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TABLE OF CONTENTS 6 LINEAR EQUATIONS IN TWO VARIABLES 1 6.1 LINEAR EQUATIONS IN TWO VARIABLES 1 6.2 SOLUTION OF A LINEAR EQUATION IN TWO VARIABLES 3 6.3 GRAPH OF LINEAR EQUATION IN TWO VARIABLES 5 6.4 EQUATIONS OF LINES PARALLEL TO X–AXIS AND Y–AXIS 9 10 SURFACE AREAS AND VOLUMES 12 10.1 SURFACE AREA OF A CUBOID 12 10.2 RIGHT CIRCULAR CYLINDER 16 10.3 RIGHT CIRCULAR CONE 19 10.4 SPHERE 21 11 AREAS 23 11.1 AREA OF PLANAR REGIONS 23 11.2 PARALLELOGRAMS ON THE SAME BASE AND BETWEEN THE SAME PARALLELS 26 11.3 TRIANGLES ON THE SAME BASE AND BETWEEN THE SAME PARALLELS 29 12 CIRCLES 32 12.1 INTRODUCTION 32 12.2 ANGLE SUBTENDED BY A CHORD AT A POINT ON THE CIRCLE 36 12.3 PERPENDICULAR FROM THE CENTRE TO A CHORD 38 12.4 ANGLE SUBTENDED BY AN ARC OF A CIRCLE 42 12.5 CYCLIC QUADRILATERALS 49 13 GEOMETRICAL CONSTRUCTIONS 56 13.1 BASIC CONSTRUCTIONS 56 13.2 CONSTRUCTION OF TRIANGLES (SPECIAL CASES) 58
TABLE OF CONTENTS 14 PROBABILITY 60 14.1 PROBABILITY/USES OF PROBABILITY IN REAL LIFE 60 15 PROOFS IN MATHEMATICS 67 15.1 MATHEMATICAL STATEMENTS 67 15.2 REASONING IN MATHEMATICS 70 15.3 THEOREMS, CONJECTURES AND AXIOMS 72 15.4 WHAT IS A MATHEMATICAL PROOF? 74 PROJECT BASED QUESTIONS 76 ADDITIONAL AS BASED PRACTICE QUESTIONS 78
CHAPTER 6 LINEAR EQUATIONS IN TWO VARIABLES EXERCISE 6.1 LINEAR EQUATIONS IN TWO VARIABLES 6.1.1 Key Concepts i. An equation of degree one is called a linear equation. ii. A linear equation with two variables is called a linear equation in two variables. iii. The general form of a linear equation in two variables is ax + by + c = 0. 6.1.2 Additional Questions Objective Questions 1. [AS1] The cost of a pen is Rs. 2 more than the cost of 3 pencils. Its equation form is . (A) y = 2x + 3 (B) y = x + 23 (C)y = 3x + 2 (D)None of these 2. [AS1] The cost of 5 balls and 8 corks is Rs. 85. Its equation form is . (A) 5x + 8y = 85 (B) 5x –8 y = 85 (C) 5x × 8y = 85 (D) None of these 3. [AS1] The sum of twice a number and thrice another number is 70. Its equation form is . (A) 2a × 3b = 70 (B) 2a + 3b = 70 (C) 2a + 3b + 70 = 0 (D) None of these EXERCISE 6.1. LINEAR EQUATIONS IN TWO VARIABLES 1
4. [AS1] The difference of two numbers is 10. The equation form for this is . (A) 10 – b = a (B) 10 – a = b (C)a + b = 0 (D)a – b = 10 5. [AS1] Three times a number subtracted from two times another number gives 27. The equation for this is . (A) 2x – 3y = 27 (B) 3x – 2y = 27 (C)2x + 3y = 27 (D)2x – 3y + 27 = 0 Short Answer Type Questions 6 [AS3] Express the linear equation 24 = 4x in the form ax + by + c = 0 and write the values 5 of a, b and c. 7 [AS1] Express the linear equation 3x = 4y – 17 in the general form ax + by + c = 0 and find the values of a, b and c. 8(i) [AS4] Latha’s age is 3 times the age of Lalitha. Write a linear equation in two variables to represent the above information. (ii) [AS4] The total cost of 3 pens and 5 note books is 155. Express this as a linear equation in two variables. Long Answer Type Questions 9 [AS1] Express the equation 3x = 5y −4 as a linear equation of the form ax + by + c = 0 and also 4 2 find the coefficients a, b and c. 10 [AS4] The sum of a two–digit number and the number obtained by reversing its digits is 132. Express this as a linear equation in two variables x and y. EXERCISE 6.1. LINEAR EQUATIONS IN TWO VARIABLES 2
EXERCISE 6.2 SOLUTION OF A LINEAR EQUATION IN TWO VARIABLES 6.2.1 Key Concepts i. Any pair of values of ‘x’ and ‘y’ which satisfy the linear equation in two variables is called its solution. ii. There will be many solutions for a linear equation in two variables. 6.2.2 Additional Questions Objective Questions 1. [AS1] A solution of the equation 3x + y = 6 is . (A) (1, 6) (B) (0, 6) (C)(2, 6) (D)(3, 3) 2. [AS1] A solution of the equation x + 2y = 4 is . (A) (2, 4) (B) (2, 0) (C)(–1, 2) (D)(0, 2) 3. [AS1] An equation that has x = 2 and y = 1 as a solution is . (A) x + y + 4 = 0 (B) 5x + 3y = 14 (C)2x + 5y = 9 (D)None of these 4. [AS1] A solution of 4x – 5y = 5 is . (A) (5, 3) (B) (3, 5) (C)(0, 1) (D)(2, –1) EXERCISE 6.2. SOLUTION OF A LINEAR EQUATION IN TWO VARIABLES 3
5. [AS1] A solution of 7x + 3y = 27 is . (A) (7, 3) (B) (3, 2) (C)(4, –1) (D)(5, –3) Long Answer Type Questions 6 [AS1] Find three different solutions of the linear equation, 4x + 5y = 23. 7 [AS1] If (x, y) = (2, 5) is a solution of the equation 3x – 4y = k, find the value of k and rewrite the equation. 8 [AS1] If (a, 0) and (0, b) are the solutions of the linear equation 2x – 3y + 6 = 0, find the values of a and b. 9 [AS2] Check which of the following are the solutions of the equation 3x + y = 13. (i) (3, 4) (ii) (2, 5) (iii) (5, –2) (iv) (7, 2) (v) (4, 1) EXERCISE 6.2. SOLUTION OF A LINEAR EQUATION IN TWO VARIABLES 4
EXERCISE 6.3 GRAPH OF LINEAR EQUATION IN TWO VARIABLES 6.3.1 Key Concepts i. There can be many solutions for a linear equation in two variables. ii. These solutions can be plotted on a graph sheet to obtain the graph of that equation. iii. The graph of a linear equation in two variables is a straight line. 6.3.2 Additional Questions Objective Questions 1. [AS1] The graph of y = x passes through . (A) (1, 7) (B) (–2, –5) (C) (0, 0) (D)(3, 5) 2. [AS1] A point on the graph of 3x + y = 7 is . (A) (5, 2) (B) (4, 1) (C)(1, 3) (D)(2, 1) 3. [AS1] The graph of 2x – 5y = 8 passes through . (A) (–1, –2) (B) (1, 2) (C)(3, 2) (D)(5, 4) 4. [AS1] The line passing through (0, –3) is . (A) 3x + y = 2 (B) 4x – y = 3 (C)2x – y = 5 (D)x + y = 3 EXERCISE 6.3. GRAPH OF LINEAR EQUATION IN TWO VARIABLES 5
5. The graph of a linear equation is a . (A) curve (B) straight line (C) parabola (D) none Short Answer Type Questions 6 [AS2] Of the given four linear equations, find the equation that represents the given graph. (i) x = y (ii) x + y = 2 (iii) y = 2x (iv) x = 2y EXERCISE 6.3. GRAPH OF LINEAR EQUATION IN TWO VARIABLES 6
7 [AS2] Of the given four linear equations, find the equation that represents the given graph. (i) x + y = 0 (ii) 2x – 3y + 6 = 0 (iii) 2x + 3y + 6 = 0 (iv) 3x – 2y + 6 = 0 8(i) [AS4] The sum of two numbers is 8. Frame a linear equation in two variables. (ii) [AS4] Rita’s age is thrice the age of Gita. Frame the linear equation in two variable and represent it on a graph. Find Rita’s age when Gita’s age is 3 years. 9(i) [AS5] Draw the graph of y = x. (ii) [AS5] Draw the graph of y = 4x. 10 [AS4] When Raju was born, his father was 25 years old. Form an equation and draw a graph for this data. From the graph, find the age of his father when Raju is 15 years old. 11 [AS5] Draw the graph of x + 2y = 3 on a graph sheet. EXERCISE 6.3. GRAPH OF LINEAR EQUATION IN TWO VARIABLES 7
12 [AS5] Draw the graph of the equation 2x + 3y = 6. From the graph find (i) The solution (x, y) where x = 6. (ii) The solution (x, y) where y = 4. 13 [AS5] Draw the graph of the equation 2x + y = 5. From the graph find (i) The solution (x, y) such that x = 5. (ii) The solution (x, y) such that y = 3. EXERCISE 6.3. GRAPH OF LINEAR EQUATION IN TWO VARIABLES 8
EXERCISE 6.4 EQUATIONS OF LINES PARALLEL TO X–AXIS AND Y–AXIS 6.4.1 Key Concepts i. An equation of the type y = mx represents a line passing through the origin. ii. The graph of x = k is a line parallel to Y–axis and at a distance of ‘k’ units from Y–axis. iii. The graph of x = k passes through the point (k, 0) where k ∈ Q. iv. The graph of y = k is a line parallel to X–axis and at a distance of ‘k’ units from X–axis. v. The graph of y = k passes through the point (0, k) where k ∈ Q. vi. The equation of X–axis is y = 0. vii. The equation of Y–axis is x = 0. 6.4.2 Additional Questions Objective Questions 1. [AS1] The equation of a line parallel to X–axis is . (A) y = 5 (B) x = 5 (C) x – 3 = 0 (D) y = x 2. [AS1] The equation of a line perpendicular to X–axis is . (A) y = 5 (B) x = 5 (C)x –y = 3 (D)x = y 3. [AS1] The equation of a line passing through (3, 4) is . (A) x – y = 2 (B) 2x – 3y = 0 (C)3x + 4y = 10 (D)4x – 3y = 0 4. [AS1] The equation y = mx + c passes through the point . (A) (c, 0) (B) (c, m) (C)(0, c) (D)(m, c) EXERCISE 6.4. EQUATIONS OF LINES PARALLEL TO X –AXIS AND Y –. . . 9
5. [AS1] The point of intersection of x = a and y = b is . (A) (a, b) (B) (b, a) (C)(ab, 0) (D)(0, ab) Very Short Answer Type Questions 6 [AS2] Fill in the blanks. to the X–axis. (i) The line x – 1 = 0 is (ii) The equation of a line parallel to X–axis at a distance of 3 units in the positive direction is . . (iii) The line x = 3 is to the Y–axis. (iv) The equation of X –axis is . (v) The equation of a line parallel to Y–axis at a distance of 4 units in the positive direction is Short Answer Type Questions 7(i) [AS1] Write the equations of any two lines that are parallel to X –axis. (ii) [AS1] Write the equations of any three lines that are parallel to Y–axis. 8(i) [AS5] Give the graphical representation of x − 2 = 0. (ii) [AS5] Give the graphical representation of 2y = 5. Long Answer Type Questions 9 [AS1] Write the equations of the lines parallel to X–axis and Y–axis passing through the points: (i) (1, 2) (ii) (–3, 4) (iii) (–2, –5) (iv) (3, –2) (v) (0, 0) EXERCISE 6.4. EQUATIONS OF LINES PARALLEL TO X –AXIS AND Y –. . . 10
10 [AS5] Solve the equation 6x + 4 = 3x + 10 and represent the solution on (i) A number line. (ii) A Cartesian plane. 11 [AS5] Solve the equation 4x – 3 = 3x + 1 and represent the solution on (i) A number line. (ii) A Cartesian plane. EXERCISE 6.4. EQUATIONS OF LINES PARALLEL TO X –AXIS AND Y –. . . 11
CHAPTER 10 SURFACE AREAS AND VOLUMES EXERCISE 10.1 SURFACE AREA OF A CUBOID 10.1.1 Key Concepts i. The objects having only length and breadth are called 2 –D objects. ii. The objects having length, breadth and height (thickness) are called 3 –D objects. These are also known as solid figures. iii. The cuboid and cube both have six faces, of which four are lateral faces and there is a base and a top. iv. If length (l), breadth (b) and height (h) are the measurements of a cuboid, then a) The lateral surface area of cuboid = 2h(l + b) b) The total surface area of cuboid = 2(lb + bh + hl) c) The volume of cuboid = lbh v. If the length of the edge of the cube is ‘a’ units, then a) The LSA of cube = 4a2 b) The TSA of cube = 6a2 c) The volume of cube = a3 vi. The lateral surface area (LSA) of a regular prism is the product of its perimeter and height = 2h(l + b) vii. The total surface area (TSA) of a regular prism is the sum of the lateral surface area (LSA) and twice the base area. TSA of a rectangular prism = 2(lb + bh + lh) √ viii. If the base of a right prism is an equilateral triangle, its volume is 3 a2h cubic units, where ‘a’ is 4 the side of the base and ‘h’ is its height. ix. The lateral surface area of a pyramid = Perimeter of its base × slant height 2 x. The total surface area of a pyramid = LSA + Area of its base xi. The volume of a pyramid = 1 × Area of its base × height 3 EXERCISE 10.1. SURFACE AREA OF A CUBOID 12
10.1.2 Additional Questions Objective Questions 1. [AS1] The lateral surface area of a cuboid of length 6 m, breadth 4 m and height 3 m is sq. m. (A) 54 (B) 108 (C) 72 (D) 60 2. [AS1] The total surface area of a cube of side 20 m is sq. m. (A) 3200 (B) 2400 (C) 800 (D) 1600 3. [AS1] The volume of a cube of side 20 cm is cu. cm. (A) 3200 (B) 2400 (C) 8000 (D) 1600 4. [AS1] The volume of a cuboid of dimensions 50 m, 30 m and 10 m is cu. m. (A) 15000 (B) 1600 (C) 4600 (D) 2300 5. [AS1] If L = 3B = 4H where L, B and H are the dimensions of a cuboid and L = 60 m then the area of four walls of the cuboid is sq. m. (A) 2400 (B) 2600 (C) 2200 (D) 2300 Short Answer Type Questions 6(i) [AS1] a) The side of a cube is 18 cm. Find its lateral surface area and total surface area. b) Find the surface area of the given rectangular prism in which each = 1 cm2 . EXERCISE 10.1. SURFACE AREA OF A CUBOID 13
(ii) [AS1] A thick metallic box 3 m long, 2.5 m wide and 2 m depth is to be made. If it is open at the top, find the a) area of the sheet required for making the box. b) cost of sheet if the sheet measuring 1 m2 costs Rs. 20. 7(i) [AS1] The volume of a cube is 1000 cubic centimetres. What is the length of its edge? (ii) [AS1] A cube of a metal of 6 cm edge is melted and recast into a cuboid whose base is 3.60 cm × 0.60 cm. Find the height of the cuboid. Find also the surface areas of the cuboid and the cube. 8(i) [AS1] A pyramid has a square base of side 4 cm and a height of 9 cm. Find its volume. (ii) [AS1] Find the volume of the given pyramid, if its height is 15 cm. 9(i) [AS1] The total surface area of a cube is 346.56 sq. cm. Find its side. (ii) [AS1] The dimensions of a rectangular solid are in the ratio 4 : 3 : 2 and its total surface area is1300 cm2. Find its length, breadth and height. EXERCISE 10.1. SURFACE AREA OF A CUBOID 14
Long Answer Type Questions 10 [AS1] Find the surface areas of the following prisms. 11 [AS4] The length of a cold storage is double its breadth. Its height is 3 m. The area of its four walls is 180 m2 . Find its volume. 12 [AS4] A teak wood log is cut first in the form of a cuboid of length 2.3 m, width 0.75 m and of a certain thickness. Its volume is 1.104 m3. Find its thickness. Also find the number of rectangular planks of size 2.3 m x 0.75 m x 0.04 m that can be cut from the cuboid. 13 [AS4] Hameed has built a cuboidal water tank with lid for his house, with each outer edge 1.5 m long. He gets the outer surface of the tank excluding the base covered with square tiles of side 25 cm. Find the amount he would spend for the tiles, if the cost of the tiles is Rs. 360 per dozen. EXERCISE 10.1. SURFACE AREA OF A CUBOID 15
EXERCISE 10.2 RIGHT CIRCULAR CYLINDER 10.2.1 Key Concepts i. The volume of a pyramid is 1 rd the volume of a right prism if both have the same base and 3 height. ii. A cylinder is a solid having two circular ends with a curved surface as lateral surface. iii. If the line segment joining the centres of the base and top is perpendicular to the base, then it is called a right circular cylinder. iv. If ‘r’ is the radius of the right circular cylinder, then a) Curved surface area of the cylinder = 2πrh b) Total surface area of the cylinder = 2πr(h + r) c) Volume of the cylinder = πr2h 10.2.2 Additional Questions Objective Questions 1. [AS1] The radius and height of a right circular cylinder are in the ratio 2 : 3 and its curved surface area is 462 sq. m. Then its height is m. (A) 7 (B) 10.5 (C) 21 (D) 3.56 2. [AS1] The total surface area of the right circular cylinder whose base radius is 3 cm and height is 4 cm is sq. cm. (A) 132 (B) 528 7 (C) 528 (D) 132 7 3. [AS1] The volume of the cylinder with base radius 4 cm and height 14 cm is cu. cm. (A) 1408 (B) 176 (C) 352 (D) 704 EXERCISE 10.2. RIGHT CIRCULAR CYLINDER 16
4. [AS1] The volume of a cylinder is 10692 cu. cm. and its base area is 1782 sq. cm. 7 7 Then its height is cm. (A) 6 (B) 42 7 (D) 12 (C) 6 5. [AS3] The C.S.A of a cylinder of radius ‘r’ and height ‘h’ is given by . (A) 2πrh (B) 2πr2h (C) πrh (D) πr2 Short Answer Type Questions 6 [AS1] The radius of a solid cylinder is 14 cm and total surface area is 5623 cm2. Find its height. 7 [AS1] The curved surface area of a right circular cylinder is 396 cm2 . Its radius is 9 cm. Find the height and volume of the cylinder. 8 [AS4] A water tank, 21 m deep is of radius 2 m. Find the cost of cementing the inner curved surface at the rate of Rs. 5 per square metre. Long Answer Type Questions 9 [AS1] The curved surface area of a cylinder is 5500 sq. cm and the circumference of its base is 55 cm. Find the height of the cylinder and volume of the cylinder. 10 [AS1] The total surface area of a right circular cylinder is 231 sq. cm. Its curved surface area is 2 of the total surface area. Find the radius of its base and height. 3 11 [AS1] The curved surface area of a right circular cylinder is 4.4 m2. If the radius of the base of thecylinder is 0.7 m, find its height and also find the cost of painting its total surface area at Rs. 5 per sq. m. 12 [AS1] The cylinder has a diameter 200 cm. Its curved surface area is 88000 cm2 . Find the volume of the cylinder (in m3). 13 [AS1] The area of the base of a right circular cylinder is 17600 sq. cm. Its volume is 140800 cu.cm. Find the area of the curved surface of the cylinder. EXERCISE 10.2. RIGHT CIRCULAR CYLINDER 17
14 [AS1] The volume of a cylinder is 648 π cu. cm. Its height is 8 cm. Find its curved surface area. 15 [AS4] The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions to move once over to level a play ground. Find the area of the play ground in square metres. 16 [AS4] The inner diameter of a cylindrical wooden pipe is 24 cm and its outer diameter is 28 cm. The length of the pipe is 35 cm. Find the mass of the pipe, if 1cm3of wood has a mass of 0.6 g. 17 [AS4] A lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior. The diameter of the pencil is 7 mm and the diameter of the graphite is 1 mm. If the length of the pencil is 14 cm, find the volume of the wood and that of the graphite. EXERCISE 10.2. RIGHT CIRCULAR CYLINDER 18
EXERCISE 10.3 RIGHT CIRCULAR CONE 10.3.1 Key Concepts i. A cone is a geometrical shape with a circular base and having a vertex at the top. ii. If the line segment joining the vertex to the centre of the base is perpendicular to the base, then it is called a right circular cone. iii. The line segment joining the vertex to any point on the circumference on the circular base is called the slant height (l), which is equal to r2 + h2. iv. C.S.A of a cone = πrl v. T.S.A of a cone = πr(l + r) vi. Volume of a cone = 1 πr2h 3 10.3.2 Additional Questions Objective Questions 1. [AS1] The base radius of a cone is 7 cm and its height is 24 cm. Its slant height is cm. (A) 25 (B) 31 (C) 17 (D) 625 2. [AS1] The radius of a cone is 4 cm and its slant height is 5 cm. Then its curved surface area is . (A) 220 sq. cm (B) 220 sq. cm 7 (C)440 sq. cm (D) 440 sq. cm 7 3. [AS1] The volume of cone whose base radius is 3 cm and height 7 cm is cu. cm. (A) 198 (B) 132 (C) 66 (D) 33 EXERCISE 10.3. RIGHT CIRCULAR CONE 19
4. [AS1] The base radius and height of a cylinder and cone are equal, then the ratio of their volumes in order is . (A) 1 : 3 (B) 3 : 1 (C)1 : 1 (D)1 : 9 5. [AS1] The base area of the cone is 195 sq. cm and height is 5 cm then its volume is cu. cm. (A) 39 (B) 2925 (C) 975 (D) 325 Short Answer Type Questions 6 [AS1] The volume of a conical vessel is 37680 cu. cm. Find the height of the cone, if the diameter of its base is 60 cm. Long Answer Type Questions 7 [AS1] Derive the formula for curved surface area of right circular cone whose radius is r and slant height is l. 8 [AS1] The curved surface area of a cone is 4070 sq. cm and its diameter is 70 cm. Find the slant height and the total surface area of the cone. 9 [AS1] A right circular cone is 36 cm high and radius of its base is 16 cm. It is melted and recast into a right circular cone with base radius 12 cm. Find its height. 10 [AS1] Find the volume of the largest right circular cone that can be cut out of a cube whose edge is 9 cm. 11 [AS1] A semi circular sheet of metal of diameter 56 cm is bent into an open conical cup. Find the depth of the cup. 12 [AS4] Monica has a piece of canvas of area is 551 sq. m. She used it to have a conical tent made,with a base radius 7 m. Assuming that all the stitching margins and the wastage incurred while cutting, amounts to approximately 1 sq. m, find the volume of the tent that can be made with it. 13 [AS4] A heap of wheat is in the form of a cone whose diameter is 10.5 m and height 3 m. Find itsvolume. The heap is to be covered by canvas to protect it from rain. Find the cost of canvas if it costs Rs.12 per sq. m. EXERCISE 10.3. RIGHT CIRCULAR CONE 20
EXERCISE 10.4 SPHERE 10.4.1 Key Concepts i. A sphere is a geometrical object formed where the set of points in space are equidistant from a fixed point. ii. If ‘r’ is the radius of the sphere, then a) Surface area of the sphere = 4πr2 b) Volume of the sphere = 4 πr3 3 iii. A plane through the centre of a solid sphere divides it into two equal parts each of which is called a hemisphere. iv. a) Area of hemisphere (curved) = 2πr2 b) Total surface area of hemisphere = 3πr2 c) Volume of hemisphere = 2 πr3 3 10.4.2 Additional Questions Objective Questions 1. [AS1] If the largest possible sphere is made from a cube of side 15 cm, its radius is . (A) 15 cm (B) 7.5 cm (C)3.75 cm (D)30 cm 2. [AS1] The surface area of sphere of radius 7 cm is . (A) 2464 sq. cm (B) 1232 sq. cm (C)308 sq. cm (D)616 sq. cm 3. [AS1] The quantity of milk that a hemispherical bowl of diameter 10.5 cm can hold is . (A) 302.1875 cm3 (B) 303.1875 cm3 (C)103.1875 cm3 (D)203.1875 cm3 EXERCISE 10.4. SPHERE 21
4. [AS1] The volume of a sphere whose radius is 7 cm is . (A) 1437.3 cm3 (B) 1347.3 cm3 (C)1427.3 cm3 (D)2437.3 cm3 5. [AS3] The total surface area of hemisphere of radius r cm is sq. cm. (A) 2πr2 (B) 4πr2 (C) 3πr2 (D) πr2 Long Answer Type Questions 6 [AS1] Find the volume and surface area of a sphere of radius 21 cm. 7 [AS1] A vessel is in the form of a hemispherical bowl mounted by a hollow cylinder. The diameter of the sphere is 14 cm and the total height of the vessel is 13 cm. Find its capacity. 8 [AS1] Find the total surface area of a hemisphere of radius 5.25 cm. 9 [AS1] If the radius of a sphere is tripled, what is the ratio of the volume of the first sphere to that of the second? 10 [AS1] The volume of a sphere is 4851 cu. cm . Find its radius and surface area. 11 [AS1] Twenty seven solid iron spheres, each of radius r and surface area S are melted to form a sphere with surface area S’. (i) Find the radius r’ of the new sphere. (ii) Ratio of S and S’. 12 [AS2] A sphere of diameter 6 cm is dropped in a right circular cylindrical vessel partly filled with water. The diameter of the cylindrical vessel is 12 cm. If the sphere is completely immersed in water, by how much will the level of water rise in the cylindrical vessel? 13 [AS2] The volumes of two spheres are in the ratio 64 : 27. Find their radii, if sum of their radii is 28 cm. 14 [AS2] How many lead shots, each of diameter 0.3 cm can be made from a cuboid with dimensions 18 cm x 11 cm x 6 cm? EXERCISE 10.4. SPHERE 22
CHAPTER 11 AREAS EXERCISE 11.1 AREA OF PLANAR REGIONS 11.1.1 Key Concepts i. Area of a figure is a number (in some unit) associated with the part of the plane enclosed by that figure. ii. Two congruent figures have equal areas, but the converse need not be true. iii. If X is a plane region formed by two non–overlapping plane regions of figures P and Q, then ar(X) = ar(P) + ar(Q). 11.1.2 Additional Questions Objective Questions 1. [AS1] The base BC of a gm ABCD is 15 cm and its area is 90 sq.cm. Then its corresponding altitude is _______. (A) 15 cm (B) 10 cm (C) 6 cm (D) 75 cm 2. [AS1] In a parallelogram ABCD, AB = 12 cm and AD = 18 cm. AE and AF are the altitudes from A on to CD and BC respectively such that AE = 15 cm. Then AF = . (A) 10 cm (B) 227 cm (C) 30 cm (D) 15 cm 3. [AS1] gm ABCD and gm APQD lie on the same base. The common base of these parallelograms is _______. (A) BC (B) PQ (C) AB (D) AD EXERCISE 11.1. AREA OF PLANAR REGIONS 23
4. [AS1] The ratio of the area of a parallelogram and the area of the parallelogram formed by joining themid–points of the sides of the given parallelogram in order is . (A) 2 : 1 (B) 1 : 2 (C) 1 : 3 (D) 3 : 1 5. [AS1] In ABC, i f AB = 5 cm; BC = 8 cm; ∠A = 40◦and ∠C = 50◦ then ar( ABC) = . (A) 40 sq. cm (B) 20 sq. cm (C) 10 sq. cm (D) 26 sq. cm Long Answer Type Questions 6 [AS1] In the following figures find out which figures lie on the same base and between the same parallels. In each case write two parallels and the common base. 7 [AS2] If E, F, G and H are respectively the mid –points of the sides of a parallelogram ABCD, show that ar ( gm EFGH) = 1 ar ( gm ABCD) . 2 EXERCISE 11.1. AREA OF PLANAR REGIONS 24
8 [AS2] ABCD is a parallelogram and O is a point in its interior. Prove that ar(4 AOB) + ar(4 COD) = ar(△ AOD) + ar(△ BOC). EXERCISE 11.1. AREA OF PLANAR REGIONS 25
EXERCISE 11.2 PARALLELOGRAMS ON THE SAME BASE AND BETWEEN THE SAME PARALLELS 11.2.1 Key Concepts i. Area of a parallelogram is the product of its base and the corresponding altitude. ii. Two figures are said to be on the same base and between the same parallels, if they have a common base (side) and the vertices opposite to the common base of each figure lie on a line parallel to the base. iii. Parallelograms on the same base (or equal bases) and between the same parallels are equal in area. iv. Parallelograms on the same base (or equal bases) and having equal areas lie between the same parallels. 11.2.2 Additional Questions Objective Questions 1. [AS1] Two parallelograms have equal area and they lie between the same parallel lines, then they lie on. (A) The same base (B) Equal bases (C)Both (A) and (B) (D)None of these 2. [AS1] The ratio of the areas of a triangle and a parallelogram which lie on the same base and between the same parallel lines is . (A) 1 : 2 (B) 2 : 1 (C) 1 : 4 (D) 1 : 3 3. [AS1] Two parallelograms are on the same base and between the same parallels. The ratio of their areas is . (A) 1 : 2 (B) 2 : 1 (C) 3 : 1 (D) 1 : 1 EXERCISE 11.2. PARALLELOGRAMS ON THE SAME BASE AND BETWEEN THE . . . 26
4. [AS1] A, B, C and D are the midpoints of the sides of gm PQRS . If ar( gmPQRS ) = 36 sq. cm, ar( gm ABCD) = . (A) 72 sq. cm (B) 18 sq. cm (C) 36 sq. cm (D) 9 sq. cm 5. [AS1] In the figure, PQRS is a parallelogram. If X and Y are the mid–points of PQ and S R respectively and diagonal S Q is drawn, the ratio ar( XQRY) : ar( QS R) = . (A) 1 : 4 (B) 2 : 1 (C)1 : 2 (D)1 : 1 Short Answer Type Questions 6(i) [AS1] Find the area of the parallelogram ABCD. (ii) [AS1] PQRS is a parallelogram. PM is the height from P to S R and PN is the height from P to QR. If S R = 12 cm and PM = 7.6 cm, a) Find the area of parallelogram PQRS. b) Find PN, if QR = 8 cm. EXERCISE 11.2. PARALLELOGRAMS ON THE SAME BASE AND BETWEEN THE . . . 27
7(i) [AS1] The height of a parallelogram is half its base. If the area of the parallelogram is 200 sq. cm, find its base and height. (ii) [AS1] In a parallelogram the base and height are in the ratio 5 : 2. If the area of the parallelogram is 360 sq. m, find its base and height. 8(i) [AS1] In the given figure, name the two parallelograms that lie on the same base and between the same parallel lines. (ii) [AS1] Which of the following figures lie(s) on the same base and between the same parallels? a) b) 9(i) [AS2] In a parallelogram ABCD, prove that the sum of any two consecutive angles is 180◦. (ii) [AS2] In a parallelogram ABCD, ∠D = 115◦. Determine ∠A and ∠B. EXERCISE 11.2. PARALLELOGRAMS ON THE SAME BASE AND BETWEEN THE . . . 28
EXERCISE 11.3 TRIANGLES ON THE SAME BASE AND BETWEEN THE SAME PARALLELS 11.3.1 Key Concepts i. If a parallelogram and a triangle are on the same base and between the same parallels, then the area of the triangle is half the area of the parallelogram. ii. Triangles on the same base (or equal bases) and between the same parallels are equal in area. iii. Triangles on the same base (or equal bases) and having equal areas lie between the same parallels. 11.3.2 Additional Questions Objective Questions 1. [AS1] △ ABC and △ ABD lie on the same base and between the same parallels. If AB = 10 cm and ar(△ ABC) = 60 cm 2 , then the altitude of △ ABC = . (A) 12 cm (B) 6 cm (C) 30 cm (D) 15 cm 2. [AS1] ABC and PQR lie on equal bases and also lie between the same parallels. Then the ratio of their areas is . (A) 1 : 2 (B) 2 : 1 (C)1 : 3 (D)1 : 1 3. [AS1] ABCD is a trapezium in which AB CD. If ar( ABC) = 24 sq. cm and AB = 8 cm then the height of ABC is . (A) 3 cm (B) 6 cm (C) 4 cm (D) 8 cm EXERCISE 11.3. TRIANGLES ON THE SAME BASE AND BETWEEN THE SAME. . . 29
4. [AS1] In ABC, D, E and F are the mid–points of AB, BC and CA respectively. The ratio of the areas ar( DBE) : ar( EFC) is . (A) 1 : 4 (B) 4 : 1 (C)1 : 1 (D) 1 : 3 5. [AS3] The median of a triangle divides it into two . (A) Congruent triangles (B) Isosceles triangles (C)Right triangles (D)Triangles of equal area Short Answer Type Questions 6(i) [AS2] △ABC is a triangle right angled at B, P is the midpoint of AC and Q is the midpoint of AB. Prove that PQ ⊥ AB. (ii) [AS2] In ABC, AD is the median through A and E is the mid–point of AD. BE produced meets AC in F. Prove that AF = 1 AC. 3 EXERCISE 11.3. TRIANGLES ON THE SAME BASE AND BETWEEN THE SAME. . . 30
Long Answer Type Questions 7 [AS2] Show that the area of a triangle is half the product of its base (or any side) and the corresponding altitude (height). 8 [AS2] In the adjoining figure, ar (△ DRC) = ar (△ DPC) and ar (△ BDP) = ar (△ ARC) . Show that both DCPR and ABCD are trapeziums. EXERCISE 11.3. TRIANGLES ON THE SAME BASE AND BETWEEN THE SAME. . . 31
CHAPTER 12 CIRCLES EXERCISE 12.1 INTRODUCTION 12.1.1 Key Concepts i. A circle is a set of all points in a plane which are at a fixed distance from a fixed point in the same plane. ii. The fixed point is called the centre of the circle and the fixed distance is called the radius. iii. A chord is a line segment joining any two points on a circle. iv. The longest of all chords is the diameter. v. A line segment joining the centre and any point on a circle is called its radius. vi. The length of a diameter of a circle is double its radius. vii. Circles with the same radii are called congruent circles. viii. Circles with the same centre and different radii are called concentric circles. ix. The diameter of a circle divides the circle into two semi–circles. 12.1.2 Additional Questions angles at the centre of the circle. Objective Questions (B) Acute (D) Equal 1. [AS3] Equal arcs of a circle subtend (A) Right (C) Obtuse EXERCISE 12.1. INTRODUCTION 32
2. [AS3] The major segment of a circle is the part of a circle which of the circle. (A) Contains the centre (B) Does not contain the centre (C)Contains the exterior (D)None of these 3. [AS3] The diameter is the longest of the circle. (A) Arc (B) Radius (C)Chord (D)Sector 4. [AS3] Two circles are said to be congruent if they have . (A) Equal radii (B) The same centre (C)Different radii (D)None of these 5. [AS3] The diameter of a circle divides the circle into equal parts. (A) 4 (B) 3 (C) 8 (D) 2 Very Short Answer Type Questions [ ] 6 [AS3] State true or false. (i) A circle divides the plane on which it lies into three parts. EXERCISE 12.1. INTRODUCTION 33
[AS3] Choose the correct answer. . (ii) In the following figure AC is a (A) Diameter (B) Chord (C)Radius (D)None of these Short Answer Type Questions 7(i) [AS5] Represent the following in a circle: a) Major sector b) Semicircle (ii) [AS5] Name the following parts. EXERCISE 12.1. INTRODUCTION 34
a. AB b. AQ c. AQP Long Answer Type Questions 8 [AS5] Name the following parts from the adjacent figure where ’O ’ is the centre of the circle. a) AO b) AB c) BxD d) AC e) ADC f) ADB g) AYC h) S haded Area EXERCISE 12.1. INTRODUCTION 35
EXERCISE 12.2 ANGLE SUBTENDED BY A CHORD AT A POINT ON THE CIRCLE 12.2.1 Key Concepts i. The area enclosed by a chord and an arc is called a segment. ii. The area enclosed by an arc and two radii joining the end points of the arc with the centre is called a sector. iii. Equal chords of a circle subtend equal angles at the centre and vice versa. 12.2.2 Additional Questions Objective Questions 1. [AS1] AB and CD are two equal chords of a circle and AB subtends an angle of 80◦ at the centre of the circle. Then the angle subtended by CD at the centre is . (A) 40◦ (B) 80◦ (C) 120◦ (D) 0◦ 2. [AS1] PQ and RS are two chords of a circle which are equidistant from the centre. If PQ = 7 cm then RS = . (A) 7 cm (B) 14 cm (C)3.5 cm (D)Cannot be said 3. [AS1] A chord AB of length 8 cm is drawn in a circle of radius 5 cm. The distance of the chord from the centre is . (A) 8 cm (B) 4 cm (C) 5 cm (D) 3 cm EXERCISE 12.2. ANGLE SUBTENDED BY A CHORD AT A POINT ON THE CI. . . 36
4. [AS3] The angle in a semi–circle is . (B) 45◦ (A) ◦ (D) 180◦ 0 (C) 90◦ 5. [AS3] The angle in a semi–circle at the centre of the circle is . (A) 180◦ (B) 45◦ (C) 90◦ (D) 0◦ Short Answer Type Questions 6 [AS1] A chord is at a distance of 12 cm from the centre of a circle of radius 13 cm. Find the length of the chord. Long Answer Type Questions 7 [AS1] AB and CD are two parallel chords of a circle such that AB = 10 cm and CD = 24 cm. If they are on the opposite sides of the centre of the circle and the distance between them is 17 cm, find the radius of the circle. EXERCISE 12.2. ANGLE SUBTENDED BY A CHORD AT A POINT ON THE CI. . . 37
EXERCISE 12.3 PERPENDICULAR FROM THE CENTRE TO A CHORD 12.3.1 Key Concepts i. The angle in a minor segment of a circle is obtuse and the angle in a major segment is acute. ii. Angles in the same segment are equal. iii. Any angle in a semi–circle is 90°. iv. The perpendicular drawn from the centre of a circle to a chord bisects the chord and vice versa. v. The angle subtended by a minor arc at the centre is acute and the angle subtended by a major arc is obtuse. vi. Equal chords are equidistant from the centre of the circle and vice–versa. 12.3.2 Additional Questions Objective Questions 1. [AS1] A chord of length 14 cm is drawn in a circle of diameter 50 cm. The distance between the chord and the centre of the circle is . (A) 7 cm (B) 25 cm (C)14 cm (D)24 cm 2. [AS1] A chord AB of a circle of radius 13 cm is at a distance of 5 cm from the centre. The length of another chord PQ which is at the same distance from the centre of the circle is . (A) 12 cm (B) 24 cm (C)10 cm (D)26 cm 3. [AS1] In a circle of radius 5 cm, a chord AB bisects another chord CD perpendicularly. Then CD = . (A) 10 cm (B) 5 cm (C)20 cm (D)None of these EXERCISE 12.3. PERPENDICULAR FROM THE CENTRE TO A CHORD 38
4. [AS3] In a circle AB and BC are two chords such that AB ⊥ BC, then arc ABC is a . (A) Diameter (B) Radius (C) Semicircle (D)None of these 5. [AS3] The perpendicular drawn from the centre of a circle to a chord the chord. (A) Bisects (B) Trisects (C) Intersects (D)None of these Very Short Answer Type Questions 6 [AS1] Fill in the blanks. (i) circle(s) can be drawn from three non–collinear points. [AS1] Choose the correct answer. (ii) If a line drawn from the centre of a circle bisects a chord then the line is to that chord. (A) Parallel (B) Adjacent (C) Perpendicular (D)None of these (iii) A chord of length 14 cm is at a distance of 6 cm from the centre. The length of another chord at a distance of 2 cm from the centre of the circle is . (A) 12 cm (B) 14 cm (C)16 cm (D)18 cm (iv) An equilateral triangle ABC is inscribed in a circle with centre O. The measurement of ∠BOC is . (A) 120◦ (B) 30◦ (C) 60◦ (D) 90◦ EXERCISE 12.3. PERPENDICULAR FROM THE CENTRE TO A CHORD 39
(v) The angle formed in a minor segment of a circle is angle. (A) An acute (B) An obtuse (C)A right (D)A straight angle Short Answer Type Questions 7(i) [AS2] In the given figure, ∠ABC = 45◦. Show that OA ⊥ OC. (ii) [AS2] AOC is a diameter of the circle given Arc(AX B) = 1 Arc(BYC). Find ∠BOC. 2 8(i) [AS2] In a circle with centre O, OT ⊥ PQ, OS ⊥ PR and OT = 5, PQ = 24 and PR = 25. Find OS. EXERCISE 12.3. PERPENDICULAR FROM THE CENTRE TO A CHORD 40
(ii) [AS2] In given the figure, OD is perpendicular to the chord AB of a circle whose centre is O and BC is a diameter. Prove that CA = 2 OD. 9(i) [AS1] The radius of a circle is 13 cm and the length of one of its chords is 24 cm. Find the distance of the chord from the centre. (ii) [AS2] In the figure, there are two circles intersecting each other at A and B. Prove that the line joining their centres bisects the common chord AB. EXERCISE 12.3. PERPENDICULAR FROM THE CENTRE TO A CHORD 41
EXERCISE 12.4 ANGLE SUBTENDED BY AN ARC OF A CIRCLE 12.4.1 Key Concepts i. A part of a circle is called an arc. ii. The area enclosed by a chord and an arc is called a segment. iii. The angle subtended by a minor arc at the centre is acute and the angle subtended by a major arc is obtuse. iv. There is exactly one circle that passes through three non–collinear points. v. The circle passing through all the three vertices of a triangle is called a circumcircle. vi. The angle subtended by an arc at the centre is twice that of the angle it subtends at the remaining part of the circle. 12.4.2 Additional Questions Objective Questions 1. [AS3] The angle in a minor segment is angle. (A) A zero (B) An acute (C) A right (D) An obtuse 2. [AS3] The angle subtended by a major segment at the centre of the circle is angle. (A) A reflex (B) An acute (C) A right (D) An obtuse 3. [AS1] In a circle with centre ‘O’, an arc (AB) subtends an angle of 158◦ at the centre ‘O’ then the angle subtended by arc (AB) at point C on the remaining circle is . (A) 208◦ (B) 79◦ (C) 158◦ (D)None of these EXERCISE 12.4. ANGLE SUBTENDED BY AN ARC OF A CIRCLE 42
4. [AS1] In the given figure, if O is the centre of the circle and ∠ACB = 30◦, then ∠ AOB is . (A) 30◦ (B) 15◦ (C) 60◦ (D) 90◦ 5. [AS1] In the given figure, if △ ABC and △ DBC are inscribed in a circle such that ∠BAC = 60◦ and ∠DBC = 50◦ then ∠BCD = . (A) 40◦ (B) 50◦ (C) 60◦ (D) 70◦ Very Short Answer Type Questions [] 6 [AS1] State true or false. (i) If ∠ABC = 45◦ then ∠AOC = 90◦. EXERCISE 12.4. ANGLE SUBTENDED BY AN ARC OF A CIRCLE 43
[AS1] Fill in the blanks. (ii) Angles in the same segment of a circle are . (iii) The angle subtended by an arc at the centre is the angle it subtends at the remaining part of the circle. [AS1] Choose the correct answer. (iv) If two diameters of a circle intersect each other at right angles, then the quadrilateral formed by joining their end points is a . (A) Rhombus (B) Rectangle (C)Parallelogram (D)Square (v) If the angle subtended by an arc at the centre is 210◦ then it subtends at the remaining part of the circle. (A) 41◦ (B) 23◦ (C) 105◦ (D) 108◦ Short Answer Type Questions 7(i) [AS1] “O ” is the centre of circle and ∠AOC = 120◦. Find ∠ABC. EXERCISE 12.4. ANGLE SUBTENDED BY AN ARC OF A CIRCLE 44
(ii) [AS1] In the given figure “O ” is the centre. ∠POQ = ◦ and PS ⊥ OQ. Find ∠MQS. 70 8(i) [AS1] In a circle with centre O, QO ⊥ PR, PR = 12 units and S Q = 2 units. Find x. (ii) [AS1] In the given circle with centre O, OV ⊥ S Q, OV ⊥ PR, OP = 10, S T = 5, PU = 8. Find T U. 9 [AS1] In a circle with centre ’O ’ and radius 10 units, OQ⊥PR and PR = 8. Find x. EXERCISE 12.4. ANGLE SUBTENDED BY AN ARC OF A CIRCLE 45
10(i) [AS1] In the given figure, “O ” is the centre of the circle and AD bisects ∠BAC. Find ∠BCD. (ii) [AS1] In the given figure, “O ” is the centre of circle and ∠PAQ = 35◦. Find ∠OPQ. 11(i) [AS2] Let “O ” be the centre of a circle. If PQ is its diameter then prove that ∠PRQ = 90 ◦. (ii) [AS2] Angles in the same segment of a circle are equal. Prove. 12(i) [AS2] ’O’ is the centre of the given circle. Find the length of CD, if AB = 7 cm. EXERCISE 12.4. ANGLE SUBTENDED BY AN ARC OF A CIRCLE 46
(ii) [AS2] In the given figure, prove AB = CD. Long Answer Type Questions 13 [AS1] Find ∠AOC in the figure given. 14 [AS1] The radius of a circle is 13 cm and the length of one of its chords is 10 cm. Find the distance of the chord from the centre. 15 [AS1] In a circle of radius 5 cm, AB and AC are two chords of 6 cm each. Find the length of chord BC. EXERCISE 12.4. ANGLE SUBTENDED BY AN ARC OF A CIRCLE 47
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