Short Answer Type Questions 10(i) [AS1] Find the common factors and factorize: 4xy + 4x + 3y + 3 (ii) [AS1] Factorize: 12xy − 2y2 − 4yz + 24xz 11(i) [AS1] Factorize: 4x2y + 6xy + 8y2x3 (ii) [AS1] Factorize: 9x4y4 + 18x3y3 + 27x2y2 + 36xy 12 [AS1] Factorize: x2− xz + xy − xz 13 [AS1] Factorize: (y − x)a + (x − y)b 14(i)[AS1] If (a − 2b) is one of the factors of 9(a − 2b)2+ 6(2b − a), find the other factors. (ii) [AS1] If (a + b) is a factor of ax2 + by2 + bx2 + ay2 , then find the other factors. Long Answer Type Questions 15 [AS1] Factorize by grouping the common factors: (i) 3x 2y + 6xy + 9xy2 (ii) 8x4y4 + 6x3y3 + 4x2y2 + 2xy EXERCISE 12.1. FACTORS OF ALGEBRAIC EXPRESSIONS 49
EXERCISE 12.2 FACTORISATION USING IDENTITIES 12.2.1 Key Concepts i. The greatest common factor of two or more monomials is the product of the greatest common factors of the numerical coefficients and the common letters with the smallest powers. ii. If the given expression is the difference of two squares, then to factorize it, we use the formula (a 2– b2 ) = (a + b)(a – b). iii. If the given expression is a complete square, we use one of the following formulae to factorise it: i) a2 + 2ab + b2 = (a + b)2 = (a + b)(a + b) ii) a2 – 2ab + b2 = (a – b)2 = (a – b)(a – b) iv. If the given expression is in the form x2 + x(a + b) + ab, then we factorize it into the form (x + a)(x + b). 12.2.2 Additional Questions Objective Questions 1. [AS1] The factorisation of a4b4 − 16c4 is_______. (A) (ab − 2c)(ab + 2c)(a2b 2+ 4c 2) (B) (ab + 2c)(ab + 2c)(a2b2 + 4c2) (C)(ab − 2c)(ab + 2c)(a2b2 − 4c2) (D)(ab − c)(ab + 2c)(a2b2 + 4c2) 2. [AS1] If one of the factors of 6x3 − 9x is 3x then the other factor is ______. (A) 3x2 − 3 (B) 2x2 − 3 (C)2x3 − 3x (D)2x2 − 3x 3. [AS4] The area of a square field is (x2− 6x + 9) sq. units. The length of its side is _______units. (A) (x + 3) (B) (x − 3) (C)(2x + 3) (D)(x − 6) EXERCISE 12.2. FACTORISATION USING IDENTITIES 50
4. [AS1] A factor of 4x2 + 20x + 25 is . (A) 2x − 5 (B) 2x + 4 (C)2x + 3 (D)2x + 5 5. [AS1] If a factor of x2 − 4 is (x + 2), then the other factor is . (A) x − 4 (B) x + 4 (C) x + 3 (D) x − 2 Very Short Answer Type Questions 6 [AS1] Answer the following questions in one sentence. Factorize: 9a2− 2 16b 7 [AS1] Answer the following questions in one sentence. (i) If one of the factors of x4 − 16y4 is x2 + 4y2 , then find the other factors. (ii) If one of the factors of 4x2− 4x − 15 is 2x − 5, then find the other factor. 8 [AS4] Answer the following questions in one sentence. Raju wants to distribute Rs. (21x3 + 42x2 − 84x + 63 equally among some persons such that each person gets Rs.(x3 + 2x2 − 4x + 3 . Find the number of persons. Short Answer Type Questions 9(i) [AS1] The product of two equal factors is 9x 4 + 30x2y2 + 25y4 . Write the identity to be used to find the equal factors and also find them. (ii) [AS3] The square of an expression is 49m2– 112mn + 64n 2. Write the identity to be used to simplify and also find the expressions. EXERCISE 12.2. FACTORISATION USING IDENTITIES 51
Long Answer Type Questions 10 [AS1] Find the products using identities. (i) (x + 4)(x + 5) (ii) (3x + 4)(4x + 6) (iii) 32a2− 74b2 11 [AS1] Factorize: a2 + 15a + 36 12 [AS1] The area of a parallelogram is (x2 + 5x – 24) square units. Find its base and height. 13 [AS5] Factorize: a4 − 16(b − c)4 and draw a cuboid with these factors as dimensions. EXERCISE 12.2. FACTORISATION USING IDENTITIES 52
EXERCISE 12.3 DIVISION OF ALGEBRAIC EXPRESSIONS 12.3.1 Key Concepts i. Division is the inverse of multiplication. ii. Division of algebraic expressions is similar to the division of numbers. 12.3.2 Additional Questions Objective Questions 1. [AS1] 12x3y3 ÷ 3x2y = . (A) 4xy (B) 4x2y (C) 4 xy2 (D) 4 x2y2 2. [AS1] If the quotient of −15a2bc3 and x is −5ac2 , then x is . (A) 75a3bc5 (B) 3ab (C) 3 (D) 3abc 3. [AS1] If 14x3 − 5x2 + 9x − 1 is divided by 2x − 1 then the remainder is . (A) 1 (B) 2 (C) 3 (D) 4 4. [AS1] 72xyz2 ÷ (− 9xz) = (B) 8yz (A) −8yz (D) −8 xy2 (C) −8 xy EXERCISE 12.3. DIVISION OF ALGEBRAIC EXPRESSIONS 53
5. [AS4] The area of a blackboard in a class is 15m2n3 sq. units and its length is 5m2n 2 units. Its breadth is units. (A) 3mn (B) 3m (C) 3n (D) 3m2n Very Short Answer Type Questions 6 [AS1] Answer the following questions in one sentence. (i) Divide −72x 2yz by −12xyz. (ii) Divide 72a4b5c8 by −9a2b2c3 7 [AS2] Answer the following questions in one sentence. Using division of polynomials, state whether x + 6 is a factor of x2 − x − 42 or not. 8 [AS4] Answer the following questions in one sentence. The marks scored by John for 7x questions is 49x2 − 14x. Find the marks for each question. Short Answer Type Questions 9 [AS1] Divide and write the quotient in each of the following. (i) 64x4 ÷ 4x3 (ii) 100000x5 ÷ (10)3x4 (iii) 16x3y3z3 ÷ 2xyz 10 [AS1] Divide: 6x 4 + 12x3 + 10x2 ÷ 2x2 11(i) [AS1] Divide: (9x2 + 63x) ÷ (x + 7) (ii) [AS1] Divide 36y3(50y2 − 72) by 24y2(5y + 6). 12(i) [AS4] A car travels 33(p4+ 5p3 − 24p2) km in 11p(p + 8) h. Find the speed of the car. (ii) [AS4] The cost of a cloth of length 3xyz m is 3x 2yz + 6xy2 z + 9xyz2, then find the cost of 1 m of cloth. EXERCISE 12.3. DIVISION OF ALGEBRAIC EXPRESSIONS 54
Long Answer Type Questions 13 [AS1] Divide: (i) 24xy2z3 by 6yz2 (ii) 63a2b4c6 by 7a2b2c3 EXERCISE 12.3. DIVISION OF ALGEBRAIC EXPRESSIONS 55
EXERCISE 12.4 ERROR ANALYSIS 12.4.1 Key Concepts i. One must remember to solve problems containing algebraic expressions in different operations,keeping in mind the errors with the concepts of factorization and operations. 12.4.2 Additional Questions Objective Questions 1. [AS1] Sunil simplified (x + 2)(x − 2) as x2 + 4. The correct product is . (A) x2 + 2 (B) x2 − 2 (C) x2 − 4 (D) x2 + 3 2. [AS1] (3x)2 + 7x = (B) 16x (A) 10x (D) 21 x3 (C)9x2 + 7x 3. [AS1] If x = −3, the value of 2x2+ 12x − 9 is . (A) 9 (B) –27 (C) 36 (D) −36 4. [AS1](x − 7)2 = (B) x2 + 49 (A) x2 − 49 (D) x2 − 14x + 49 (C) x2 + 14x + 49 EXERCISE 12.4. ERROR ANALYSIS 56
5. [AS3] The correct statement among the following is . (A) 9(x − 8) = 9x − 17 (B) 3x + 3 = 3 x+1 (C)(x − 3)2 = x2 − 6x + 6 (D)(x − 7)(x + 7) = x2 + 49 Long Answer Type Questions 6 [AS2] James and Sam have done the following multiplication. Verify whose multiplication is correct. James Sam i) 9(a − b) = 9a − 9b i) 9(a − b) = 9a − 9b ii) (4x)2 = 16x2 ii) (4x)2 = 4x2 iii) (a + b)(a − b) = a2 − b2 iii) (a + b)(a − b) = a2 + b2 iv) (9x+6y)2 = 81x2+36y2+108xy iv) (9x+6y)2 = 81x2+6y2+81xy v) (3x − 6y)2 = 9x2 + 36y2 − 36xy v) 9x2 + 6y2 + 36xy = (3x − 6y)2 7 [AS2] Amina solved an equation as follows: 6x 2+ 35xy − 6y2 = 6x2 + 36xy − xy − 6y2 = 6x(x + 6y) − y(x + 6y) = (x + 6y)(6x − y) What can you say about the correctness of the solution? Identify where she has gone wrong and correct it. EXERCISE 12.4. ERROR ANALYSIS 57
8 [AS2] Find out the error and correct it. x2 − 4x + 4y − y2 = (x2 − y2) − 4(x + y) = (x − y)(x + y) − 4(x + y) = (x + y)(x − y − 4) 9 [AS5] In each of the following, the value of x = −5 is substituted and are simplified. Find out the errors (if any) in the substitution and represent them correctly. (i) x2 + 5x + 25 = 25 + 25 + 25 = 75 (ii) x2 + 13x + 16xy + 14x = 25 − 65 + 16y(−5) − 70 (iii) x2 + 6x2 + 7x + 12 = 25 − 6(−5)2 + 7(−5)y + 12 (iv) x3 − 3x2 + 6xy + 7xy + 18 = 125 − 3(25) − 30y + 18 (v) x2 − 7x2 + 8x + 14 = 25 + 155 + 8(5) + 14 EXERCISE 12.4. ERROR ANALYSIS 58
CHAPTER 14 SURFACE AREA AND VOLUME (CUBE-CUBOID) EXERCISE 14.1 SURFACE AREA OF CUBE AND CUBOID 14.1.1 Key Concepts i. If l, b and h are the dimensions of a cuboid, then: a. its lateral surface area is 2h(l + b) sq. units. b. its total surface area is 2(lb + bh + hl) sq. units. ii. Lateral surface area of a cube is 4a2 sq. units. iii. Total surface area of a cube is 6a 2 sq. units. 14.1.2 Additional Questions Objective Questions 1. [AS3] L.S.A. of a cuboid is sq. units. (A) 2h(l + b) (C) 4a2 (B) 2(l + b) (D) 6a2 2. [AS1] If s = 5 cm, then T.S.A. of a cube is cm2. (A) 125 (B) 150 (C) 100 (D) 175 3. [AS3] T.S.A. of a cube is sq. units. (A) a3 (C) 4a2 (B) 2h (l +b) (D) 6a2 EXERCISE 14.1. SURFACE AREA OF CUBE AND CUBOID 59
4. [AS1] If the side of a cube is doubled then the change in its T.S.A is . (A) 2 times (B) 3 times (C)4 times (D)5 times 5. [AS1] The lateral surface area of a cuboid of length 3 cm and breadth 5 cm is 112 cm2. Then its height is cm. (A) 4 (B) 5 (C) 6 (D) 7 6. [AS1] The dimensions of a room are 16 m × 13 m × 10 m. Area of its 4 walls is (A) 2080 m2 (B) 580 m2 (C)1040 m 2 (D)720 m 2 7. [AS1] If the lateral surface area of a cube is 400 m2then the length of its side is ______. (A) 10 m (B) 25 m (C)600 m (D)20 m 8. [AS4] An iron shelf of dimensions 2 m × 1 1 m × 1 m was to be made. The area of iron sheet 2 2 required to make it is ______. (A) 9.50 m2 (B) 95.0 m2 (C)0.95 m2 (D)950 m2 9. [AS4] Raju purchased a box with dimensions 25 cm × 15 cm × 10 cm. Raju wants to paint it. The area to be painted by Raju is_____. (A) 1500 cm2 (B) 1550 cm2 (C)3750 cm2 (D)800 cm2 EXERCISE 14.1. SURFACE AREA OF CUBE AND CUBOID 60
10. [AS5] A cuboid among the following is ________. (A) (B) (C) (D) Very Short Answer Type Questions 11 [AS1] Answer the following questions in one sentence. (i) Find the L.S.A of a cuboid of dimensions 24 cm × 18 cm × 12 cm. (ii) Find T.S.A. of a cube of side 12 cm. (iii) Find the L.S.A. of a cube whose side is 11 cm. 12 [AS1] Answer the following questions in one sentence. (i) If the area of four walls of a room of length 15 m and breadth 10 m is 350 m2 , find its height. (ii) If the total surface area of a cube is 600 m2, find its side. EXERCISE 14.1. SURFACE AREA OF CUBE AND CUBOID 61
13 [AS1] Answer the following questions in one sentence. (i) If the length and breadth of a cuboid are doubled, how many times does its L.S.A increase? (ii) The side of a cube is tripled. Find the increase in its T.S.A. 14 [AS4] Answer the following questions in one sentence. (i) The perimeter of the floor of a room is 30 m and its height is 3 m. Find the area of its four walls. (ii) Raju made a cube of edge 15 cm with cardboard. Find how much cardboard is required to make it. 15 [AS5] Answer the following questions in one sentence. (i) Four cubes of side 1 cm are joined together such that two surfaces of each cube are coincident with two other cubes. Represent this with a diagram. Also find the L.S.A of the new cube formed. (ii) If six cubes of 1 cm are arranged in such a way that two surfaces of each cube are coincident with two other cubes. Draw a figure for this and find its L.S.A. Short Answer Type Questions 16 [AS1] A cuboid is to be made using a cardboard. The dimensions of the cuboid are If the cardboard costs Rs. 5 per sq. cm, find the total cost of making the cuboid. 17 [AS4] If the cost of painting the T.S.A. of a cuboid of length 50 cm and breadth 35 cm at the rate of Rs. 2.50 / sq. cm is Rs.17250, find the height of the cuboid. 18 [AS4] The measures of a textbook are 22 cm × 11 cm × 3 cm. It is to be covered with a brown paper. If each book requires 164 cm2 of more paper for folding, how much paper is required to wrap 85 such books? 19 [AS4] An aquarium is in the form of a cuboid whose external measures are 80 cm × 30 cm × 40 cm. The base, side and back faces are to be covered with a coloured paper. Find the area of the paper needed. Long Answer Type Questions 20 [AS1] Find the surface area of a cuboid whose height is h, length is l and breadth is b. 21 (i) [AS1] Find the surface area of a cube whose edge is 6 cm. (ii) [AS1] Find the total surface area of a cuboid formed by combining two cubes each of edge 8 cm. 22 [AS4] Find the cost of painting the outer surface of a closed box which is 50 cm long, 40 cm broad and 20 cm high at the rate of 80 paise per cm2. EXERCISE 14.1. SURFACE AREA OF CUBE AND CUBOID 62
23 [AS1] Find the total surface area of the following cylinders. 24 [AS1] The dimensions of a cuboid are in the ratio of 1 : 2 : 3 and its total surface area is 88 m2. Find the dimensions and also its lateral surface area. 25 [AS5] Draw a cube with height ‘h’, length ‘l’ and breadth ‘b’. Find its total surface area and lateral surface area. Compare them. What do you understand? 26 [AS1] If the edge of a cube is increased by 25%, then by what percentage does the surface area increase? 27 [AS1] If the surface area of the cuboid is 1300 cm2 and its height and length are 20 cm and 10 cm, find its breadth. 28 [AS2] Two cubes each of side 'b' units are joined to form a cuboid as shown in the figure. What is the surface area of this cuboid? Is it 12 b2? Is the surface area of cuboid formed by joining three such cubes 18 b2? Why? 29 [AS4] The dimensions of an oil tin are 26 cm × 26 cm × 45 cm. Find the area of the tin sheet required for making 20 such tins. If 1 square metre of the tin sheet costs Rs.10, find the cost of the tin sheet used for these 20 tins. 30 [AS4] A rectangular block of ice measures 40 cm by 25 cm by 15 cm. Calculate its weight in kg, 9 of the weight of the same volume of water and 1 cm3 of water weighs 1g. if ice weighs 10 31 [AS4] A swimming pool is 20 m in length, 15 m in breadth and 4 m in depth. Find the cost of cementing its floor and walls at the rate of Rs.12 per square metre. 32 [AS4] Arrange 12 cubes of edges of equal length to form a cuboid of the smallest surface area. EXERCISE 14.1. SURFACE AREA OF CUBE AND CUBOID 63
EXERCISE 14.2 VOLUME OF CUBE AND CUBOID 14.2.1 Key Concepts i. Objects that have a definite shape are called solids. ii. The space occupied by a solid body is called its volume. iii. The standard unit of volume is cu.cm or cm3 1000 cu.mm = 1 cu.cm 1000 cu.cm = 1 cu.dm 1000 cu.dm = 1 cu.m 1 l = 1 cu.dm 1 kl = 1000 l iv. For a cuboid of length l, breadth b and height h: volume = l × b × h cubic units. v. For a cube of side ‘a’ , volume = a3 cubic units. 14.2.2 Additional Questions Objective Questions 1. [AS3] Volume of a cuboid = cu. units. (A) lb (B) lbh h (C) s3 (D) lh b 2. [AS1] The volume of a piece of wood measuring 20 cm × 10 cm × 8 cm is . (A) 1700 cu. cm (B) 1600 cu. cm (C)1800 cu. cm (D)1900 cu. cm 3. [AS3] 1 litre = . (A) 1000 cu. cm (B) 100 cu. cm (C)10000 cu. cm (D)10 cu. cm EXERCISE 14.2. VOLUME OF CUBE AND CUBOID 64
4. [AS1] The volume of a cube is 125 cu. cm. Then, the length of its edge is . (A) 3 cm (B) 4 cm (C)5 m (D)5 cm 5. [AS1] Two cubes each of edge 3 cm are joined end to end. The volume of the resulting cuboid is . (A) 27 cu. cm (B) 72 cu. cm (C)54 cu. cm (D)45 cu. cm 6. [AS1] If the volume of a cuboid of breadth 15 m and height 8 m is 2400 cu. m, then its length is _____. (A) 10 m (B) 15 m (C) 20 m (D)25 m 7. [AS1] If the volume of a cube is 42875 cu. m, then the length of its side is ______. (A) 25 m (B) 35 m (C)45 m (D)15 m 8. [AS4] The volume of a brick which is 20 cm long and 8 cm wide is 640 cu. cm. The height of the brick is _____. (A) 32 cm (B) 16 cm (C)8 cm (D)4 cm 9. [AS4] The dimensions of a cuboid are 15 cm × 10 cm × 8 cm. The number of cubes of 1 cm3 required to make this cuboid is _____. (A) 1200 (B) 200 (C) 270 (D) 540 EXERCISE 14.2. VOLUME OF CUBE AND CUBOID 65
Very Short Answer Type Questions 10 [AS1] Answer the following questions in one sentence. (i) Find the volume of a cuboid whose length is1.5 m, breadth is 25 cm and height is 15 cm. (ii) Find the volume of a cube of side 7 cm. 11 [AS1] Answer the following questions in one sentence. (i) If the area of the base of a cuboid is 216 sq. m and its volume is 2376 cu. m, then find its height. (ii) If the length and height of cuboid are 24 m and 8 m and its volume is 3072 m , then find its breadth. (iii) If the volume of a cube is 19683 cu. m, then find its side. 12 [AS1] Answer the following questions in one sentence. (i) The length of a cuboid is doubled and its height is reduced by12 . What is the change in its volume? (ii) The side of a cube is doubled. Find the change in its volume. 13 Answer the following questions in one sentence. [AS4] Akhilesh wants to find out volume of air contained in his room and he found measurements of room as l = 15m, b = 12m, h = 4m. Help him to find it out. Short Answer Type Questions 14 [AS1] The length of the a side of a cube is 15 cm. Find the volume and T.S.A of the cube. 15 [AS4] The volume of a cuboidal room is 25346 cu. m. Find the height of the room if the cost of polishing its floor at Rs.7.50 per sq. m is Rs.8265. 16(i) [AS4] How many soap cakes each measuring 7 cm × 5 cm × 2.5 cm can be placed in a box of size 56 cm × 40 cm × 25 cm? (ii) [AS1] If equal cubes, each of edge 5 cm are placed adjacent to each other, find the volume of the cuboid so formed. [AS4] A match box measures 4 cm × 2.5 cm × 1.5 cm. What will be the volume of a packet containing 12 such match boxes? How many such packets can be placed in a card board box whose size is 60 cm × 30 cm × 24 cm? 18 [AS4] A field is 80 m long and 50 m broad. In one of its corners, a pit 10 m long, 7.5 m broad and 8 m deep is dug. The earth taken out of it is evenly spread over the remaining part of the field. Find the rise in the level of the field. EXERCISE 14.2. VOLUME OF CUBE AND CUBOID 66
Long Answer Type Questions 19 [AS4] A box is 80 cm long, 40 cm wide and 100 cm high. Soap cakes measuring 8 cm × 5 cm × 2.5 cm are to be packed in the box, so that no space is left. Find the number of soap cakes that can be packed in each box. 20 [AS4] How many cubes each of side 4 cm can be cut from a metal block in the form of a cuboid whose length, breadth and height are 12 cm, 8 cm and 10 cm respectively? What volume of the metal is wasted? 21 [AS4] Water is pouring into a cuboidal reservoir at the rate of 40 litres per minute. If the length of the reservoir is 6 m, breadth is 4 m and height is 2 m, find the number of hours it will take to fill the reservoir. 22 [AS4] A water tank built by a municipality of a town to supply water to its 25000 inhabitants at 125 litres per day per person is 40 m long and 31.25 m broad. The tank, when it is full, can supply water for two days to the inhabitants of the town. Find the depth of the tank. EXERCISE 14.2. VOLUME OF CUBE AND CUBOID 67
—— Project Based Questions —— (i) Conduct a survey in your locality, find out the age of the people living in your locality. For the data you collect, make a grouped frequency distribution table. Also represent the data using a histogram and frequency polygon. Calculate the mean, median and mode for the data. From the data collected, find out which age group has the highest number of people. (ii) Collect pictures of different objects, structures, flowers etc. In these, show how many lines of symmetry exist and also mention what is the order of rotation for each picture. Stick the pictures and represent all the information. (iii) Take an object which can cast a shadow. Place the object at a point and flash light on the object using a torch. Measure the length of the shadow and the distance of the torch from the object. Repeat your observation, by moving closer or far away from the object. Tabulate your observations. How do the length of the shadow and the distance of the torch from the object vary? Is it a case of inverse proportion or a direct proportion? Explain your answer with reasons. PROJECT BASED QUESTIONS 68
Additional AS Based Practice Questions Q1 [AS2] The score obtained by a student in his exams in 5 subjects are 92, 95, 90, 88 and 94, all out of 100. Calculate and verify if the mean of his marks is 91.8 or 93.8. Q2 [AS2] The mean of 11 observations is 12. If one observation 17 is deleted, then show that the new mean is 11.5. Q3 [AS2] The perimeters of two circles are equal. Are their areas also equal? Check and verify. Q4 [AS2] The ratio of the bases of two triangles and the ratio of their areas are given as x : y and a : b respectively. Find the ratio of their corresponding altitudes. Q5 [AS5] Find the length of the side of the given square if area of the square is 625 square units and then find the value of ‘x’. Q6 [AS5] The figure shows the dimensions of a wall having a window and a door of a room. Write an algebraic expression for the area of the wall to be painted. ADDITIONAL AS BASED PRACTICE QUESTIONS 69
Q7 [AS2] If two cubes each of edge 2 cm are joined end to end, then verify if the volume of the resulting cuboid is 16 cm3 or 32 cm3. Q8 [AS2] If the edge of a cube is increased 3 times, then show that its T.S.A. is 9 times that of the original T.S.A. Q9 [AS5] 36 cubes of the same size are arranged to form a cuboid as shown in the given figure. Write the length, breadth and height and find the volume of the cuboid. Q10 [AS5] The surface area of the given cube is: A) 600 cm2 B) 100 cm2 C) 300 cm2 D) 120 cm2 ADDITIONAL AS BASED PRACTICE QUESTIONS 70
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