11. Perimeter and Area Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET PS – 1 Perimeter and Area • Perimeter and Area of Squares and Rectangles PS – 2 PS – 3 • Area of triangles and Parallelograms ----- • Numerical PS – 4 PS – 5 • Circumference and Area of Circles Self Evaluation Sheet • Conversion of Units • Application problems Worksheet for “Perimeter and Area” Evaluation with Self Check or ---- Peer Check* 88

PRACTICE SHEET - 1 (PS-1) 1. Find the perimeter of a square of side 10 m. 2. The perimeter of a rectangular field is 50 m. If the length of the field is 13 m, find the area of the field. 3. A man wants to cover his floor with carpet which costs Rs. 100 for every square meter. His floor is in the form of a square which has a perimeter of 24 m. What will be the cost of carpeting the floor? 4. Given a square of side 10 m and rectangle of 11 m and breadth 9 cm, which one has a larger area? 5. The area of a given square and rectangle are same. If the length of each side of the square is 4 m and the ratio of length and breadth of rectangle is 2, find the length of the rectangle. 6. The length of a rectangle is 12 cm and its width is 5 cm smaller. What is the area of the rectangle? 7. A rectangle has a length of 12 cm and its width is 3 times smaller. What is the area of the rectangle? 8. On a field with length 25 m and width 12 m a house was built in the shape of a square whose side is 9 m. What is the area of the remaining field? 9. A square-shaped park has side of 30 m and an alley with width of 2 m. What is the area of the shaded region? 30 m 2m 10. How many square tiles of side 9 cm will be needed to fit in a square floor of a bathroom of side 720 cm. Find the cost of tiling at the rate of Rs. 75 per tile. 11. If it costs Rs. 2400 to fence a square field at the rate of Rs.6/m, find the length of the side and area of the field. 89

PRACTICE SHEET - 2 (PS-2) 1. Find the area of a parallelogram whose side is 3 m with an altitude of 3 m. 8 2. Find the height ‘X’ if the area of the parallelogram is 24 cm2 and the base is 4 cm. 3. Given that the height of the triangle is 5 cm and it divides the base into 2 equal parts of 3 cm each. Calculate the area of the triangle. 4. Determine the area of the parallelogram shown in the figure. 8 km 7 km 5. Determine the area of the triangle shown in the figure. 8m 6m 6. Determine the area of the shaded portion. 7. Determine the area of the parallelogram shown in the figure. 11 m 18 m 90

PRACTICE SHEET - 2 (PS-2) 8. DL and BM are the heights on the sides AB and AD respectively of parallelogram ABCD. If the area of parallelogram is 1470 cm2, AB = 35 cm and AD = 49 cm, find the length of BM and DL. DC M LB A 9. A parallelogram has side length of 10 m, base length of 9 m and height of 5 m. Find area and perimeter. 10. Find the missing values. Base Height Area of Triangle i) 15 cm x 87 cm2 ii) x 31.4 mm 1256 mm2 iii) 22 cm x 170.5 cm2 11. A parallelogram of area 48 cm2 is divided into 2 congruent triangles. If height of the triangle is 4 cm then find the base of the triangle. 12. A parallelogram has sides of length 12 cm and 8 cm. If the distance between the 12 cm sides is 5 cm, find the distance between 8 cm sides. 13. Find the base length of a parallelogram if the area is 24 m2 and the corresponding height is 8 m. 91

PRACTICE SHEET - 3 (PS-3) 1. A circular dinner plate has a radius of 6 cm. What is the area of the plate? 2. The distance around the wheel of a truck is 9.42 ft. What is the diameter of the wheel? 3. A lawn sprinkler sprays water 5 ft in every direction as it rotates. What is the area of the sprinkled lawn? 4. What is the circumference of a 12 inch pizza? 5. An asteroid hit the Earth and created a huge round crater. Scientists measured the distance around the crater as 78.5 miles. What is the diameter of the crater? 6. A semicircle shaped rug has a diameter of 2 ft. What is the area of the rug? 7. A spinner has 6 sectors, half of which are red and half of which are black. If the radius of the spinner is 8 in, what is the area of the red sectors? 8. From a circular sheet of radius 5 cm, a circle of radius 3 cm is removed. Find the area of the remaining sheet. 9. The diameter of a wheel is 70 cm. How many times the wheel will revolve in order to cover a distance of 110 m? 10. A well of diameter 150 cm has a stone parapet around it. If the length of the outer edge of the parapet is 616 cm, find the width of the parapet. 11. A thin wire is in the form of an equilateral triangle of side 11 cm. Find the area of a circle whose circumference is equal to the length of wire. 12. Find the area of a circle whose circumference is same as the perimeter of square of side 22 cm. 13. From a rectangular metal sheet of size 20 cm by 30 cm a circular sheet as big as possible is cut. Find the area of the remaining sheet. 14. Find the circumference of a wheel whose radius is 35 cm. Find the distance covered in 60 seconds, if it revolves 5 times per second. 15. A 3 m wide road runs around a circular park whose circumference is 132 m. Find the cost of fencing the outer boundary of the road at a rate of Rs. 10/m. 16. A circular flower bed is surrounded by a path of 2.5 m wide. The diameter of the flower bed is 40 m. Find the area of the path. 17. A square metallic frame has a perimeter of 208 cm. It is bent in shape of a circle. Find the area of the circle. 92

PRACTICE SHEET - 4 (PS-4) 1. A rectangular part is 40 m wide and 25 m long. A path 2.5 m wide is constructed outside the path. Find the area of the path. 2. A man wants to paint 2 of his walls. The cost of painting is Rs. 10/cm2. How much money is required to paint the rectangular wall of length 400 cm and breadth 900 cm and a circular wall of diameter of 126 cm? 3. Find the area of the shaded portion. A circle is inscribed in a square such that both their centers coincide. BDE is an isosceles triangle. The side of the square is 10 m. AB 10 m CDE 4. David has a square garden whose one of the sides is covered by the wall of his house. The length of each side of the garden is 25 ft. If the cost of fencing is Rs. 3 per feet, find the total cost of fencing the garden. 5. Find the area of the shaded portions: 15 cm 15 cm i) 15 cm 30 cm 15 cm 10 cm 10 cm 10 cm ii) 10 cm 20 cm 5 cm 5 cm 15 cm 93

PRACTICE SHEET - 5 (PS-5) I. Choose the correct option: (B) Both A and B are true 1. The area is measured in ________. (C) A is true, B is false (A) Square units (B) Cube units (D) A is false, B is true (C) Units 4 (D) None of the above 10. Statement (A): If the diameter of a circle is 2. The side of an equilateral triangle of area equal to the side of a square, then the ratio of their area is 4 : π 64 3 cm2 is: (B) 14 cm (A) 12 cm Statement (B): The circumference of a circle is (C) 16 cm (D) 18 cm equal to the perimeter of a square, then their areas are in the ratio π : 4 3. The distance covered along the boundary forming a closed figure is: (A) Both A and B are true (A) Perimeter (B) Area (B) Both A and B are false (C) Length (D) Breadth (C) A is true, B is false (D) A is false, B is true 4. The perimeter of the n-sided polygon whose II. Short Answer Questions: side is 'n' cm is: 1. In the following figure in which 3 semi circles (A) 2n (B) n2 (C) n + 2 (D) n - 2 are drawn taking the mid-point of each side of the square as the center. From that figure show area of the figure = 280 cm2. 5. The perimeter of the equilateral triangle whose Use π = 22 length of each 'a' cm is: 7 (A) 3a cm (B) 3a sq.cm (C) a3 cm (D) a3 sq.cm 6. If the side of a square is halved, then the area becomes part of the original area. (A) 1 (B) 1 2 3 (C) 1 (D) 1 2. By applying appropriate formula. Find the area 4 8 of following figure. 3. Divide the following figure in to 2 shapes then 7. If a parallelogram and a rectangle are on the find the area of shaded region. same base and between the same parallel lines, then the perimeter of the rectangle is: (A) Equal to the perimeter of the parallelogram (B) Greater than the perimeter of the parallelogram (C) Less then the perimeter of the parallelogram (D) Can's say 8. The ratio of area of a sector and area of circle is: (A) 360º : xº (B) xº : 360º (C) xº : 90º (D) xº : 180º 9. Statement (A): In reference to a circle the value of ‘π’ is equal to Circumference Diameter Statement (B): Circumference of a circle is always more than three times of its diameter. (A) Both A and B are false 94

PRACTICE SHEET - 5 (PS-5) III. Long Answer Questions: 1. Find the area of following figure. 2. Divide the given shapes as instructed. (i) into 4 rectangle (ii) into 2 trapeziums (iii) into 3 triangles (iv) into 2 triangles and a rectangle 95

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. Find the area: (1 Mark) 5. A wire in the shape of rectangle whose width is 22cm is bent to form a square of side 31 cm. Find 4 cm the length of the rectangle. Also find which shape 10 cm encloses more area. (3 Marks) 2. What is the circumference of a circular disc of radius 12 ft? (1 Mark) 6. Convert the following units: (3 Marks) i) 17 m2 to cm2 3. Find the perimeter of the given shape of an ii) 180000 m2 to hectares equilateral triangle and a semi circle. iii) 6.3 m2 to mm2 (Use π = 3.14 ) (2 Marks) 12 cm 7. Area of a square field is 81 hectares. Find the cost of fencing the field with a wire at Rs. 25/ metre. (3 Marks) 4. Find the area of the shaded portion. (2 Marks) A 30 m B 20 m D 10 m C 96

12. Algebraic Expressions Learning Outcome • Form algebraic expressions using variables, constants & arithmetic operations. By the end of this chapter, a student will be able to: • Identify the various elements of an expression. • Simplify, add, subtract & solve algebraic • Differentiate between like and unlike terms. equations. • Define monomials, binomials, trinomials & • Apply formulae & rules using algebraic polynomials. expressions. Concept Map Key Points • Terms which have the same algebraic factors are like terms. Eg: 4xy − 3xy • A variable can take any value. Its value is not fixed. • A constant has a fixed value. Eg: 2, -2000 • Terms which have different algebraic factors are • Variables & constants are combined to form unlike terms. Eg: 4xy − 3x algebraic expressions • Sum of 2 like terms is a like term with coefficient • Terms are added to make an expression. Eg.: equal to the sum of the coefficients of 2 like terms. addition of terms 2xy & 6 gives the expression Thus, 5xy + 2xy = (5 + 2) xy = 7xy 2xy + 6 • Difference of 2 like terms is a like term with • A term is a product of factors. The term 2xy in the coefficient equal to the difference of the coefficient of the 2 like terms. Thus, expression 2xy + 6 is a product of factors 2, x and y . 5xy − 2xy = (5 − 2) xy = 3xy Factors containing variables are said to be algebraic factors. • While adding 2 algebraic expression, the like terms • Coefficient is the numerical factor in the term. are added & the unlike terms are left as they are. At times, any one factor in a term is called the Eg: 3x2 + 2x and 6x + 3 = 3x2 + 8x + 3 . Like terms coefficient of the remaining part of the term. • An expression with one term is called a monomial. 2x & 6x are added. Unlike terms 3x2 &3 are left as • A two term expression is called a binomial. is. • A three term expression is called a trinomial. • Value of an expression depends on the value of the • An expression with one or more term is called a variable from which the expression is formed. polynomial. Expression ------------- 3x2 − 2xy Terms-------------- 3x2 −2x Factors 3 xx −2 x y 97

12. Algebraic Expressions Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Algebraic Expressions • Identify the various elements of an PS-1 expression, like and unlike terms, PS-2 monomials, binomials, trinomials and polynomials • Add and subtract algebraic expressions • Simplifying the algebraic expressions and finding PS-3 their value by substituting the variable value Worksheet for “ Algebraic Expressions” PS-4 Evaluation with Self ---- Self Evaluation Sheet Check or Peer Check* 98

PRACTICE SHEET - 1 (PS-1) 1. Write the following statements in the form of the algebraic expression. (a) x is multiplied by itself and then added to the product of x & y (b) Three times of x� and four times of y are multiplied and then subtracted from z (c) Cube of s subtracted from cube of t (d) Sum of square of x and cube of y (e) Two times q� subtracted from square of q 2. Identify the terms and their factors in the following expression. Show the terms and factors by tree diagram. (i) y − 5 (ii) 5 + x2 + x (iii) x3 − x2 (iv) 5xy + 8x2 y2 (v) a2 + 2ab + b2 3. Identify the numerical coefficients of terms (other than constants) in the following expressions: (i) 6 − 3x2 (ii) x + 2xy + 3z (iii) 5( x + y + z) (iv) x2 − 2x − 3 y − y2 4. Identify the coefficient of x2 . (i) yx2 − x + 2 (ii) 13x2 + 2xy + y2 (iii) 3x − 5x2 − 2 (iv) 5 − y2 + 3x 5. Classify the following expression as monomials, binomials and trinomials. (i) x2 + y2 (ii) x + y + z (iii) 5 xy (iv) 7 x2 (v) 1 + x + x3 (vi) x2 + 2xy + y2 (vii) 4a2b + 4b2a 6. Identify the like terms in the following (i) x3 y2z ,10x3 y2z , xyz , xy2 (ii) −7 x2 y ,5x2 y ,5xy , x3 y3 (iii) ab, 8a , 9b,− a2b2 , 5ab,− 10b, 20a2b2 , a3 , −100a , 13a2b, 14ab2 99

PRACTICE SHEET - 2 (PS-2) 1. Simplify the following like terms. (i) 50x3 − 21x + 107 + 41x3 − x + 1 − 93 + 71x − 31x3 (ii) 3 x2 yz2 + 4 xy2 z2 − 3 x2 y2 + 2 x2 y2 z2 + 10 x2 yz2 −10 xy2 z2 + 10 x2 y2 z2 (iii) 10xyz − 5xy2z2 + 5xy2z − 20xyz − 5x2 yz + 10xy2z (iv) x4 + 4x3 y + 3xy3 + 3x2 y2 − 3x3 y − 2xy2 − y4 −2 x2 y2 2. Add the following expression (i) p2 − 7 pq − q2 and −3 p2 − 2 pq + 7q2 (ii) x3 y2 + x2 y2 + x2 y3 and −3x3 y2 − 2x2 y2 +3 x2 y3 (iii) 5 x4 + 6 x3 + 2x2 + 5 , 1 + 17 x2 − 6 x3 + x4 and x4 − x3 + x2 + 1 87 88 7 (iv) x2 + x2 + y2 , 10x2 − 10 y2 − 10z2 and x4 + x2 + y2 3. Subtract the following expression. (i) −7 p2q from −3 p2q (ii) −2a2 − 2ab2 from −a2 − b2 + 2ab (iii) 4.5 x3 − 3.5x2 + 6.5 from 6.2 x3 − 6.5x2 + 2x (iv) x3 y3 − x2 y2 + 5x3 y2 − 5x2 y3 from 3 x3 y3 + 2 x2 y2 − 8 x3 y2 + 2 x2 y3 4. (a) What should be subtracted from 2x3 + 2x2 + 5xy to get x3 + x2 − 3xy ? (b) What should be added to 5a2 + b2 − 6 to get 3a2 − 6b2 + 12 ? 5. (i) What should be taken away from 3x3 − 4 y3 − 2x2 y2 + 5xy to obtain −3x3 − 6 y3 − 5x2 y2 + 10xy (ii) What should be taken away from? x2 + y2 + 2xy + 10 to obtain −2x2 − y2 + 4xy 100

PRACTICE SHEET - 3 (PS-3) 1. If x = 5 , find the value of: (i) x + 5 (ii) x2 − 2x + 2 (iii) − x3 + 2x2 + x − 5 2. Simply the expression and find the value when x = 10 , y = −5 , z = −2 (i) 5x − 2 y + 3z − 6x + y + z (ii) x2 + y2 − 2x − 2 y + 5 + z2 (iii) 3x − 2 y + x2 − y2 − 5x − 5 y + z2 101

PRACTICE SHEET - 4 (PS-4) I. Choose the correct option: 1. The product of each of literal or numerical value is called: (A) Expression (B) Factor (C) Term (D) Variable 2. The product of a monomial and a binomial is: (A) A monomial (B) A Binomial (C) A trinomial (D) Can't be determined 3. Equations which are used frequently to solve problems are called ______. (A) Formulae (B) Statement (C) Subject (D) None of the above Ans: A 4. The number of terms in ax3 + bx2 + cx + dx + e: (A) 1 (B) 2 (C) 4 (D) 5 5. The degree of 5x2y3z10 is: (A) 5 (B) 10 (C) 15 (D) 20 6. The factors of -2 a2b2c are: (A) -2×a×b×c (B) -2×a2×b×c (C) -2×a×a×b×b×c (D) -2×a2×b2×c2 7. The length of a side of square is given as 2x + 5, which expression represents the perimeter of the square? (A) 2x + 20 (B) 6x + 15 (C) 8x + 5 (D) 8x + 20 8. 123x2y - 138x2y is a like term of: (A) 10 xy (B) -15 xy (C) -15 xy2 (D) -10 x2y 9. Statement (A): Each monomial is a polynomial Statement (B): Each term is a monomial (A) Both A and B are true (B) Both A and B are false (C) A is true, B is false (D) A is false, B is true 10. Statement (A): LCM × HCF = P(x) × q(x), if P(x) and q(x) are two polynomials. Statement (B): Unlike terms can be added (A) Both A and B are true (B) Both A and B are false (C) A is true, B is false (D) A is false, B is true II. Short Answer Questions: 1. Show (−12cd ) × −4 c 2 d × −15 c 2d 2 = −120c5d 4 3 2 2. Subtract 3x2 - 5x + 4 from 5x2 + 6x - 8 by applying vertical method. 3. If A = 5p2 - 3q2 + r2, B = -2q2 + 3P2 - 4r2 and C = -7r2 + 3P2 + 2q2 then simplify A + B + C. 102

PRACTICE SHEET - 4 (PS-4) III. Long Answer Questions: 1. Each symbol given below represent an algebraic expression. = 2x2 + 3y = 5x2 + 3x = 8y2 - 3x2 = 2x + 3y then simplify the following representation by using above expresssions. +- - 2. Amisha has a square plot of side m and another triangular plot with base and height each equal to m. What is the total area of both plots? 103

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. Write the statement in form of algebraic 5. From the sum of 5 − 4x − 3x2 and 5x + x2, expression. Sum of square of x and cube of y is subtracted from square of z. (1 Mark) subtract the sum of −5x2 + 2 and 4x2 − 5x + 2. (3 Marks) 6. Find the sum of the following equation. 2. Evaluate the below expression when x = −5 x3 + 3x − 2x2 − 5 , −3x2 + 2x − 2 , x + 2x2 + 3x3 Evaluate expression when x = 2 , x = 5 . (i) 2x − 5 (ii) x2 − x + 2 (1 Mark) (3 Marks) 3. Simply the expression x3 + x2 y2 − xy3 + x3 y + 5 x2 y2 + 3 xy3 − 3 x3 y + 10 and find the value when x = −3, y = 3. (2 Marks) 7. Subtract the following equation x3 − x2 − x + 10, x2 + 5x + 100 , −10x3 − 10x2 and evaluate when x = −2 , x = −3. (3 Marks) 4. What should be take away from 2x3 − 3 y3 − x2 y2 to obtain 5x3 + y3 + 2x2 y2. (2 Marks) 104

13. Exponents and Powers Learning Outcome • Write numbers in the standard form. • Solve numerical problems on exponents. By the end of this chapter, a student will be able to: • Understand and write numbers in exponential form. • Explain laws of exponents. Concept Map am × an = am+n am = amn an ( ) am n = amn am × bm = (ab)m am = am bm b a0 = 1 Key Points raised to the power of 6 or sixth power of 2 ) • a4, a is the base and 4 is the exponent (read as a Generally exponents are used to read, understand and compare very large numbers. raised to the power of 4) Ex: computers memory, strength of earth quakes, • x3 y2, x & y are the bases and 3 & 2 are the distances between planetary bodies etc. The exponent of a number says how many times to exponents (read as x cubed into y squared) use the number in a multiplication. Note that x3 y2 & y2 x3 are same since the powers Consider number 1000 = 10 ×10 ×10 = 103. of x and y are same. The short notation103 stands for the product 10×10×10 • x y , x is the base and y is the exponent (read as x . “10” is called the base and “ 3 ” is called the exponent. 103 is read as 10 raised to the power of 3 or simply third raised to the power of y) power of 10. 103 is the exponential form of 1000. 103 • (−3)7 , base is -3 (negative number) and 7 is the means number 10 has to be multiplied 3 times. Exponents are also called as power or indices. exponent (read as -3 raised to the power of 7) Examples Use of exponents while writing numbers in expanded • 45 , 4 is the base and 5 is the exponent (read as 4 form: raised to the power of 5 or fifth power of 4 ) Consider number 123456. The expanded form is • 26 , 2 is the base and 6 is the exponent (read as 2 123456 = 1×100000 + 2 ×10000 + 3×1000 + 4 ×100 + 5 ×10 + 6 ×1 which can be written as 105

13. Exponents and Powers 1×105 + 2 ×104 + 3×103 + 4 ×102 + 5×101 + 6 using 6. Numbers with exponents zero: exponents. For a non-zero integer a, a0 = 1 i.e. any number (except Laws of exponents zero) raised to the power 0 is 1. Ex: 30 ,1000 , 250 are equal to 1. 1. Multiplying Powers with the same base: For a non-zero integer a, am × an = am+n where m� & n Remember: ( )−1 even�number =� 1 & ( )−1 odd �number =�− 1 are whole numbers. Expressing large numbers in standard form (using Examples: decimal system) Any number can be expressed as a decimal number • 33 × 34 = (3× 3× 3)(3× 3× 3× 3) = 3× 3× 3× 3× 3× 3× 3 between 1.0 and 10.0 including 1.0 multiplied by a power of 10. Such a form of number is called its = 37 = 33+4 standard form. For ex: The standard form of the number 34786529 is • (−5)2 × (−5)3 = (−5)× (−5) × (−5)× (−5)× (−5) 3.4786529 ×107. = (−5)5 = (−5)2+3 Note that 34.786529×106 or 0.34786529×108 are not the standard forms. • x2 × x4 = ( x × x)× ( x × x × x × x) = x2+4 = x6 Consider number 3456.8, the standard form is 3.4568×103. There are 4 digits to the left of the decimal Thus when the bases of the exponents are same, the point and hence the exponent of 10 in the standard product of exponents will be addition of the powers form is 4 – 1 = 3. with the same base. So, it is important to remember that, one less than the digit count to the left of the decimal point in a given 2. Dividing powers with the same base: number is the exponent of 10 in the standard form. For a non-zero integer a, am ÷ an = am−n where m� & n 106 are whole numbers and m > n i.e. the division of exponents with same bases is equal to difference between the powers with the same base. Examples: • 26 ÷ 24 = 26 = 2×2×2×2×2×2 = 26−4 = 22 24 2×2×2×2 • x23 ÷ x12 = x23 = x23−12 = x13 x12 3. Power of a power: For a non-zero integer a, (am )n = amn where m &�n are whole numbers. Examples: • (43 )5 = 43×5 = 415 • (a3 )z = a3×z = a3z 4. Multiplying powers with same exponents: For a non-zero integer a , am × bm = (ab)m where m is a whole number. Examples: • 202 × 22 = (20× 20)× (2× 2) = (20× 2)× (20× 2) = 40 × 40 = 402 • 820 × 920 = (8× 9)20 = 7220 • 512 × q12 = (5× q)12 = 5q12 5. Dividing powers with the same exponents: For non-zero integers a & b am ÷ bm = am = ( a )m where bm b m is a whole number. Example: 36 ÷ 46 = 36 = ( 3)6 46 4

13. Exponents and Powers Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Exponents & Powers • Introduction to exponents PS – 1 • Laws of exponents PS - 2 • Expanded form of a number using PS - 3 exponents Self evaluation sheet • Standard form of large numbers Worksheet for “Exponents and Powers” Evaluation with self check or peer -------- test 107

PRACTICE SHEET - 1 (PS-1) 1. In 1725 ,� _____ is the base and ____ is the exponent. 2. Express the following in exponential form. (a) 7 × 7 × 7 × 7 × 7 × 7 (b) 9× 9× d × d × d × d (c) x × x × y × y × z 3. Find the value of (i) 35 (ii) 210 4. Simplify the following (a) 53 ×102 (b) (−6)2 × 23 (c) 25 ÷16 5. Identify the greatest number in each of the following (a) 28 or 82 (b) 34 or 43 6. Simplify and write the answer in exponential form. (use laws of exponents) (i) 218 × 217 (ii) 79 ÷77 (iii) �(83 )9 7. Write the number 76192 in expnaded form using exponentials. 8. Fill in the blanks. (a) 2x × 5x = _____ (b) 58 ÷ 78 = ______ (c) 750 =� � � � ________ 9. Find the value of (i) 6p89qp263q6291 (ii) 281×622 10 10. Write the exponential form for 9× 9× 9 taking base as 3. 11. Say true or false. (a) 57 ÷ 75 = 52 (b) (−1)8 × (−1)6 = (−1)14 (c) x100x×25x50 = x102 12. Express the following numbers in standard form. (i) 9876512 (ii) 500000000 108

PRACTICE SHEET - 2 (PS-2) 1. Which of the following is greater? ( ) ( ) 234 23 ×4 or 2. Find the value of m in the following. (a) 441m7 = 256 (b) 38 3×6m2 = 81 3. Express the following in standard form. (a) 78� crore (b) The distance between sun and mars is 227.9 milion kilometers 4. Find the value of m in the following. n (5 ÷ 4)3 (25 ÷16)0 = m ÷ n 5. The product of two numbers is 7123. If one of the number is 745, then find the other number. 109

PRACTICE SHEET - 3 (PS-3) I. Choose the correct option. 3. Simplify x3n+1. x3n−1 1. Find the value of (-1)8462+(-1)8463 x2n (A) 8462 (B) 8463 (C) 0 (D) 16925 III. Long answer questions. 2. What is the value of ‘X’, if 27x-1 = 3x+3 1. Simplify 3 − 1 −2 + 1 1 −2 (A) 1 (B) 2 1024 2 (7776)5 5 643 (C) 4 (D) 3 2. Show that 3. Choose the incorrect answer from following xa a2 +ab+b2 . xb b2 +bc+c2 . xc c2 +ac+a2 = 1. xb xc xa (A) aᵐ.aⁿ = am+n (B) am = a m bm b (C) a0 = 0 (D) a−m = 1 am 4. Which one is greater from following (A) 45 (B) 54 (C) 64 (D) 46 5. Write exponential form of 4x16x64 (A) 212 (B) 84 (C) 163 (D) All the above 6. Use a = 103 × 0.0099 , b = 10-2 × 110 find sqrt(a/b) = (A) 3 (B) 0.3 (C) 9 (D) 0 7. The indirect form of aⁿ/aᵐ (A) am-n (B) 1 am−n 1 m (C) an−m (D) an 8. Solve my. nx, by using m = ax, n = ay (A) a2(xy) (B) a 2xy (C) a2xy (D) 2xy a 9. Simplified form of (256)0.16. (256)0.09 (A) 22 (B) 24 (C) 25 (D) 23 10. Dissect form (Exponential form) of 2606 (A) 2000 + 60 + 6 (B) 2 x 1000 + 6 x 10 + 6 (C) 2 x 103 + 6 x 102 + 0 + 6 x 100 (D) none of the above II. Short answer questions. 1. If a + b = 0, show that (1+ xa)-1 + (1 + xb)-1 = 1. 2. Solve 22x : 2x2 = 1: 8 110

Self-Evaluation Sheet Marks: 15 Time: 30 Mins (2 Marks) 1. Write in exponential form. 5. Simplify the following. (a) x� raised to power of y (b) −234× −234× −234× −234 (2 Marks) 73 × 55 × 94 × 32 35 × 52 × 38 2. Express each of the following as product of powers of their prime factors. (i) 1000 (ii) 600 (2 Marks) 3. Match the following 6. Find the number from the expanded form given below. (i) (−2)4 × (−)28 (a) −1 7 ×106 + 2 ×104 + 3 ×102 + 1×101 + 9 ×100 (2 Marks) (ii) (−2)14 ÷ (−2)4 (b) �(−2)12 (4 Marks) (iii) (−1)146 (c) �(−2)10 (iv) (−1)1097 (d) �1 7. Say true or false. The standard form of number 500987642 is 500.987642×106 (1 Mark) 4. Compare the following numbers (2 Marks) (a) 76 ×10200 and 76 ×10199 (b) 9.7 ×107 and 9.8 ×107 111

14. Symmetry Learning Outcomes At the end of this chapter, student will be able to: • Draw symmetrical figures. • Identify line of symmetry for regular polygons. • Identify conditions of line symmetry and rotational symmetry. Concept Map Key Points • Line symmetry and mirror reflection are naturally related to each other with mirror reflection, lift, • A figure has line symmetry if this is a line about right changes in orientation has to be accounted which the figure may be folder so that 2 points of for. the figure will coincide. • Rotation turns an object about a fixed point. The line of symmetry point about which the object rotates is the angle of rotation. • Regular polygons have equal sides and equal angles eg: - Equilateral triangle, square. • A quarter turn rotation is 900. • A half turn rotation is 1800. • Regular polygons have more than one i.e. multiple • A full turn rotation is 3600. lines of symmetry. • Rotation can be clockwise or anti-clockwise. • If an object looks exactly the same after a rotation, • The number of lines of symmetry in a regular polygon is equal to the number of sides of the the object is said to have rotational symmetry. polygon i.e. • Order of rotational symmetry:- The number of times an object looks exactly the same is a complete turn of 3600 is called the order of rotational symmetry. Polygon No. of sides No. of lines of symmetry Equilateral 03 03 triangle Square 04 04 Regular pentagon 05 05 Regular hexagon 06 06 112

14. Symmetry PRACTICE SHEET Work Plan PS – 01 COVERAGE DETAILS PS – 02 • Introduction PS – 03 • Lines of symmetry for regular polygon PS – 04 • Reflection symmetry Self-evaluation • Rotational symmetry • Order of symmetry • Line symmetry and Rotational Symmetry Worksheet for “Symmetry” Evaluation with self-check or Peer check* 113

PRACTICE SHEET - 1 (PS-1) 1. How many lines of symmetry are there in an equilateral triangle? 2. How many lines of symmetry are there in a square. 3. A regular polygon has only one line of symmetry. True or false 4. A shape is said to have line symmetry when one half of it is the mirror image of the other half. True or false? 5. Line AB is a line of symmetry. True or false? 6. How many lines of symmetry does a regular hexagon have? 7. Which letters of the English alphabet have a vertical mirror reflectional symmetry? 8. Which letters of the English alphabet have a horizontal mirror reflectional symmetry? 9. Which letters of the English alphabet have both horizontal and vertical mirror reflectional symmetry? 10. After drawing all the lines of symmetry in a regular polygon having ‘n’ sides, it is divided into how many triangles? 11. What are the possible numbers of lines of symmetry for a triangle? 12. What are the 2 names of the line of symmetry of an isosceles triangle? 13. Which capital letter of English alphabet have no lines of symmetry? 14. Find lines of symmetry of the figure shown? 15. How many lines of symmetry do the following figures have? 114

PRACTICE SHEET - 1 (PS-1) 16. Complete the following diagrams by drawing the mirror images. a) b) c) d) e) f) g) h) 115

60°PRACTICE SHEET - 2 (PS-2) 1. What is the order of rotational symmetry of a square about its centre? 2. What is the order of rotational symmetry of letter S? 3. What is the order of rotational symmetry for a scalene triangle? 4. What is the order of rotational symmetry of letter X about its centre? 5. Order of rotational symmetry of an equilateral triangle about the centroid is? 6. Letter ‘c’ has rotational symmetry. True or False? 7. Give order of rotational symmetry. i) 180 0, 3600 - Order 2 ii) 3600 - Order 1 iii) 180 0, 3600 - Order 2 iv) 120 0, 240 0, 3600 - Order 3 v) 180 0, 3600 - Order 2 vi) 180 0, 3600 - Order 2 vii) 3600 - Order 1 viii) 180 0, 3600 - Order 2 116

PRACTICE SHEET - 3 (PS-3) 1. Which of the following figures has/have both line symmetry and rotational symmetry? i) Square ii) Rectangle iii) Pentagon iv) Isosceles triangle 2. Which of the following letters have both line symmetry and rotational symmetry? H, O, Z 117

PRACTICE SHEET - 4 (PS-4) I. Choose the correct option: 1. A line segment is symmetrical about its (A) End point (B) Any point (C) Midpoint (D) Point of tri section 2. If a symmetrical shape is folded on its axis of (A) (B) symmetry, then the two portions obtained by the folding are (A) Similar (B) Regular (C) Congruent (D) Equivalent (C) (D) None of the above 3. The line symmetry is also called as (A) Congruent symmetry (B) Point symmetry 9. Statement (A): The angle of rotational (C) Linear symmetry (D) None of the above symmetry for a regular hexagon is 6º. Statement (B): The order of rotational 4. The number of lines of symmetry for a circle is symmetry for a regular hexagon is 60º. (A) 4 (B) 8 (A) Both A and B are true (C) 16 (D) Infinite (B) Both A and B are false. (C) A is true, B is false 5. The number of points of symmetry of a semicir- (D) A is false, B is true cle is (A) 0 (B) 1 10. Statement (A): A circle has a point symmetry (C) 2 (D) 3 about its centre. Statement (B): The angle of turning during rota- 6. The mirror image of the letter 'B' is tion is called the angle of rotation. (A) Both A and B are true (A) (B) (B) Both A and B are false. (C) (D) (C) A is true, B is false (D) A is false, B is true 7. In the adjacent figure l is line of symmetry, then II. Short Answer Questions: 1. Show the rotational symmetry of the following the remaining part to make as symmetric is figure. (A) (B) 2. Apply the rotational symmetry and check the following figure have rotational symmetry. (C) (D) (a) 8. Among the following _______ figure shows line (b) 118 of symmetry.

PRACTICE SHEET - 4 (PS-4) 3. Divide the letters of English alphabet A to Z and find out which have (i) Vertical lines of symmetry (ii) Horizontal lines of symmetry (iii) No lines of symmetry III. Long Answer Questions: 1. Find the order of rotational symmetry of each figure. (a) (b) (c) (d) 2. Analyse and draw the line symmetry of above figure. (a) (b) (c) (d) 119

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. Find the number of lines of symmetry for the 3. Find the number of lines of symmetry of the following figure. (3 Marks) following figure. (2 Marks) ABC DEF G XW DEF ABC 4. In a square, red balls are arranged along one diagonal and yellow balls are arranged aalong with another diagonal. Find the rotational symmetry of the arrangement. (2 Marks) 2. Draw the mirror image of the figure and then make a final image after overlapping it on the mirror image. (3 Marks) 120

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 5. Which capital letter of English alphabets have 7. If a fan has a rotational symmetry or order 4, no line of symmetry? (2 Marks) how many blades are there in the fan? (1 Mark) 6. Which of the following letters have horizontal symmetry? E, G, C, B, O, W (2 Marks) 121

15. Visualising Solid Shapes Learning Outcomes • Identify nets which can be used to make various shapes. At the end of this chapter, students will be able to: • Identify 2 -D and 3-D figures. • Draw oblique sketches of shapes. • Identify edges, vertices and faces of shapes. • Draw isometric sketches of shapes. • Visualise 3-D shapes on a 2-D surface. • View different sections of solids. Concept Map Key Points b. An isometric sketch is drawn on an isometric dot paper, a sample of which is given at the • The circle, the square, the rectangle, the end of this book. In an isometric sketch of the quadrilateral and the triangle are examples of solid the measurements kept proportional. plane figures; the cube, the cuboid, the sphere, the • Different sections of a solid can be viewed in cylinder, the cone and the pyramid are examples many ways: of solid shapes. a. One way is to view by cutting or slicing the • Plane figures are of two-dimensions (2-D) and the shape, which would result in the cross-section solid shapes are of three-dimensions (3-D). of the solid. • The corners of a solid shape are called its vertices; b. Another way is by observing a 2-D shadow of a the line segments of its skeleton are its edges; and 3-D shape. its flat surfaces are its faces. c. A third way is to look at the shape from • A net is a skeleton-outline of a solid that can be different angles; the front-view, the side- folded to make it. The same solid can have several view and the top-view can provide a lot of types of nets. information about the shape observed. • Solid shapes can be drawn on a flat surface (like paper) realistically. We call this 2-D representation of a 3-D solid. • Two types of sketches of a solid are possible: a. An oblique sketch does not have proportional lengths. Still it conveys all important aspects of the appearance of the solid. 122

15. Visualising Solid Shapes PRACTICE SHEETS Work Plan PS – 1 COVERAGE DETAILS PS – 2 PS – 3 (i) Introduction PS – 4 (ii) 2-D and 3-D shapes, Self-evaluation Sheet (iii) Edges, faces and vertices of shapes (iv) Nets of various shapes (i) Oblique sketches (ii) Isometric sketches (i) Visualising solids Worksheet for “Visualising Solid Shapes” Evaluation with self- check or Peer check* 123

PRACTICE SHEET - 1 (PS-1) 3. The given nets are of a dice. Insert the suitable numbers in the blanks. 1. Find the number of vertices in the following figures. 12 a. b. a. 4 5 b. 3 c. d. 6 e. f. c. 6 4 d. 1 5 32 2. Draw the nets for the following. 124

PRACTICE SHEET - 2 (PS-2) 1. The dimensions of a cuboid are 5 cm, 4 cm and 2 cm. Draw three different isometric sketches of this cuboid. 2. Three cubes each with 1 cm edge are placed side by side to form a cuboid. Sketch an oblique and isometric sketch of this cuboid. 3. Use isometric dot paper and make an isometric sketch for each one of the given shapes: a. a. b. 125

PRACTICE SHEET - 3 (PS-3) 2. For each of the solids, identify the corresponding top, front and side views. 1. For each of the given solid, the two views are given. Match for each solid the corresponding top and front views. 126

PRACTICE SHEET - 3 (PS-3) 3. For each given solid, identify the top view, front 4. Draw the front view, side view and top view of view and side view. the given objects. 127

PRACTICE SHEET - 4 (PS-4) I. Choose the correct option: (B) Both A and B are true 1. A sheet of paper is divided into small equilater- (C) A is true, B is false al triangles made up of dots is ____. (A) Isometric dot sheet (B) Graph sheet (D) A is false, B is true. (C) Grid paper (D) None of the above II. Short Answer Questions: 2. The Euler's formula 1. Explain tetrahedron with a diagram. (A) F + V = E + 2 (B) F + 2 + V = E 2. Complete the table by applying Euler's formula. (C) -E + V + F - 2 = 0 (D) All the above Ans: D F85? 3. The shape of a base in a cylinder is V 6 ? 12 (A) Triangle (B) Ellipse E ? 9 30 (C) Square (D) Circle 3. Analyse the following figure and give the names Ans: D of the objects. 4. Among the following polygons, the regular polygon is (A) Parallelogram (B) Right angled triangle (C) Square (D) Rectangle (a) (b) 5. Example for regular polyhedra is ____. (A) Cube (B) Rectangle (C) Cuboid (D) Triangle 6. The number of square shapes in a cube is (A) 4 (B) 8 III. Long Answer Questions: (C) 6 (D) 7 1. Draw a solid using the top side and front views as given below. 7. The square prism and cube are same because, they have same (A) height (B) base (C) length (D) all 8. The number of faces of a book is: (A) 3 (B) 4 2. Draw the shapes that can be formed from the (C) 5 (D) 6 following nets and name it. 9. Statement (A): The name of the polyhedron which has four vertices and four faces is tetrahedron. Statement (B): Each face of a tetrahedron is triangular. (A) Both A and B are false (B) Both A and B are true (C) A is true, B is false (a) (b) (D) A is false, B is true. 10. Statement (A): The solid objects with flat surfaces are called 'regular polyhedra'. Statement (B): The solid objects having congruent face and edges are equal are called polyhedra. (A) Both A and B are false 128

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. What shape does the given nets generate? 4. Draw the front, top and side view of the figure (1 Mark) below. (2 Marks) 2. What shape does the given nets generate? (1 Mark) 3. The given nets are of a dice. Insert the suitable numbers in the blanks. (2 Marks) 2 13 129

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 5. Draw the front, top and side view of the figure 7. For the given isometric figure, draw the oblique below. (3 Marks) figure. (3 Marks) 6. Find the number of edges, vertexes and faces of the given figure. (3 Marks) 130

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