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 –. . . 48
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 –. . . 49
CHAPTER 7 TRIANGLES EXERCISE 7.1 CRITERIA FOR CONGRUENCE OF TRIANGLES 7.1.1 Key Concepts i. If the lengths of two line segments are equal, then they are congruent. ii. Two angles are congruent if their angle measures are the same. iii. The figures that are same in shape and size are called congruent figures. iv. Two triangles are congruent if their corresponding sides and angles are equal. v. In congruent triangles, corresponding parts are equal and written in short as “CPCT” for ‘corresponding parts of congruent triangles’. vi. Two congruent triangles coincide exactly with one another. vii. There are three ways to make congruent figures coincide: a slide, a flip, or a turn. viii. SAS axiom: Two triangles are congruent if two sides and the included angle of one triangle are respectively equal to the two sides and the included angle of the other triangle. ix. SSA or ASS axioms do not exist because the triangles cannot be congruent unless the angle is between the equal sides. x. ASA axiom: Two triangles are congruent, if two angles and the included side of one triangle are respectively equal to two angles and the included side of the other triangle. xi. We may call ASA congruency axiom as the AAS congruency axiom because two triangles are congruent if any two pairs of angles and pair of corresponding sides are equal. 7.1.2 Additional Questions Objective Questions 1. [AS1] In ABC and DEF, ∠B = ∠E, ∠C = ∠F, AB = DE and AC = DF. Then ABC and ∆DEF are . (A) Similar (B) Congruent (C) Equal (D)None of these EXERCISE 7.1. CRITERIA FOR CONGRUENCE OF TRIANGLES 50
2. [AS1] If PQR XYZ and ∠P = ∠X, then PR = . (A) XY (B) YZ (C) XZ (D)None of these 3. [AS1] In PQR and S QR, if PQ = S R and ∠PQR = ∠QRS , then . (A) PQR ∼ S RQ (B) PQR S RQ (C) PQR S RQ (D) None of these 4. [AS1] LMN ◦ . (A) 70◦ (B) 90◦ XYZ, ∠M = 70 then ∠Y = (D) 20◦ (C) 110◦ 5. [AS1] If ABC LMN by SAS congruency, AB = 8 cm, ∠M = 80◦ and MN = 10 cm, then BC = . (A) 8 cm (B) 16 cm (C)18 cm (D)10 cm Very Short Answer Type Questions [] 6 [AS2] State true or false. The two triangles given are congruent. EXERCISE 7.1. CRITERIA FOR CONGRUENCE OF TRIANGLES 51
7 [AS2] Fill in the blanks. and the same . (i) Two plane figures are congruent, if they have exactly the same (ii) Two angles are congruent if they have the . (iii) The criteria for two squares to be congruent is . (iv) These two circles are congruent if . Short Answer Type Questions 8(i) [AS2] In the adjacent figure, OA = OB and OD = OC. Show that ∆AOD ∆BOC. (ii) [AS2] In the adjacent figure, AB = CF, EF = BD and ∠AFE = ∠DBC prove that AFE CBD. EXERCISE 7.1. CRITERIA FOR CONGRUENCE OF TRIANGLES 52
9(i) [AS2] The diagonal MO of the quadrilateral MNOP bisects the angles M and O. Prove that MN = MP and ON = OP. (ii) [AS2] In PQR the bisector PS of ∠P is perpendicular to the side QR. Show that PQ = PR. 10(i) [AS2] Line segment AB is parallel to another line segment CD. O is the mid–point of AD. Show that AOB DOC. (ii) [AS2] If PQR is an isosceles triangle with PQ = PR, show that RX = QY. EXERCISE 7.1. CRITERIA FOR CONGRUENCE OF TRIANGLES 53
EXERCISE 7.2 SOME PROPERTIES OF A TRIANGLE 7.2.1 Key Concepts i. Angles opposite to equal sides of an isosceles triangle are equal. Conversely, sides opposite to equal angles are also equal. 7.2.2 Additional Questions Objective Questions 1. [AS1] One acute angle of a right triangle is 25◦, then the other angle is . (A) 60 ◦ (B) 65◦ (C) 25◦ (D) 55◦ 2. [AS1] In LMN, if ∠L = ∠M, then LN = . (A) MN (B) LN (C) LM (D)None of these 3. [AS1] In an isosceles triangle, the vertical angle is 80 ◦. Then each of the base angles is . (A) 80◦ (B) 100◦ (C) 50◦ (D)None of these 4. [AS1] In an isosceles triangle one of the base angles (equal angles) is 65.◦ Then the angle at the vertex of the triangle is . (A) 50◦ (B) 65◦ (C) 130◦ (D) 115◦ EXERCISE 7.2. SOME PROPERTIES OF A TRIANGLE 54
5. [AS3] If two medians of a triangle are equal, then the triangle is . (A) Equilateral (B) Scalene (C)Right angled (D) Isosceles Long Answer Type Questions 6 [AS2] In ABC, ∠A = 100◦ and AB = AC. Find ∠B and ∠C. 7 [AS2] If two altitudes of a triangle are equal, then prove that it is an isosceles triangle. EXERCISE 7.2. SOME PROPERTIES OF A TRIANGLE 55
EXERCISE 7.3 SOME MORE CRITERIA FOR CONGRUENCY OF TRIANGLES 7.3.1 Key Concepts i. RHS congruency rule: If in two right triangles, the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent. ii. SSS congruency rule: If three sides of one triangle are respectively equal to the three sides of another triangle, then the two triangles are congruent. 7.3.2 Additional Questions Objective Questions congruency axiom. 1. [AS1] The axiom under which ABC PQR is the (A) S.A.S (B) A.S.A (C) S.S.S (D) R.H.S EXERCISE 7.3. SOME MORE CRITERIA FOR CONGRUENCY OF TRIANGLES 56
2. [AS1] In the given figure, PQ = SR and PR = SQ. Then . (A) PQR S RQ axiom. (B) PQR RS Q (C)PQ = SQ (D)∠PQR = ∠QS R 3. [AS1] The two triangles given in the figure are congruent by the (A) A.S.A (B) S.A.S (C) S.S.S (D) R.H.S EXERCISE 7.3. SOME MORE CRITERIA FOR CONGRUENCY OF TRIANGLES 57
4. [AS1] In ABC and XYZ, ∠A = ∠X = 90◦, XY = AB and AC = XZ then ABC XYZ by the congruency axiom. (A) R .H.S (B) A.S.A (C) S.A.S (D) S.S.S 5. [AS1] If PQR is an equilateral triangle such that PO bisects ∠P. Also, ∠PQO = ∠PRO, then QPO . (A) OQR (B) OQP (C) RPO (D) ROQ Long Answer Type Questions 6 [AS2] In an isosceles triangle PQR with PQ = PR, point O is in the interior of PQR such that ∠OQR = ∠ORQ. Prove that PO bisects angle QPR. EXERCISE 7.3. SOME MORE CRITERIA FOR CONGRUENCY OF TRIANGLES 58
7 [AS2] In quadrilateral ABCD, BC = CD and ∠B = ∠D = 90◦. Prove that CA bisects ∠BAD. EXERCISE 7.3. SOME MORE CRITERIA FOR CONGRUENCY OF TRIANGLES 59
EXERCISE 7.4 INEQUALITIES IN A TRIANGLE 7.4.1 Key Concepts i. If two sides of a triangle are unequal, the angle opposite to the longer side is greater. ii. The sum of any two sides of a triangle is greater than the third side. iii. The difference of any two sides of a triangle is lesser than the third side. iv. To construct a unique triangle, three independent measurements are needed. 7.4.2 Additional Questions Objective Questions 1. [AS1] The greatest angle in ABC where AB = 5.5 cm, BC = 7 cm and AC = 4 cm is . . (A) ∠A (B) ∠B (C) ∠C (D)None of these 2. [AS1] The set of measurements that does not represent the three sides of a triangle is (A) 7 cm, 7 cm, 7 cm (B) 13 cm, 15 cm, 28 cm (C)3 cm, 4 cm, 5 cm (D)5 cm, 5 cm, 4 cm 3. [AS1] In PQR, PQ − QR PR. (A) < (B) > (C) = (D)None of these 4. [AS1] In ABC, ∠B is obtuse, AB = 6 cm and BC = 8 cm. The statement that is true is . (A) 8 < AC ≤ 10 (B) 10 < AC < 12 (C)10 < AC < 14 (D)12 < AC < 14 EXERCISE 7.4. INEQUALITIES IN A TRIANGLE 60
5. [AS3] In a triangle, the angle opposite to the longer side is the . (A) Smallest angle (B) Right angle (C)Greatest angle (D)None of these Very Short Answer Type Questions 6 [AS2] Answer the following questions in one sentence. In the given triangle PQR, identify the longest side. Long Answer Type Questions 7 [AS2] In a given ABC, arrange the angles in ascending order if the lengths of the sides AB, BC and CA are x, y and z respectively, where y < z < x. 8 [AS2] Find the minimum value of x, such that ABC exists, where AB = 22 cm; BC = 33 cm and AC = x cm. EXERCISE 7.4. INEQUALITIES IN A TRIANGLE 61
CHAPTER 8 QUADRILATERALS EXERCISE 8.1 PROPERTIES OF A QUADRILATERAL 8.1.1 Key Concepts i. A quadrilateral is a simple closed figure formed by four line segments in a plane. ii. The sum of four angles in a quadrilateral is 360° or 4 right angles. iii. Trapezium, parallelogram, rhombus, rectangle, square and kite are special types of quadrilaterals. iv. A quadrilateral with only one pair of opposite sides parallel is a trapezium. v. A quadrilateral with both the pairs of opposite sides parallel is a parallelogram. vi. A parallelogram whose each angle is a right angle is a rectangle. vii. A parallelogram with all adjacent sides equal is a rhombus. viii. A parallelogram with both pairs of opposite sides parallel and each angle a right angle is a square. ix. A quadrilateral with two pairs of adjacent sides equal is a kite. 8.1.2 Additional Questions Objective Questions 1. [AS1] In a quadrilateral, if the sum of three angles is ◦ then the fourth angle is . 250, (A) 70◦ (B) 130◦ (C) 160◦ (D) 110◦ 2. [AS1] If the angles of a quadrilateral are in the ratio 2 : 3 : 7 : 6, then the measure of the largest angleis . (A) 40◦ (B) 60◦ (C) 140◦ (D) 120◦ EXERCISE 8.1. PROPERTIES OF A QUADRILATERAL 62
3. [AS1] The diagonals of a rhombus are 6 cm and 8 cm. Then the length of the side of the rhombus is . (A) 4 cm (B) 5 cm (C)6 cm (D)7 cm 4. [AS1] If one of the angles of a quadrilateral is 2 times the other and the sum of the other 3 two angles is 185◦ , then the two angles are . (A) 100◦, 75◦ (B) 120◦, 80◦ (C)105◦, 70◦ (D)100◦, 85◦ 5. [AS3] A quadrilateral in which only one pair of opposite sides are parallel is called a . (A) Rhombus (B) Trapezium (C) Rectangle (D) Parallelogram Short Answer Type Questions 6(i) [AS1] In quadrilateral PQRS, if ∠ P = 60◦ and ∠ Q : ∠ R : ∠ S = 2 : 3 : 7, then find the measure of ∠S. (ii) [AS1] ABCD is a parallelogram and ∠A = 80◦. Find the remaining angles. 7(i) [AS2] Name the figure whose diagonals are perpendicular bisectors of each other and also are equal in length. (ii) [AS2] Identify the quadrilaterals that have: a) Four sides of equal length b) Four right angles c) Both (a) and (b) EXERCISE 8.1. PROPERTIES OF A QUADRILATERAL 63
Long Answer Type Questions 8 [AS1] ABCD is a rectangle. AC is its diagonal. Find the angles of ∆ACD. Give reasons. 9 [AS1] In the figure given, ABCD is a rhombus whose diagonals meet at O. Find the values of x and y. EXERCISE 8.1. PROPERTIES OF A QUADRILATERAL 64
EXERCISE 8.2 PARALLELOGRAM AND ITS PROPERTIES 8.2.1 Key Concepts i. A parallelogram is a special type of quadrilateral with special properties. ii. If each pair of opposite sides of a quadrilateral are equal, then it is a parallelogram. iii. If each pair of opposite angles are equal, then it is a parallelogram. iv. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. v. The diagonal of a parallelogram divides it into two congruent triangles. 8.2.2 Additional Questions Objective Questions 1. [AS1] PQRS is a parallelogram, in which ∠S PQ = 65 ◦ . Then ∠RS P = . (A) 65◦ (B) 50◦ (C) 115◦ (D) 40◦ 2. [AS1] In a parallelogram PQRS, QR is 3 cm longer than PQ. If its perimeter is 70 cm, then QR = . (A) 16 cm (B) 17 cm (C)18 cm (D)19 cm 3. [AS1] In a parallelogram ABCD, ∠A = (3x − 10◦) and ∠B = (5x + ◦ then ∠D is . 30 ) (A) 50◦ (B) 130◦ (C) 60◦ (D) 120◦ 4. [AS1] In a parallelogram ABCD, AB = 5 cm and BC is 7 cm more than AB. Then the perimeter of the parallelogram ABCD is . (A) 34 cm (B) 35 cm (C)70 cm (D)68 cm EXERCISE 8.2. PARALLELOGRAM AND ITS PROPERTIES 65
5. [AS3] A parallelogram in which adjacent sides are equal is called a . (A) Kite (B) Rectangle (C) Rhombus (D) Trapezium Very Short Answer Type Questions 6 [AS1] Answer the following questions in one sentence. In the given quadrilateral, find the value of ’y’. Short Answer Type Questions 7(i) [AS1] Find the values of ’k’ and ’l’ in the given parallelogram. (ii) [AS1] In the given parallelogram, EXERCISE 8.2. PARALLELOGRAM AND ITS PROPERTIES 66
a) If ACB = 60◦, and ∠ CAB = 35◦, find the measure of ∠ ABC. b) Find the measures of ∠ BAD, ∠ BCD and ∠ ADC using properties of parallel lines. Long Answer Type Questions 8 [AS2] Given a parallelogram EFGH, with diagonals EG and HF intersecting at K. Prove that ∆EFK ∆GHK. EXERCISE 8.2. PARALLELOGRAM AND ITS PROPERTIES 67
EXERCISE 8.3 DIAGONALS OF A PARALLELOGRAM 8.3.1 Key Concepts i. A diagonal of a quadrilateral is the line segment joining a pair of opposite vertices. ii. If both the pairs of opposite sides of a quadrilateral are parallel, then it is called a parallelogram. iii. In a parallelogram, (a) the opposite angles are equal. b) the sum of two adjacent angles is supplementa◦ry (i.e., equal to 180 ). c) a diagonal divides it into two congruent triangles. d) the diagonals are not equal and they bisect each other. iv. A parallelogram becomes (a) a rectangle when each of its angles is a right angle. (b) a rhombus when all its sides are equal. (c) a square when all sides are equal and all angles are equal to 90°. v. If the diagonals of a quadrilateral bisect each other, it becomes a parallelogram. vi. In a rectangle (a) opposite sides are equal. (b) each of its angles is a right angle. (c) the diagonals are of the same length. (d) the diagonals bisect each other. vii. A rhombus (a) has all its four sides equal. (b) has unequal diagonals which bisect each other perpendicularly. (c) has diagonals that divide it into four congruent triangles. viii. If an angle of a rhombus is 90°, then it is called a square. ix. A square (a) has all its sides equal. (b) has all its diagonals equal and they bisect each other perpendicularly. (c) each of its angles is a right angle. (d) has diagonals that divide it into four congruent triangles. x. The angle bisectors of a parallelogram form a rectangle. 8.3.2 Additional Questions Objective Questions 1. [AS1] In a parallelogram ABCD, diagonals AC and BD bisect each other at ‘O’. If AO = 8x − 3 and OC = 3x + 7 then the length of the diagonal AC is . (A) 20 cm (B) 26 cm (C)30 cm (D)36 cm EXERCISE 8.3. DIAGONALS OF A PARALLELOGRAM 68
2. [AS1] In the parallelogram PQRS, OS = 5 cm and PR is 7 cm more than QS. The measure of OP is . (A) 8.5 cm (B) 8 cm (C) 10 cm (D) 9 cm 3. [AS1] In parallelogram ABCD, O is the mid point of the diagonals. If AO = 17 − 2x, OC = 5x − 11 and BD = 7x − 6, then OB = . (A) 22 cm (B) 20 cm (C)28 cm (D)11 cm 4. [AS3] The diagonal of a parallelogram divides it into two triangles. (A) Congruent (B) Similar (C) Right (D) None of these 5. [AS3] A figure in which the longer diagonal bisects the shorter diagonal is a . (A) Square (B) Rhombus (C) Parallelogram (D) Kite Long Answer Type Questions 6 [AS2] Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus. EXERCISE 8.3. DIAGONALS OF A PARALLELOGRAM 69
7 [AS2] In a parallelogram, the bisectors of any two adjacent angles intersect at right angles. Prove. 8 [AS2] In a parallelogram ABCD, the bisector of ∠A also bisects BC at P. Prove that AD = 2AB. 9 [AS2] In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ. Show that i) APD CQB ii) AP = CQ iii) AQB CPD iv) AQ = CP v) APCQ is a parallelogram. EXERCISE 8.3. DIAGONALS OF A PARALLELOGRAM 70
EXERCISE 8.4 THE MIDPOINT THEOREM OF A TRIANGLE 8.4.1 Key Concepts i. A line segment joining the midpoints of two sides of a triangle is parallel to the third side and also half of it. ii. A line through the midpoint of a side of a triangle parallel to another side bisects the third side. 8.4.2 Additional Questions Objective Questions 1. [AS1] In ABC, P and Q are midpoints of AC and BC respectively. If AB = 8.6 cm, then PQ = . (A) 4 cm (B) 4.3 cm (C)4.5 cm (D)8.6 cm 2. [AS1] In ABC, D, E and F are the midpoints of BC, CA and AB respectively. Then the perimeter of ABC is times the perimeter of DEF. (A) 4 (B) 3 (C) 2 (D) 1 3. [AS1] In PQR , PQ = 7 cm, QR = 10 cm and RP = 11 cm. X is the mid point of QR, Y is a point on PRand XY PQ , then the measure of XY is . (A) 3.5 cm (B) 5 cm (C)5.5 cm (D)9 cm 4. [AS1] In ABC, D, E and F are the midpoints of sides BC, CA and AB respectively. By joining D, E and F the ABC is divided into congruent triangles. (A) 1 (B) 2 (C) 3 (D) 4 EXERCISE 8.4. THE MIDPOINT THEOREM OF A TRIANGLE 71
5. [AS3] The figure formed by joining the midpoints of the sides of a parallelogram in order is . (A) Rectangle (B) Parallelogram (C) Square (D) Rhombus Short Answer Type Questions 6(i) [AS1] In ∆LMN, P is the mid–point of LM and Q is the mid –point of LN. Find PQ if MN = 10 cm and ∠LPQ, if ∠PMN = 60◦. (ii) [AS1] XYZ is an equilateral triangle, L and M are mid–points of XY and XZ respectively. LM = 6cm. Find the perimeter of XYZ. Long Answer Type Questions 7 [AS1] ∆ABC is an isosceles triangle in which ∠B = ∠C and LM BC. If ∠A = 50 ◦, find ∠LMC. If L and M are mid –points of sides AB and AC respectively, find the length of LM given that BC = 22 cm. 8 [AS2] (i) Prove that the line joining the mid–points of the two sides of a triangle is parallel to the third side. (ii) In ABC, D, E and F are the mid –points of sides AB, BC and CA respectively. Show that ABC is divided into four congruent triangles by joining D, E and F. EXERCISE 8.4. THE MIDPOINT THEOREM OF A TRIANGLE 72
CHAPTER 9 STATISTICS EXERCISE 9.1 COLLECTION OF DATA 9.1.1 Key Concepts i. The facts or figures collected with a definite purpose are called data. For example, census information, temperatures of various cities, production of cars in a financial year etc. ii. Extraction of meaning (purpose) of data is studied in the branch of mathematics called Statistics. iii. If the information is collected by enquiring directly or visiting personally, then the data is called Primary data. iv. The information (data) collected from a source, which had already collected or recorded is called Secondary data. v. The difference between the minimum and the maximum value of the data is called the Range. ∴ Range = max. value – min. value vi. An ungrouped data is a raw data (actual observations). vii. Representation of large data in the form of frequency distribution table can be easily analyzed and interpreted. viii. Frequency of a certain class is the number of items that are found in that class. ix. In a frequency table the entire data is grouped into different classes like 0 –9, 10 –19, 20 –29, . . . etc. x. The least number of each class is called its lower limit, e.g.: In the classes in point (ix), 0, 10, and 20 . . . are lower limits. xi. The greatest number of each class is called its upper limit, e.g.: In the above classes in point (ix), 9, 19, and 29 . . . are upper limits. xii. The average of the upper limit of class and the lower limit of the very next class is called the upper boundary of that class. The upper boundary of the class becomes the lower boundary of the succeeding class. xiii. The difference between the upper boundary and the lower boundary of the class is called the class–interval of the class. xiv. The classes 0 –9, 10 –19, 20 –29, which are non–overlapping are called inclusive classes. xv. The classes 0 –10, 10 –20, 20 –30, which are overlapping are called exclusive classes. EXERCISE 9.1. COLLECTION OF DATA 73
9.1.2 Additional Questions Objective Questions 1. [AS1] The range of the data 98.1, 98.6, 99.2, 90.3, 86.5, 95.3, 92.9, 96.3, 94.2 and 95.1 is . (A) 13.3 (B) 3.9 (C) 12.7 (D)None of these 2. [AS3] The data collected from a survey is known as data. (A) Primary (B) Secondary (C) Grouped (D)None of these 3. [AS3] The difference between the maximum value and the minimum value of a data is called . (A) Mean (B) Median (C) Mode (D) Range 4. [AS3] The data collected rearranged into small groups is called . (A) Raw data (B) Grouped data (C)Ungrouped data (D)None of these 5. [AS3] In a grouped data, the number of observations in one particular class is called . (A) Class interval (B) Class boundaries (C) Midvalue (D) Frequency Short Answer Type Questions 6 [AS3] What are the two types of statistical data? Define them. 7 [AS3] Give 2 examples for each: primary data and secondary data. EXERCISE 9.1. COLLECTION OF DATA 74
8(i) [AS5] Given are the marks obtained in a mathematical test by 30 students of class IX of a school. Prepare a discrete frequency distribution. 20, 24, 23, 22, 25, 18, 16, 20, 21, 22, 16, 15, 20, 18, 21, 23, 25, 24, 22, 20, 22, 21, 16, 24, 25, 24, 20, 21, 23, 25. (ii) [AS5] Make a grouped frequency distribution table of the following data about the weights (in kg) of some bags of grain. 180, 155, 178, 90, 101, 105, 124, 118, 126, 176, 135, 157, 134, 99, 112, 115, 104, 108, 178, 159, 175, 147, 129, 117, 128, 151, 140, 169, 120, 165, 98, 107, 119, 93, 170, 144, 154, 164, 174, 160, 95, 105, 113, 123, 133, 168, 172, 149, 179, 97. Arrange the above in a frequency distribution with class length 10. 9(i) [AS5] The following graph depicts the daily wages of workers in a factory. Construct the frequency distribution table. EXERCISE 9.1. COLLECTION OF DATA 75
(ii) [AS5] Represent the data given in the following graph as a frequency distribution table. Long Answer Type Questions 10 [AS5] Represent the data from the frequency distribution as a bar graph and answer the following questions. Height 145 154 158 162 Frequency 5631 (i) Find the number of students whose height is more than 154 cm. (ii) Find the number of students whose height is 158 cm. (iii) Find the percentage of the number of students whose height is 158 cm. EXERCISE 9.1. COLLECTION OF DATA 76
EXERCISE 9.2 MEASURES OF CENTRAL TENDENCY 9.2.1 Key Concepts i. A typical value of the data around which the observations are concentrated is called a measure of central tendency. ii. The types of central tendencies are mean, mode and median. iii. The nature of the data and its purpose will be the criteria to go for mean or median or mode among the measures of central tendency. iv. Mean is the quotient obtained by dividing the sum of all the observations of a data by the number of observations in the data, Mean = S um o f observations or x = xi Number o f observations n v. For a grouped frequency distribution, the arithmetic mean, x = fi xi . fi vi. By the method of deviation, the A.M. of the data = A + fid where A is the assumed mean, fi fi is the sum of the frequencies. vii. Median is the middle–most value of all observations of the data, when they are arranged in order (ascending or descending). viii. When the number of observations (n) is odd, then median = n+1 th observation. 2 ix. When 'n' is even, median is the average of the n th n + 1 th observations of the data. 2 2 and x. Mode is the value which occurs most frequently among all the observations. xi. For some data mode may not exist. 9.2.2 Additional Questions Objective Questions 1. [AS1] The mean of the data 90, 95, 90, 70, 86, 80, 100, 95, 80, 85, 90 and 75 is . (A) 86.3 (B) 83.6 (C) 85.6 (D) 83 2. [AS1] The median of the scores 70, 90, 86, 95, 100, 85, 80, 90, 75, 90, 80 and 95 is . (A) 90 (B) 86 (C) 88 (D) 85 EXERCISE 9.2. MEASURES OF CENTRAL TENDENCY 77
3. [AS1] The mode of the scores 90, 95, 90, 70, 86, 80, 100, 95, 80, 85, 90 and 75 is . (A) 100 (B) 90 (C) 75 (D) 95 4. [AS1] The sum of all the observations of a data is 975 and its mean is 39, then the number of observations of the data is . (A) 936 (B) 1014 (C) 25 (D) 39 5. [AS3] Average of the given data is nothing but its . (A) Mode (B) Median (C) Mean (D) None Short Answer Type Questions 6(i) [AS1] Find the arithmetic mean of 20, 25, 28, 30 and 32. (ii) [AS1] The arithmetic mean of 6, 10, x and 12 is 8. Find the value of x. 7(i) [AS1] Find the median of the first ten prime numbers. (ii) [AS1] The median of the following observations arranged in ascending order 5, 9, 12, 18, x + 2, x + 4, 30, 31, 34, 39 is 24. Find x. 8(i) [AS1] A cricket player scored the following runs in 12 one day matches. 70, 50, 30, 52, 80, 70, 48, 70, 40, 70, 47, 55 Find his modal score. EXERCISE 9.2. MEASURES OF CENTRAL TENDENCY 78
(ii) [AS1] The marks scored by number of students in a class are as given. Number of students 15 8 6 10 12 Marks scored 35 45 50 40 30 Find the mode of the given data. Long Answer Type Questions 9 [AS1] Find the mean salary of workers of a factory from the following table. Salary 3000 4000 5000 6000 7000 8000 9000 10000 Total (in Rs) 16 12 10 8 6 4 3 1 60 Number of workers 10 [AS1] If the mean of 5 observations x, x + 3, x + 6, x + 9 and x + 12 is 16, find the mean of the first 3 observations. 11 [AS1] Find the arithmetic mean of the following frequency distribution. x1 2 3 4 5 67 f 4 10 12 15 12 10 8 12 [AS1] Find out the value of the mode from the following data. 50, 60, 70, 80, 70, 70, 50, 70, 60, 50. EXERCISE 9.2. MEASURES OF CENTRAL TENDENCY 79
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 80
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 81
(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 82
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 83
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 84
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 85
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 86
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 87
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 88
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 89
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 90
—— Project Based Questions —— (i) Let x = p be a rational number, such that the prime factorization of q is of the form 2n.5m, q where n , m are non–negative integers. Then x has a decimal expansion which terminates. Prove it. (ii) Constructing the ‘square root spiral’: Take a large sheet of paper and construct the square root spiral for the number 8. (iii) Prove these identities geometrically. a) (a + b) (a − b) = a2 − b2 b) (x + a) (x + b) = x 2 + (a + b) x + ab (iv) Give a brief note on Euclidean Geometry and list the postulates given by Euclid. (v) Draw 10 different triangles on a sheet of paper. Produce one of the sides of each of these triangles. Find out the sum of the three angles of the triangle and also, measure the exterior angle formed. Note down the observations in the following table and write your conclusion at the end. PROJECT BASED QUESTIONS 91
S.No. Name Angle 1 Angle 2 Angle 3 Sum of Exterior of the the Angle 1 triangle three formed 2 Angles 3 4 5 6 7 8 9 10 (vi) On a graph sheet draw the coordinate axes and plot the following points in pairs. Join them by line segments. (9, 0) , (0, 1) ; (8, 0) , (0, 2) ; (7, 0) , (0, 3) ; (6, 0) , (0, 4) ; (5, 0) , (0, 5) ; (4, 0) , (0, 6) ; (3, 0) , (0, 7) ; (2, 0) , (0, 8) ; (1, 0) , (0, 9) . What do you observe? (vii) On a graph paper, mark the points with ordered pairs (a, 0) where 0 < a < 10 for different values of a and join all these points by means of a straight line. What do you observe? On another graph paper, mark the points with ordered pairs (0, b) where −10 < b < 10 for different values of b and join all these points by means of a straight line. What do you observe? From these two graphs, can you give a general form of equation of the straight lines parallel to the co–ordinate axes? PROJECT BASED QUESTIONS 92
(viii) In countries like USA and CANADA, temperature is measured in Fahrenheit, where as in coun- tries like India, it is measured in Celsius. Here is a relation between Fahrenheit (F) and Celsius (C) by which we can convert the given temperature from one unit to another. F= 9 C + 32 5 Prepare a graph of the above relation by considering it as a linear equation in two variables using x – axis for Celsius and y – axis for Fahrenheit. From the graph identify: a) The temperature in Fahrenheit if the temperature in Celsius is 30◦C. b) The temperature in Celsius if the temperature in Fahrenheit is 95◦ F. c) If the temperature is 0◦ C , what is the temperature in Fahrenheit and if the temperature is 0◦ F , what is the temperature in Celsius? d) Is there a temperature which is numerically equal in both Fahrenheit and Celsius? If yes, find it. PROJECT BASED QUESTIONS 93
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