Math All chapters MCQ with answers

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Number System QUESTIONS 1. For P  N, 34P  24P is always divisible by ____ (a) 15 (b) 5 (c) 13 (d) Both (b) and (c) (e) None of these 2. The greatest number of 5 digits exactly divisible by 15, 24 and 36 is ____ (a) 99620 (b) 99720 (c) 99968 (d) 99960 (e) None of these 3. The greatest number of 6 digits exactly divisible by all the numbers between 1 and 10 (both inclusive) is _________ (a) 997920 (b) 999768 (c) 999660 (d) 999760 (e) None of these 4. The smallest three digit number which leaves remainders 8 and 12 when divided by 28 and 32 respectively, is _________ (b) 222 JOIN @iitwale in Telegram (a) 102 (c) 202 (d) 204 (e) None of these 5. If P = n - 1+ n+1 where n is a positive integer then the value of P is (a) a rational number (b) not a rational number (c) an integer (d) a natural number (e) None of these 6. The largest number that will divide 398, 606 and 474 leaving remainders 7, 11 and 15 respectively is ________ (a) 52 (b) 26 (c) 17 (d) 18 (e) None of these 7. Which one among the following statements is true? (a) The remainder when the square of any number is divided by 4 is 1 or 0. (b) There is no natural number for which 4 ends with digit zero. (c) A positive integer n is prime, if no prime p less than or equal to n divides n. (d) All the above (e) None of these

8. The unit value of 6100 - 5100 is ______ (b) 1 (a) 0 (d) 3 (c) 2 (e) None of these 9. For any odd natural number n, ( 3)4n +( 2)4n is always divisible by ______ (a) 5 (b) 7 (c) 17 (d) 13 (e) None of these 10. If I is a positive integer then (I)2 will be in the form of ———— (a) 4m for some integer m (b) 8m for some integer m (c) 4m+1 for some integer m (d) Both (a) and (c) (e) None of these 11. Which among the following statements is not true? (a) The square of any odd integer is of the form 4q + 1, for some integer q. (b) For any odd integer p, p2 - 1 is divisible by 8. (c) If p and q are both oddJpoOsiItiNve i@ntegiietrws/ athelen pin2 Tqe2 liseegvernam and divisible by 4. (d) For any natural number n, 12n cannot end with the digit 0 or 5. (e) None of these 12. For any natural number n, (2n  1)2  1 is always divisible by ______ (a) 2 (b) 4 (c) 8 (d) All the above (e) None of these 13. Which of the following statements is always true? (a) The sum or difference of a rational and an irrational number is rational. (b) Every irrational number is a surd. (c) The product or quotient of a non-zero rational number and an irrational number is irrational. (d) All the above (e) None of these 14. The value of (27)3p - (13)3p ends in ______ (where p is a natural number) (a) 0 (b) 4 (c) 6 (d) Either (b) or (c) (e) None of these 3

15. If P  1 3  5  7.............21 and Q  2  4  6  8 10............22 then HCF of P and Q is ________ (a) 12375 (b) 14175 (c) 825 (d) 925 (e) None of these 16. In a seminar, the numbers of participants in science, English and Mathematics are 144, 180 and 192 respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject. (a) 38 (b) 40 (c) 43 (d) 45 (e) None of these 17. Without actually performing the Song division, choose which among the following rational numbers will not have a terminating decimal expansion. (a) 123 (b) 351 16 27  58  718 (c) 32 (d) 833 28  59 49  27 (e) None of these 18. The largest number that divides 588, 1999 and 1650 leaving 3, 10 JanOd I1N2 r@espiietcwtiavelley iisn__T__e_l_egram (a) 117 (b) 109 (c) 27 (d) 43 (e) None of these 19. The decimal expansion of the rational number 12879 will terminate after: 1250 (a) One decimal places (b) Two decimal places (c) Three decimal places (d) Four decimal places (e) None of these 20. Find the greatest prime factor in 527527. (a) 17 (b) 11 (c) 13 (d) 31 (e) None of these 21. If p is a single digit natural number and the unit digits of p4 and p are same, then how many possibilities p can assume? (b) 3 (a) 2 (d) 5 (c) 4 (e) None of these 4

22. The sum of LCM and HCF of two numbers is 29610. If their LCM is 140 times v the HCF of the numbers then which among the following can be one of the numbers? (a) 330 (b) 1470 (c) 525 (d) 462 (e) None of these 23. The value of 223m  283 m ends in ______ {M  N} . (a) 8 (b) 2 (c) 6 (d) 0 (e) None of these 24. If LCM and HCF of two numbers are 324 and 18 respectively, then how many such pairs of numbers are possible? (a) 0 (b) 1 (c) 2 (d) 3 (e) None of these 25. If p = 11+ 5, q = 14 + 2 and r = 13 + 3 then which one of the following holds true? (a) p > q > r (b) p < q < r (c) p > r > q (d) p < r < q JOIN @iitwale in Telegram (e) None of these 26. The number of ways, in which 360 can be resolved in two factors, is ______ (a) 24 (b) 18 (c) 12 (d) 15 (e) None of these 27. If u  16 7  16 5, v  7  5, w  8 7  8 5, x  16 7  16 5, and y  4 7  4 5, then which one of the following is a rational number? (a) uvxy (b) uvwxy (c) uxwy (d) vwxy (e) None of these 28. A mason has to fit two bathrooms with square marble tiles of the largest possible size. The dimensions of each such bathroom are 12 fts and 10 fts. If the size of the tiles in inches has to be taken then number of such tiles required is ____ (a) 15 (b) 30 (c) 60 (d) 80 (e) None of these 29. If HCF of 374 and 255 is H and H = 255m + 374n then the value of m - n is equal to _____ (a) 3 (b) 4 (c) 5 (d) 1 (e) None of these 5

30. Choose which one among the following statement is incorrect? (a) HCF of two co-primes a and b is 1. (b) LCM of two co-primes m and n is mn. (c) By using Euclid's division lemma for two numbers 155 and 345, we get 345 = 155 × 2 + 35. (d) The remainder, when the square of any prime number greater than 3 is divided by 6, is 1. (e) None of these 31. If LCM and HCF of two numbers are 3003 and 21 respectively, then how many such numbers are possible? (a) 0 (b) 1 (c) 2 (d) 3 (e) None of these 32. The largest number which divides 1288 and 2915 and leaves the remainders 1 and 8 respectively, is H and it satisfies the expression, H = 45m + 288n. Find the value of m + n. (a) 11 (b) 15 (c) 13 (d) 10 (e) None of these 33. The smallest number, which when increased by 19 is exactly divisible by both 2079 and 1404, is _______ JOIN @iitwale in Telegram (a) 6200 (b) 625 (c) 6218 (d) 3208 (e) None of these 34. 7+4 3 equals to ________ 2 (a) 2  6 (b) 2 2 6 2 (c) 2 6 (d) 32 2 4 (e) None of these 35. If HCF of the numbers (3600, x) = 20 then how many values are possible for x? (where it is assumed that x is a product of a power of 2 and a power of 5 only) (a) One (b) Two (c) Three (d) Four (e) None of these 36. The number of ways, in which 576 can be resolved into two factors, is ________ (a) 8 (b) 9 (c) 10 (d) 11 (e) None of these 6

37. Four runners P, Q, R and S start running around a circular track simultaneously. If they complete one round in 16, 12, 24, 18 minutes respectively, after how much time they will meet next? (a) 2 hours 20 minutes (b) 2 hours (c) 3 hours 18 minutes (d) 2 hours 24 minutes (e) None of these 38. If LCM and HCF of two numbers are equal, then the numbers will be _________ (a) Composite (b) Prime (c) Equal (d) Co-prime (e) None of these 39. If the product of two numbers is 149058 and HCF of these numbers is 21 then how many pairs of these numbers are possible? (a) 1 (b) 2 (c) 3 (d) 4 (e) None of these 40. If A  14  (1 2 3  4  5  ................10 14) and B  19  (1 2 3  4  5 ................10 19) then which one of the following is/are correct? (i) B  A is a prime number. JOIN @iitwale in Telegram (ii) B  A is a composite number. (iii) A is a composite number. (iv) B is a prime number. (a) Both (i) and (ii) (b) Both (ii) and (iii) (c) Both (iii) and (iv) (d) All (i), (ii), (iii) and (iv) (e) None of these 7

1. (d) 2. (b) ANSWER - KEY 4. (d) 5. (b) 6. (c) 7. (d) 11. (c) 12. (d) 3. (a) 9. (d) 10. (d) 16. (c) 17. (b) 8. (b) 21. (b) 22. (b) 13. (c) 14. (d) 15. (b) 26. (c) 27. (b) 18. (a) 31. (c) 32. (a) 23. (d) 19. (d) 20. (d) 36. (d) 37. (d) 28. (c) 33. (c) 24. (c) 25. (c) 38. (c) 29. (c) 30. (e) 34. (b) 35. (b) 39. (b) 40. (b) JOIN @iitwale in Telegram 8

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Polynomials QUESTIONS 1. If the zeroes of the quadratic polynomial x2  a  3 x  b are 3 and - 4, then (a) a  2, b  6 (b) a   2, b  12 (c) a  3, b  4 (d) a  4, b   3 (e) None of these 2. See the graph given below: (i) (ii) (iii) JOIN @iitwale in Telegram Based on these graphs, identify the correct statement among the following: (a) In case ii, the quadratic polynomial a x2  bx  c has two distinct zeroes. (b) In case i, the quadratic polynomial ax2  bx  c has two equal zeroes. (c) In case iii, the quadratic polynomial has no zero. (d) All the above (e) None of these 3. Which one among the following statements is incorrect? (a) Graph of a linear polynomial is a straight line whereas the graph of a quadratic polynomial has one of the two shapes of parabola either open upwards or open downwards . (b) The shape of the parabola depends on the value of 'a' of the quadratic polynomial ax2  bx  c . (c) The zeroes of a quadratic polynomial ax2  bx  c , a  0 are y coordinates of the points where the parabola y  ax2  bx  c intersects the y-axis. (d) A real number m is a zero of the polynomial p(x) if p (m) = 0 (e) None of these 4. A polynomial of degree n has ________ (a) two zeroes (b) n zeroes (c) atleast n zeroes (d) atmost n zeroes (e) None of these

5. If one zero of the quadratic polynomial x2  5x  k is 3 then second zero of this polynomial is ______ (a) 5 (b)  3 (c)  5 (d)  8 (e) None of these 6. If the zeroes of a quadratic polynomial ax2  bx  c are both negative, then (a) a is positive., b and c are negative (b) a is negative/ b and c are positive (c) a and c are negative, b is positive (d) a, b and c all have the same sign (e) None of these 7. If (3 + 3 ) is one of the zeroes of the quadratic polynomial x2 + mx + 6 then find the second zero. (a)  3 (b) 3  3 (c) 3  3 (d) 3 (e) None of these 8. For a quadratic polynomial 2 x2 - 8x + b, sum of its roots is 4 and one of the roots is 4+ 2 , then 2 the value of b is______ JOIN @iitwale in Telegram (a) 3 (b) 6 (c) 7 (d) 8 (e) None of these 9. If the zeroes of the quadratic polynomial p(x)  abx2  (b2  ac)x bc are  &  , then (a)   b2 and   c2 (b)   a and   b a b b c (c)   b and   c (d)   a and   c a b b b (e) None of these 10. If  and  are zeroes of the quadratic polynomial p x  ax2  bx  c, then the value of    is  _________ . (a) b2  ac (b) b2  ac ac ac (c) b2  2ac (d) c2  2ac ac ac (e) None of these 3

11. If the zeroes of the quadratic polynomial ax2  x  b are 3 and 5 , then 2 3 (a) a = 15, b = 6 (b) a = 6, b = 15 (c) a = 12, b = 4 (d) a = 4, b = 12 (e) None of these 12. If p(x)  25x2  15x  a where  and  are the zeroes of the polynomial, also if it is given that 3  3  63 , then 125 (a) a = 5 (b) roots are 1 and 4 5 5 (c) a = 3 (d) roots are 1 and 2 5 5 (e) None of these 13. If two zeroes of the cubic polynomial px2  qx2  rx  s are 0, then the third zero is _________ (a) p (b) p q q (c)  p (d) 0 q JOIN @iitwale in Telegram (e) None of these 14. If one of the zeroes of a cubic polynomial of the form x3  ax2  bx  c is the negative of the other, then (a) a is of negative sign and b and c are of positive sign (b) b is of negative sign and a and c are of positive sign (c) a and c are of opposite signs and b is of negative sign (d) a and b are of opposite signs and c is of positive sign (e) None of these 15. If all the zeroes of the cubic polynomial x3  c x2  d x  b are equal, then (a) cd = 9 b (b) bd = 8 b (c) cd = 6 b (d) bd = 8 b (e) None of these 16. If p and q are the of the polynomial bx2  cx  a, value of 1 + 1 p3 q3 (a) 3abc  c3 (b) 3abc  c3 ab2 ab2 (c) 3abc  c3 (d) 3abc  c3 a2b a2b (e) None of these 4

17. If the zeroes of the polynomial 6x2  7 3x  15  0 are  &  , then (a)   3 &   53 2 3 (b)    3 &   5 3 (c)   3 &   5 3 2 3 (d)   5 3 &    3 (e) None of these 18. If  &  are the zeroes of the quadratic polynomial 3x2  11x  6, then find the polynomial whose zeroes are (2  ) and (  2) (a) k  x2  5x  270  , k is any non-zero real number  9  (b) k  x2  11x  260  , k is any non-zero real number  9  (c) k(3x2  3x  26), k is any non-zero real number (d) k(2x2  5x  27), k is any non-zero real number JOIN @iitwale in Telegram (e) None of these 19. If ,  &  are the roots of the equation x3  4x2  53x  168 then the relation between their roots is _______ (a) 3    2 (b) 3  4  4 (c) 3    4 (d)   2  (e) None of these 20. What must be subtracted from 6x4  16x3  15x2  8x  9, so that it is exactly divisible by 3x2  5x  2 ? (a) 19x  15 (b) 19x  16 (b) 13x  19 (d) 19x  15 (e) None of these 21. If p(x)  x3  10x2  31x  30 and q(x)  x3  12x2  41x  42 , then find the LCM of the polynomials p(x) and q(x). (a) x4  17x3  101x2  247x  210 (b) x3  36x2  90x  105 (c) x4  18x3  95x2  234x  119 (d) x3  18x2  108x  114 (e) None of these 5

22. What should be added to 1 to get 1 x2 - 12x + 32 x2 - 11x + 30 (a) 2x2  25x  96 (b) 2x2  25x  66 (x  6)(x  5)(x 4)(x 8) (x  6)(x  5)(x 4)(x 8) (c) 2x2  25x  66 (d) 2 (x  6)(x  5)(x 4)(x 8) (x  6)(x  5)(x 4)(x 8) (e) None of these 23. If p(x)  x2  x  1 and q(x)  x3  x  1, then the HCF of p(a) - p(b) and q(a) (a) a  b  1 (b) a – b  1 (c) a  b (d) a  b (e) None of these 24. If (x2  x  1) is a factor of x4  9x3  qx2  8x  5 then find the values of p and q. (a) p   3, q  4 (b) p  4, q   3 (c) p  2, q   4 (d) p   4, q  2 (e) None of these 25. If the zeroes of the algebraic expression 3ax2  x(3b  5a)  5b are 3 aJnOd I3N5 , @theinitfwinad lteheinvalTueeolef bagr.am 7 (a) 1 (b) 4 3 5 (c) 7 (d) 3 3 (e) None of these 26. If degree of both p(x) and [p(x) + q(x)] is 15 then degree of q(x) can be (a) 12 (b) 10 (c) 15 (d) any one of the above (e) None of these 27. If the LCM of p(x) and q(x) is a9  b9 then their HCF can be (a) (a - b)  (b) a2  b2  ab (c) a6  b6  a3b3 (d) All the above (e) None of these 28. If m a 1 and n a  1 , then m2  n2  3mn is equal to a 1 a  1 (a) a4  18a2  1 (b) a4  9a2  3 a4  2a2  1 a4  2a2  1 (c) a4  9a2  3 (d) a4  16a2  1 a4  2a2  1 a4  2a2  1 (e) None of these 6

29. Solve p2(q - r)2 + q2 - (p - r)2 + r2 - (p - q)2 (p+ r)2 - q2 (p  q)2 - r2 (q+ r)2 - p2 (a) 1 (b) p + q + r pqr (c) 0 (d) 1 (e) None of these 30. Find the value of a - b so that 8x4  14x3  ax2  bx  2 is exactly divisible by 4x2  3x  2 . (a) 4 (b) 6 (c) 9 (d)  3 (e) None of these 31. If two zeroes of the polynomial f(x)  x4  2x3  18x2  6x  45 are  3 and 3 , then find the sum of other two zeroes. (b)  1 (a) 0 (d) 1 (c)  2 (e) None of these 32. If the zeroes of the polynomial x3  15x2  66x  80 are ,  &  and it is also given that 2     then (b)   3 JOIN @iitwale in Telegram (a)   4 (c)   7 (d)   2 (e) None of these 33. If  &  are the zeroes of the polynomial x2  6x  k such that 2    11 then k is equal to (a) 18 (b)  23 (c) 391 (d)  391 (e) None of these 34. If p and q are the zeroes of the quadratic polynomial f(x)  cx2  ax  b then the value of p4  q4 is ______ (a) (a2  2bc)2  b2c2 (b) (a2  2bc)2  2b2c2 c4 c4 (c) (b2  2ac)2  a2c2 (d) (b2  2ac)2  2a2c2 c4 c4 (e) None of these 35. If on dividing the polynomial f(x)= x3 - 4x2 +7x - 9 by a polynomial g(x), the quotient q(x) and the remainder r(x) are (x  3) and (2x  3) respectively, the polynomial g(x) is ____ (a) x2  x  1 (b) x2  x  2 (c) 2x2  x  1 (d) 2x2  x  2 (e) None of these 7

36. If the zeroes of the polynomial f(x)  ax3  3bx2  3cx  d are in A.P. then 2b3  a2d is equal to _______ (a) a2bc (b) 3abc (c) 2b2ac (d) abc (e) None of these 37. If f(x) = 3x4 + 6x3 - 2x2 - l0x - 5 and two of its zeroes are - 1, - 1, then the other two zeroes are _______ (a) 3 ,  3 (b) 2 ,  2 5 5 5 5 (c) 5 ,  5 (d) 5 ,  5 3 3 4 4 (e) None of these 38. If , ,  are the zeroes of the polynomial p(x)  x3  ax2  bx  c, then 1  1  1  __________    (a) a (b) b b c (c) a (d) c c d (e) None of these 39. The graph of a polynomial f(x) is shown below: JOIN @iitwale in Telegram The number of real zeroes of the polynomial f(x) is _________ (a) 1 (b) 2 (c) 3 (d) 4 (e) None of these 40. The graph of a polynomial p(x)  ax2  bx  c is shown below: Based on the above graph which one is correct? 8

(a) a > 0, b > 0, c > 0 (b) a < 0, b < 0, c > 0 (c) a < 0, b < 0, c < 0 (d) a > 0, b < 0, c > 0 (e) None of these ANSWER – KEY JOIN @iitwale in Telegram 1. (b) 2. (c) 3. (c) 4. (d) 5. (d) 6. (d) 7. (b) 8. (c) 11. (b) 12. (b) 13. (c) 9. (c) 10. (b) 16. (e) 17. (c) 18. (b) 21. (a) 22. (c) 23. (c) 14. (c) 15. (a) 26. (d) 27. (d) 28. (a) 31. (c) 32. (d) 33. (c) 19. (c) 20. (a) 36. (b) 37. (c) 38. (c) 24. (b) 25. (b) 29. (d) 30. (c) 34. (b) 35. (b) 39. (d) 40. (d) 9

Surface Area and Volume 1. A cuboidal metal of dimensions 9. From a circle of radius 15 cm, a sector 44 cm30 cm 15 cm was melted and cast central angle 216 is cut and its bounding radii are joined without overlap so as to into a cylinder of height 28 cm. Find its form a cone. Find its volume. radius. (a) 20cm (b) 15cm (c) 10cm (d) 18cm (a) 1081.3cm3 (b) 1071.3cm3 (c) 1018.3cm3 (d) 1061.9cm3 2. The length of the longest rod which can, be 10. A cylindrical vessel of diameter 9 cm has kept inside a rectangular box is 27 cm. If the length and the breadth of the box are 23 cm some water in it. A cylindrical iron piece of diameter 6 cm and height 4.5 cm is dropped and 10 cm respectively, find its height. in it. After it was completely immersed, what (a) 8cm (b) 10cm (c) 12cm (d) 14cm is the rise in the level of water? (a) 0.8 cm (b) 0.5 cm (c) 0.1 cm (d) 0.3 cm 3. The volume of a cube is numerically equal to its surface area. What is the length of its side? 11. The diameter of a copper sphere is 6 cm. It is beaten and drawn into a wire of diameter (a) 6 units (b) 8 units 0.2 cm. What is the length of the wire? (c) 9 units (d) 10 units (a) 36cm (b) 360cm (c) 3600 cm (d) 306 cm 4. The total surface area of a cylinder is 220 sq cm with height 6.5 cm. Find its volume. 12. A cylindrJicOal IvNess@el iiotfwdaialemeitnerT4elcemgrias m (a) 25.025 cm3 (b) 2.5025 cm3 partly filled with water. 300 lead balls are (c) 2502.5 cm3 (d) 250.25 cm3 dropped in it. The raise in water level is 0.8 5. The lateral surface area of a cylinder is 176 cm. Find the diameter of each ball. cm2 and its base area is 38.5cm2 .What is its (a) 0.8 cm (b) 0.4 cm (c) 0.2 cm (d) 0.6 cm volume? 13. Three solid glass balls of radii 'r' cm, 6 cm and 8 cm are melted into a solid sphere of (a) 830 cm3 (b) 380 cm3 radius 9 cm. Find the value of 'r'. (c) 308 cm3 (d) 803 cm3 6. Identify the ratio of lateral surface areas of (a) 1 cm (b) 1 cm two cylinders with equal heights. 4 3 (a) H: h (b) R: r (c) 1 cm (d) 1 cm (c) 1: 2 (d) 2 : 3 2 7. A cone and a hemisphere have equal bases and equal volumes. Identify the ratio of their heights. 14. A hemispherical bowl is made of steel of (a) 1 : 3 (b) 1 : 2 0.25 cm thickness. The inner radius of the bowl is 5 cm. What is the volume of steel (c) 2 : 1 (d) 3 : 1 used? (a) 42.15 cm3 8. The area of the base of a cone is 616 sq cm. (b) 41.52 cm3 Its height is 48 cm. Find its total surface (c) 41.25cm3 area. (d) 43.21cm3 (a) 2816cm2 (b) 2861 cm2 (c) 2618cm2 (d) 2681 cm2

15. A hollow sphere of internal and external 21. A cylindrical vessel contains 49.896 litres of diameters 4 cm and 8 cm respectively is liquid. The cost of painting its curved melted into a cone of base diameter 8 cm. surface area at 2 paise/sq cm is ` 95.04. Find the height of the cone. Find its total surface area. (a) 14 cm (b) 12 cm (c) 16 cm (d) 18 cm (a) 5724 cm2 (b) 7524 cm2 (c) 5742 cm2 (d) 7254 cm2 16. A circus tent is in the form of a cone over a 22. The ratio of base radius and height of a cylinder. The diameter of the base is 9 m, the height of cylindrical part is 4.8 m and cone is 3 :4. If the cost of smoothening the curved surface area at 5 paise/sq cm is ` the total height of the tent is 10.8m. What 115.50, find the volume of cone. area of canvas is required for the tent? (a) 24.184 sq. m (b) 2418.4 sq. m (a) 12963cm3 (b) 12693cm3 (c) 241.84sq.rn (d) 24.164 sq. m (c) 12936cm3 (d) 12639cm3 17. If the radii of the circular ends of a conical 23. A vessel is conical in shape. If its volume is 33.264 litres and height is 72 cm, find the bucket of height 45 cm are 28 cm and 7 cm, cost of repairing its curved surface area at? what is the capacity of the bucket? 12/sq m. (a) 48150cm3 (b) 48510cm3 (a) ` 5.94 (b) ` 6.94 (c) 48105cm3 (d) 48015cm3 (c) ` 7.95 (d) ` 9.65 18. The height of a cone is 30 cm. A small cone 24. The cost of painting the curved surface area is cut off at the top by a plane parallel to the ovof luamceonoeJf OathteI5Ncposn@/ecmiiift2wiitssa`sllae3n5ti.n2h0eT,igefhintledisgt2hr5eam base. If its volume is 1 . the volume of 27 cm. cone, what is the height at which the section is made? (a) 1223cm3 (b) 1232cm3 (a) 10 cm (b) 15 cm (c) 1323cm3 (d) 1332cm3 (c) 20 cm (d) 19 cm 25. A piece of metal pipe is 77 cm long with 19. A top is of the shape of a cone over a inside diameter of the cross section as 4 cm. hemisphere. The radius of the hemisphere is If the outer diameter is 4.5 cm and the metal 3.5 cm. The total height of the top is 15.5 weighs 8 gm/cu cm, find the weight of pipe. cm. Find the total area of the top. (a) 2.057 kg (b) 20.57 kg (c) 205.7 kg (d) 2.097 kg (a) 214.5cm2 (b) 21.45cm2 (c) 215.4cm2 (d) 21.43cm2 26. A rectangular solid metallic cuboid 20. The base radius of a cylinder is1 2 . times its 9 cm8 cm  2 cm is melted and recast into 3 solid cubes each of side 2 cm. How many height. The cost of painting its curved such solid cubes can be made? surface area at 2 paise / cm2 is ` 92.40. Find (a) 20 (b) 18 (c) 12 (d) 15 the volume of the liquid. (a) 80850 cm3 27. The side of a solid metallic cube is 50 cm. It is melted and recast into 8000 similar solid (b) 80580 cm3 cubical dice. Find the side of each die. (a) 2.5 cm (b) 6 cm (c) 80508 cm3 (d) 800508 cm3 (c) 1.9cm (d) 5.2cm 3

28. Water flows at the rate of 5 km per hour 34. 56 circular plates, each of radius 5 cm and through a pipe of diameter 14 cm into a thickness 0.25 cm, are placed one above the rectangular tank which is 25 m long and 22 other to form a solid right circular cylinder. m wide. Find the time in which the level of Find the volume of the cylinder so formed. water in the tank rises by 21 cm. [Take (a) 900 cm3 (b) 885 cm3   22 .] 7 (c) 1100 cm3 (d) 1250cm3 (a) 2 hours 35. A semicircular piece of paper of radius 14 (b) 5 hours cm is rolled to form a cone of the largest (c) 3 hours possible size. Find the capacity of the cone. (d) 1.5 hours (a) 721.5cm3 (b) 645.10cm3 29. Two right circular cones X and Y are made (c) 449.64 cm3 (d) 622.37 cm3 such that X has a radius three times the 36. In a solid cylinder of height 10 cm and radius of Y and Y has a volume half the volume of X. What is the ratio of heights of diameter 8 cm, two equal conical cavities have been made at both its ends. If the X and Y? diameter of the cavity is 6 cm and height 4 (a) 5 : 8 (b) 4 : 7 (c) 2 : 9 (d) 3 : 5 cm, find the volume of the remaining solid in terms of  . 30. Rain water from a roof 22 m  20 m drains (a) 95 cm3 (b) 136 cm3 into a cylindrical vessel with diameter of (c) 216 cm3 (d)192  cm3 base 2 m and height 3.5 m. If the vessel is 37. A solid spJhOereINand@aisitowlidahleemiinspTheerelehgavream just full, find the amount of rainfall in cm. (a) 4.5 cm (b) 2.5 cm the same total surface area. Find the ratio of (c) 3 cm (d) 1 cm their volumes. 31. The diameter of a solid metallic sphere is 16 (a) 3 :1 (b) 3 3 : 5 cm. The sphere is melted and recast into 8 (c) 3 3 : 4 (d) 1: 3 similar solid spherical balls. What is the 38. A sphere, of diameter 12 cm, is dropped radius of the each of the balls? into a right circular cylindrical vessel, partly (a) 2 cm (b) 3 cm filled with water. If the sphere is completely (c) 4 cm (d) 6 cm immersed in water, the water level in the 32. A hemispherical bowl of internal diameter cylindrical vessel rises by 3 5 cm. Find the 36 cm is full of liquid. The liquid is to be 9 filled into cylindrical shaped bottles each of diameter of the cylindrical vessel. (a) 16cm (b) 18cm radius 3 cm and height 9 cm. How many (c) 12cm (d) 13cm bottles are required to empty the bowl? (a) 24 (b) 35 (c) 12 (d) 48 39. The largest right circular cone is carved out of a cube whose edge is of length 'p' units. 33. If the radius of the base of a right circular Find the volume of the cone. cylinder is halved, keeping the height same, (a)  p3 cu.units find the ratio of the volume of the reduced 12 cylinder to that of the original cylinder. (a) 1 : 4 (b) 2 : 3 (b)  p3 cu.units (c) 1 : 2 (d) 5 : 7 4 (c)  p3cu.units (d)  p3 cu.units 5 4

40. A right-angled triangle ABC, where B  90o 44. The radii of circular ends of a solid frustum , is rotated about BC. If BC = 16 cm and of a cone are 33 cm and 27 cm and its slant AC = 20 cm, find the volume of the right height is 10 cm. Find its total surface area. circular cone traced out by the triangle. (a) 1564.15cm2 (b) 7599.43cm2 (c) 6542.1 cm2 (d) 8265.14cm2 45. The ratio of the radii of two spheres is 3 : 2. Find the ratio of their volumes. (a) 25 : 4 (b) 5 : 16 (c) 9 :14 (d) 27 : 8 (a) 2413cm3 (b) 2311 cm3 46. A cylindrical can whose base is horizontal (c) 1254cm3 (d) 1725cm3 and of internal radius 3.5 cm contains sufficient water so that when a solid sphere is placed in the can, water just covers the 41. A bucket of height 16 cm made up of metal sphere. Given that the sphere just fits in the sheet is in the form of frustum of a right circular cone with radii of its lower and can, find the depth of water in the can before the sphere was put into it. upper ends as 3 cm and 15 cm respectively. (a) 2.3 cm (b) 5.1 cm What is the slant height of the bucket? (a) 15 cm (b) 20 cm (c) 1.5 cm (d) 3.2 cm (c) 5 cm (d) 12 cm 47. The internal and external radii of a 42. A right-angled triangular plate of smaller hemispherical metallic vessel are 7 cm and sides measuring 5 cm and 12 cm is rotated 10.5 cm [email protected] 1aclem3ionf Tthee lemgetraal m about the longest side to trace a double cone as shown in the figure. Find the ratio weighs 10 g, find the weight of the vessel. of the curved surface areas of the two cones. (a) 25.15 kg (b) 32.1 kg (c) 17.07 kg (d) 16.27 kg 48. A tent is made in the form of a frustrum A of a right circular cone surmounted by another right circular cone B. The diameter of the ends of the frustrum A are 8 m and 4 m, its height is 3 m and the height of the cone B is (a) 6 : 1 (b) 5 : 12 2 m. What is the area of the canvas required (c) 3 : 14 (d) 6 : 5 for the tent? 43. The largest possible cube is made from a (a) 86.75 m2 (b) 85.77 m2 wooden sphere of radius 6 3 cm cm. Find (c) 86.77 m2 (d) 87.67 m2 the surface area of the cube. 49. What is the maximum distance between any two points of a cube of side T units? (a) 864cm2 (b) 542cm2 (c) 735cm2 (d) 625cm2 (a) ( 2 1)l units (b) 2 l units (c) 3 l units (d) 2l units 5

Answer - Keys 1. B 2. B 3. A 4. D 5. C 6. B 7. C 8. A 9. C 10. B 11. C 12. B 13. D 14. C 15. A 16. C 17. B 18. C 19. A 20. A 21. B 22. C 23. A 24. B 25. A 26. B 27. A 28. D JO2I9N. @Ciitwale i3n0T. eleBgram 31. C 32. D 33. A 34. C 35. D 36. B 37. C 38. B 39. A 40. A 41. B 42. B 43. A 44. B 45. D 46. A 47. C 48. B 49. C 6

Solution 1. (b) Volume of cuboid = Volume of cylinder r  7 2  lbh   r2h  V   r2h  250.25cm3  r2  443015 7 22 28 Hence, volume of the cylinder  250.25 cm3 5. (c) Base area of a cylinder  38.5 cm2  r 15 cm Hence radius of the cylinder is equal to 15   r2  38.5cm2  r  7 cm cm. 2 2. (b) Given that the length of the longest rod Lateral surface area 176 cm2 = 27 cm Length of the box, l  23 cm  2 rh 176cm2  h  8cm Breadth of the box, b = 10 cm Let 'h' be the height of the box Volume   r2h  22   7 2 8 Length of the longest rod = Length of the 7  2  diagonal  l2  b2  h2  308 cm3 Hence, volume of the cylinder  308 cm3  27  (23)2  (10)2  h2 Squaring on both sides 6. (b) Not available  (27)2  (23)2  (10)2  h2 7. (c) Not available  h  100 10cm Hence, the height of the box = 10 cm JOIN @iitwale in Telegram 8. (a) Not available 9. (c) Not available 3. (a) Volume of a cube  l 3cu. units 10. (b) Not available Surface area of a cube  6l2sq. units, 11. (c) Diameter of sphere = 6 cm Diameter of wire =0.2 cm where 'l ' is the length of each side of a Volume of cylindrical wire = Volume of cube. sphere Given that the volume of a cube is numerically equal to its surface area  l3cu.units  6l2 sq.units   r2h  4 R3 3  l3 6  l  6units l2  h  4  22  33  7  1 The length of each side of the cube = 6 37 22 0.01 units.  3600 cm  Length of wire is 3600 cm. 4. (d) TSA  2 r(r  h) 220  2  22  r  r  13  12. (b) Volume of 300 spherical balls 7  2  = Volume of rise in water level  r  10, 7  300  4  22   d 3  22  2 2 4 2 3 7  2  7 5 Since radius cannot be — ve, 7

 d  2  0.4cm   r2h  49896 cm3 5 C.S.A.  9504 cm2 13. (d) Given that the radii of three solid glass 2 balls is 'r' cm, 6 cm and 8 cm, sum of the  2 rh  4752 volumes of the three glass balls   r2h  49896  4  r3  4  (6)3  4  (8)3 2 rh 4752 33 3  r  21and h  36 T.S.A.  2 r(h  r)  7524 cm2  4  (r3  63  83)cm3 3 22. (c) Not available The volume of the solid sphere of radius 23. (a) Not available 9cm 24. (b) Not available   4  (9)3  243 4 3 25. (a) Not available  243 4  4  (r3  728) 26. (b) Not available 3 27. (a) The side of the given metallic cube = 50  r 1 cm JOIN @iitwale in Telegram 14. (c) Not available Its volume  (50)3cm3 125000 cm3 15. (a) Not available Let ' x ' cm be the side of a cubical die. Then volume of 8000 dice 16. (c) Not available  8000 (x)3cm3 According to the problem, the volume of 17. (b) Not available 8000 cubical dice = The volume of the given solid metallic cube 18. (c) Not available   8000(x)3 125000  x  5  2.5 19. (a) Height of cone 15.5  3.5 12 cm 2 l  h2  r2 Hence, the side of the cubical die is 2.5 cm.  (3.5)2  (12)2 12.5cm 28. (d) Not available C.S.A.   rl 137.5 sq cm C.S.A. of hemisphere  2 r2  77 cm2 29. (c) Not available Total area of the top = 214.5 sq cm. 30. (b) Not available 20. (a) C.S.A.  2 22  h 5h  220h2 7 3 21 31. (c) Not available  9240  220h2 32. (d) Not available 2 21  V   r2h  80850cm3 21. (b) V  49.89610000 cm3 8

33. (a) Not available 45. (d) Not available 34. (c) Not available 35. (d) Not available 46. (a) Let the initial depth of water in the can 36. (b) Not available be xcm 37. (c) Not available 38. (b) Not available Then, volume of initial water column 39. (a)   r2h   (3.5)2  cm3 Volume of the sphere Volume of the cone  1  r2h  4  r3  4  (3.5)2 x cm3 3 33 2 Now, volume of initial water column +  Volume of the sphere = Volume of the cylinder up to the height of water level  1    p  p   p3 cu.units   (3.5)J2O xIcNm3@ 34iitw(3.a5)l3ecmin3 Telegram 3  2 12 40. (a) Given, h = 16 cm and l  20 cm.    7 2 7 cm3  2   x  4  3.5  7  x  14  7 33  r  l2  h2  202 162 cm  x  7  14  7  12 cm 33 Volume  1  r2h The depth of water in the can before the 3 sphere was put into it =2.3 cm  1  22 122 16cm3 47. (c) Not available 37  16896 cm3  2413cm3 48. (b) Not available 7 49. (c) Not available 41. (b) Not available 42. (b) Not available 43. (a) Not available 44. (b) Not available 9

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Co-ordinate Geometry QUESTIONS 1. PQOR is a rectangle whose vertices are Q (0, 7), O (0, 0) and R (- 5, 0). Find the length of its diagonal. (a) 84 (b) 8 (c) 12 (d) 74 (e) None of these 2. The point which lies on the perpendicular bisector of the line segmen joining the points P (- 3, - 5) and Q (3, 5) is (a) (-3, 4) (b) (2, -5) (c) (0, 0) (d) (-1, 4) (e) None of these 3. The points (- 5, 0), (0, 4) and (5, 1) are the vertices of a _______ (a) right triangle (b) isosceles triangle (c) equilateral triangle (d) scalene triangle (e) None of these 4. The points (4, 5), (6, 7) and (8, 9) are ________ (a) collinear JOIN @iitwale in Telegram (b) not collinear (c) the vertices of a right triangle (d) vertices of an isosceles triangle (e) None of these 5. If a point (x, y) is equidistant from the points (p + q, q - p) and (p - q, p + q), then ________ (a) qx  yp  0 (b) qx  yp (c) px  qy (d) px  qy  0 (e) None of these  6.  a , 2a  If M and N are the points whose coordinates are ap2, 2ap and  p2 p  respectively and S is the point (a, 0), then 1 + 1 is equal to _________ SM SN (a) a (b) ap (c) 1 (d) 1 a ap (e) None of these 7. The coordinates of circumcentre of the triangle whose vertices are (3, 6),(4, -4) and (3, -5) are (a)  3 , 0  (b)  3 , 1  2   2 2 

(c)  1 ,  3 (d)  1 , 3   2  3 2  (e) None of these 8. The ratio in which the line 2x  3y  9  0 the points (2, 3) and (4, 7) is _________ (a) 3 : 1 (b)  5 : 4 (c)  1 : 5 (d) 2 : 5 (e) None of these 9. The area of a triangle, whose vertices are (a, a - 2), (a + 2, a + 2) and (a + 3, a), is _________ (a) 1 sq unit (b) 2 sq units (c) 3 sq units (d) 4 sq units (e) None of these 10. If the points (p, 0), (0, q) and (1, 1) are collinear, then 1 + 1 equals to _______ p q (a) 1 unit (b) 2 unit (c) 3 unit (d) 4 unit (e) None of these 11. The distance between the points (cos ,sin,) and (cos , sin,) is _______ (a) 3 (b) 2 JOIN @iitwale in Telegram (c) 2 (d) 1 (e) None of these 12. If A  7 , m  is the midpoint of the line segment joining the points B (3, -2) and C (4, 6), then the value  3 5  of m is _______ (a) 4 (b) 10 (c)  4 (d)  8 (e) None of these 13. If the point A (-3, m) divides the line segment joining the points B (-8, -5) and C (1, - 4) in the ratio a : b, then m equals to (a) 5 (b) 40 3 3 (c) 40 (d) 30 9 7 (e) None of these 14. The ratio, in which the line 5x + 3y = - 9 divides the segment joining the points (2, 3) and (- 3, 4), is ________ (a) 4 : 3 (b)  4 : 3 (c)  14 : 5 (d)  14 : 3 (e) None of these 3

15. If a point A (-1, 3) divides internally the line segment joining B (3, 4) and ( in ratio 2 : 3, then the coordinates of point C is ______ (a) (2,-3) (b)  2, 3   2  (c)  1, 3  (d) (-2, 3)  2  (e) None of these 16. The points (6, 1), (8, 2), (9, 4) and (7, 3) are the vertices of a (a) trapezium (b) rectangle (c) square (d) rhombus (e) None of these 17. If the coordinates of mid-point of the sides of a triangle are (21, 7), (- 5, 8, 5) and (10, 8.5), then which among the following is not a vertex of this triangle? (a) (  16, 10) (b) (7, 6) (c) (36, 7) (d) (6, 7) (e) None of these 18. The third vertex of a triangle, if two of its vertices are (- 5, 2) and (1, - 3) and the centroid is at (- 2, 3), is JOIN @iitwale in Telegram (a) (- 2, 5) (b) (- 2, 10) (c) (0, 5) (d) (10, - 2) (e) None of these 19. The vertices of a  PQR are P (1, 2), Q (- 3, 2) and R (5 - 6). Which among the following is a true statement for this triangle PQR? (a) Circumcentre of  PQR is (- 1, - 4) (b) Circumradius of  PQR is 740 unit (c) The point where perpendicular bisectors of PQ and PR intersects, is the circumcentre of the triangle PQR. (d) All the above (e) None of these 20. Find the orthecentre of the triangle ABC having its vertices as A (- 2, - 3), B (2, 1) and C (5, - 2). (a) (  3, 5) (b) (  4, 6) (c) (2, 1) (d) (0, 0) (e) None of these 21. The equation of a line whose inclination is 60 an of 5 units on x-axis, is (a) 3x  y  3 (b) 3x  y  5 3 (c) x  3y  3 (d) x  3y  5 3 (e) None of these 4

22. The area of a triangle formed by the line 4x  3y  12  0 with the coordinate axes is __________ (a) 3 sq unit (b) 4 sq unit (c) 5 sq unit (d) 6 sq unit (e) None of these 23. The equation of a line which is parallel to the line 3x  5y  8  0 and making an intercept - 3 on x- axis is (a) 5y  3x  9 (b) 5y  3x  9 (c) 3x  5y  9 (d) 3x  5y  9 (e) None of these 24. Which among the following can be the vertices of an equilateral triangle? (a) (2, 3), (5, 4), (7, 8) (b) (3, - 6), (5,-9), (6, - 12) (c) (3,-3), (-3, 3), (3 3,3 3) (d) All the above (e) None of these 25. If the vertices of a triangle have integral coordinates, then the triangle cannot be (a) an equilateral triangle JOIN @iitwale in Telegram (b) a right triangle (c) an isosceles triangle (d) All the above (e) None of these 26. If three vertices (in order) of a parallelogram are (p + q, p - q), (2p + q, 2p q) and (p - q, p + q), then its fourth vertex is _________ (a) q,  q (b) 2q,  2q (c) q, q (d)  p, p (e) None of these 27. The equation of median drawn from the vertex P to the side QR of a  PQR, whose vertices are P (2, 3), Q (- 3, - 5) and R (6, 2), is _______. (a) 9x  y  18 (b) 9x  y  15 (c) 3x  8y  18 (d) 3x  8y  15 (e) None of these 28. A line 6x  5y  30 intersects the coordinate axes. Find the length of the smallest side of the triangle so formed by the axis. (a) 3 units (b) 4 units (c) 5 units (d) 6 units (e) None of these 5

29. The line segment joining the points (3, - 2) and (4, - 6) is divided by the line 2x  3y 11  0 in the ratio m: n externally at point (a, b). Which among the following is true for the given information? (a) m = 11 (b) n = 21 (c) a b  89 (d) All the above 10 (e) None of these 30. If the mid-points of sides of a  ABC are P (2, 5), Q (- 3, 8) and R (6, 12), then its vertices are ________ (a) (13, 10), (  9, 2), (2, 14) (b) (12, 9), (  8, 1), (1, 15) (c) (11, 9), (  7, 1), (1, 15) (d) (  12, 8), (  7, 2), (  2, 15) (e) None of these 31. The three medians of a  PQR intersect at a point S. If the medians are PM, QN & RT and the area of  PQR is 90 cm2 , then the area of the quadrilateral MSRN is ________ (a) 45 cm2 (b) 30 cm2 (c) 15 cm2 (d) 60 cm2 (e) None of these 32. The length of the side of a at the of the angle of Pythagorean trianJglOe oINf @8icitmw,a1l5ecimn 1T7eclmegthream hypotenuse is ________ (a) 5 5 cm (b) 6 4 cm 23 23 (c) 7 3 cm (d) 4 2 cm 23 23 (e) None of these 33. The length of the diagonal of a square, which can be inscribed in a right triangle of sides 5 cm, 12 cm and 13 cm, is ______ (a) 30 2 cm (b) 60 2 cm 17 17 (c) 15 cm (d) 10 2 cm (e) None of these 34. The equation of a line, which passes through (3, 4) and the product of whose intercepts on the coordinate axes is 48, is _________ (a) 6x  9y  48 (b) 8x  9y  48 (c) 4x  3y  24 (d) 3x  4y  24 (e) None of these 6

35. If the equations of three concurrent lines are px  6y  8  0, qx  5y  8  0 and rx  4y  8  0, then the value of p + r is __________ (a) q (b) 2q (c) 3q (d) 4q (e) None of these 36. The point (p, q) divides the line formed by joining the points (p + q, p + q) and (p - q, q - p) in the ratio _______ (a) p : q internally (b) q : p externally (c) 2 : 1 (d) 1 : 1 (e) None of these 37. The radius of the circle which passes through the origin (0, 6) and (6, 0) is (a) 12cm (b) 6 cm (c) 3 2 cm (d) 4 2 (e) None of these 38. Area of the region formed by the lines 3| x| 2|y| 6 is (a) 8 cm2 (b) 10 cm2 (c) 12 cm2 (d) 14 cm2 JOIN @iitwale in Telegram (e) None of these 39. The orthecentre of the triangle formed by the lines 5x  8y  10, 16x  10y  13 and y-axis, is _____ (a)  102 , 95  (b)  51 , 89   89 178   89 95  (c)  102 , 95  (d)  51 , 95   89 178   89 89  (e) None of these 40. The line which is perpendicular to 5x  2y  3  0 is _________ (a) 2x  5y  8  0 (b) 3x  5y  9 (c) 4x  3y  7 (d) 5x  2y  3 (e) None of these 7

1. (d) 2. (c) ANSWER - KEY 4. (a) 5. (b) 6. (c) 7. (b) 11. (b) 12. (b) 3. (d) 9. (d) 10. (a) 16. (d) 17. (b) 8. (c) 21. (b) 22. (d) 13. (c) 14. (c) 15. (b) 26. (c) 27. (b) 18. (b) 31. (b) 32. (a) 23. (a) 19. (d) 20. (c) 36. (d) 37. (c) 28. (c) 33. (b) 24. (a) 25. (b) 38. (c) 29. (d) 30. (c) 34. (c) 35. (b) 39. (a) 40. (a) JOIN @iitwale in Telegram 8

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Circles 1. PT is a tangent and PAB is a secant of the 6. If the four sides of a quadrilateral ABCD are circle meeting the circle at A and B. If PA = tangential to a circle, which of the following 4 cm and AB = 5 cm. What is the length of is true? PT? (a) AC  AD  BD  CD (a) 8 cm (b) 6 cm (b) AB  CD  BC  AD (c) 3 cm (d) 1 cm (c) AB  CD  AC  BD (d) AC  AD  BC  AB 2. In the given figure, AB is a tangent at P. 7. In the given figure, OA and OB are two radii of the circle. If PA and PB are tangents to the circle at A and B respectively which is the correct statement? Find APD . (b) EPB (a) PDE (d) PED (c) DPE 3. Two concentric circle have their centres at (a) AOB and APB are the O. OP = 4 cm and OQ = 5 cm. AB is a supplementary angles. chord of the circle and a tangent to the inner (b) PBO and APB are complementary circle at P. angles. JOIN @iitwale in Telegram (c) PBO and AOB are complementary angles. (d) Both (a) and (c) Find the length of AB. (b) 5 cm 8. AP is a secant and PT is a tangent to a (a) 6 cm circle. Lf PT=15cm and AB = 8BP, find (c) 8 cm (d) 9 cm AP. 4. Find the value of x. (a) 45 cm (b) 36 cm (c) 25 cm (d) 62 cm (a) 5 (b) 8 9. In the figure, find ABC . 2 3 (c) 6 (d) 9 5 4 5. What is the number of common tangents that can be drawn to two intersecting circles touching at two points? (a) 2 (b) 4 (a) 60 (b) 120 (c) 3 (d) 1 (c) 180 (d) 90

10. If OA and OB are radii and PA and PB are 14. AB is a diameter of a circle with centre O. tangents to the circle, which of the following Tangent ST touches the circle at C. If is quadrilateral OAPB? BAC  40 , find ACT . (a) Rectangle (b) Square (a) 50° (b) 30° (c) Parallelogram (c) 60° (d) 40° (d) Kite 11. In the given figure, a circle touches the side BC of ABC at P and touches AB and AC produced at Q and R respectively. If AQ = 5 cm, find the perimeter of ABC . 15. In the given figure, OA is a radius and AB is a tangent to the circle. (Note that BC is the shortest distance between B and the circle). (a) 6cm (b) 10cm (c) 12cm (d) 15cm 12. A tangent ST at P to a circle at P is parallel lf OA= 12cm and AB = 16 cm, find the to a chord QR of the circle. distance bJetOweINen B@ainitd(wbC)a.6lcemin (a) 7cm Telegram (c) 8cm (d) 12cm 16. An isosceles triangle in which AB = AC is inscribed in a circle. Which of the following is correct statement? Which of the following is true? (a) P is equidistant from the extremities of (a) Tangent at A is parallel to BC. the chord. (b) Tangent at A is perpendicular to AB. (b) PT is the tangent parallel to the chord (c) Tangent at A is parallel to AC. PQ. (d) Tangent at A is perpendicular to BC. (c) PT is the tangent parallel to the cord PR. (d) S is equidistant from the extremities of 17. If the distance between two circles of radii R the chord. and r is d, find the length of the direct 13. In the given figure.TP and TQ are tangents common tangent. to the circle. If PAQ  70 , what is PTQ ? (a) d 2  (R  r)2 (b) d 2  (R  r)2 (a) 30 (b) 45 (c) (R  r)2  d 2 (c) 60 (d) 40 (d) d 2  (R  r)2 3

18. A tangent ST, touches a circle at B.AB is a (22-25): OA and OB are radii and PA chord such that ABT  65 and PB are tangents of the circle, as shown in the given figure. Find AOB , where O is the centre of the circle. (b) 145 (a) 120 (c) 110 (d) 130 22. Find APB if OPB  25 . 19. In the given figure, two circles with centres A (a) 50 (b) 40 and B of radii 5 cm and 3 cm touch each other internally. (c) 70 (d) 25 23. If AP= 15 cm and the radius is 8 cm, find the distance of P from the centre of the circle. (a) 12cm (b) 15cm (c) 17cm (d) 20cm 24. If PB = 20 cm and OA = 15 cm, find the (sah)o2rt5esctmdiJstOanIcNe b@etwiiete(wbn)at1hl5eeccmiirncleTaenldePg.ram If the perpendicular bisector of segment AB (c) 30cm (d) 10cm meets the bigger circle in P and Q, find the length of PQ. (a) 4 6cm (b) 24 cm 25. If APO 18 , find AOB . (c) 8 3 cm (d) 4 3 cm (a) 24 (b) 144 (c) 108 (d) 175 20. In the given figure, 0 is the centre of the 26. PQ is a chord of a circle, which on circle. PQ is the tangent to the circle and producing, meets the tangent TR at R. If PQ secant PAB passes through the centre O. = 5 cm and QR = 4 cm, find TR. If PQ =5 cm and PA = 1 cm, find the (a) 3 cm (b) 6 cm radius of the circle. (c) 9cm (d) 10cm (a) 8cm (b) 12cm (c) 10cm (d) 6cm 27. Two circles of radii 8 cm and 2 cm have direct common tangent of length 10 cm, 21. A If all the sides of a parallelogram touch a find the distance between their centres. circle, what is the parallelogram? (a) A rectangle (b) A rhombus (a) 8cm (b) 10cm (c) 21 cm (d) 200 cm (c) A square (d) A trapezium 4

28. BD and CD are two tangents drawn to a 32. With the vertices of a ABC as centres, circle with centre O. three circles are described, each touching the other two externally. If the sides of the triangle are 9 cm, 7 cm and 6 cm, find the radii of the circles. If BDC  44 ,find BAC . (a) 68 (b) 32 (c) 44 (d) 96 29. If an equilateral triangle ABC is inscribed in (a) 4 cm, 7 cm and 3 cm a circle, which of the following is formed by (b) 7 cm, 5 cm and 2 cm the tangents at their vertices? (c) 5 cm, 4 cm and 3 cm (a) A scalene triangle (d) 4 cm, 5 cm and 2 cm (b) An equilateral triangle (c) An isosceles triangle 33. A, B and C are three points on a circle. The (d) A right triangle tangent at C meets BA produced at T. Given that ATC  36 and ACT  48 , 30. In the given figure, ABCD is a quadrilateral find the angle subtended by AB at the in which Q  90 . A circle C (O, r) centre of the circle. touches the sides AB, BC, CD and DA at P, Q, R, S respectively. If BC = 38 cm, CD = JOIN @iitwale in Telegram 25 cm, and BP = 27 cm, find the value of r. (a) 14cm (b) 15cm (a) 90 (b) 96 (c) 10cm (d) 16cm (c) 108 (d) 120 31. In the given circle, O is the centre, QP is a 34. Tangents AP and AQ are drawn to a circle diameter and PT is a tangent. with centre 0 from an external point A. If QOS 120 , find PTS . Identify the correct statement. (a) PAQ  2OPQ (a) 30 (b) 40 (b) PAQ  OPQ (c) PQA  QPA (c) 60 (d) 80 (d) PQA  2OPA 5

35. PT and QT are tangents drawn to the circle 39. A circle with centre 0 and radius 5 cm is with centre O. If OT= 25 cm and OP = inscribed in an equilateral triangle ABC. 15cm, find QT. (a) 20cm (b) 10cm Find the perimeter of ABC . (c) 5cm (d) 15cm (a) 15 3 cm (b) 25 2 cm 36. Two circles with centres 0 and P, and radii 8 (c) 14 2 cm (d) 30 3 cm cm and 4 cm touch each other externally. Find the length of their common tangent 40. ABC is a right - angled triangle with QR. B  90 , AB=12cm and AC=13cm.A circle with centre 0 is inscribed inside the triangle. (a) 8cm (b) 7cm JOIN @iitwale in Telegram (c) 8 2 cm (d) 7 3 cm 37. PQ is a chord of a circle .the tangent XR at Find the radius of the circle. X on the circle cuts PQ produced at R. If (a) 8cm (b) 7cm (c) 4 cm (d) 2 cm XR 12 cm , PQ  x cm, QR  (x  2)cm , find x in cm. 41. Three circles with centres A,B and C touch one another externally. The three centres (a) 6 (b) 7 are joined to one another to form a triangle. (c) 10 (d) 14 38. A circle with centre 0 is inscribed in an isosceles triangle PQR in which PQ = PR= 17 cm and QR = 16 cm. Find the radius of the circle. If AB = 7 cm, BC = 11 cm and AC = 6 cm, (a) 6.2 cm (b) 4.8 cm find the radii of the three circles. (c) 5.6 cm (d) 7.4 cm (a) 2 cm, 7 cm, 9 cm (b) 1 cm, 6 cm, 5 cm (c) 12 cm, 5 cm, 6 cm (d) 3cm, 10 cm, 7 cm 6

42. In the given figure, ABC is an isosceles 44. In the given figure, 0 is the centre of in circle triangle in which AB = AC.A circle through inscribed in ABC . B touches AC at its mid-point D and intersects AB at P. Which of the following is correct? Find BOC if BAC  40. (a) 100 (a) AP  3 AB (b) AP  2 AB (b) 120 4 3 (c) 90 (d) 110 (c) AP  4 AB (d) AP  1 AB 45. In ABC, B  90 .If a circle drawn with 5 4 AB as diameter intersects the hypotenuse AC at P, which of the following is true? 43. A tangent CQ touches a circle with centre O at P. Diameter AB is produced to meet the JOIN @iitwale in Telegram tangent at C. If ACP  a and BPC  b , find the relation connecting a and b. (a) The tangent drawn to the circle at P bisects the side BC. (a) a  b 180 (b) a  2b  90 (b) The tangent drawn to the circle at A (c) a  b  60 (d) 2a  b 100 bisects the side AB. (c) The tangent drawn to the circle at B bisects the side AC. (d) The tangent drawn to the circle at C bisects the side BC. 7

Answer - Keys 1. B 2. D 3. A 4. D 5. A 6. B 7. A 8. A 9. A 10. D 11. B 12. A 13. D 14. A 15. C 16. A 17. D 18. D 19. B 20. B 21. B 22. A 23. C 24. D 25. B 26. B 27. A 28. A JO2I9N. @Biitwale i3n0T. eleAgram 31. C 32. D 33. B 34. A 35. A 36. C 37. C 38. B 39. D 40. D 41. B 42. D 43. B 44. D 45. A 8

Solutions 1. (b) Not available 17. (d) Not available 2. (d) Not available 18. (d) Not available 3. (a) Not available 19. (b) Not available 4. (d) Not available 20. (b) Not available 5. (a) Not available 21. (b) Not available 6. (b) Given sides of a quadrilateral ABCD are 22. (a) Not available tangential to a circle.  AB  DC  AD  BC 23. (c) OP2  OA2  AP2  82 152 7. (a) Not available OP  289 cm 17 cm 8. (a) Not available 24. (d) OP2  OA2  PA2 (since, PA= PB.) OP  25 cm 9. (a) Not available Hence, the shortest distance between the 10. (d) Not available circle is 25JO15I[email protected] in Telegram 11. (b) Not available 25. (b) AOP 180 [90 18o]  72 AOB  72 2 144 12. (a) SPQ  PRQ .....(1) 26. (b) PR  PQ  QR  5cm  4cm  9cm (Angle between tangent and chord is equal PR  QR  TR2 to the angle in the alternate segment.)  9cm 4cm  TR2  TR  6cm SPQ  PRQ 27. (a) Not available ......(2) (Alternate angles as ST | | QR .) 28. (a) Not available  PRQ  PQR ....... (2) 29. (b) Not available [From statements 1 and 2.] 30. (a) Not available  PQ = PR  P is equidistant from Q 31. (c) Not available and R, the extremities of the chord. 32. (d) Not available 13. (d) Not available 33. (b) Not available 14. (a) Not available 34. (a) Not available 15. (c) Not available 35. (a) Not available 16. (a) Not available 9

36. (c) Join 0 to P and Q. Join P to R. Draw OFA  90 SP  OQ . In AOF,AOF=70  y Now SP = QR, as they are opposite sides of From the figure, rectangle PRQS. x  x  y  y  z  z  360o OP  8 cm  4 cm  12 cm OS  8 cm  4 cm  4 cm BOC 110  SP  OP2  OS 2  122  42cm  8 2 cm 45. (a) Join P to B. Now, BPC  90  QR  8 2cm BPM  MPC  90 But BPM  A 37. (c) RX 2  RP  RQ A  MPC  90  122  (2x  2)(x  2)  A  90  MPC  x 10  7 (from ABC ) Hence, the value of  (in cms) = 10 cm. A  90  C  MPC  C 38. (b) Not available  MP=MC but, MP=MB 39. (d) Not available MHeCn=ce,MthBJe OstaItNem@entiiintwopatlioeni(na)Tisetrlueeg. ram 40. (d) Not available 41. (b) Not available 42. (d) Not available 43. (b) Given BPC  band ACP  a . Also, OPA  OAP  b (Angles in an isosceles triangle OAP, angle in deternate segment.) CPO  90  CPA  90  b In ACP, ACP 180 [(b  90)  b]  a  2b  90 44. (d) AO is joined. Since the circle is the in circle for ABC , AO, BO and CO are the angle bisectors of A,B and C respectively. DAO  FAO  1 BAC  20o 2 10

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Arithmetic Progressions 1. Find the general term of the series 4, 7, 10, 9. What is the sum of first 'n' odd numbers 13...... starting from 11? (a) 3n - 7 (b) 3n + 7 (c) 3n +1 (d) 3n – 1 (a) n2 10n (b) n2 10n (c) 10n  n2 (d) n2 2. Which is the first negative term of the 10. How many two digit numbers are divisible arithmetic progression 35, 30, 25,.....? (a) 7th term (b) 5th term by 4? (a) 20 (b) 16 (c) 9th term (d) 11th term (c) 25 (d) 22 3. Find the next term of the arithmetic 11. How many terms of the arithmetic progression 12, 27, 48 progression 1, 9, 17, .... must be taken to (a) 75 (b) 60 give a sum of 1540? (a) 10 (b) 40 (c) 80 (d) 90 (c) 20 (d) 15 4. What is the term of an arithmetic 12. What is the sum of 36 terms of the series progression whose sum to 'n' terms is whose nth term is 5n + 4? 2n2  3n ? (a) 4347 (b) 3474 (a) 4n - 3 (b) 4n - 5 (c) 4374 (d) 3447 (c) 4n + 5 (d) 5 - 4n JOIN @iitwale in Telegram 13. Divide 21 into three parts that are in 5. Which term of the arithmetic progression arithmetic progression and whose product is 30, 27, 24,.... is O? (a) 8th term (b) 10th term 168. (a) 3, 6, 9 (c) 13th term (d) 11th term (b) 14, 17, 11 6. The numbers 28, 22, 'x', 'y', 4 are in (c) 2, 6, 14 (d) 2, 7, 12 arithmetic progression. What are the respective values of 'x' and 'y'? (a) 10, 16 (b) 20, 18 14. Which term of the arithmetic progression 3, (c) 18, 16 (d) 16, 10 8, 13, ..... is 55 more than its 20th term? (a) 29th term 7. What is the sum to 'n' terms of the series (b) 31st term (c) 25th term 5, 20, 45, 80,.....? (d) 27th term (a) n(n 1) 5 (b) n(n 1) 15. Identify the formula for then nth term of the 2 2 arithmetic progression whose first term, (a) is 2 - x, and the common difference, (c) n  (n 1)  (d) n(n 1) 5 (d) is 'x'. 2  5  (a) 2  x  nx (b) 2  (n  2)x 8. What is the sum of the first 'n' even numbers? (c) 2  x  nx (d) 2  (n 1)x (a) n2 1 (b) n(n 1) (c) n2  n (d) n(n 1)

16. Find the number of terms of the arithmetic 23. A piece of wire is used to make circles of the following pattern. progression 1 , 1 , 2 ,......11 323 6 (a) 8 (b) 10 (c) 12 (d) 13 17. The first, second and last terms of an arithmetic progression are 5, 9 and 101 respectively. Find the number of terms in the arithmetic progression. The radius of the first circle is 'r' cm and it is (a) 30 (b) 50 increased by 1 cm successively. Find the (c) 25 (d) 75 length of the wire that is required to make 5 such circles. 18. Find the sum of the first 18 terms, of the (a) (10  r)cm arithmetic progression 12b, 8b, 4b,...... (b) 2 cm (a) 126b (c) 10 (r  2)cm (b) -56b (d) 2 (r  2)cm (c) 256b (d) - 288b 24. An arithmetic progression has a 3rd term of 13 and a last term of 148. If the common 19. Find the sum of the arithmetic progression, difference is 5, find the number of terms of x  2y,2x  y,3x......11x  8y . the progression. (a) 33(2x  y) (a) 30 (b) 22(2x  y) (c) 28(x  y) (b) 40 JOIN @iitwale in Telegram (d) 33x 11y (c) 50 (d) 75 20. Find the sum from the sixth term to the 25. Given that the sum of the first 'n' terms of an twelfth term of the arithmetic progression 6, arithmetic progression is 2n2 + 3n, find the 10, 14, .... twelfth term. (a) 176 (a) 72 (b) 232 (b) 36 (c) 266 (d) 184 (c) 625 (d) 56 21. The first term and the common difference of 26. The first two terms of an arithmetic an arithmetic progression are 13 and 5 progression are 27 and 24 respectively. respectively. Find the sum from the 6th term How many terms of the progression are to to the 20th term. be added to get - 30? (a) 876 (a) 15 (b) 20 (b) 146 (c) 25 (d) 18 (c) 1095 (d) 1080 27. A secondary school had an enrolment of 1620 students in the year 2009 which increases by 150 students per year. What 22. Find the value of 'n' if the sum of the first 'n' was the enrolment in the year 2013? terms of the A.P.I 5, 23, 31, ....... is 708. (a) 9 (b) 8 (a) 2170 (b) 2070 (c) 2220 (d) 2150 (c) 12 (d) 10 3

28. A piece of wire is used to make rectangles 33. The given figure shows a pattern of several with the same height of 'h' cm. The lengths squares drawn consecutively. of the rectangles are increased by 1 cm successively. Find the total length of wire required to make 7 rectangles in this pattern. (a) 2[h  b  7] (b) 2b + h + 7 The perimeters of the squares in figure form (c) 2n + h + 6 (d) 2[b  h  6] an arithmetic progression. The length of the first square is x cm and the length of the 29. The postal rates in India from town M to other consecutive squares differ from each town N are tabulated as shown. other by 1 cm. Given the sum of the perimeters of the first five squares is 160 cm, Weight 1-60 61- 81- 101- 121- find the sum of the perimeters of the first 10 (g) 80 100 120 140 squares. (a) 280cm Postal 3.00 3.50 4.00 4.50 5.00 (b) 240cm Rate(`) (c) 420 cm (d) 360 cm Amir posts a 230 g parcel from town M to town N, what is the postal charge? (a) ` 8.00 (b) ` 7.50 34. In an arithmetic progression, the fourth term (c) ` 7.00 (d) ` 6.00 is 8 and tvhJaeOluseuImNof o'@pf'1if2iitthtweermapsltheitseir1nm56Ti.s e10le00g.ram Find the 30. A cineplex has 13 rows of seats with 10 seats in the first row, 12 in the second, 14 in (a) 200 (b) 500 the third and so on. What is the total (c) 300 number of seats in the cineplex? (d) 100 (a) 252 (b) 256 (c) 258 (d) 286 35. A roll of thread 907i cm long is cut into 5 31. Sulekha intends to read a 200 page book parts to make up 5 circles as shown in the within a specific period-She starts by figure. reading 11 pages on the first day and increases her rate of reading by reading an The radii of the circles increase by 1 cm extra 2 pages for each subsequent day How consecutively. Find the radius of the smallest long will she take to finish reading the book? circle. (a) 20 (a) 7 cm (b) 5 cm (b) 10 (c) 6 cm (d) 9 cm (c) 15 (d) 25 32. The 6th term of an arithmetic progression is 36. A circle is divided into 18 sectors such that the angles subtended at the centre form an 30 and the sum of the first six terms is 210. arithmetic progression. Given that the angle Find the sum of the next 6 consecutive terms. of the smallest sector is11.5 , find the angle (a) 100 (b) 50 of the biggest sector. (c) 138 (d) 204 (a) 30 (b) 15.5 (c) 45 (d) 28.5 4

37. An arithmetic progression has 10 terms. The 45. The sum to 'n' natural numbers is S1 sum of sum of the odd terms is 245 whereas the the squares of 'n' natural numbers is S2 and sum of the even terms is 305. Find the sum of the cubes of 'n' natural numbers is common difference. (a) 2 (b) 7 S3 Which of the following is equal to 9S22 ? (c) 3 (d) 1 (a) (1 8S1)S2 (b) S3(1 8S1) 38. Given an arithmetic progression -10,-5, (c) S1(1 8S2) 0,..... Identify three consecutive terms of the (d) S1(1 8S3) progression whose sum is 90. (a) 26, 31, 36 (b) 25, 30, 35 (c) 20, 30, 40 (d) 18, 23, 28 46. In an arithmetic progression if 7 times the 39. The first three terms of a sequence are 8, 'y' 7th term is equal to 11 times the 11th term, find its 18th term. and 18. Find the positive value of 'y' so that (a) 1 (b) 0 (c) -1 (d) 2 the sequence is an arithmetic progression. (a) 26 (b) 18 (c) 12 (d) 13 47. Find the sum of the numbers in between 1 and 1000 which are divisible by 9. (a) 55944 (b) 54954 40. The sum of the first 10 terms of an (c) 99994 (d) 99894 arithmetic progression is four times the sum of its first five terms. Find the ratio of the first term to the common difference. 48. There are 'n' arithmetic means in between 'a' and 'b'. Find the common difference. (a) 1:4 (b) 4:1 (c) 2:1 (d) 1:2 (a) b  a JOIN @iit(wb)able ain Telegram n 1 n 1 41. How many numbers between 100 and 1000 (c) b  a (d) b  a are divisible by 7? n 1 n 1 (a) 7 (b) 128 49. What is the sum of 'n' arithmetic means (c) 132 (d) 127 between 'a' and 'b'? 42. Find the sum of 14 AM's between 5 and 8. (a) 3 (a  2b) (b) n [a  b] (a) 91 (b) 89 2 2 (c) 90 (d) 85 (c) n [a  b] (d) n [a  b] 43. If13  23 .....  m3  3025, find m. 2 4 (a) 12 (b) 10 50. Find the sum of 1  1   1  2   1 3 .... (c) 17 n  n  n (d) 12 terms. (a) n (b) n 1 2 2 44. If mth term of an arithmetic progression is 'n' (c) n 1 (d) n(n 1) and nth term is 'm', find its (m  n)th term. 2 n2 (a) 0 (b) m  n  p 51. In an arithmetic progression, if (c) m  n tp =q, tq  p, find tpq . (d) mn mn (a) p  q  pq (b) p  q (c) p  q  pq (d) pq  p  q 5

52. Find three numbers in arithmetic progression whose sum is 3 and product is- 35. (a) 1,5,7 (b) 5,7,1 (c) 5,1,7 (d) 7,5,1 53. Which of the following is incorrect? (a) Sn of the arithmetic progression 3, 13,23,....is 5n2  8n . (b) Sn  3n2  8n is the sum of an arithmetic progression whose common difference is 6. (c) nth term, tn  sn  sn1 (d) The nth term of an arithmetic progression is tn  a  (n 1)d . 54. Given P=3+5+7+ .... (n terms) and Q = 5 + 8 +11 +..... (10 terms).What is the value of 'n' if P  7 ? (b) 35 Q (d) 20 (a) 25 (c) 30 55. What is the value of 'x' if 2x, x 10 and 3x  2 JOIN @iitwale in Telegram are in arithmetic progression? (a) 5 (b) 6 (c) 4 (d) -6 6

Answer - Keys 1. C 2. C 3. A 4. B 5. D 6. D 7. A 8. D 9. A 10. D 11. C 12. B 13. D 14. B 15. B 16. B 17. C 18. D 19. A 20. C 21. C 22. C 23. C 24. A 25. A 26. B 27. C 28. D JO2I9N. @Biitwale i3n0T. eleDgram 31. B 32. C 33. C 34. B 35. A 36. D 37. D 38. B 39. D 40. D 41. B 42. A 43. B 44. A 45. B 46. B 47. A 48. A 49. B 50. B 51. A 52. C 53. A 54. B 55. B 7