Point Abscissa Ordinate Coordinate Quadrant A B C D E F 9(i) [AS5] Plot the following points in the Cartesian plane: (3, –2), (4, 4), (–5, 3), (–2, –1). (ii) [AS5] Plot the following points in the Cartesian plane whose abscissa and ordinates are given. Also write the coordinates of the points in the form of ordered pairs in the given table. xy (x, y) 04 21 –3 2 –4 –5 4 –2 –2 0 EXERCISE 5.2. CARTESIAN SYSTEM 48
EXERCISE 5.3 PLOTTING A POINT ON THE CARTESIAN PLANE WHEN ITS CO–ORDINATES ARE GIVEN 5.3.1 Key Concepts i. The process of marking a point on a Cartesian plane using its coordinates is called “plotting the point”. ii. Draw the X–axis and Y–axis perpendicular to each other. They intersect at a point “O” called origin. iii. If a point A (x, y) is to be marked, we start from the origin, move ‘x’ units along X–axis, ‘y’ units along Y–axis and plot the point A. 5.3.2 Additional Questions Objective Questions 1. [AS3] The point (–3, –3) lies the Y–axis. (A) To the left of (B)To the right of (C) Above (D) Below 2. [AS3] The point (4, –7) lies the X–axis. (A) To the left of (B)To the right of (C) Above (D) Below 3. [AS3] The point (10, 0) lies on the . (A) Y–axis (B) X–axis (C)Origin (D)None of these 4. [AS3] If both the coordinates of a point are negative then the point lies in . (A) Q1 (B) Q2 (C) Q3 (D) Q4 EXERCISE 5.3. PLOTTING A POINT ON THE CARTESIAN PLANE WHEN IT. . . 49
5. [AS3] If the abscissa is of a point is positive and its ordinate is negative, then it lies in . (A) Q1 (B) Q2 (C) Q3 (D) Q4 Very Short Answer Type Questions side of the Y–axis. 6 [AS3] Fill in the blanks. quadrant. (i) The point (5, –3) lies to the (ii) The point (–2, 7) lies in the (iii) The point (–3, 7) lies the X–axis. (iv) The point (6, 0) lies on the . (v) The point (0, –4) lies on the . 7 [AS5] Fill in the blanks. Observe the given graph carefully and fill in the blanks. (i) The coordinates of the point H are ______________. (ii) The coordinates of the point which lies on the X–axis are _____________. (iii) The coordinates of the point K are . (iv) The coordinates of the point which lies on the Y–axis are ________________. (v) The coordinates of the point B are . EXERCISE 5.3. PLOTTING A POINT ON THE CARTESIAN PLANE WHEN IT. . . 50
Long Answer Type Questions 8 [AS2] What can you say about the positions of the points (3, 2), (–4, 2), (0, 2), (1, 2), (5, 2) and (–1, 2)? Locate these points on a graph paper. 9 [AS5] Plot the points (1, 2), (7, 2) and (4, 5) on a graph sheet. Join them using straight lines to form a triangle and find its area. 10 [AS5] Plot the points P(2, 1), Q(6, 1), S(0, 5) and R(4, 5) on a graph sheet. Join all the points in order to form a parallelogram and find its area. EXERCISE 5.3. PLOTTING A POINT ON THE CARTESIAN PLANE WHEN IT. . . 51
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 52
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 53
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 54
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 55
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 56
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 57
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 58
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 59
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 60
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 61
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 62
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 63
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 64
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 65
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 66
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 67
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 68
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 69
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 supplement◦ary (i.e., equal to 180 degrees). (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 70
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 71
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 72
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 73
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 74
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 75
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 76
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 77
(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 78
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 79
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 80
(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 81
—— 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 82
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 83
Additional AS Based Practice Questions Q1 [AS4] Identify which of the variables x, y, z and u represent rational numbers and which irrational numbers. i) x2 = 5 ii) y2 = 9 iii) z2 = 0.04 iv) u2 = 17 4 Q2 [AS4] If a = 3 + √5 , then find the value of a2 + 1 . 2 a2 Q3 [AS4] Using a suitable identity, evaluate the following: i) 1033 ii) 101 x 102 iii) 9992 Q4 [AS1] In the given figure, LM is a line parallel to the y axis at a distance of 3 units. i) What are the coordinates of the points P, R and Q. ii) What is the difference between the abscissa of the points L and M? ADDITIONAL AS BASED PRACTICE QUESTIONS 84
Q5 [AS1] A point lies on the x-axis at a distance of 7 units from the y-axis. What are its coordinates? What will be its coordinates if it lies on y-axis at a distance of -7 units from the x-axis? Q6 [AS4] Write the coordinates of the vertices of a rectangle whose length and breadth are 5 and 3 units respectively, one vertex at the origin, the longer side lies on the x-axis and one of the vertices lies in the third quadrant. Q7 [AS4] On plotting the points O (0, 0), A (3, 0), B (3, 4), C (0, 4) and joining OA, AB, BC and CO, which of the following figures is obtained? A) Square B) Rectangle C) Trapezium D) Rhombus Q8 [AS3] Any point on the x-axis is of the form A) (x, y) B) (0, y) C) (x, 0) D) (x, x) Q9 [AS3] Any point on the line y = x is of the form A) (a, a) B) (0, a) C) (a, 0) D) (a, -a) Q10 [AS2] Plot the following points and check whether they are collinear or not: i) (1, 3), (-1, -1), (-2, -3) ii) (1, 1), (2, -3), (-1, -2) iii) (0, 0), (2, 2), (5, 5) Q11 [AS2] Check which of the following points lie on the y-axis? A(1, 1), B(1, 0), C(0, 1), D(0, 0), E(0, -1), F(-1, 0), G(0, 5), H(-7, 0), I(3, 3) Q12 [AS1] In the given figure, the point identified by the coordinates (-5, 3) is ADDITIONAL AS BASED PRACTICE QUESTIONS 85
A) T B) R C) L D) S Q13 [AS1] The perpendicular distance of the point P (3, 4) from the y-axis is A) 3 B) 4 C) 5 D) 7 Q14 [AS4] A frisbee lands on top of a 15 feet concrete wall. To retrieve the frisbee, a ladder must be placed such that the foot of the ladder is 6 feet from the base of the wall and the top of the ladder rests on top of the wall. The shortest length of of the ladder that can be used is: A) 15 feet B) 16 feet C) 17 feet D) 18 feet Q15 [AS4] In the given triangle ABC, find the longest side using the correct property. ADDITIONAL AS BASED PRACTICE QUESTIONS 86
Q16 [AS5] In the given figure, CDE is an equilateral triangle formed on a side CD of a square ABCD. Show that ∆ADE ̴̳ ∆BCE. Q17 [AS2] 30 children were asked about the number of hours they watched TV programmes in the last week. The results are recorded as under: Number of hours 0-5 5 - 10 10 - 15 15 - 20 Frequency 8 16 4 2 Can we say that the number of students who watched TV for 10 or more hours a week is 22? Justify your answer. Q18 [AS2] The following frequency distribution has been represented graphically as given: Marks 0 - 20 20 - 40 40 - 60 60 - 100 Number of students 10 15 20 25 ADDITIONAL AS BASED PRACTICE QUESTIONS 87
Do you think this representation is correct? Why? Q19 [AS4] A class consists of 50 students out of which 30 are girls. The mean of marks scored by girls in a test is 73 (out of 100) and that of boys is 71. Determine the mean score of the whole class. Q20 [AS4] The points scored by a basketball team in a series of matches are as follows: 17, 2, 7, 27, 25, 5, 14, 18, 10, 24, 48, 10, 8, 7, 10, 28 Find the median and mode for the data. ADDITIONAL AS BASED PRACTICE QUESTIONS 88
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