t (d) \u22121 , 55 \t\t(a) 3\t\t (b) 28 12 8 12 18 \t\t(c) 13\t\t (d) 18 1\t 9.\t The roots of the equation x2 + 5x + 1 = 0 are \t26.\t If the roots of the equation ax2 \u2212 bx + 5c = 0 are in \t\t(a) 5 +2 21 , 5 \u2212 21 the ratio of 4 : 5, then 2 \t\t(a) ab = 18c2 \t\t(b) \u22125 \u2212 21 , 5+ 21 \t\t(b) 81b2 = 4ac 2 2 \t\t(c) bc = a2 \t\t(d) 4b2 = 81ac \t\t(c) \u22125 + 21 , \u22125 \u2212 21 2 2 \t27.\t The speed of Uday is 5 km\/h more than that \t\t(d) \u22125 + 29 , \u22125 \u2212 29 of Subash. Subash reaches his home from office 2 2 2 hours earlier than Uday. If Subash and Uday stay 12 km and 48 km from their respective offices, 2\t 0.\t If a and b are the roots of the equation 3x2 \u2212 2x find the speed of Uday. \u2212 8 = 0, then a2 \u2212 ab + b\u20092 = _____. \t\t(a) 76 \t\t (b) 25 \t\t(a) 10 km\/h 93 \t\t(b) 4 km\/h \t\t(c) 16 \t\t (d) 32 33 \t\t(c) 9 km\/h \t21.\t If 2x2 + (2p \u2212 13) x + 2 = 0 is exactly divisible by \t\t(d) 8 km\/h x \u2212 3, then the value of p is 2\t 8.\t If the roots of the quadratic equation c(a \u2212 b)x2 + \t\t(a) \u221216 \t\t (b) 19 a(b \u2212 c)x + b(c \u2212 a) = 0 are equal, then 66 \t\t(a) 2b\u22121 = a\u22121 + c\u22121 PRACTICE QUESTIONS \t\t(c) 16 \t\t (d) \u221219 \t\t(b) 2c\u22121 = a\u22121 + b\u22121 66 \t\t(c) 2a\u22121 = b\u22121 + c\u22121 \t22.\t If x2 \u2212 ax \u2212 6 = 0 and x2 + ax \u2212 2 = 0 have one common root, then a can be _____. \t\t(d) None of these \t\t(a) \u22121\t\t (b) 2 \t29.\t If the roots of the quadratic equation x2 \u2212 3x \u2212 304 = 0 are a and b, then the quadratic equation with \t\t(c) \u22123\t\t (d) 0 roots 3a and 3b is \t23.\t The root of the equation x 2 1 + x 1 2 + \t\t(a) x2 + 9x \u2212 2736 = 0 \u2212 + \t\t(b) x2 \u2212 9x \u2212 2736 = 0 3x(x + 1) \t\t(c) x2 \u2212 9x + 2736 = 0 (x \u2212 1)(x + 2) = 0 among the following is _____. \t\t(d) x2 + 9x + 2736 = 0 \t\t(a) 2\t\t (b) 3 \t\t(c) \u22121\t\t (d) 0 \t30.\t If x2 + a1x + b1 = 0 and x2 + a2x + b2 = 0 have a common root (x \u2212 k), then find k. 2\t 4.\t If a and b are the roots of the equation, 2x2 \u2212 5x + 2 = 0, then (a \u2212 1)b\u22121 = _____. where (a > b). \t\t(a) k = \u03b12 \u2212 \u03b11 \t(b) k = \u03b12 \u2212 \u03b11 \t\t(a) 1 \t\t (b) \u2212 1 \u03b22 \u2212 \u03b21 \u03b22 \u2212 \u03b21 2 2 \t\t(c) 1 \t\t (d) 1 \t\t(c) k = \u03b12 \u2212 \u03b11 \t(d) k = \u03b12 \u2212 \u03b11 2 \u03b22 \u2212 \u03b21 \u03b22 \u2212 \u03b21","Quadratic Expressions and Equations 5.19 Level 2 \t31.\t If one of the roots of x2 + (1 + k)x + 2k = 0 is twice \t38.\t If (x + 2) is a common factor of the expressions x2 + ax \u2212 6, x2 + bx + 2 and kx2 \u2212 ax \u2212 (a + b), then the other, then a2 +b2 ______ . k = ______. ab \t\t(a) 2\t\t (b) 1 \t\t(a) 2\t\t (b) 3 \t\t(c) 4\t\t (d) 7 \t\t(c) 1\t\t (d) \u22122 \t32. \tIf \u03b1 and \u03b2 are the roots of 2x2 \u2212 x \u2212 2 = 0, then 3\t 9.\t The roots of a pure quadratic equation exists only (\u03b1\u22123 + \u03b2\u2009\u22123 + 2\u03b1\u2009\u22121\u03b2\u2009\u22121) is equal to if ______. \t\t(a) \u2212 17 \t\t (b) 23 \t\t(a) a > 0, c < 0\t (b) c > 0, a < 0 8 6 \t\t(c) a > 0, c \u2264 0\t (d) Both (a) and (b) \t\t(c) 37 \t\t 29 9 (d) \u2212 8 \t40.\t The roots of the equation x \u22121 + x \u2212 3 = 2 1 , x \u22122 x \u2212 4 3 \t33.\t In a right angled triangle, one of the perpendicular (where x \u2260 2, 4) are sides is 4 cm greater than the other and 4 cm lesser than the hypotenuse. Find the area of triangle \t\t(a) 6 + 10, 6 \u2212 10 in cm2. \t\t(b) 6 + 2 10, 6 \u2212 10 \t\t(a) 72\t\t (b) 48 \t\t(c) 36\t\t (d) 96 \t\t(c) 6 + 6 10, 6 \u2212 6 10 \t34.\t In a fraction, the denominator is 1 less than \t\t(d) 2 + 2 10, 6 \u2212 2 10 the numerator. The sum of the fraction and its 1 reciprocal is 2 56 . Find the fraction. 4\t 1.\t If x + 3 is the common factor of the expres- sions ax2 + bx + 1 and px2 + qx \u22123, then \t\t(a) 3 \t\t (b) 13 \u2212 (9a + 3p) = ______ . 2 12 3b + q \t\t(c) 18 \t\t (d) 8 \t\t(a) \u22122\t\t (b) 2 PRACTICE QUESTIONS 17 7 \t\t(c) 3\t\t (d) \u22121 \t35.\t The length of the rectangular surface of a table is \t42.\t If the sum of the roots of an equation x2 + px + 1 = 0 (p > 0) is twice the difference between them, 10 m more than its breadth. If the area of the sur- then p = ______. face is 96 m2, its perimeter is (in m) _______. \t\t(a) 64\t\t (b) 44 \t\t(a) 1 \t\t (b) 3 4 4 \t\t(c) 52\t\t (d) 48 \u2212 \t36.\t If \u03b1 and \u03b2 are the roots of the equation x2 + 9x \t\t(d) 4 \t\t (d) 3 + 18 = 0, then the quadratic equation having the 3 2 roots \u03b1 + \u03b2 and \u03b1 \u2212 \u03b2 is ______, where (\u03b1 > \u03b2). \t\t(a) x2 + 6x \u2212 27 = 0\u2003\u2003(b) x2 \u2212 9x + 27 = 0 \t43.\t The equation x + 5 = 3 + 5 has \u2212 \u2212 \t\t(c) x2 \u2212 9x + 7 = 0\u2003\u2003\u2009\u2009\u2009\u2009(d) x2 + 6x + 27 = 0 3 x 3 x 3\t 7.\t Find the minimum value of the quadratic \t\t(a) no real root. expression 4x2 \u2212 3x + 4. \t\t(b) one real root. \t\t(a) \u221255 \t\t (b) 55 \t\t(c) two equal roots. 16 16 \t\t(d) infinite roots. \t\t(c) 16 \t\t (d) 161 \t44.\t The roots of the equation 1 3 \u2212 1 5 = 8 are 15 22 2x \u2212 2x +","5.20 Chapter 5 \t\t(a) 2 \u2212 1 , 2 \u2212 1 4\t 7.\t The roots of x2 \u2212 2x \u2212 1 = 0 are ______. 22 \t\t(a) 2 + 1, 2 \u2212 1 \t\t(b) \u22121 + 17 , \u22121 \u2212 1 17 \t\t(b) 1, 2 2 2 \t\t(c) 1 + 2, 1 \u2212 2 \t\t(c) 2 + 1 , 2 \u2212 1 22 \t\t(d) 2, 1 1 + 2 17 , 1 \u2212 17 4\t 8.\t If 2x2 + 4x \u2212 k = 0 is same as (x \u2212 5) \uf8eb x + k \uf8f6 = 0, 2 \uf8ed\uf8ec 10 \uf8f8\uf8f7 \t\t(d) then find the value of k. \t45.\t If the quadratic expression x2 + (a \u2212 4)x + (a + 4) \t\t(a) 100\t\t (b) 90 is a perfect square, then a = ______. \t\t(c) 70\t\t (d) 35 \t\t(a) 0 and \u22124\t (b) 0 and 6 \t49.\t If the numerically smaller root of x2 + mx = 2 \t\t(c) 0 and 12\t\t (d) 6 and 12 is 3 more than the other one, find the value of m. \t46.\t The minimum value of 2x2 \u2212 3x + 2 is ______. \t\t(a) 7 \t(b) 4 \t (c) 4\t (d) \u22123 \t\t(a) \u22121\t\t (b) 1 8 7 \t\t(c) \u22122 \t\t (d) 2 Level 3 \t50.\t Two persons A and B solved a quadratic equation the number are in the office and the remaining six of the form x2 + bx + c = 0. A made a mistake in employees are on leave. What is the number of noting down the coefficient of x and obtained the employees in the group? roots as 18 and 2, where as B obtained the roots as \u22129 and \u22123 by misreading the constant term. The \t\t(a) 49\t\t (b) 64 correct roots of the equation are \t\t(c) 36\t\t (d) 100 \t\t(a) \u22126, \u22123\t\t (b) \u22126, 6 PRACTICE QUESTIONS 5\t 5.\t Find the quadratic equation whose roots are 2 \t\t(c) \u22126, \u22125\t\t (d) \u22126, \u22126 times the roots of x2 \u2212 12x \u2212 13 = 0. 5\t 1.\t If a and b are the roots of x2 \u2212 x + 2 = 0, then find \t\t(a) x2 \u2212 24x \u2212 52 = 0 the value of (a\u22126 + b\u2009\u22126 + 2a\u2009\u22123b\u22123)a6b6. \t\t(b) x2 \u2212 24x \u2212 26 = 0 \t\t(c) x2 \u2212 14x \u2212 15 = 0 \t\t(a) 16\t\t (b) 25 \t\t(d) None of these \t\t(c) 30\t\t (d) 36 \t56.\t If one of the roots of ax2 + bx + c = 0 is thrice that 5\t 2.\t If b1, b2, b3, \u2026, bn are positive, then the least value of the other root, then b can be of (b1 + b2 + b3 + \u2219\u2219\u2219 + bn) \uf8eb 1 + 1 + + 1 \uf8f6 is \t\t(a) 4ac \t\t (b) 16ac \uf8ec b1 b2 bn \uf8f7 3 9 \uf8ed \uf8f8 \t\t(a) b1b2 \u2219\u2219\u2219 bn\t(b) n2 + 1 \t\t(c) 4 ac \t\t (d) 4ac 3 3 \t\t(c) n(n + 1)\t\t (d) n2 \t53.\t The equation x +1 \u2212 4x \u22121 = x \u2212 1 has \t57.\t If \u03b1, \u03b2 are the roots of px2 + qx + r = 0, then \u03b13 + \u03b2\u20093 = _____. \t\t(a) no solution.\t (b) one solution. \t\t(c) two solutions.\t (d) more than two solutions. \t\t(a) 3qpr \u2212 q3 \t(b) 3pqr \u2212 3q p3 p3 5\t 4.\t Out of the group of employees, twice the square root of the number of the employees are on a trip \t\t(c) pqr \u2212 3q \t(d) 3pqr \u2212 q to attend a conference held by the company, half p3 p3","Quadratic Expressions and Equations 5.21 \t58.\t If \u03b1 and \u03b2 are the roots of x2 \u2212 (a + 1) x + 1 (a2 + \t\t(a) ax2 + (ab \u2212 ac)x \u2212 c = 0 2 \t\t(b) ax2 + (b \u2212 c)x \u2212 bc = 0 a + 1) = 0 then \u03b1\u20092 + \u03b2\u20092 = ______. \t\t(c) a2x2 + (b \u2212 c)x \u2212 ac = 0 \t\t(d) a2x2 + (ab \u2212 ac)x \u2212 bc = 0 \t\t(a) a\t\t (b) a2 \t \t (c) 2a\t\t (d) 1 \t66.\t Ramu swims a distance of 3 km each upstream and downstream. The total time taken is one hour. If \t59.\t The number of roots of the equation the speed of the stream is 4 km\/h, then find the 2 x 2 \u22127 x +6 =0 speed of Ramu in still water. \t\t(a) 4\t\t (b) 3 \t\t(c) 2\t\t (d) 1 \t\t(a) 12 km\/h\t\t (b) 9 km\/h \t60.\t In a quadratic equation ax2 \u2212 bx + c = 0, a, b, c \t\t(c) 8 km\/h\t\t (d) 6 km\/h are distinct primes and the product of the sum of \t67.\t In solving a quadratic equation x2 + px + q = 0 a student made a mistake in copying the coefficient the roots and product of the roots is ro991ot.s Find the of x and obtained the roots as 4, \u22123 but one of the difference between the sum of the and the actual roots is 2 what is the difference between the actual and wrong values of the coefficients of x? product of the roots. \t\t(a) 2\t\t (b) 3 \t\t(a) 5\t\t (b) 4 \t\t(c) 4\t\t (d) Cannot be determined \t\t(c) 7\t\t (d) +6 \t61.\t Maximum value of 2 +12x \u2212 3x2 is _____. \t68.\t The roots of ax2 \u2212 bx + 2c = 0 are in the ratio of 2x2 \u2212 8x + 9 2 : 3, then _____. \t\t(a) 14\t\t (b) 17 \t\t(a) a2 = bc\t\t (b) 3b2 = 25ac \t\t(c) 2b2 = 75c\t (d) 5b2 = ac \t\t(c) 11\t\t (d) Cannot be determined 6\t 9.\t If the roots of 9x2 \u2212 2x + 7 = 0 are 2 more than the 6\t 2.\t If x2 + ax + b and x2 + bx + c have a common factor roots of ax2 + bx + c = 0, then 4a \u2212 2b + c can be (x \u2212 k), then k = _______. a \u2212 b b\u2212c \t\t(a) \u22122\t\t (b) 7 PRACTICE QUESTIONS b \u2212 c c \u2212a \t\t(a) \t\t (b) \t\t(c) 9\t\t (d) 10 \t\t(c) c \u2212b \t\t (d) c\u2212b \t70.\t If the roots of ax2 + bx + c = 0 are 2 more than b \u2212a a\u2212b the roots of px2 + qx + r = 0, then the value of c in terms of p, q and r is \t63.\t If 9x \u2212 3y + z = 0, then the value of y y2 \u2212 4xz \t\t(a) p + q + r\t\t (b) 4p \u2212 2q + r 2x 4x2 + \t\t(c) 3p \u2212 q + 2r\t (d) 2p + q \u2212 r (where x, y, z are constants). \t71.\t If the roots of 2x2 + 7x + 5 = 0 are the reciprocal roots of ax2 + bx + c = 0, then a \u2212 c = _____. \t\t(a) 9\t\t (b) 2 \t\t(c) 3\t\t (d) 6 \t\t(a) 3\t\t (b) \u22123 \t\t(c) \u22122\t\t (d) \u22125 6\t 4.\t If the roots of 3x2 \u2212 12x + k = 0 are complex, then 7\t 2.\t If the roots of the equation ax2 + bx + c =0 is 1 find the range of k. k times the roots of px2 + qx + r = 0, then which of \t\t(a) k < 22\t\t (b) k < \u221210 the following is true? \t\t(c) k > 11\t\t (d) k > 12 \t\t(a) a = pk\t\t (b) a = p \t\t(c) aq = pbk\t\t b q 6\t 5.\t If a, b are the roots of ax2 + bx + c = 0, then find the quadratic equation whose roots are a + b, ab. (d) ab = pqk2","5.22 Chapter 5 TEST YOUR CONCEPTS Very Short Answer Type Questions \t1.\t non-zero real numbers \t11.\t Yes \t12.\t \u22121 \t2.\t root \t13.\t real and distinct \t14.\t 4ac \t3.\t 3 1\t 5.\t Zero 1\t 6.\t zero \t4.\t quadratic \t17.\t \u00b16 \t18.\t positive \t5.\t 3 1\t 9.\t a + c 2 \t6.\t x2 + x(a + b) + ab = 0 \t7.\t (x \u2212 2) \t8.\t real numbers \t9.\t 2, \u22123 1\t 0.\t \u2212a \u00b1 a2 \u2212 4b 2 Short Answer Type Questions \t20.\t (a) (x + 2)(x + 3) \t32.\t 11, 13 \t\t(b) (x \u2212 9)(x + 4) \t\t(c) (x \u2212 2)(2x + 9) 3\t 3.\t 6x2 + 5x \u2212 59 = \u22123 \t21.\t (a) Complex conjugates \t\t(b) Real and distinct 3\t 4.\t x2 + 5x \u2212 24 = 0 \t\t(c) Real and equal \t22.\t \u22124 and \u221233 3\t 5.\t zero 3\t 6.\t rational or irrational, but unequal 3\t 7.\t 1 , 1 23 53 \t23.\t \u00b1 2 3\t 8.\t 3 4 \t24.\t \u22122 3 \t39.\t 3, \u22121 2 ANSWER KEYS \t25.\t 6 \t40.\t p < 16 \t26.\t qx2 \u2212 (p2 \u2212 2q)x + q = 0 \t27.\t 8 km\/h 4\t 1.\t x2 \u2212 30x + 1 = 0 \t28.\t 12 or 24 2\t 9.\t x2 \u2212 (6ab \u2212 2a \u2212 3b)x \u2212 (2a + 3b)(6ab) = 0 \t42.\t 3b2 = 49ac 4\t 3.\t 4 \t30.\t (a) m2 \u2212 2ln \u2003\u2002(b) m2 \u2212 2ln \u2002\u2003 (c) 3lmn \u2212 m3 4\t 4.\t 0 b c l2 ln n3 \t45.\u2002 \u22121, \u2212 a \t31.\t `16 \t46.\t \u22127 Essay Type Questions \t49.\t 4 \t47.\t x = 1, x = \u221211 4\t 8.\t \u22121, \u22122, 3 and \u22126 \t50.\t 1 \u00b1 37 , 3 \u00b1 13 2 6","Quadratic Expressions and Equations 5.23 CONCEPT APPLICATION Level 1 \t 1.\u2002(c)\t 2.\u2002(c)\t 3.\u2002(b)\t 4.\u2002(c)\t 5.\u2002(d)\t 6.\u2002(a)\t 7.\u2002(c)\t 8.\u2002(c)\t 9.\u2002(a)\t 10.\u2002 (c) \t11.\u2002 (a)\t 12.\u2002 (d)\t 13.\u2002 (c)\t 14.\u2002 (d)\t 15.\u2002 (a)\t 16.\u2002 (c)\t 17.\u2002 (c)\t 18.\u2002 (d)\t 19.\u2002 (c)\t 20.\u2002 (a) \t21.\u2002 (b)\t 22.\u2002 (a)\t 23.\u2002 (c)\t 24.\u2002 (d)\t 25.\u2002 (c)\t 26.\u2002 (d)\t 27.\u2002 (d)\t 28.\u2002 (c)\t 29.\u2002 (b)\t 30.\u2002 (a) Level 2 33.\u2002 (d)\t 34.\u2002 (d)\t 35.\u2002 (b)\t 36.\u2002 (a)\t 37.\u2002 (b)\t 38.\u2002 (c)\t 39.\u2002 (d)\t 40.\u2002 (a) 43.\u2002 (a)\t 44.\u2002 (b)\t 45.\u2002 (c)\t 46.\u2002 (a)\t 47.\u2002 (c)\t 48.\u2002 (c)\t 49.\u2002 (b) \t31.\u2002 (d)\t 32.\u2002 (d)\t \t41.\u2002 (d)\t 42.\u2002 (c)\t Level 3 \t50.\u2002 (d)\t 51.\u2002 (b)\t 52.\u2002 (d)\t 53.\u2002 (a)\t 54.\u2002 (c)\t 55.\u2002 (a)\t 56.\u2002 (c)\t 57.\u2002 (a)\t 58.\u2002 (a)\t 59.\u2002 (a) \t60.\u2002 (a)\t 61.\u2002 (a)\t 62.\u2002 (d)\t 63.\u2002 (c)\t 64.\u2002 (d)\t 65.\u2002 (d)\t 66.\u2002 (c)\t 67.\u2002 (a)\t 68.\u2002 (b)\t 69.\u2002 (b) \t70.\u2002 (b)\t 71.\u2002 (a)\t 72.\u2002 (c) ANSWER KEYS","5.24 Chapter 5 CONCEPT APPLICATION Level 1 \t1.\t Simplify and factorize. \t17.\t Let (x \u2212 a)(x \u2212 b)(x \u2212 c) = 0 and expand. 1\t 8.\t Substitute x = \u22122 and x = 3 to get to equation in \t2.\t Use the formula to find the discriminant. a\u00a0and b and then solve for a and b. \t3.\t Use the formula to find the maximum value. \t4.\t Find the sum and product of the roots. Let a2b + 1\t 9.\t Apply formula x = \u2212b \u00b1 b2 \u2212 4ac . ab\u20092 = ab(a + b). 2a \t5.\t Let the roots be a and 3a. 2\t 0.\t Find ab and (a + b\u2009) and use, a2 \u2212 ab + b2 = (a + b\u2009)2 \u2212 3ab. \t6.\t Solve for x. 2\t 1.\t Substitute x = 3 in the given equation and simplify. \t7.\t (a \u2212 b)2 = (a + b)2 \u2212 4ab. 2\t 2.\t Find x in terms of a and substitute x = a in either \t8.\t Find the value of the discriminant for each of the of the equations. equations. \t9.\t If one root is 4 \u2212 5, then the other root is 2\t 3.\t Simplify and solve for x. 4 + 5, because the coefficients of the xn terms are rational. \t24.\t Factorize LHS of the given equation to find a and b. \t10.\t Solve for x. \t25.\t (i)\tForm a quadratic equation by assuming the \t11.\t The quadratic equation with reciprocals of the age of son as r aynedarsa.c The roots of quadratic equation are 1 only if sum of all the \uf8eb 1 \uf8f6 Hints and Explanation roots of the equation f\u2009(x) = 0 is f \uf8ec\uf8ed x \uf8f8\uf8f7 = 0. coefficient s = 0. 1\t 2.\t (i)\tUse the concept of perfect square of a number. \t\t(ii)\tAssume the ages of father and son as x and (25\u00a0\u2212 x) years. \t\t(ii)\tIf a2 + b2 + c2 = 0 is true only when a = b = c = 0. \t\t(iii)\tWrite the relation in terms of x according to the data and then solve the equation. \t13.\t (i)\tSolve the equation to find the number of real roots. 2\t 6.\t (i)\tUse the concept of sum and product of the roots of a quadratic equation. \t\t(ii)\tReplace |x| by y and solve for y. \t\t(iii)\tNow, x = \u00b1y. \t\t(ii)\tTake the roots as 4a, 5a. 1\t 4.\t (i)\tUse the concept |x| and find the roots. \t\t(iii)\tUsing the sum of the roots and product of the roots eliminate a. \t\t(ii)\tReplace |x| by y and solve for y. \t\t(iii)\tNow, x = \u00b1y. 2\t 7.\t (i)\tFrame the quadratic equation from the given data. 1\t 5.\t (i)\tUse the concept of sum and product of the roots of a quadratic equation. \t\t(ii)\tAssume the speed of Subhash as x and Subhash reaches his home in t hours and Uday reaches \t\t(ii)\tThe product of the roots obtained by the first his home in (t + 2) hours. student is product of the roots of the required quadratic equation. \t\t(iii)\tNow, use time = dsispteaendce, then solve the equa- tion for x. \t\t(iii)\tThe sum of the roots obtained by the second student is sum of the roots of the required qua- 2\t 8.\t (i)\tIf sum of all coefficients is zero, then 1, c are dratic equation. a 1\t 6.\t Take 2 as common and then factorize. the roots of ax2 + bx + c = 0.","Quadratic Expressions and Equations 5.25 \t\t(ii)\tIf the sum of the coefficients is 0, then 1 and \t\t(ii)\tThe quadratic equation whose roots are c a are the roots of the equation. m times of the roots of the equation f\u2009(x) \t\t(iii)\tUse product of the roots concept. =0 is \u22121 , 55 = 0. 12 18 \t29.\t (i)\tThe quadratic equation with thrice the roots \t30.\t (i)\tUse the concept of common root of given 1 equations. x of f\u2009(x) = 0 as roots is f \uf8eb \uf8f6 = 0. \t\t(ii)\tPut x = k in the given equations and solve \uf8ec\uf8ed \uf8f8\uf8f7 for k. Level 2 \t31.\t (i)\tUse the concept of sum and product of the \t\t(iii)\tFind the value of \u2018l\u2019, using l(l \u2212 10) = 96. roots of a quadratic equation. \t\t(iv)\tCalculate the perimeter of the rectangle using \t\t(ii)\tAssume the roots as a and 2a. 2(l + l \u2212 10). \t36.\t Find a + b, ab and using these values find a \u2212 b. \t\t(iii)\tFind the sum of the roots and product of the 3\t 7.\t Use the formula to find the minimum value. roots. 3\t 8.\t Substitute x = \u22122 in the first two expressions, \t\t(iv)\tFrom the above equation eliminate \u2018a\u2019. equated to zero. \t\t(v)\tThen obtain the value of k2 + 1 . \t39.\t (i)\tA pure quadratic equation is ax2 + c = 0. k \t\t(ii)\tPure quadratic equation is ax2 + c = 0. \t32.\t (i)\tSimplify the required expression and find \t40.\t (i)\tSimplify the equation 1. a\u00a0+\u00a0b and ab. \t\t(ii)\tTake the LCM of the equation. \t\t(ii)\tFind the sum of the roots and product of the Hints and Explanation roots. \t\t(iii)\tConvert it into quadratic equation. \t\t(iii)\tUse relation, a3 + b3 = (a + b)3 \u2212 3ab(a + b). \t\t(iv)\tSolve the equation for x. \t33.\t (i)\tUse Pythagorean theorem to find the sides of \t41.\t (i)\tIf x + k is the common root, then x = \u2212k the triangle. \u00adsatisfies both the equations. \t\t(ii)\tAssume the sides as x, x \u2212 4 and hypotenuse as \t\t(ii)\tIf x + a is factor of f\u2009(x), then f\u2009(\u2212a) = 0. x + 4. \t\t(iii)\tWrite p in terms of q and b in terms of a. \t\t(iii)\tFind the value of x using the relation \t\t(iv)\tNow substitute these values in the given expression and simplify. (hypotenuse)2 = sum of the squares of the other two sides. 4\t 2.\t Find the sum and product of the roots and form the equation as per the condition given in the \t\t(iv)\tThe area of triangle = 1 \u00d7 base \u00d7 height. problem. 2 \t34.\t (i)\tForm the quadratic equation and solve for x. 4\t 3.\t (i)\tSimplify the equation. \t\t(ii)\tA rational function f (x) is defined only \t\t(ii)\tAssume the fraction as x x 1 . \u2212 g(x ) when g(x) > 0. \t\t(iii)\t x x + x \u2212 1 = 1261 . 4\t 4.\t (i)\tSimplify the equation. \u22121 x \t\t(ii)\tTake the LCM. \t\t(iii)\tConvert it into quadratic equation. \t\t(iv)\tSolve the above equation. \t35.\t (i)\tUse the formula to find the area of the rectangle. \t\t(ii)\tAssume the length and breadth as lm and (l \u2212 10) m.","5.26 Chapter 5 \t\t(iv)\tSolve the equation by using formula x \t48.\t 2x2 + 4x \u2212 k = 0\b (1) = \u2212b \u00b1 b2 \u2212 4ac . \t\t\\\\ \u21d2 (x \u2212 5) is a factor of Eq. (1). 2a \t\t \u21d2 x \u2212 5 = 0 \u21d2 x = 5 \t45.\t (i)\tIf the equation is a perfect square, then it has \t\t\\\\ 2(5)2 + 4(5) \u2212 k = 0 equal roots. \t\t(ii)\tQuadratic equation is a perfect square, if b2 \u2212 \t\t50 + 20 \u2212 k = 0 4ac = 0. \t\t\\\\ k = 70. \t\t(iii)\tSubstitute the value of b and c in the above \t49.\t The difference of the roots of ax2 + bx + c = 0 is equation and obtained the value of a. \t46.\t The minimum value of ax2 + bx + c is b2 \u2212 4ac for a > 0. a 4ac \u2212 b2 . (a > 0) 4a \t\t\\\\ For x2 + mx \u2212 2 = 0. It is m2 + 8 = 3 \t\tThe minimum value of 2x2 \u2212 3x + 2 = \u21d2 m = \u00b11. 4 \u00d7 2 \u00d7 2 \u2212 (\u22123)2 = 7 . \t\tThat is, the equation could be x2 + x \u2212 2 = 0 or 4\u00d72 8 x2 \u2212 x \u2212 2 = 0. \t47.\t Given x2 \u2212 2x \u2212 1 = 0 \t\tThat is, (x + 2)(x \u2212 1) = 0 or (x \u2212 2)(x + 1) = 0 x= 2\u00b1 4 \u2212 4 \u00d7 1\u00d7 (\u22121) \t\tThe roots are \u22122, 1 or \u22121, 2. 2\u00d71 \t\tAs the numerically smaller root is greater, the roots \t\tx = 2 \u00b1 8 = 2\u00b12 2 =1\u00b1 2 are \u22122, 1 and m = 1. 2 2 x = 1 + 2 or 1 \u2212 2. Hints and Explanation Level 3 \t50.\t (i)\tUse the concept of sum of the roots and \t\t(ii)\tSquare the given expression twice and then product of the roots of a quadratic equation. solve for x. \t\t(ii)\tThe product of the roots obtained by A and \t54.\t (i)\tForm the quadratic equation and solve sum of the roots obtained by B is equal to the for x. product and sum of the roots of the required equation respectively. \t\t(ii)\tAssume number of employees in the group as x. Then write the quadratic equation in x 5\t 1.\t (i)\tSimplify the expression and find (a + b) and according to the data and solve it. ab. \t55.\t f(x) = x2 \u2212 12x \u2212 13 = 0 \t\t(ii)\tFirst find the sum of the roots and product of \t\tIf the roots of g(x) are 2 times the roots of f(x), then the roots. x \t\t(iii)\t\u03b1 \u22126 + \u03b2 \u22126 2 (\u03b1 3 +\u03b2 3 ) g(x) = f \uf8eb 2 \uf8f6 = 0. 3\u03b2 \u03b1 6\u03b2 6 \uf8ec\uf8ed \uf8f7\uf8f8 + \u03b1 3 = \t52.\t (i)\tTake b1 = b2 = b3 \u2026 bn = k and find the value. \u21d2 \uf8eb x \uf8f62 \u2212 12 \uf8eb x \uf8f6 \u2212 13 = 0 \uf8ec\uf8ed 2 \uf8f7\uf8f8 \uf8ec\uf8ed 2 \uf8f8\uf8f7 \t\t(ii)\tAM (a1, a2, \u2026, an) \u2265 HM (a1, a2, \u2026, an). x2 12x \u2212 13 =0 \t\t(iii)\ta1 + a2 + + an n \u21d2 4 \u2212 2 n 1 \u2265 1 1 . \t\t\u21d2 x2 \u2212 24x \u2212 52 = 0. a1 a2 a2 + + 5\t 6.\t Let the roots be k and 3k. \t\tSum of the roots = k + 3k 5\t 3.\t (i)\tSimplify the equation.","Quadratic Expressions and Equations 5.27 \t\t\u21d2 4k = \u2212b \u21d2 k = \u2212b . \t\t2y \u2212 3 = 0 or y \u2212 2 = 0 a 4a \t\tProduct of the roots = k \u00d7 3k \t\ty = 3 or y = 2 2 \u21d2 3k2 = c \t\t|x| = 3 or |x| = 2 a 2 \u21d2 3 \uf8eb \u2212b \uf8f62 = c \t\tx = \u00b1 3 or x = \u00b12. \uf8ed\uf8ec 4a \uf8f8\uf8f7 a 2 \u21d2 3b2 = c \t\t\\\\ x has 4 real solutions. 16a2 a 6\t 0.\t Product of the sum of the roots and product of the \u21d2 3b2 =16ac 91 \t\t\u21d2 b = \u00b14 ac . roots is 9 , i.e., 3 b c 91 \t57.\t \u03b1 +\u03b2 = \u2212q \uf8eb a \u00d7 a \uf8f6 = 9 p \uf8ec\uf8ed \uf8f8\uf8f7 \u03b1 \u22c5\u03b2 = r bc = 91 p a2 9 \t\tbc \u03b1 3 + \u03b2 3 = (\u03b1 + \u03b2 )3 \u2212 3\u03b1\u03b2 (\u03b1 + \u03b2 ) a\u00d7a = 13 \u00d7 7 3\u00d73 \u2212q \uf8f63 r \u2212q = \uf8eb p \uf8f7 \u2212 3 \uf8eb p \uf8f6\uf8eb p \uf8f6 \u21d2 b = 13 , c = 7 or b = 7 , c = 13 \uf8ec \uf8f8 \uf8ec \uf8f7\uf8ec \uf8f7 a 3 a 3 a 3 a 3 \uf8ed \uf8ed \uf8f8\uf8ed \uf8f8 = \u2212q3 + 3pqr 13 7 Hints and Explanation p3 3 3 \t\tThe required difference is \u2212 = 2. \u2234\u03b1 3 + \u03b23 = 3pqr \u2212 q3 . 2 +12x \u2212 3x2 p3 2x2 \u2212 8x + 9 \t61.\t For the maximum value of , 5\t 8.\t x2 \u2212 (a + 1)x + 1 (a2 + a + 1) = 0 2 + 12x \u2212 3x2 is maximum and 2x2 \u2212 8x + 9 is 2 minimum. \t\t a + b = a + 1 \t\tThe maximum value of 2 + 12x \u2212 3x2 and mini- \t\tab = 1 (a2 + a + 1) mum value of 2x2 \u2212 8x + 9 occurs at x = \u2212b , 2 2a i.e., 2. \t\ta2 + b2 = (a + b)2 \u2212 2ab \t\tWhen x = 2, 1 \t\t \t = (a + 1)2 \u2212 2 \uf8ee 2 (a2 + a + 1)\uf8fb\uf8f9\uf8fa 2 +12x \u2212 3x2 = 2 + 24 \u221212 = 14. \uf8f0\uf8ef \t\t2x2 \u2212 8x + 9 8 \u221216 + 9 \t\t\t = a2 + 2a + 1 \u2212 a2 \u2212 a \u2212 1 = a. 5\t 9.\t 2|x|2 \u2212 7|x| + 6 = 0 \t62.\t x2 + ax + b and x2 + bx + c have a common factor \t\tLet |x| = y (x \u2212 k) \t\t2y2 \u2212 7y + 6 = 0 \t\t2y2 \u2212 3y \u2212 4y + 6 = 0 \t\t \u21d2 k2 + ak + b = 0 and k2 + bk + c = 0 \t\ty(2y \u2212 3) \u2212 2(2y \u2212 3) = 0 \t\t(2y \u2212 3)(y \u2212 2) = 0 \t\t \u21d2 k2 + ak + b = k2 + bk + c \t\tak + b = bk + c \t\tk = c \u2212 b . a \u2212 b","5.28 Chapter 5 \t63.\t 9x \u2212 3y + z = 0 consider xa2 \u2212 ya + z = 0, a quadratic \t\t(x \u2212 8) (x + 2) = 0 equation in a. Where x, y, z are constants. \t\tx = 8 km\/h ( speed cannot be \u22122 km\/h). \t\tLet a = 3 \u21d2 (3)2 x \u2212 3y + z = 0 is a quadratic equa- tion in 3. 6\t 7.\t Quadratic equation with 4, \u22123 as roots is x2 \u2212 1x \u2212 12 = 0, quadratic equation whose product 3 = \u2212(\u2212y) + y2 \u2212 4 \u22c5 x \u22c5 z of the roots is \u221212.\b(1) 2.x \t\tAs one of the actual roots is 2, the other root is \u22126. \b (from (1)) 3 y y 2 \u2212 4xz . \t\tThe quadratic equation is x2 \u2212 (\u22126 + 2) x \u2212 12 = 0 2x 4x2 \u21d2 = + \t\tx2 + 4x \u2212 12 = 0. \t\t 6\t 4.\t Given the roots of the given equation are complex \t\t\\\\ The difference between the coefficients of x = 4 \u2212 (\u22121) = 5. \t\t \u21d2 b2 \u2212 4ac < 0 \t\t \u21d2 (\u221212)2 \u2212 4(3) k < 0 \t68.\t Let a, b be the roots of ax2 \u2212 bx + 2c = 0 \t\t144 \u2212 12 k < 0 \t\t \u221212k < \u2212144 \t\tGiven = \u03b1 = 2 \t\t12 k > 144 \u03b2 3 \t\tk > 12. \t\t\u21d2 \u03b1 = 2\u03b2 3 \t\tProduct of the roots \t65.\t a, b are the roots of ax2 + bx + c = 0 \u03b1 \u22c5 \u03b2 = 2c a \u2212b \u21d2\u03b1 + \u03b2 = a 2\u03b2 \u00d7 \u03b2 = 2c 3 a c . Hints and Explanation \t\t \u03b1 \u22c5\u03b2 = a \t\t\u03b2 2 = 3c \b(1) a \t\tQuadratic equation whose roots are a + b, and ab \t\tSum of the roots = a + b is x2 \u2212 \uf8eb \u2212b + c \uf8f6 x + \u2212b \u00d7 c = 0. \t\t\u21d2 2\u03b2 + \u03b2 = b \uf8ec\uf8ed c a \uf8f8\uf8f7 a a 3 a \t\tx 2 + \uf8eb b \u2212 c \uf8f6 x \u2212 bc = 0 \t\t\u21d2 5\u03b2 = b \uf8ed\uf8ec a \uf8f7\uf8f8 a2 3 a \t\ta2 x2 + (ab \u2212 ac)x \u2212 bc = 0. \t\t\u03b2 = 3b 5a 6\t 6.\t Let the speed of Ramu = x km\/h \t\tTotal time taken is = 1 hour \t\t\u03b2 2 = 9b2 \b(2) 25a2 \t\tThat is, x 3 4 + x 3 4 = 1 \t\tFrom Eqs. (1) and (2), 3c = 9b2 3b2 = 25ac. \u2212 + a 25a2 3x + 12 + 3x \u2212 12 = 1 \t69.\t f(x) = ax2 + bx + c = 0 \t\t x2 \u2212 16 \t\tf(x \u2212 2) = 9x2 \u2212 2x + 7 = 0 \t\t6x = x2 \u2212 16 \t\t a(x \u2212 2)2 + b(x \u2212 2) + c = 9(x)2 \u2212 2x + 7\\\\ \t\tx2 \u2212 6x \u2212 16 = 0 \t\ta(x2 \u2212 4x + 4) + bx \u2212 2b + c = 9x2 \u2212 2x + 7 \t\tx2 \u2212 8x + 2x \u2212 16 = 0 \t\tax2 \u2212 (4a \u2212 b)x + 4a \u2212 2b + c = 9x2 \u2212 2x + 7 \t\tx(x \u2212 8) + 2(x \u2212 8) = 0 \t\t \u21d2 4a \u2212 2b + c = 7.","Quadratic Expressions and Equations 5.29 \t70.\t Let f(x) \u2261 px2 + qx + r = 0 \t\tThat is, 2 \uf8eb 1 \uf8f62 + 7 \uf8eb 1 \uf8f6 + 5 = 0 \uf8ec\uf8ed x \uf8f8\uf8f7 \uf8ed\uf8ec x \uf8f7\uf8f8 \t\tGiven the quadratic equation whose roots are 2 more than the roots of f(x) as ax2 + bx + c = 0. \t\t \u21d2 2 + 7x + 5x2 = 0 \u21d2 5x2 + 7x + 2 = 0 \t\t \u21d2 f(x \u2212 2) = ax2 + bx + c \t\ta = 5, b = 7, c = 2 \t\t \u21d2 p(x \u2212 2)2 + q(x \u2212 2) + r = ax2 + bx + c \t\ta \u2212 c = 5 \u2212 2 = 3. \t\t \u21d2 px2 \u2212 4px + 4p + qx \u2212 2q + r = ax2 + bx + c \t72.\t ax2 + bx + c \t\t = p(kx)2 + q(kx) + r = 0 \t\t \u21d2 px2 + (q \u2212 4p)x + (4p \u2212 2q + r) = ax2 + bx + c \t\t \u21d2 a = pk2, b = qk, c = r \t\t \u21d2 c = 4p \u2212 2q + r. 7\t 1.\t If the roots of 2x2 + 7x + 5 = 0\b (1) \t\tba pk2 qk \t\tare the reciprocal roots of ax2 + bx + c = 0, then ax2 = + bx + c = 0 is obtained by substituting 1 in Eq. (1). \t\t \u21d2 aq = pbk. x Hints and Explanation","162CChhaapptteerr SKeitnsemanatdics Relations REMEMBER Before beginning this chapter, you should be able to: \u2022 Understand the basic definitions of sets \u2022 Apply basic operations on sets \u2022 Understand basic concept of Venn diagrams KEY IDEAS After completing this chapter, you should be able to: \u2022 learn about sets, representation of sets and some definitions related to sets \u2022 Apply operations on sets \u2022 Understand the Venn diagrams \u2022 Obtain cartesian products of sets \u2022 Know about relation of sets, its representation, types, properties and to find its domain and range \u2022 Understand the functions of sets Figure 1.1","6.2 Chapter 6 INTRODUCTION In everyday life we come across different collections of objects. For example, A herd of sheep, a cluster of stars, a posse of policemen, etc. In mathematics, we call such collections as sets. The objects are referred to as the elements of the sets. SET A set is a well-defined collection of objects. Let us understand what we mean by a well-defined collection of objects. We say that a collection of objects is well-defined if there is some reason or rule by which we can say whether a given object of the universe belongs to or does not belong to the collection. Elements of a Set The objects in a set are called elements or members of the set. We usually denote the sets by capital letters A, B, C or X, Y, Z, etc. If a is an element of a set A, then we say that a belongs to A and we write, a \u2208\u2009A. If a is not an element of A, then we say that a does not belong to A and we write, a \u2209\u2009A. To understand the concept of a set, let us look at some examples. Examples: 1.\t L\u0007 et us consider the collection of odd natural numbers less than or equal to 15. \t \u0007In this example, we can definitely say what the collection is. The collection comprises the numbers 1, 3, 5, 7, 9, 11, 13 and 15. 2.\t \u0007Let us consider the collection of students in a class who are good at painting. In this example, we cannot say precisely which students of the class belong to our collection. So, this collection is not well-defined. Hence, the first collection is a set where as the second collection is not a set. In the first example given, the set of odd natural numbers less than or equal to 15 can be represented as set A = {1, 3, 5, 7, 9, 11, 13, 15}. Some Sets of Numbers and Their Notations N = Set of all natural numbers = {1, 2, 3, 4, 5, \u2026}. W = Set of all whole numbers = {0, 1, 2, 3, 4, 5, \u2026}. Z or I = Set of all integers = {0, \u00b11, \u00b12, \u00b13, \u2026}. Q = Set of all rational numbers = \uf8f1p where p, q \u2208 Z and q \u2260 0\uf8fd\uf8fc. \uf8f2 \uf8fe \uf8f3 q Cardinal Number of a Set The number of elements in a set A is called its cardinal number. It is denoted by n(A). A set which has finite number of elements is a finite set and a set which has infinite number of elements is an infinite set. Examples: 1.\t Set of English alphabets is a finite set. 2.\t Set of number of days in a month is a finite set.","Sets and Relations 6.3 3.\t The set of all even natural numbers is an infinite set. 4.\t Set of all the lines passing through a point is an infinite set. 5.\t \u0007The cardinal number of the set X = {a, c, c, a, b, a} is n(X) = 3 as in sets only distinct elements are counted. Representation of Sets We represent sets by the following methods: Roster or List Method In this method, a set is described by listing out all the elements in the set. Examples: 1.\t \u0007Let W be the set of all letters in the word JANUARY. Then we represent W as, W = {A, J, N, R, U, Y}. 2.\t L\u0007 et M be the set of all multiples of 3 less than 20. Then we represent the set M as, M = {3, 6, 9, 12, 15, 18}. Set Builder Method In this method, a set is described by using a representative and stating the property or properties which the elements of the set satisfy, through the representative. Examples: 1.\t \u0007Let D be the set of all days in a week. Then we represent D as, \t D = {x: x is a day in a week}. 2.\t \u0007Let N be the set of all natural numbers between 10 and 20, then we represent the set N as, N = {x: 10 < x < 20 and x \u2208 N}. Some Simple Definitions of Sets Empty Set or Null Set or Void Set A set with no elements in it is called an empty set (or) void set (or) null set. It is denoted by { } or f. (read as \u2018phi\u2019) Examples: 1.\t Set of all positive integers less than 1 is an empty set. 2.\t Set of all mango trees with apples is an empty set. Singleton Set A set consisting of only one element is called a singleton set. Examples: 1.\t \u0007The set of all vowels in the word MARCH is a singleton, as A is the only vowel in the word. 2.\t T\u0007 he set of whole numbers which are not natural numbers is a singleton, as 0 is the only whole number which is not a natural number.","6.4 Chapter 6 Equivalent Sets Two sets A and B are said to be equivalent if their cardinal numbers are equal. We write this symbolically as A ~ B or A \u2194 B. Examples: 1.\t Sets, X = {2, 4, 6, 8} and \t Y = {a, b, c, d} are equivalent as n(X) = n(Y) = 3. 2.\t Sets, X = {Dog, Cat, Rat} and \t Y = { , , } are equivalent. 3.\t Sets, X = {-1, -7, -5} and \t \tY = {Delhi, Hyderabad} are not equivalent as n(X) \u2260 n(Y\u2009). \u2002Note\u2002\u2002 If the sets A and B are equivalent, we can establish a one-to-one correspondence between the two sets. That is, we can pair up elements in A and B such that every element of set A is paired with a distinct element of set B and every element of B is paired with a distinct element of A. Equal Sets Two sets A and B are said to be equal if they have the same elements. Examples: 1.\t Set, A = {a, e, i, o, u} and B = {x: x is a vowel in the English alphabet} are equal sets. 2.\t Set, A = {1, 2, 3} and B = {x, y, z} are not equal sets. 3.\t Set, A = {1, 2, 3, 4, \u2026} and B = {x: x is a natural number} are equal sets. \u2002Note\u2002\u2002 If A and B are equal sets, then they are equivalent but the converse need not be true. Disjoint Sets Two sets A and B are said to be disjoint, if they have no elements in common. Examples: 1.\t \u0007Sets X = {3, 6, 9, 12} and Y = {5, 10, 15, 20} are disjoint as they have no elements in common. 2.\t Sets A = {a, e, i, o, u} and B = {e, i, j} are not disjoint as they have elements e and i in common. Subset Let A and B be two sets. If every elements of set A is also an element of set B, then A is said to be a subset of B or B is said to be a superset of A. If A is a subset of B, then we write A \u2286 B or B \u2287 A. Examples: 1.\t Set A = {2, 4, 6, 8} is a subset of set B = {1, 2, 3, 4, 5, 6, 7, 8}. 2.\t Set of all primes except 2 is a subset of the set of all odd natural numbers. 3.\t Set A = {1, 2, 3, 4, 5, 6, 7, 8} is a superset of set B = {1, 3, 5, 7}.","Sets and Relations 6.5 \u2002Notes\u2002 1.\t Empty set is a subset of every set. 2.\t Every set is a subset of itself. 3.\t If a set A has n elements, then the number of subsets of A is 2n. 4.\t If a set A has n elements, then the number of non-empty subsets of A is 2n - 1. Proper Subset and Superset If A \u2286 B and A \u2260 B, then A is called a proper subset of B and is denoted by A \u2282 B. If A \u2282 B, then B is called a superset of A and is denoted as B \u2283 A. Power Set The set of all subsets of a set A is called power set of A. It is of A denoted by P(A). Example: Let A = {x, y, z}. Then the subsets of A are f, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}. So, P(A) = {f, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}. We observe that the cardinality of P(A) is 8 = 23. \u2002Notes\u2002 1.\t I\u0007f a set A has n elements, then the number of subsets of A is 2n, i.e., the cardinality of the power set is 2n. 2.\t If a set A has n elements, then the number of proper subsets of A is 2n - 1. 3.\t If a set A has n elements, then the number of non-empty proper subsets is 2n - 2. Universal Set A set which consists of all the sets under consideration or discussion is called the universal set. It is usually denoted by \u222a or \u00b5. Example: L\u0007 et A = {a, b, c}, B = {c, d, e} and C = {a, e, f, g, h}. Then, the set {a, b, c, d, e, f, g, h} can be taken as the universal set here. \u2234 \u00b5 = {a, b, c, d, e, f, g, h}. Complement of a Set Let \u00b5 be the universal set and A \u2286 \u00b5. Then, the set of all those elements of \u00b5 which are not in set A is called the complement of the set A. It is denoted by A\u2032 or Ac. A\u2032 = {x : x \u2208 \u00b5 and x \u2209 A}. Examples: 1.\t Let \u00b5 = {3, 6, 9, 12, 15, 18, 21, 24} and A = {6, 12, 18, 24}. Then, A\u2032 = {3, 9, 15, 21}. 2.\t Let \u00b5 = {x: x is a student and x \u2208 class X}. \t And B = {x: x is a boy and x \u2208 class X}. \t Then, B\u2032 = {x: x is a girl and x \u2208 class X}.","6.6 Chapter 6 \u2002Notes\u2002 1.\t A and A\u2032 are disjoint sets. 2.\t \u00b5\u2032 = f and f\u2032 = \u00b5. Operations on Sets Union of Sets Let A and B be two sets. Then, the union of A and B, denoted by A \u222a B, is the set of all those elements which are either in A or in B or in both A and B. That is, A \u222a B = {x: x \u2208 A or x \u2208 B}. Examples: 1.\t Let A = {-1, -3, -5, 0} and B = {-1, 0, 3, 5} then, A \u222a B = {-5, -3, -1, 0, 3, 5}. 2.\t \u0007Let A = {x: 5 \u2264 5x < 25 and x \u2208 N} and B = {x: 5 \u2264 (10x) \u2264 20 and x \u2208 N}, then, A \u222a B = {x: 5 \u2264 5x \u2264 20 and x \u2208 N}. \u2002Notes\u2002 1.\t If A \u2286 B, then A \u222a B = B. 2.\t A \u222a \u00b5 = \u00b5 and A \u222a f = A. 3.\t A \u222a A\u2032 = \u00b5. Intersection of Sets Let A and B be two sets. Then the intersection of A and B, denoted by A \u2229 B, is the set of all those elements which are common to both A and B. That is, A \u2229 B = {x: x \u2208 A and x \u2208 B}. Examples: 1.\t Let A = {1, 2, 3, 5, 6, 7, 8} and B = {1, 3, 5, 7}. A \u2229 B = {1, 3, 5, 7}. 2.\t \u0007Let A be the set of all English alphabet and B be the set of all consonants. A \u2229 B is the set of all consonants in the English alphabet. 3.\t \u0007Let E be the set of all even natural numbers and O be the set of all odd natural numbers. E \u2229 O = { } or f. \u2002Notes\u2002 1.\t If A and B are disjoint sets, then A \u2229 B = f and n(A \u2229 B) = 0. 2.\t If A \u2286 B, then A \u2229 B = A. 3.\t 3A \u2229 \u00b5 = A and A \u2229 f = f. 4.\t A \u2229 A\u2032 = f Difference of Sets Let A and B be two sets. Then the difference A - B is the set of all those elements which are in A but not in B. That is, A - B = {x: x \u2208 A and x \u2209 B}. Example: Let A = {3, 6, 9, 12, 15, 18} and B = {2, 6, 8, 10, 14, 18}. A - B = {3, 9, 12, 15} and B - A = {2, 8, 10, 14}.","Sets and Relations 6.7 \u2002Notes\u2002 1.\t A - B \u2260 B - A unless A = B. 2.\t For any set A \u00b7 A\u2032 = \u00b5 - A. Symmetric Difference of Sets Let A and B be two sets. Then the symmetric difference of A and B, denoted by A D B, is the set of all those elements which are either in A or in B but not in both, i.e., A D B = {x: x \u2208 A and x \u2209 B or x \u2208 B and x \u2209 A}. \u2002Note\u2002\u2002 A D B = (A - B) \u222a (B - A) (or) A D B = (A \u222a B) - (A \u2229 B). \u0007Example: Let A = {1, 2, 4, 6, 8, 10, 12} and B = {3, 6, 12}. A D B = (A - B) \u222a (B - A) = {1, 2, 4, 8, 10} \u222a {3} = {1, 2, 3, 4, 8, 10}. EXAMPLE 6.1 If A = {3, 5, 7, 8} and B = {7, 8, 9, 10}, then find the value of (A \u222a B) - (A \u2229 B). The following are the steps involved in solving the above problem. Arrange them in sequential order. (A) A \u222a B = {3, 5, 7, 8, 9, 10} and A \u2229 B = {7, 8}. (B) Given A = {3, 5, 7, 8} and B = {7, 8, 9, 10}. (C) (A \u222a B) - (A \u2229 B) = {3, 5, 9, 10}. (D) (A \u222a B) - (A \u2229 B) = {3, 5, 7, 8, 9, 10} - {7, 8}. SOLUTION The sequential order is BADC. Some Results\u2002 For any three sets A, B and C, we have the following results: 1.\t Commutative law: \t(i)\tA \u222a B = B \u222a A \t(ii)\t A \u2229 B = B \u2229 A \t(iii)\t A D B = B D A 2.\t Associative law: \t(i)\t(A \u222a B) \u222a C = A \u222a (B \u222a C) \t(ii)\t (A \u2229 B) \u2229 C = A \u2229 (B \u2229 C) \t(iii)\t (A D B) D C = A D (B D C) 3.\t Distributive law: \t(i)\tA \u222a (B \u2229 C) = (A \u222a B) \u2229 (A \u222a C) \t(ii)\t A \u2229 (B \u222a C) = (A \u2229 B) \u222a (A \u2229 C)","6.8 Chapter 6 4.\t De-Morgan\u2019s law: \t(i)\t(A \u222a B)\u2032 = A\u2032 \u2229 B\u2032 \t(ii)\t (A \u2229 B)\u2032 = A\u2032 \u222a B\u2032 \t(iii)\t A - (B \u222a C) = (A - B) \u2229 (A - C) \t(iv)\t A - (B \u2229 C) = (A - B) \u222a (A - C) 5.\t Identity law: \t(i)\tA \u222a f = f \u222a A = A \t(ii)\t A \u2229 \u00b5 = \u00b5 \u2229 A = A 6.\t Idempotent law: \t(i)\tA \u222a A = A \t(ii)\t A \u2229 A = A 7.\t Complement law: \t(i)\t(A\u2032)\u2032 = A \t(ii)\t A \u222a A\u2032 = \u00b5 \t(iii)\t A \u2229 A\u2032 = f Dual of an Identity An identity obtained by interchanging \u222a and \u2229, and \u03d5 and \u00b5 in the given identity is called the dual of that identity. Examples: 1.\t Consider the identity, A \u222a B = B \u222a A. Dual of the identity is, A \u2229 B = B \u2229 A. 2.\t Consider the identity, A \u222a \u00b5 = \u00b5. Dual of the identity is, A \u2229 f = f. Venn Diagrams We also represent sets pictorially by means of diagrams called Venn diagrams. In Venn diagrams, the universal set is usually represented by a rectangular region and its subsets by closed regions inside the rectangular region. The elements of the set are written in the closed regions and the elements which belong to the universal set are written in the rectangular region. \u0007Example: Let \u00b5 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 4, 6, AB \u03bc 7, 8} and B = {2, 3, 4, 5, 9}. We represent these sets in the form 10 of Venn diagram as in Fig. 6.1. 12 3 We can also represent the sets in Venn diagrams by shaded 6 8 4 9 5 regions. 7 Examples: Figure 6.1 1.\t V\u0007enn diagram of A \u222a B, where A and B are two overlapping sets and it is shown in Fig. 6.2: AB \u03bc Figure 6.2","Sets and Relations 6.9 2.\t \u0007Let A and B be two overlapping sets. Then, the Venn diagram of A \u2229 B is as shown in Fig. 6.3: AB \u03bc Figure 6.3 3.\t For a non-empty set A, Venn diagram of A\u2032 is as shown in Fig. 6.4: A\u2032 \u03bc A Figure 6.4 4.\t \u0007Let A and B be two overlapping sets. Then, the Venn diagram of A - B is as shown in Fig. 6.5: AB \u03bc Figure 6.5 5.\t L\u0007 et A and B be two sets such A \u2286 B. We can represent this relation using Venn diagram as shown in Fig. 6.6: B\u03bc A Figure 6.6 Some Formulae on the Cardinality of Sets Let A = {1, 2, 3, 5, 6, 7} and B = {3, 4, 5, 8, 10, 11}. Then, A \u222a B = {1, 2, 3, 4, 5, 6, 7, 8, 10, 11} and A \u2229 B = {3, 5}. In terms of the cardinal numbers, n(A) = 6, n(B) = 6, n(A \u2229 B) = 2 and n(A \u222a B) = 10. So, n(A) + n(B) - n(A \u2229 B) = 6 + 6 - 2 = 10 = n(A \u222a B). We have the following formulae:","6.10 Chapter 6 For any three sets A, B and C. 1.\t n(A \u222a B) = n(A) + n(B) - n(A \u2229 B). 2.\t n(A \u222a B \u222a C) = n(A) + n(B) + n(C) - n(A \u2229 B) - n(B \u2229 C) - n(C \u2229 A) + n(A \u2229 B \u2229 C). Example 6.2 If n(A) = 7, n(B) = 5 and n(A \u222a B) = 10, then find n(A \u2229 B). Solution Given, n(A) = 7, n(B) = 5 and n(A \u222a B) = 10. We know that, n(A \u222a B) = n(A) + n(B) - n(A \u2229 B). So, 10 = 7 + 5 - n(A \u2229 B) \u21d2 n(A \u2229 B) = 2. Example 6.3 If n(A) = 8 and n(B) = 6 and the sets A and B are disjoint, then find n(A \u222a B). Solution Given, n(A) = 8 and n(B) = 6. A and B are disjoint. \u21d2 A \u2229 B = f \u21d2 n(A \u2229 B) = 0. \u2234 n(A \u222a B) = n(A) + n(B) - n(A \u2229 B) = 8 + 6 - 0 = 14. \u2002Note\u2002\u2002 If A and B are two disjoint sets, then n(A \u222a B) = n(A) + n(B). Example 6.4 In a locality, the number of residents who read only The Hindu, only The Times of India, both the newspapers and neither of the newspapers are in the ratio 2 : 3 : 4 : 1. The number of residents who read at least one of these newspapers is 160 more than those who read neither of these newspapers. Find the number of residents in the locality. Solution Let the number of residents who read only The Hindu be 2x. \\\\ Number of residents who read The Times of India, both the newspapers and neither of the newspapers are 3x, 4x and x respectively. 2x + 3x + 4x = x + 160 8x = 160 \u21d2 x = 20 Number of residents in the locality = 2x + 3x + 4x + x = 10x = 200.","Sets and Relations 6.11 Ordered Pair Let A be a non-empty set and a, b \u2208 A. The elements a and b written in the form (a, b) is called an ordered pair. In the ordered pair (a, b), a is called the first coordinate and b is called the second coordinate. \u2002Note\u2002\u2002 Two ordered pairs are said to be equal only when their first as well as the second coordinates are equal, i.e., (a, b) = (c, d) \u21d4 a = c and b = d. So, (1, 2) \u2260 (2, 1) and if (a, 5) = (3, b) \u21d2 a = 3 and b = 5. CARTESIAN PRODUCT OF SETS Let A and B be two non-empty sets. The Cartesian product of A and B, denoted by A \u00d7 B, is the set of all ordered pairs (a, b), such that a \u2208 A and b \u2208 B. That is, A \u00d7 B = {(a, b): a \u2208 A, b \u2208 B}. E\u0007 xample: Let A = {1, 2, 3} and B = {2, 4}. A \u00d7 B = {(1, 2), (1, 4), (2, 2), (2, 4), (3, 2), (3, 4)} and B \u00d7 A = {(2, 1), (2, 2), (2, 3), (4, 1), (4, 2), (4, 3)}. We observe that, A \u00d7 B \u2260 B \u00d7 A and n(A \u00d7 B) = 6 = n(B \u00d7 A). \u2002Notes\u2002 1.\t A \u00d7 B \u2260 B \u00d7 A, unless A = B. 2.\t For any two sets A and B, n(A \u00d7 B) = n(B \u00d7 A). 3.\t If n(A) = p and n(B) = q, then n(A \u00d7 B) = pq. Some Results on Cartesian Product 1.\t A \u00d7 (B \u222a C) = (A \u00d7 B) \u222a (A \u00d7 C) (or) (A \u222a B) \u00d7 (A \u222a C). 2.\t A \u00d7 (B \u2229 C) = (A \u00d7 B) \u2229 (A \u00d7 C) (or) (A \u2229 B) \u00d7 (A \u2229 C). 3.\t If A \u00d7 B = f, then either \t(i)\tA = f \t(ii)\t B = f \t(iii)\t both A = f and B = f. \t Cartesian product of sets can be represented in following ways: 1.\t Arrow diagram 2.\t Tree diagram 3.\t Graphical representation Representation of A \u00d7 B Using Arrow Diagram Example 6.5 If A = {a, b, c} and B = {1, 2, 3}, then find A \u00d7 B. Solution In order to find A \u00d7 B, represent the elements of A and B as shown in the diagram; 6.8. Now draw the arrows from each element of A to each element of B.","6.12 Chapter 6 Now, represent all the elements related by arrows in ordered pairs in a set, which is the required A \u00d7 B. That is, A \u00d7 B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c, 1), (c, 2), (c, 3)}. AB a1 b2 3 c Figure 6.8 Representation of A \u00d7 B Using a Tree Diagram Example 6.6 If A = {a, b, c} and B = {1, 2, 3}, then find A \u00d7 B. Solution To represent A \u00d7 B using tree diagram, write all the elements of A vertically and then for each element of A, write all the elements of B and draw arrows as shown in the diagram, \u2234 A \u00d7 B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c 1), (c, 2), (c, 3)}. A B A\u00d7B 1 (a, 1) a 2 (a, 2) 3 (a, 3) 1 (b, 1) b 2 (b, 2) 3 (b, 3) 1 (c, 1) c 2 (c, 2) 3 (c, 3) Figure 6.9 Graphical Representation of A \u00d7 B Example 6.7 If A = {1, 2, 3} and B = {3, 4, 5}, then find A \u00d7 B. Solution Consider the elements of A on the X-axis and the elements of B on the Y-axis and mark the points (see Fig. 6.10). \u2234 A \u00d7 B = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 3), (3, 4), (3, 5)}.","Sets and Relations 6.13 Y 6 B 5 \u2022 \u2022 \u2022 A\u00d7B 4 \u2022\u2022\u2022 3 \u2022\u2022\u2022 2 1 0 1 2 3 4 5 6X A Figure 6.10 RELATION We come across certain relations in real life and also in basic geometry, like is father of, is a student of, is parallel to, is similar to, etc. Definition Let A and B be two non-empty sets and R \u2286 A \u00d7 B. R is called a relation from the set A to B. (Any subset of A \u00d7 B is called a relation from A to B). \u2234 A relation contains ordered pairs as elements. Hence, \u2018A relation is a set of ordered pairs\u2019. Examples: Let A = {1, 2, 4} and B = {2, 3}. Then, A \u00d7 B = {(1, 2), (1, 3), (2, 2), (2, 3), (4, 2), (4, 3)}. 1.\t Let R1 = {(1, 2), (1, 3), (2, 3)} \t Clearly, R1 \u2286 A \u00d7 B and we also notice that, for every ordered pair (a, b) \u2208 R1, a < b. \t So, R1 is the relation is less than from A to B. 2.\t Let R2 = {(4, 2), (4, 3)}. \t Clearly, R2 \u2286 A \u00d7 B and we also notice that, for every ordered pair (a, b) \u2208 R2, a > b. \t So, R2 is the relation is greater than from A to B. \u2002Notes\u2002 1.\t If n(A) = p and n(B) = q, then the number of relations possible from A to B is 2pq. 2.\t If (x, y) \u2208R, then we write xRy and read as x is related to y. Domain and Range of a Relation Let A and B be two non-empty sets and R be a relation from A to B, we note the followings: 1.\t The set of first coordinates of all ordered pairs in R is called the domain of R. 2.\t The set of second coordinates of all ordered pairs in R is called the range of R.","6.14 Chapter 6 Example: Let A = {1, 2, 4}, B = {1, 2, 3} and R = {(1, 1), (1, 2), (2, 1), (2, 3), (4, 3)} be a relation from A to B. Then, domain of R = {1, 2, 4} and range of R = {1, 2, 3}. Representation of Relations We represent the relations by the following methods: Roster-Method (or) List Method In this method, we list all the ordered pairs that satisfy the rule or property given in the relation. Example: Let A = {1, 2, 3}. If R is a relation on the set A having the property is less than, then the roster form of R is, R = {(1, 2), (1, 3), (2, 3)}. Set-builder Method In this method, a relation is described by using a representative and stating the property or properties, which the first and second coordinates of every ordered pair of the relation satisfy, through the representative. Example: Let A = {1, 2, 3}. If R is a relation on the set A having the property is greater than or equal to, then the set builder form of R is, R = {(x, y)\/x, y \u2208 A and x \u2265 y}. Arrow Diagram In this method, a relation is described by drawing arrows between the elements which satisfy the property or properties given in the relation. Example: Let A = {1, 2, 4} and B = {2, 3}. Let R be a relation from A to B with the property is less than. Then, the arrow diagram of R is as shown in Fig. 6.11. ARB 12 2 43 Figure 6.11 Inverse of a Relation Let R be a relation from A to B. The inverse relation of R, denoted by R-1, is defined as, R-1 = {(y, x)\/(x, y) \u2208 R}. E\u0007 xample: Let R = {(1, 1), (1, 2), (2, 1), (2, 3), (4, 3)} be a relation from A to B, where A = {1, 2, 4} and B = {1, 2, 3}. Then, R-1 = {(1, 1), (1, 2), (2, 1), (3, 2), (3, 4)}. \u2002Notes\u2002 1.\t Domain of R-1 = Range of R. 2.\t Range of R-1 = Domain of R. 3.\t If R is a relation from A to B, then R-1 is a relation from B to A. 4.\t If R \u2286 A \u00d7 A, then R is called a binary relation or simply a relation on the set A. 5.\t For any relation R, (R-1)-1 = R.","Sets and Relations 6.15 Types of Relations 1.\t O\u0007 ne-one relation: A relation R: A \u2192 B is said to be one-one relation if different elements of A are paired with different elements of B, i.e., x \u2260 y in A \u21d2 f(x) \u2260 f(y) in B. \t Example: AR B 13 24 35 4 Figure 6.12 \t Here, the relation R = {(1, 3), (2, 4), (3, 5)}. 2.\t O\u0007 ne-many relation: A relation R: A \u2192 B is said to be one-many relation if at least one element of A is paired with two or more elements of B. \t Example: ARB 13 24 35 4 Figure 6.13 \t Here, the relation R = {(1, 3), (1, 4), (3, 5)}. 3.\t M\u0007 any-one relation: A relation R: A \u2192 B is said to be many-one relation if two or more elements of A are paired with an element of B. \t Example: AR B 12 23 34 5 Figure 6.14 \t Here, the relation R = {(1, 2), (2, 4), (3, 2), (5, 3)}. 4.\t \u0007Many-many relation: A relation R: A \u2192 B is said to be many-many relation if two or more elements of A are paired with two or more elements of B.","6.16 Chapter 6 Example: A RB 13 24 35 Figure 6.15 Here, the relation R = {(1, 3), (1, 4), (2, 3), (2, 5), (3, 4), (3, 5)}. Properties of Relations Reflexive Relation A relation R on a set A is said to be reflexive if for every x \u2208 A, (x, x) \u2208 R. Examples: 1.\t Let A = {1, 2, 3} then, R = {(1, 1), (1, 2), (2, 2), (3, 3), (2, 3)} is a reflexive on A. 2.\t L\u0007 et A = {1, 2, 3} then, R = {(1, 1), (2, 3), (1, 2), (1, 3), (2, 2)} is not a reflexive relation as 3 \u2208 A but (3, 3) \u2209 R. \u2002Note\u2002\u2002 Number of reflexive relations defined on a set having n elements is 2n2 \u2212n. Symmetric Relation A relation R on a set A is said to be symmetric, if for every (x, y) \u2208 R, (y, x) \u2208 R. Examples: 1.\t Let A = {1, 2, 3}. Then, R = {(1, 1), (1, 2), (2, 2), (2, 1), (3, 3)} is a symmetric relation on A. 2.\t \u0007Let A = {1, 2, 3}. Then, R = {(1, 1), (1, 2), (3, 1), (2, 2), (3, 3)} is not a symmetric relation as, (1, 2) \u2208 R but (2, 1) \u2209 R. \u2002Note\u2002\u2002 A relation R on a set A is symmetric iff R = R-1, i.e., R is symmetric if R = R-1. Transitive Relation A relation R on a set A is said to be transitive if (x, y) \u2208 R and (y, z) \u2208 R, then (x, z) \u2208 R. That is, R is said to be transitive, whenever (x, y) \u2208 R and (y, z) \u2208 R \u21d2 (x, z) \u2208 R. Examples: 1.\t I\u0007 f A = {1, 2, 3}, then, R = {(1, 1), (1, 2), (1, 3), (2, 1), (3, 1), (2, 2), (3, 3)} is a transitive relation. 2.\t L\u0007 et A = {1, 2, 3}. Then, R = {(1, 2), (2, 2), (2, 1), (3, 3), (1, 3)} is not a transitive relation as (1, 2) \u2208 R and (2, 1) \u2208 R but (1, 1) \u2209 R. Anti-symmetric Relation A relation R on a set A is said to be anti-symmetric if (x, y) \u2208 R and (y, x) \u2208 R, then x = y., i.e., R is said to be anti-symmetric if for x \u2260 y, (x, y) \u2208 R \u21d2 (y, x) \u2209 R.","Sets and Relations 6.17 Examples: 1.\t Let A = {1, 2, 3}. Then, R = {(1, 1), (1, 2), (3, 3)} is an anti-symmetric relation. 2.\t \u0007Let A = {1, 2, 3}. Then, R = {(1, 1), (1, 2), (1, 3), (2, 1), (3, 3)} is not an anti-symmetric relation as (1, 2) \u2208 R and (2, 1) \u2208 R but 1 \u2260 2. Equivalence Relation A relation R on a set A is said to be an equivalence relation if it is, 1.\t Reflexive 2.\t Symmetric 3.\t Transitive \u2002Note\u2002\u2002 For any set A, A \u00d7 A is an equivalence relation. In fact it is the largest equivalence relation. Identity Relation A relation R on a set A defined as, R = {(x, x)\/x \u2208 A} is called an identity relation on A. It is denoted by IA. Example: Let A = {1, 2, 3}. Then, R = {(1, 1), (2, 2), (3, 3)} is the identity relation on A. \u2002Note\u2002\u2002 Identity relation is the smallest equivalence relation on a set A. FUNCTION Let A and B be two non-empty sets and f is a relation from A to B. If f is such that 1.\t for every a \u2208 A, there is b \u2208 B such that (a, b) \u2208 f and 2.\t \u0007no two ordered pairs in f have the same first element, then f is called a function from set A to set B and is denoted as f: A \u2192 B. \u2002Notes\u2002 1.\t If (a, b) \u2208 f then f(a) = b and b is called the f image of a and a is called the preimage of b. 2.\t I\u0007 f f: A \u2192 B is a function, then A is called the domain of f and B is called the co-domain of f. 3.\t T\u0007 he set f(A) which is all the images of elements of A under the mapping f is called the range of f. Few examples of functions are listed below in the Fig. 6.16: 1. A fB 2. A gB 1 5a 1 2 6 b2 3 7 c3 4 4 d5 Figure 6.16","6.18 Chapter 6 Example 6.8 If R = {(x, y): x \u2208 W, y \u2208 W and (x + 2y)2 = 36}, then R-1 is ______. (a) {(0, 3), (2, 2), (1, 4), (0, 6)} (b) {(0, 6), (0, 3), (2, 2), (4, 1)} (c) {(3, 0), (2, 2), (1, 4), (0, 6)} (d) {(3, 0), (2, 2), (1, 4), (6, 0)} SOLUTION Given (x + 2y)2 = 36 and (x, y) \u2208\u2009W. \u21d2 x + 2y = 6 \\\\ The possible values of (x, y) are (0, 3), (2, 2), (4, 1), (6, 0). \u21d2 R = {(0, 3), (2, 2), (4, 1), (6, 0)} \\\\ R-1 = {(3, 0), (2, 2), (1, 4), (0, 6)}.","Sets and Relations 6.19 TEST YOUR CONCEPTS Very Short Answer Type Questions \t1.\t If P = {1, 2, 3, 4, 5, 6, 7} and Q = {2, 5, 8, 9}, 1\t 7.\t If n(P) = 2439 and Q = f then n(P \u00d7 Q) = PRACTICE QUESTIONS then find P \u222a Q. ________. \t2.\t If P = {1, 2, 3, 4, 5, 6, 7} and Q = {2, 5, 8, 9}, \t18.\t If P = {a, b, c, d} and Q = {1, 2, 3, 4, 5} then then find P - Q. n(P \u00d7 Q) = ________. \t3.\t If P = {1, 2, 3, 4, 5, 6, 7} and Q = {2, 5, 8, 9}, 1\t 9.\t Let A = {a, b, c} and B = {p, q}. Draw the arrow then find P \u2229 Q. diagram of A \u00d7 B. \t4.\t If X \u2282 Y and Y \u2282 X, then ________. \t20.\t If n(A) = 6 and n(B) = 3, then find the number of subsets of A \u00d7 B. \t5.\t The complement of f\u2032 is ________. 2\t 1.\t Let A = {x, y, z} and B = {p, q}, then draw the tree \t6.\t A \u2229 A\u2032 = ________. diagram of A \u00d7 B and B \u00d7 A. \t7.\t The symmetric difference of A and B is 2\t 2.\t If n(P) = 17, n(Q) = 10 and n(P \u2229 Q) = 7, then commutative. (True\/False) n(P D Q) is ________. \t8.\t The cardinal number of a set is 5. Find the cardinal \t23.\t If (x, 2p + q) = (y, p + 2q) then p - q = ________. number of the power set. \t24.\t If n(A) = 40 and n(B) = 23, then find n(A - B) and \t9.\t The order in which the elements are placed plays n(B - A) when B \u2282 A. an important role in sets. (True\/False) 2\t 5.\t Find n(P \u00d7 Q), if n(Q - P) = 10 and n(P - Q) = 13 \t10.\t If n(A \u222a B) = 16 and n(A \u2229 B) = 4, then the num- and n(P \u2229 Q) = 8. ber of elements in the symmetric difference of A 2\t 6.\t Find the number of relations from A to A, where and B is ________. A = {1, 2, 3, 4}. 1\t 1.\t If P and Q are disjoint, then (P \u2229 Q)\u2032 is ________. \t27.\t A relation R = {(a, b), (a, a), (a, c), (x, x), (x, y), \t12.\t If P and Q are disjoint, then P - Q = ________ (y, y), (d, d), (d, c)}.Write a relation E \u2282 R such and Q - P = ________. that x is equal to y. \t13.\t If V = {a, e, i, o, u}, then find the number of non- \t28.\t A = {1, 2, 3, 4}, B = {3, 4} and R = {(3, 1), empty proper subsets of V. (3, 2), (4, 1), (4, 2), (4, 3)} is a relation from B into A. Write R in set-builder form. \t14.\t If a set has 510 non-empty proper subsets, then find the cardinal number of the set. 2\t 9.\t P = {3, 5, 6, 8, 9}, Q = {6, 10, 12, 16, 17} and R = {(x, y)\/(x, y) \u2208 P \u00d7 Q, 2x = y} is a relation from \t15.\t If A is universal set, then ((A\u2032)\u2032)\u2032 is ________. P into Q, write R in list form. 1\t 6.\t If n(P) = 2 and n(Q) = 5000, then n(P \u00d7 Q) = 3\t 0.\t If n(X \u2229 Y\u2009\u2032) = 9, n(Y \u2229 X\u2009\u2032) = 10 and n(X \u222a Y) = ________. 25, then find n(X \u00d7 Y\u2009). Short Answer Type Questions \t31.\t If A = {2, 3, 4, 6, 7, 9, 10, 12}, B = {1, 3, 5, 8, 9, \t33.\t If n(X - Y) = 30 + a, n(Y - X) = 20 + 2a, n(X \u222a Y) 10, 11, 15}, C = {3, 4, 7, 10, 11, 13, 15} and \u00b5 = = 100 and n(X \u2229 Y) = 15 + 2a, then find a. {1, 2, 3, \u2026, 15}. Then, find (A \u222a B)\u2032. 3\t 4.\t If n(P \u0394 Q) = n(P \u222a Q), then P and Q are 3\t 2.\t If A = {2, 3, 4, 6, 7, 9, 10, 12}, B = {1, 3, 5, 8, 9, ________. 10, 11, 15}, C = {3, 4, 7, 10, 11, 13, 15} and \u00b5 = {1, 2, 3, \u2026, 15}. Then, find (A \u222a B \u222a C)\u2032. \t35.\t If n(A - B) = 25, n(B - A) = 15 and n(A \u222a B) = 60, then n(A \u2229 B) = ________.","6.20 Chapter 6 \t36.\t If n(P \u2229 Q) = 12 and n(Q) = 37, then find the 4\t 1.\t Given that R = {(1, 1), (3, 3), (2, 3), (3, 2), (2, 2)} value of n(P\u2009\u2032 \u2229 Q) on the set A = {1, 2, 3}. Which property is not satisfied by R on A? 3\t 7.\t In a colony of 170 members, 70 subscribe Deccan Chronicle and 120 subscribe Times of India. How \t42.\t What type of relation does R define on the set of many subscribe only Deccan Chronicle? (Each integers, if x + y = 8? subscribes at least one.) \t43.\t In a class of 50 students 20 take Sanskrit but not 3\t 8.\t A = {1, 2, 3, 4} and f(x) = 2x2, x \u2208 A. If f(x) = 18, Hindi and 37 take Sanskrit. How many students then find x. take Hindi but not Sanskrit? (Each student takes either Sanskrit or Hindi.) \t39.\t Write the following sets in the roster form. 4\t 4.\t In a class of 60 students, 25 speak Hindi, 45 speak \t\t(i)\tP\u0007 = {x\/x \u2208 W and x \u2209 N} (ii) S = {\u2009f\/f is a English. How many of them speak both English factor of 13} and Hindi, if each student speaks either English or Hindi? 4\t 0.\t A, B and C are three different sets and A \u00d7 (B \u2229 C\u2009) = (A \u00d7 B) \u2229 (A \u00d7 C). Judge the given statements \t45.\t In term examination 40% students failed in by taking any three non empty sets A, B and C. English, 32% failed in Physical Science. What is (True\/False). the pass percentage, if 10% failed in both? Essay Type Questions PRACTICE QUESTIONS \t46.\t The following figure depicts the number of fami- \t47.\t On the set of all colleges in a state, a relation R is lies subscribed for three different newspapers, i.e., defined such that \u2018two colleges are related if they Eenadu (E), Hindu (H) and Vaartha (V). belong to the same district\u2019. Find the properties satisfied by R. \t\tFind the number of people who read \t\t(i)\tatleast two papers \t48.\t In a set of students studying in the same class, two \t\t(ii)\tatmost two papers students are related \u2018if their weights are not equal\u2019. \t\t(iii)\tatleast three papers Find the properties satisfied by it. \t\t(iv)\tatmost three papers \t\t(v)\tatleast one paper. 4\t 9.\t If a set A has 4 elements and a reflexive relation R \t\t(vi)\tatmost one paper. defined in set A has x elements, then what is the range of x? E H\u03bc \t50.\t In a club 45% plays cricket, 20% plays only 9 22 15 football. Find the percentage of members who 17 6 18 plays only cricket if 10% play both. (Each plays at least one.) 33 V None = 10 CONCEPT APPLICATION Level 1 \t1.\t Which of the following cannot be the cardinal \t2.\t Consider the following statements number of the power set of any finite set? \t\tp: 3 \u2208 {1, {3}, 5, 7} \t\t(a) 26\t\t (b) 32 \t\tq: 2 \u2208 {1, {2, 4}, 5} \t\tWhich of the following is true? \t\t(d) 8\t\t (d) 16","Sets and Relations 6.21 \t\t(a) p alone \t\t(a) {2, 3, 5, 7, 11, 13, 17, 19} \t\t(b) q alone \t\t(b) {3, 5} \t\t(c) Both (p) and (q) \t\t(c) {2, 3, 5, 7, 11, 13, 17, 19, 23, 29} \t\t(d) Neither (p) nor (q) \t\t(d) {2, 3} \t3.\t If A = {1, 2, 3} and B = {2, 6, 7}, then (A - B) \u222a \u222910 (B - A) = \t11.\t If Pn is the set of first n prime numbers, then Pn \t\t(a) f\t\t (b) \u00b5 n=3 is \t\t(c) {1, 2, 3, 6, 7}\t (d) {1, 3, 6, 7} \t\t(a) {3, 5, 7, 11, 13, 17, 19} \t\t(b) {2, 3, 5} \t4.\t If (x \u2212 y, x + y) = (2, 8), then the values of x and y \t\t(c) {2, 3, 5, 7, 11, 13, 17} are respectively \t\t(d) {3, 5, 7} \t\t(a) 5, 3\t\t (b ) 7, 5 1\t 2.\t If n(\u00b5) = 100, n(A) = 50, n(B) = 20 and n(A \u2229 B) \t\t(c) 4, 2\t\t (d) 10, 8 = 10, then n[(A \u222a B)c] = \t5.\t If X = {x: x2 - 12x + 20 = 0} and Y = {x: \t\t(a) 60\t\t (b) 30 x2 + 5x - 14 = 0}, then X - Y = \t\t(c) 40\t\t (d) 20 \t\t(a) {2}\t\t (b) {10} \t13.\t Let Z denote the set of integers, then {x \u2208\u2009Z: \t\t(c) {-7}\t\t (d) { } |x - 3| < 4 \u2229 {x \u2208 Z:|x \u2212 4|< 5} = \t6.\t The number of subsets of A \u00d7 B if n(A) = 3 and \t\t(a) {-1, 0, 1, 2, 3, 4} n(B) = 3 is \t\t(b) {-1, 0, 1, 2, 3, 4, 5} \t\t(c) {0, 1, 2, 3, 4, 5, 6} \t\t(a) 512\t\t (b) 256 \t\t(d) {-1, 0, 1, 2, 3, 5, 6, 7, 8, 9} \t\t(c) 511\t\t (d) 255 \t7.\t If A = {1, 2, 3, 4}, then how many subsets of A 1\t 4.\t If A = \uf8f1\uf8f2n : n3 + 5n2 +2 is an integer and n itself contain the element 3? \uf8f3 n PRACTICE QUESTIONS \t\t(a) 24\t\t (b) 28 is an integer \uf8fc , then the number of elements in the \uf8fd \t\t(c) 8\t\t (d) 16 \uf8fe set A is \t8.\t In \u2018aRb\u2019 if \u2018a and b have the same teacher\u2019, then R is ________. \t\t(a) 1\t\t (b) 2 \t\t(a) reflexive\t\t (b) symmetric \t\t(c) 3\t\t (d) 4 \t\t(c) transitive\t (d) equivalence 1\t 5.\t If A = {1, 2, 3}, then the relation R = {(1, 1) (2, 2), (3, 1), (1, 3)} is \t9.\t A relation R: Z \u2192 Z is such that R = {(x, y)\/y = \t\t(a) reflexive.\t (b) symmetric. 2x + 1} is a \t\t(c) transitive.\t (d) equivalence. \t\t(a) one to one relation. \t\t(b) many to one relation. 1\t 6.\t If n(A \u00d7 B) = 45, then n(A) cannot be \t\t(c) one to many relation. \t\t(d) many to many relation. \t\t(a) 15\t\t (b) 17 \t\t(c) 5\t\t (d) 9 \t10.\t If Pn is the set of first n prime numbers, then \t17.\t If A, B and C are three non-empty sets such that A and B are disjoint and the number of elements \u222a10 contained in A is equal to those contained in the set of elements common to the sets A and C, then Pn = (A \u222a B \u222a C) is necessarily equal to n=2","6.22 Chapter 6 \t\t(a) n(B \u222a C). \t\t(a) Reflexive\t (b) Symmetric \t\t(c) Transitive\t (d) All the above \t\t(b) n(A \u222a C). \t\t(c) Both (a) and (b) 2\t 5.\t A = {ONGC, BHEL, SAIL, GAIL, IOCL} and R is a relation defined as \u2018two elements of A are \t\t(d) None of these related if they share exactly one common letter\u2019. For instance, BHEL and SAIL are related as they \t18.\t R and S are two sets such that n(R) = 7 and R \u2229 have a common letter L. The relation R is _____. S \u2260 \u03d5. Further n(S) = 6 and S \u0394 R. The greatest possible value of n(R \u0394 S) is _______. \t\t(a) anti-symmetric\t (b) only transitive \t\t(a) 11\t\t (b) 12 \t\t(c) only symmetric\t (d) equivalence \t\t(c) 13\t\t (d) 10 \t26.\t X is the set of all engineering colleges in the state of A.P and R is a relation on X defined as two col- \t19.\tConsider the following statements: leges are related iff they are affiliated to the same \t\t\u2009\u2009(i) Every reflexive relation is anti-symmetric. university then R is \t\t(ii) E\u0007very symmetric relation is anti-symmetric. \t\t(a) only reflexive.\t (b) only symmetric. Which among (i) and (ii) is true? \t\t(a) (i) alone is true \t\t(c) only transitive.\t (d) equivalence. \t\t(b) (ii) alone is true \t\t(c) Both (i) and (ii) are true \t27.\t In the following figure, which of the following can \t\t(d) Neither (i) nor (ii) is true be the value of n(A \u222a B \u222a C)? In the figure, 1, 2, 3, \u2026 represents the number of elements in the \t20.\t The relation \u2018is not equal to\u2019 is defined on the set respective regions. of real numbers is satisfies which of the following? \t\t(a) 22\t\t (b) 23 \t\t(a) Reflexive only\t (b) Symmetric only \t\t(c) 24\t\t (d) 25 \t\t(c) Transitive only\t (d) Equivalence A B\u00b5 PRACTICE QUESTIONS \t21.\t If R = {(a, b)\/|a + b|=|a|+|b|} is a relation on a 1 23 set {-1, 0, 1} then R is _______. x \t\t(a) reflexive\t\t (b) symmetric 56 7 \t\t(c) anti symmetric\t (d) equivalence C \t22.\t For all p, such that 1 \u2264 p \u2264 100, if n(Ap) = p + 2 and \t28.\t In a class, each student likes either cricket or foot- ball, 40% of the students like football, 80% of the \u2229100 students like cricket. The number of students who like only cricket is 40 more than the num- A1 \u2282 A2 \u2282 A3 \u2282 \u2026 \u2282 A100 and Ap = A, then ber of students who like only football. What is the strength of the class? p=3 n(A) = \t\t(a) 3\t\t (b) 4 \t\t(c) 5\t\t (d) 6 \t\t(a) 80\t\t (b) 100 2\t 3.\t If R = {(a, b)\/a + b = 4} is a relation on N, then \t\t(c) 120\t\t (d) 150 R is _____. \t\t(a) reflexive\t\t (b) symmetric 2\t 9.\t For all p, such that 1 \u2264 p \u2264 100, n(Ap) = p + 1 and \t\t(c) anti symmetric\t (d) transitive \u222a100 A1 \u2282 A2 \u2282 \u2026 \u2282 A100. Then Ap contains _____ p=1 \t24.\t Let R be a relation defined on S, the set of squares elements. on a chess board, such that xRy, for x, y \u2208 S, if x and y share a common side. Then, which of the \t\t(a) 99\t\t (b) 100 following is false for R? \t\t(c) 101\t\t (d) 102","Sets and Relations 6.23 Level 2 \t30.\t In a class, 70 students wrote two tests, viz., Test-I \t\t(a) 1\t\t (b) 2 and Test-II. 50% of the students failed in Test-I and 40% of the students failed in Test-II. How \t\t(c) 3\t\t (d) 4 many students passed in both the tests? \t38.\t \u2018aRb\u2019 if \u2018a is the father of b\u2019. Then R is ________. \t\t(a) 21\t\t (b) 7 \t\t(a) reflexive\t\t (b) symmetric \t\t(c) 28\t\t (d) 14 \t\t(c) transitive\t (d) None of these 3\t 1.\t Every man in a group of 20 men likes either \t39.\t Let A be a set of compartments of a train. Then mangoes or an apple. Every man who likes apples the relation R is defined on A as \u2018aRb if and only if also likes mangoes. 9 men like mangoes but not a and b have the link between them\u2019. Then which apples. How many like mangoes and apples? of the following is true for R? \t\t(a) 9\t\t (b) 11 \t\t(a) Reflexive\t (b) Symmetric \t\t(c) 10\t\t (d) 12 \t\t(c) Transitive\t (d) All of these \t32.\t In an election, two contestants A and B contested. 4\t 0.\t A relation R: N \u2192 N defined by R = {(x, y)\/y x% of the total voters voted for A and (x + 20)% = x2 + 1} is for B. If 20% of the voters did not vote, then find x. \t\t(a) one to one.\t (b) one to many. \t\t(a) 30\t\t (b) 25 \t\t(c) many to one.\t (d) many to many. \t\t(c) 40\t\t (d) 35 \t41.\t If R = {(a, b)\/|a + b|= a + b} is a relation defined on a set {-1, 0, 1} then R is ________. 3\t 3.\t If A = {1, 2, 3, 4}, then how many subsets of A contain the element 1 but not 4? \t\t(a) reflexive\t\t (c) symmetric \t\t(a) 16\t\t (b) 4 \t\t(c) anti symmetric\t (d) transitive \t\t(c) 8\t\t (d) 24 \t42.\t Set-builder form of the relation R = {(\u22122, \u22127), (\u22121, \u22124), (0, \u22121), (1, 2), (2, 5)} is \t34.\t A relation R: Z \u2192 Z defined by R = {(x, y)\/y = PRACTICE QUESTIONS x2 \u2212 1} is \t\t(a) R = {(x, y)\/y = 2x \u2212 3; x, y \u2208 Z} \t\t(b) R = {(x, y)\/y = 3x \u2212 1; x, y \u2208 Z} \t\t(a) one to one relation. \t\t(c) R = {(x, y)\/y = 3x \u2212 1; x, y \u2208 N} \t\t(d) R = {(x, y)\/y = 3x \u2212 1; \u22122 \u2264 x < 3 and x \u2208 Z} \t\t(b) many to one relation. \t\t(c) one to many relation. \t\t(d) many to many relation. 3\t 5.\t If a set A has 13 elements and R is a reflexive rela- \t43.\t A group of 30 men participate in a survey on lan- tion on A with n elements, n \u2208 Z+, then guage skills. The number of men who know both English and Hindi was equal to the number of \t\t(a) 13 \u2264 n \u2264 26\t (b) 0 \u2264 n \u2264 26 men who know neither of these languages. The number of men who know English is 4 more than \t\t(c) 13 \u2264 n \u2264 169\t (d) 0 \u2264 n \u2264 169 those who know Hindi. How many know Hindi? \t36.\t Example of an equivalence relation among the fol- \t\t(a) 11\t\t (b) 12 lowing is \t\t(a) is a father of\t (b) is less than \t\t(c) 13\t\t (d) 14 \t\t(c) is congruent to\t (d) is an uncle of 4\t 4.\t In a locality, the number of people buying only The Times of India is 80% of the number of \t37.\t If A = {p \u2208 N; p is a prime and p= 7n2 + 3n + 3 people buying both The Times of India and The n Hindu. The number of people buying only The Hindu is 60% less than the number who buy both. for some n \u2208 N}, then the number of elements in the set A is","6.24 Chapter 6 The number of people buying neither of these \t\t(a) BADC\t\t (b) BDCA is 22,000 less than the number of people in the locality. How many people buy both newspapers? \t\t(c) BCAD\t\t (d) BDAC \t\t(a) 10,000\t\t (b) 20,000 \t48.\t The number of subsets of {{a}, {b, c}, d, e} is ______. \t\t(c) 25,000\t\t (d) 30,000 \t\t(a) 32\t\t (b) 16 4\t 5.\t Find the number of subsets of A \u00d7 B, if n(A) = 2 \t\t(c) 8\t\t (d) 20 and n(B) = 4. 4\t 9.\t If R = {(a, a), (a, c), (b, c), (b, b), (c, c), (a, b)} on \t\tThe following are the steps involved in solving the the set X = {a, b, c}, then how many subsets of R above problem. Arrange them in sequential order. are reflexive relations? \t\t(A) The number of elements in A \u00d7 B is 4 \u00d7 2 = 8. \t\t(a) 15\t\t (b) 16 \t\t(B) T\u0007 he number of subsets of a set with n elements \t\t(c) 8\t\t (d) 9 = 2n. \t\t(C) Given n(A) = 2 and n(B) = 4. 5\t 0.\t The relation R = {(2, 2), (1, 1), (1, 3), (3, 1)} on the set A = {1, 2, 3} is ______. \t\t(D) \\\\ Required number of subsets is 28 = 256. \t\t(a) CBAD\t\t (b) CABD \t\t(a) reflexive\t\t (b) symmetric \t\t(c) CDAB\t\t (d) CBDA \t\t(c) transitive\t (d) Both (b) and (c) \t46.\t If m = {1, 2, 3, 4, 5, 6} and A = {1, 2, 3, 4} and 5\t 1.\t Which of the following statement(s) is\/are true? B = {3, 4, 5, 6}, then find Ac \u2229 Bc. The follow- ing are the steps involved in solving the above \t\t(A) Every subset of an infinite set is infinite. problem. Arrange them in sequential order. \t\t(B) Every set has a proper subset. \t\t(A) Ac = {5, 6} and Bc = {1, 2} \t\t(C) Number of subsets of every set is even. \t\t(B) Ac \u2229 Bc = {5, 6} \u2229 {1, 2} \t\t(D) Every subset of a finite set is finite. \t\t(C) We know that Ac = m - A and Bc = m - B. \t\t(a) A and B\t\t (b) A, B and C \t\t(D) Ac \u2229 Bc = f PRACTICE QUESTIONS \t\t(c) B, C and D\t (d) D \t\t(a) CBAD\t\t (b) CDBA 5\t 2.\t If A and B are two non empty sets and n(A \u00d7 B) = 36, then which of the following cannot be equal to \t\t(c) CABD\t\t (d) CADB n(B)? 4\t 7.\t If A = {1, 2} and B = {2, 3}, then find the number \t\t(a) 9\t\t (b) 6 of elements in (A \u00d7 B) \u2229 (B \u00d7 A). The following are the steps involved in solving the above problem. \t\t(c) 8\t\t (d) 12 Arrange them in sequential order. 5\t 3.\t If the number of reflexive relations defined on a \t\t(A) (A \u00d7 B) \u2229 (B \u00d7 A) = {(2, 2)} set A is 64, then the number of elements in A is \t\t(B) Given A = {1, 2} and B = {2, 3} ______. \t\t(C) n[(A \u00d7 B) \u2229 (B \u00d7 A)] = 1 \t\t(D) \u0007A \u00d7 B = {(1, 2) (1, 3) (2, 2) (2, 3)} and B \u00d7 A \t\t(a) 3\t\t (b) 2 = {(2, 1), (2, 2), (3, 1), (3, 2)} \t\t(c) 6\t\t (d) 5 Level 3 neither of the two. The number of families who got new houses is 6 greater than the number of \t54.\t In a rehabilitation programme, a group of 50 fami- families who got compensation. How many fami- lies were assured new houses and compensation lies got houses? by the government. Number of families who got both is equal to the number of families who got","Sets and Relations 6.25 \t\t(a) 22\t\t (b) 28 5\t 8.\t There are a total of 70 ladies who watch at least \t\t(c) 23\t\t (d) 25 one of the channels, i.e., Zee TV, Sony TV and Star Plus. The total number of ladies who watch \t55.\t In an office, every employee likes at least one of Zee or Sony but not Star plus, the number of tea, coffee and milk. The number of employees ladies who watch Sony or Star Plus but not Zee who like only tea, only coffee, only milk and all and the number of ladies who watch Star Plus or the three are all equal. The number of employ- Zee but not Sony is 90. How many ladies watch at ees who like only tea and coffee, only coffee and least two of these channels if 10 ladies watch all the milk and only tea and milk are equal and each is three channels? equal to half the number of employees who like all the three. Then a possible value of the number of \t\t(a) 25 employees in the office is _________. \t\t(b) 30 \t\t(a) 65\t\t (b) 90 \t\t(c) 40 \t\t(c) 77\t\t (d) 84 \t\t(d) 35 \t56.\t In a school, on the Republic day, three dramas A, \t59.\t If R = {(x, y)\/x \u2208 W, y \u2208 W, (2x + y)2 = 49}, then B and C are performed on the dais. In a group of R\u22121 is ______. people, who attended the function and who like at least one of the three dramas, 16 people like A, 20 \t\t(a) {(5, 1), (3, 2), (1, 3)} people like B, 15 people like C, 4 people like both A and B, 3 people like both A and C, 3 people like \t\t(b) {(7, 0), (5, 1), (3, 2), (1, 3)} both B and C and 2 people like all the three. Then how many people like at most two? \t\t(c) {(7, 0), (1, 5), (2, 3), (1, 3)} \t\t(d) {(0, 7), (5, 1), (3, 2), (1, 3)} \t\t(a) 59\t\t (b) 41 6\t 0.\t Which of the following cannot be the number of reflexive relations defined on a set A? \t\t(c) 4\t\t (d) 6 \t\t(a) 1\t\t (b) 4 5\t 7.\t The students of a class like at least one of the games out of Chess, Caroms and Judo. The number of \t\t(c) 4096\t\t (d) 512 students who like only Chess and Caroms, only Caroms and Judo, only Chess and Judo and the \t61.\t In a class, the number of students who like only PRACTICE QUESTIONS number of those who like all the three are equal. Chess, only Caroms, both the games and neither The number of students who like only Chess, only of the games are in the ratio 2 : 4 : 1 : 3. The Caroms, only Judo and the number of those who number of students who like at least one of these like all the three are equal. A possible value of the games is 120 more than those who like neither number of students in the class is of the games. Find the number of students in the class. \t\t(a) 30\t\t (b) 40 \t\t(a) 300\t\t (b) 240 \t\t(c) 50\t\t (d) 70 \t\t(c) 270\t\t (d) 360","6.26 Chapter 6 TEST YOUR CONCEPTS 1\t 5.\t f \t16.\t 10000 Very Short Answer Type Questions 1\t 7.\t 0 1\t 8.\t 20 \t1.\t {1, 2, 3, 4, 5, 6, 7, 8, 9} 2\t 0.\t 218 \t2.\t {1, 3, 4, 6, 7} \t22.\t 13 \t3.\t {2, 5} \t23.\t 0 \t4.\t X = Y 2\t 4.\t 17, 0 \t5.\t f 2\t 5.\t 378 \t6.\t f \t26.\t 216 \t7.\t True \t27.\t E = {(a, a), (x, x), (y, y), (d, d)} \t8.\t 32 \t28.\t R = {(x, y)\/(x, y) \u2208 B \u00d7 A, x > y} \t9.\t False \t29.\t R = {(3, 6), (5, 10), (6, 12), (8, 16)} 1\t 0.\t 12 \t30.\t 240 1\t 1.\t \u00b5 \t12.\t P, Q \t39.\t (i) P = {0} \t13.\t 30 \t\t(ii) S = {1, 13} 1\t 4.\t 9 \t40.\t True \t41.\t anti-symmetric property ANSWER KEYS Short Answer Type Questions 4\t 2.\t one to one type of relations 4\t 3.\t 13 \t31.\t {13, 14} 4\t 4.\t 10 3\t 2.\t {14} \t45.\t 38% \t33.\t a = 7 \t34.\t disjoint 4\t 8.\t only symmetric property \t35.\t 20 \t49.\t 4 \u2264 x \u2264 216 \t36.\t 25 \t50.\t 35% \t37.\t 50 \t38.\t x = 3 Essay Type Questions \t46.\t (i)\t63\t\t (ii) 124 \t\t(iii)\t6\t\t (iv) 130 \t\t(v)\t120\t\t (vi) 67 \t47.\t reflexive, symmetric, transitive"]