9th Standard Mathematics All Chapter Important Questions-I- 2020

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Questions:

Generic 2

25 x 2 = 50

Find the probability that a leap year selected at random will contain 53 Sundays.

If 3 (tan \(\theta\)) + 4 (sec \(\theta\) \(\times\) sin 6) = 24. Then find all the trigonometric ratios of the angle \(\theta\)

Find the TSA and LSA of a cuboid whose length, breadth and height are 10 cm, 12 cm and 14 cm respectively.

Find the centroid of the triangle whose vertices are (2, -5), (5, 11) and (9, 9)

Find the value of k for which the system of linear equations has no solution
ka+ 10b = 15, 8a+5b=9

Frame two problems in calculating probability, based on the spinner shown here.

On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeeper gains Rs. 2000. But if he sells the T.V. at 10% gain and the fridge at 5% loss, he gains Rs. 1500 on the transaction. Find the actual price of the T.V. and the fridge.

An advertisement board is in the form of an isosceles triangle with perimeter 36m and each of the equal sides are 13 m. Find the cost of painting it at Rs. 17.50 per square metre.

Find the centroid of the triangle whose vertices are
(i) (2, −4), (−3, −7) and (7, 2)
(ii) (−5, −5), (1, −4) and (−4, −2)

10.

If cos \(\theta\) : sin\(\theta\) =1: 2, then find the value of = \(\frac { 8cos\theta -2sin\theta }{ 4cos\theta +2sin\theta } \)

11.

Find any 3 irrational numbers between 0.12 and 0.13.

12.

State Associative property of sets.

13.

The chord of length 32 cm is drawn at the distance of 12 cm from the centre of the circle. Find the radius of the circle

14.

In a class test in mathematics,10 students scored 75 marks,12 students scored 60 mark, 8 students scored 40 marks and 3 students scored 30 marks. Find the mean of their score.

15.

A set of numbers consists of five 4’s, four 5’s, nine 6’s,and six 9’s. What is the mode.

16.

In figure ㄥABC = 120⁰, where A, B and C are points on the circle with centre O. Find ㄥOAC?

17.

Using factor theorem, show that (x -5) is a factor of the polynomial 2x³-5x²-28x+15

18.

Divide \(\sqrt [ 9 ]{ 8 } \) by \(\sqrt [ 6 ]{ 6 } \).

19.

Without actual division classify the decimal expansion of the following numbers as terminating or non-terminating and recurring.
\(7\over 16\)

20.

you know that \(\frac { 1 }{ 7 } \)= 0.142857. Can you predict what the decimal expansion of \(\frac { 2 }{ 7 } ,\frac { 3 }{ 7 } ,\frac { 5 }{ 7 } ,\frac { 6 }{ 7 } \) are without actually doing the long division. If so, how?

21.

In the given diagram PQRS is a parallelogram.
ㄥS = 4x - 60, ㄥQ = 30 - x. Find the angles of P and R.

22.

Write the following in "Roster" form?
(a) A = set of the months having 31 days.
(b) B = {x : x is a natural number of 2 digits divisible by 13}
(c) C = {set of vowels in the word "father"}
(d) D = {x : 5 < x < 10 ; x ∈ N}
(e) E = {x: x is a square natural number less than 16}

23.

If n(A) = 25, n(B) = 40, n(A∪B) = 50 and n(B′) = 25 , find n(A∩B) and n(U).

24.

Find the value of x

25.

Are A = {x : x ∈ N, 4 ≤ x ≤ 8} and B = { 4, 5, 6, 7, 8} equal sets?

Generic 3

14 x 3 = 42

26.

The total surface area of a cube is 864 cm². Find its volume

27.

Team I and Team II play 10 cricket matches each of 20 overs. Their total scores in each match are tabulated in the table as follows:

Match numbers	1	2	3	4	5	6	7	8	9	10
Team I	200	122	111	88	156	184	99	199	121	156
Team II	143	123	156	92	164	72	100	201	98	157

What is the relative frequency of Team I winning?

28.

Find the values of
(i) tan7° tan23° tan60° tan67° tan83°
(ii) \(\frac { cos35° }{ sin55° } +\frac { sin12° }{ cos78° } -\frac { cos18° }{ sin72° } \)

29.

If U = {x : x ∈ Z, -2 ≤ x ≤ 10}, A = {x : x = 2p +1, p ∈ Z, -1 ≤ p ≤ 4}, B = {x : x = 3q + 1, q ∈ Z, -1 ≤ q < 4} verify De Morgan’s laws for complementation.

30.

Find the GCD of a^x, a^x+y, a^x+y+z

31.

Compute and give the answer in the simplest form; \(3\sqrt { 162 } \times 7\sqrt { 50 } \times 6\sqrt { 98 } \)

32.

Find the mean of the following distribution using Step Deviation Method.

Class Interval	0-8	8-16	16-24	24-32	32-40	40-48
Frequency (f)	10	20	14	16	18	22

33.

If the quotient obtained on dividing 3x³+11x²+34x+106 by x - 3 is 3x²+ ax + b then find a, b and also the remainder.

34.

ABCD is a parallelogram and AP and CQ are perpendic from vertex A and C on diagonal BD. Show that
(i) ΔAPB ≅ ΔCQD
(ii) AP = CQ

35.

Given that A = {1,3,5,7} B = {1,2,4,6,8}. Find
(i) AΔB and
(ii) BΔA

36.

Join the points taken in order and name the figure obtained. Name atleast any one point which is lying :
(i)On x-axis
(ii)On y-axis
(iii) In each of the Quadrants.

37.

Three vertices of a rectangle are (3, 2), (-4, 2) and (-4, 5). Plot the points and find the coordinates of the fourth vertex.

38.

Express the following decimal expression into rational numbers \(17.2\overline { 15 } \)

39.

Consider the given pairs of triangles and say whether each pair is that of congruent triangles. If the triangles are congruent, say ‘how’; if they are not congruent say ‘why’ and also say if a small modification would make them congruent:

Multiple Choice Question

40 x 1 = 40

40.

The capacity of a water tank of dimensions 10 m × 5 m × 1.5 m is _______.

(a)

75 litres

(b)

750 litres

(c)

7500 litres

(d)

75000 litres

41.

The lateral surface area of a cube of side 12 cm is _______.

(a)

144 cm²

(b)

196 cm²

(c)

576 cm²

(d)

664 cm²

42.

A letter is chosen at random from the word “STATISTICS”. The probability of getting a vowel is

(a)

\(\frac { 1 }{ 10 } \)

(b)

\(\frac { 2 }{ 10 } \)

(c)

\(\frac { 3 }{ 10 } \)

(d)

\(\frac { 4 }{ 10 } \)

43.

The probability based on the concept of relative frequency theory is called _______.

(a)

Empirical probability

(b)

Classical probability

(c)

Both (1) and (2)

(d)

Neither (1) nor (2)

44.

The value of \(\frac { 1-{ tan }^{ 2 }{ 45 }^{ 0 } }{ 1+{ tan }^{ 2 }{ 45 }^{ 0 } } \) is ________.

(a)

(b)

(c)

(d)

\(\frac { 1 }{ 2 } \)

45.

The value of 2tan30° tan60° is

(a)

(b)

(c)

\(2\sqrt { 3 } \)

(d)

46.

If the coordinates of the mid-points of the sides AB, BC and CA of a triangle are (3, 4), (1, 1) and (2, −3) respectively, then the vertices A and B of the triangle are ______.

(a)

(3, 2), (2, 4)

(b)

(4, 0), (2, 8)

(c)

(3, 4), (2, 0)

(d)

(4, 3), (2, 4)

47.

If A,B,C are non-overlapping sets, then n \(\left( A\cap B\cap C \right) \) is ___________

(a)

n(A) + n(B) + n(C)

(b)

\(n\left( A\cup B\cup C \right) \)

(c)

(d)

\(n\left( A\cap B \right) \)

48.

If one of the factor of x^2-9x+18 is (x-3) then the other factor is_________

(a)

x-9

(b)

x+6

(c)

(x-6)

(d)

x-18

49.

Rationalising the denominator \(\cfrac { 1 }{ \sqrt [ 3 ]{ 3 } } \) ___________

(a)

(b)

\(\cfrac { { 3 }^{ \frac { 2 }{ 3 } } }{ 3 } \)

(c)

\(\sqrt { 3 } \)

(d)

\(\sqrt [ 3 ]{ 3 } \)

50.

The distance between the longest chord of a circle and the centre is________

(a)

(b)

(c)

(d)

51.

In the figure, PQRS and PTVS are two cyclic quadrilaterals, If \(\angle\)QRS = 70°, then \(\angle\)TVS= __________

(a)

70°

(b)

110°

(c)

80°

(d)

90°

52.

The mean of set of numbers is \(\bar{x}\) If each number is multiplied by z, the mean is

(a)

\(\bar{X}+z\)

(b)

\(\bar{X}-z\)

(c)

\(z\bar{X}\)

(d)

\(\bar{X}\)

53.

The mean of the first 10 whole number is

(a)

(b)

4.5

(c)

(d)

5.5

54.

The algebraic sum of the deviations of a set of n values from their mean is ___________

(a)

(b)

n-1

(c)

(d)

n+1

55.

The mean of set of seven number is 81. If one of the nimbers is discarded,the mean of remaining number is 78. The value of discarded number is

(a)

101

(b)

100

(c)

(d)

56.

The mean of the first 10 prime numbers is ___________

(a)

12.6

(b)

12.7

(c)

12.8

(d)

12.9

57.

The mean of 5, 9, x, 17, and 21 is 13, then find the value of x _______.

(a)

(b)

(c)

(d)

58.

In the given figure, If OP = 17cm PQ = 30, cm and OS is perpendicular to PQ, then RS is ________.

(a)

10 cm

(b)

6 cm

(c)

7 cm

(d)

9 cm

59.

What is 5.92 \(\times\)10^-3 written in decimal form?

(a)

0.000592

(b)

0.00592

(c)

0.0592

(d)

0.592

60.

Divide x³-4x²+6x by "x" the result is _____________________

(a)

\(x^{ 2 }+4x-6\)

(b)

\(x^{ 2 }-4x-6\)

(c)

\(x^{ 2 }-4x+6\)

(d)

\(x^{ 2 }+4x+6\)

61.

The value of the polynomial f(x) = 6x - 3x²+9 when x = -1 is _____________________

(a)

(b)

(c)

(d)

62.

The roots of the polynominal equation \({ x }^{ 2 }+2x=0\) are_______________

(a)

x = 0, 2

(b)

x = 1, 2

(c)

x = 1, -2

(d)

x = 0, -2

63.

ABCD is a parallelogram as shown. Find x and y.

(a)

1, 7

(b)

2, 6

(c)

3, 5

(d)

4, 4

64.

Orthocentre of a triangle is the point of concurrency of _______

(a)

medians

(b)

altitudes

(c)

angle bisectors

(d)

perpendicular bisectors of side

65.

The set does not have a proper subset is __________

(a)

Finite set

(b)

Infinite set

(c)

Null set

(d)

Singleton set

66.

A = {set of odd natural numbers}, B = {set of even natural numbers}, then A and B are ___________

(a)

equal set

(b)

equivalent sets

(c)

overlapping sets

(d)

disjoint sets

67.

The product of \(2\sqrt { 5 } \) and \(6\sqrt { 5 } \) is_______________.

(a)

\(12\sqrt { 5 } \)

(b)

(c)

(d)

\(8\sqrt { 5 } \)

68.

The decimal form of -\(\frac { 3 }{ 4 } \) is_______________

(a)

- 0.75

(b)

- 0.50

(c)

-0.25

(d)

- 0.125

69.

The diagonal of a square formed by the points (1, 0), (0, 1), (-1, 0) and (0, - 1) is_______________

(a)

(b)

(c)

\(\sqrt{2}\)

(d)

70.

The point (0, -3) lies on _______________

(a)

+ ve x-axis

(b)

+ ve y-axis

(c)

- ve x-axis

(d)

- ve y-axis

71.

The distance between the points (a, 0) and (0, b) is____________

(a)

a unit

(b)

b unit

(c)

\(\sqrt{a^2+{b^2}}\ unit\)

(d)

\(\sqrt{a^2-{b^2}}\ unit\)

72.

A point on the y-axis is ________________

(a)

(1, 1)

(b)

(6,0)

(c)

(0,6)

(d)

(-1, -1)

73.

Find the odd one out of the following.

(a)

\(\sqrt { 32 } \times \sqrt { 2 } \)

(b)

\(\frac { \sqrt { 27 } }{ \sqrt { 3 } } \)

(c)

\(\sqrt { 72 } \times \sqrt { 8 } \)

(d)

\(\frac { \sqrt { 54 } }{ \sqrt { 18 } } \)

74.

If ( x+2, 4) = (5, y–2), then the coordinates (x,y) are _____.

(a)

(7, 12)

(b)

(6, 3)

(c)

(3, 6)

(d)

(2, 1)

75.

The type of the polynomial 4–3x³is ________.

(a)

constant polynomial

(b)

linear polynomial

(c)

quadratic polynomial

(d)

cubic polynomial.

76.

The root of the polynomial equation 2x + 3 = 0 is ________.

(a)

\(\frac{1}{3}\)

(b)

\(-\frac{1}{3}\)

(c)

\(-\frac{3}{2}\)

(d)

\(-\frac{2}{3}\)

77.

If X = {x : x = 4(n – 1), n ∈ N} and Y = {y : y = 3n – 2n – 1, n ∈ N}, then X∪Y is ______________

(a)

(b)

(c)

(d)

78.

The shaded region in the adjacent diagram represents ______________

(a)

(A∪B)′

(b)

(A∩B)′

(c)

A′∩B′

(d)

A∩B

79.

The correct statement out of the following is ________.

(a)

ΔABC ≅ ΔDEF

(b)

ΔABC ≅ ΔDEF

(c)

ΔABC ≅ ΔFDE

(d)

ΔABC ≅ ΔFED

Generic 5

8 x 5 = 40

80.

A farmer has a field in the shape of a rhombus. The perimeter of the field is 400 m and one of its diagonal is 120 m. He wants to divide the field into two equal parts to grow two different types of vegetables. Find the area of the field.

81.

If A, Band C are overlapping sets, draw venn diagram for:\(A\cap B\)

82.

Prove that x -1 is a factor x⁵ - 45x⁴ + 36x³ + 45x²- 36x^-1

83.

The mean of five positive integers is twice their median.If four of the integers are 3, 4, 6, 9 and median is 6, then find the fifth integer.

84.

Find the Arithmetic Mean of the following data using Step Deviation Method

Age	15-19	20-24	25-29	30-34	35-39	40-44
No.of persons	4	20	38	24	10	9

85.

Find any three rational numbers between \(\frac { 1 }{ 2 } \) and \(\frac { 1 }{ 5 } \)

86.

Find any seven rational numbers between \(\frac { 5 }{ 8 } \) and \(\frac { 5 }{ 6 } \)

87.

Find the type of triangle formed by (-1, -1), (1, 1) and (\(-\sqrt{13},\sqrt{13}\))

Generic 8

2 x 8 = 16

88.

Construct \(\triangle\)ABC in which AB = BC = 8cm and \(\angle \)B =70^o. Locate its in centre and draw the incircle

89.

Draw an equilateral triangle of side 8 cm and locate its incentre. Also draw the incircle.

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Answers Key:

Generic 2

25 x 2 = 50

S = {Sunday Monday, Monday Tuesday, Tuesday Wednesday, Wednesday Thursday, Thursday Friday, Friday Saturday, Saturday Sunday}
n(S) = 7; n (A) = 2; P(A) =\(\frac{2}{7}\).

3 tan \(\theta\) + 4 (sec \(\theta\) \(\times\) sin\(\theta\)) = 24
\(3tan\theta +4\left( \cfrac { 1 }{ cos\theta } \times sin\theta \right) =24\)
7 tan \(\theta\) = 24
\(tan\theta =\cfrac { 24 }{ 7 } \)
hypetenuse = \(\sqrt { { 24 }^{ 2 }+49 } =\sqrt { 576+49 } =\sqrt { 625 } =25\)
\(sin\theta =\cfrac { 24 }{ 25 } ;cos\theta =\cfrac { 7 }{ 25 } ;cosec\theta =\cfrac { 25 }{ 24 } ;sec\theta =\cfrac { 25 }{ 27 } ;cot\theta =\cfrac { 7 }{ 24 } \)

TSA = 2 (lb + bh + lh)
= 2 (10 \(\times\)12 + 12 \(\times\) 14 +10 \(\times\) 14)
= 2 (120 + 168 + 140)
= 856 cm²
LSA = 2 (bh + lh)
= 2 (12 \(\times\) 14 + 10 \(\times\) 14)
= 2 (168 + 140)
= 2 (308) = 616 cm².

\(G\left( \frac { { x }_{ 1 }+{ x }_{ 2 }+{ x }_{ 3 } }{ 3 } ,\frac { { y }_{ 1 }+{ y }_{ 2 }+{ y }_{ 3 } }{ 3 } \right) =\left( \frac { 2+5+9 }{ 3 } ,\frac { -5+11+9 }{ 9 } \right) =\left( \frac { 16 }{ 3 } ,5 \right) \)

ka + 10b = 15
8a + 5b = 9
\(\cfrac { k }{ 8 } =\cfrac { 10 }{ 5 } \neq \cfrac { 15 }{ 9 } \)
\(\Rightarrow \cfrac { k }{ 8 } =\cfrac { 10 }{ 5 } \Rightarrow \cfrac { k }{ 8 } =2\Rightarrow k=16\)

(i) What is the probability that the spinner will not land on a multiple of 2?
(ii) What is the probability that the spinner will land on an odd number?

Let the actual price of a T.V. = x
Let the actual price of a Fridge = y
\(\cfrac { 5 }{ 100 } x+\cfrac { 10 }{ 100 } y=2000\)
\(\cfrac { 5 }{ 100 } (x+2y)=2000\)
\(x+2y=2000\times \cfrac { 100 }{ 5 } \)
x+2y = 40000 ...(1)
\(\cfrac { 10 }{ 100 } x-\cfrac { 5 }{ 100 } y=1500\)
\(\cfrac { 5 }{ 100 } (2x-y)=1500\)
\(2x-y=1500\times \cfrac { 100 }{ 5 } \)
2x -y = 30000
(1) X 2 \(\Rightarrow\) 2x + 4y = 80000
(2) \(\Rightarrow\) \(\cfrac { 2x-y=30000 }{ 5y=50000 } \)
y = 10000
Substitute y = 10000 in (1)
x + 2(10000) = 40000
x + 20000 = 40000
x = 40000 - 20000
x = 20000
\(\therefore\) Actual price of T.V = Rs. 20000
Actual price of Fridge = Rs. 10000

Area of an isoeeles triangle
h = \(\sqrt { { 13 }^{ 2 }-{ 5 }^{ 2 } } =\sqrt { 169-25 } =\sqrt { 144 } \)=12 m
∴ Area of the triangular board
= \(\frac { 1 }{ 2 } \times bh=\frac { 1 }{ 2 } \) \(\times\) 10 \(\times\) 12 = 60 m²
cost of painting 1m²= Rs. 17.50
cost of painting 60m² = 60 \(\times\) 17.50 = Rs. 1050

x₁ y₁ x₂ y₂ x₃y₃
(2, -4) (-3, -7) (7, 2)
(i) Centroid G (x, y) \(=\left( \frac { { x }_{ 1 }+{ x }_{ 2 }+{ x }_{ 3 } }{ 3 } ,\frac { { y }_{ 1 }+{ y }_{ 2 }+{ y }_{ 3 } }{ 3 } \right) \)
\(=\left( \frac { (2)+(-3)+7 }{ 3 } ,\frac { \left( -4 \right) +\left( -7 \right) +\left( 2 \right) }{ 3 } \right) \)
\(=\left( \frac { 6 }{ 3 } ,\frac { -9 }{ 3 } \right) =(2,-3)\)
(ii) x₁ y₁ x₂ y₂ x₃y₃
(2, -4) (1, -4) (-4, -2)
Centroid G (x, y) \(=\left( \frac { (-5)+1+(-4) }{ 3 } ,\frac { \left( -5 \right) +\left( -4 \right) +\left( -2 \right) }{ 3 } \right) \)
\(=\left( \frac { -8 }{ 3 } ,\frac { -11 }{ 3 } \right) \)

10.

\(cos\theta :sin\theta =1:2\)
\(\cfrac { cos\theta }{ sin\theta } =\cfrac { 1 }{ 2 } \)
\(cos\theta =\cfrac { 1 }{ 2 } sin\theta \)
\(\sin\theta =2 \cos\theta \)

\(\therefore \cfrac { 8cos\theta -2sin\theta }{ 4cos\theta +2sin\theta } =\cfrac { 1 }{ 2 } \)

11.

Three irrational numbers between 0.12 and 0.13 are 0.12010010001…, 0.12040040004…, 0.12070070007…

12.

For any three sets A, B and C
(i) \(A\cap \left( B\cap C \right) =\left( A\cap B \right) \cap C\)
(ii) \(A\cup \left( B\cup C \right) =\left( A\cup B \right) \cup C\)

13.

Radius of the circle =\(\sqrt { { 16 }^{ 2 }+{ 12 }^{ 2 } } \)
=\(\sqrt { 256+144 } \)
=\(\sqrt { 400 } =\sqrt { 20\times 20 } =20\)

14.

Total number of students = 10 + 12 + 8 + 3 = 33
The total score of 33 students = 10\(\times\)75 +12 \(\times\) 60 + 8\(\times\)40 + 3\(\times\)30
= 750 + 720 + 32090 = 1880
Mean of their score = \(\frac { Total\ Marks }{ number\ of\ students } =\frac { 188 }{ 33 } \)
= 56.96 or 57 approximately

15.

Size of item	4	5	6	9
Frequency	5	4	9	6

6 has the maximum frequency 9. Therefore 6 is the mode.

16.

reflex \(\angle\)AOC = 2\(\angle\)ABC = 2 \(\times\)120⁰= 240⁰
\(\therefore \)\(\angle\)AOC = 360⁰- 240⁰= 120⁰
Hence\(\angle\)OAC+\(\angle\)OCA = 180⁰- 120⁰= 60⁰
\(\Rightarrow\) 2\(\angle \)AOC = 60⁰
\(\Rightarrow\) \(\angle\)OAC=\(\cfrac { { 60 }^{ 0 } }{ 2 } \) = 30⁰ [ \(\because \) \(\angle \)OAC =\(\angle \) OCA]

17.

Let P(x) = 2x³-5x²-28x+15
By factor theorem, (x-5) is a factor of P(x), if P(5) = 0
P(5) = 2(5)³-5(5)²-28(5)+15
= 2\(\times\)125 - 5\(\times\)25 - 140+15
= 250 - 125 - 140+15
= 265 - 265 = 0
ஃ (x-5) is a factor of 2x³-5x²-28x+15

18.

\(\frac { \sqrt [ 9 ]{ 8 } }{ \sqrt [ 6 ]{ 6 } } =\frac { { 8 }^{ \frac { 1 }{ 9 } } }{ 6^{ \frac { 1 }{ 6 } } } \) (Note that 18 is the LCM of 6 and 9)
=\(\frac { { 8 }^{ \frac { 2 }{ 18 } } }{ { 6 }^{ \frac { 3 }{ 18 } } } \) (How?)
=\(\left( \frac { { 8 }^{ 2 } }{ { 6 }^{ 3 } } \right) ^{ \frac { 1 }{ 18 } }\) (How?) \(=\left( \frac { 8\times }{ 6\times 6\times 6 } \right) ^{ \frac { 1 }{ 18 } }\)
=\(\left( \frac { 8 }{ 27 } \right) ^{ \frac { 1 }{ 18 } }=\left[ \left( \frac { 2 }{ 3 } \right)^3 \right] ^{ \frac { 1 }{ 18 } }=\left( \frac { 2 }{ 3 } \right) ^{ \frac { 1 }{ 6 } }=6\sqrt { \frac { 2 }{ 3 } } \).

19.

\({7\over 16}={7\over 2^4}={7\over 2^4\times 5^6}\)
\(\therefore {7\over 16}\) has a terminating decimal expansion.

20.

\(\overline { 0.285714 } ;\overline { 0.428571 } ;\overline { 0.571428 } ;\overline { 0.714285 } ;\overline { 0.857142 } \)

21.

ㄥP = ㄥR= 168⁰

22.

(a) A = {Jan, March, May, July, Aug, Oct, Dec}
(b) B = {13,26,39,52,65,78,91}
(c) C = {a,e}
(d) D = {6,7,8,9,10}
(e) E = {1,4,9}

23.

n(A) = 25, n(B) = 40, n(AUB) = 50 and n(B') = 25
n(A\(\cap\)B) = n(A) + n(B) - n(AUB)
n(A\(\cap\)B) = 25 + 40 - 50
= 65 - 50
= 15
n(U) = n(B) + n(B)'
= 40 + 25
= 65
\(\therefore\) n(A\(\cap\)B) = 15 and n(U) = 65

24.

\(\angle SOA=2x^0\) (Vertically opposite angels are equal)
\(\angle POB+\angle SOA+\angle DOQ=180^0\) (angle of a straight line)
3x° + 2x°+ 4x° = 180°
9x° = 180°
\(x^0=\frac{180}{9}=20^0\)
The value of x = 20°.

25.

A = { 4, 5, 6, 7, 8}, B = { 4, 5, 6, 7, 8}
A and B are equal sets.

Generic 3

14 x 3 = 42

26.

Let ‘a’ be the side of the cube.
Given that, total surface area = 864 cm²
6a²= 864
a²= \(\frac{864}{6}\)
a²= 144
Therefore, side (a) = 12 cm
Now, volume of the cube = a³
= 123 = 12 ×12 ×12 = 1728 cm³

27.

In this experiment, each trial is a match where Team I faces Team II.
We are concerned about the winning status of Team I.
There are 10 trials in total; out of which Team I wins in the 1^st, 6^th and 9^th matches.
The relative frequency of Team I winning the matches = \(\frac { 3 }{ 10 } \)or 0.3

28.

(i) tan 7°tan 23°tan 60°tan 67°tan 83°
= tan 7⁰tan 83⁰tan 23⁰tan 67⁰tan 60⁰ (Grouping complementary angles)
= tan 7⁰tan(90⁰ - 7⁰)tan 23⁰tan(90⁰ - 23⁰)tan 60⁰
= (tan7⁰.cot7⁰)(tan 23⁰. cot 23⁰)tan 60⁰
= (1)\(\times\) (1)\(\times\) tan 60⁰
= tan 60⁰ = \(\sqrt { 3 } \)
(ii) \(\frac { cos35° }{ sin55° } +\frac { sin12° }{ cos78° } -\frac { cos18° }{ sin72° } \)
\(=\frac { cos\left( 90°-55° \right) }{ sin55° } +\frac { sin\left( 90°-78° \right) }{ cos78° } -\frac { cos\left( 90°-72° \right) }{ sin72° } \) \(\left[ \begin{matrix} { \text Since} \\cos35°=cos\left( 90°-55° \right) \\ sin12°=sin(90°-78°) \\ cos18°=cos\left( 90°-72° \right) \end{matrix} \right] \)
= \(\frac { sin55° }{ sin55° } +\frac { cos78° }{ cos78° } -\frac { sin72° }{ sin72° } \)
= 1+1 - 1 = 1

29.

Given U = {−2,−1,0,1,2,3,4,5,6,7,8,9,10},
A = {−1,1,3,5,7,9} and B = {−2,1,4,7,10}
Law (i) \(\left( A\cup B \right)' =A'\cap B'\)
Now, AUB = {-2,-1,1,3,4,5,7,9,10}
\(\left( A\cup B \right) '\)= {0,2,6,8} ............... (1)
Then, A'= {-2,0,2,4,6,8,10} and B' = {-1,0,2,3,5,6,8,9}
\(A'\cap B'\) = {0,2,6,8} ........... (2)
From (1) and (2), it is verified that \(\left( A\cup B \right) =A'\cap B'\)
Law (ii) \(\left( A\cap B \right) '=A'\cup B'\)
Now, \(A\cap B\) = {1,7}
\((A\cap B)'\) = {-2,-1,0,2,3,4,5,6,8,9,10} ........ (3)
Then \(A'\cup B'\) = { -2, -1,0,2,3,4,5,6,8,9,10} ......... (4)
From (3) and (4), it is verified that \(\left( A\cap B \right) '=A'\cup B'\)

30.

a^x= \(\underline { { a }^{ x } } \)
a^x+y= \(\underline { { a }^{ x } } \). ay
\(\therefore\) a^x+y+z= \(\underline { { a }^{ x } } \). a^y. a^z

31.

\(3\sqrt { 162 } \times 7\sqrt { 50 } \times 6\sqrt { 98 } \) = \(\left( 3\times 9\sqrt { 2 } \times 7\times 5\sqrt { 2 } \times 6\times 7\sqrt { 2 } \right) \)
= \(3 \times 7 \times 6 \times 9 \times 5 \times 7 \times \sqrt{2} \times \sqrt{2} \times \sqrt{2}\) = 79380\(\sqrt { 2 } \)

32.

Let Assumed mean A = 28, class width c = 8

Class Interval	Mid Value x	Frequency f	\(d-{x-A\over c}\)	fd
0-8	4	10	-3	-30
8-16	12	20	-2	-40
16-24	20	14	-1	-14
24-32	28	16	0	0
32-40	36	18	1	18
40-48	44	22	2	44
		Σ f = 100		Σ fd = –22

Mean
\(\bar X=A+{Σfd\over Σf}\times c\)
\(=28+\left(-22\over100\right)\times8\)
= 28 -1.76 = 26.24

33.

Let p(x) = 3x³+11x²+ 34x + 106
p(x) in standard form
Co-efficients are 3 11 34 106
q(x) = x - 3, its zero x = 3
Synthetic division

Quotient is 3x²+ 20x + 94, it is compared with the given quotient 3x²+ ax +b
Co-efficient of x is a = 20
Constant term is b = 94
Remainder r = 388

34.

(i) In ΔAPB and ΔCQD we have
ㄥAPB = ㄥCQD (90^o each)
AB = CD (opposite sides of parallelogram ABCD)
ㄥABP = ㄥCDQ (AB II CD and AD is a transversal)
Using ASA congruency we have,
ㄥAPB ≅ ㄥCQD
(ii) Since ΔAPB ≅ ㄥCQD
∴ Their corresponding parts are equal.
∴ AP = CQ.

35.

(i) A = {1,3,5,7} ; B = {1,2,4,6,8}
A-B = {1,3,:5,7} - {1,2,4,6,8}
= {3,5,7}
B -A = {1,2,4,6,S} - {1,3,5,7}
= {2,4,6,8}
AΔB = (A - B)U(B-A)
= {3,5,7} U {2,4,6,8}
= {2,3,4,5,6,7,8}.
(ii) BM = (B-A) U (A-B)
= {2,4,6,8} u {3,5,7}
= {2,3,4,516,7,S}

36.

Joining the points we get the aeroplane shape.
(i) A (-2,0); F (10, 0) - x-axis
(ii) C(0, 2); I (0, -2) - y-axis
(iii) D(8,1); E(11,3) - I-Quadrant
B (-1, 1) - II - Quadrant
J (-1, -1) -III - Quadrant
G (11,-3); H (8, -1) - IV - Quadrant

37.

(3, 5)

38.

Let x = 17.2151515 ...... (1)
Here period of decimal is 3, multiply equation (1) by 1000
1000 x = 17215.151515
(2) - (1)
999x = 17197.936
\( x=\frac{17197.936}{999} \\ x=\frac{17197936}{999000} \\ x=\frac{5681}{330} \)

39.

In \(\triangle ABD\ and\ \triangle CDB\)
AB = CD (Given)
AD = BC (Given)
BD is common
By SSS congruency
\(\triangle ABD\cong\triangle CDB\)

Multiple Choice Question

40 x 1 = 40

40.

(d)
75000 litres

41.

42.

43.

(a)
Empirical probability

44.

45.

(b)
2

46.

(b)
(4, 0), (2, 8)

47.

48.

49.

(b)
\(\cfrac { { 3 }^{ \frac { 2 }{ 3 } } }{ 3 } \)

50.

(b)
0

51.

(a)
70°

52.

53.

(b)
4.5

54.

(a)
0

55.

(b)
100

56.

(d)
12.9

57.

(b)
13

58.

(d)
9 cm

59.

(b)
0.00592

60.

61.

(a)
0

62.

(d)
x = 0, -2

63.

64.

(b)
altitudes

65.

66.

(d)
disjoint sets

67.

(b)
60

68.

(a)
- 0.75

69.

(a)
2

70.

(d)
- ve y-axis

71.

72.

73.

(d)
\(\frac { \sqrt { 54 } }{ \sqrt { 18 } } \)

74.

75.

(d)
cubic polynomial.

76.

77.

(b)
X

78.

(b)
(A∩B)′

79.

(d)
ΔABC ≅ ΔFED

Generic 5

8 x 5 = 40

80.

Let ABCD be the rhombus.
Its perimeter = 4 × side = 400 m
Therefore, each side of the rhombus = 100 m
Given the length of the diagonal AC = 120 m
In \(\triangle\)ABC, let a =100 m, b =100 m, c = 120 m

s = \(\frac{a+b+c}{2}=\frac{100+100+120}{2}\) = 160 m
Area of \(\triangle\)ABC =\(\sqrt{160(160-100)(160-100)(160-120)}\)
= \(\sqrt{160 \times 60\times 60 \times40}\)
= \(\sqrt{40 \times 2 \times \times2\times60\times60\times40}\)
= 40 × 2 × 60 = 4800 m²
Therefore, Area of the field ABCD = 2 × Area of \(\triangle\)ABC = 2 × 4800 = 9600 m²

81.

82.

Letp (x) x⁵ - 45x⁴ + 36x³ + 45x² - 36x- 1
Sum of co-efficients = 1 - 45 + 36 + 45 - 36 - 1 = 0
Thus x -1 is a factor of p (x)

83.

The five integers are 3, 4, 6, 9, x
Mean \(\bar{x}=\cfrac\Sigma{x}=\cfrac{3+4+6+9+x}{5}\)
=\(\cfrac{22+x}{5}\)
Their Median = 6
Mean = twice of the median
\(\cfrac{22+x}{5}=2\times6\)
22+x = 5 \(\times\) 12 = 60
\(\therefore\) x = 60- 22 = 38

84.

Step Deviation Method
\(\bar { x } =A+\left[ \frac { \Sigma fd }{ \Sigma f } \times c \right] \) Where \(d=\frac { x-A }{ c } \)
Let Assumed mean A = 32
Class Interval c = 45

Age Class Interval	x	Mid x	No.of person f	d=\(\cfrac{x-A}{c}\)	fd
15-19	14.5-19.5	17	4	-3	-12
20-24	19.5-24.5	22	20	-2	-40
25-29	24.5-29.5	27	38	-1	-38
30-34	29.5-34.5	32	24	0	0
35-39	34.5-39.5	37	10	1	10
40-44	39.5-44.5	42	9	2	18
			\(\Sigma f=105\)		\(\Sigma fd=-62\)

Mean \(\bar { x } =A+\left[ \cfrac { \Sigma fd }{ \Sigma f } \times c \right] \)
\(=32+\left[ \cfrac { -62 }{ 105 } \times 5 \right] \)
= 32 + (-2.952)
= 29.05

85.

Relational between \(\frac { 1 }{ 2 } \) and \(\frac { 1 }{ 5 } \) = \(\frac { 1 }{ 2 } \left( \frac { 1 }{ 2 } +\frac { 1 }{ 5 } \right) \)
=\(\frac { 1 }{ 2 } \left( \frac { 1 }{ 2 } +\frac { 1 }{ 5 } \right) \)
= \(\frac { 1 }{ 2 } \left( \frac { 1 }{ 2 } +\frac { 1 }{ 5 } \right) \)
Rational numbers between\(\frac { 1 }{ 2 } \) and \(\frac { 7 }{ 20 } \)
=\(\frac { 1 }{ 2 } \left( \frac { 10+7 }{ 20 } \right) \)
=\(\frac { 17 }{ 40 } \)
Rational number between \(\frac { 1 }{ 2 } \) and \(\frac { 7 }{ 20 } \) =\(\frac { 1 }{ 2 } \left( \frac { 1 }{ 2 } \times \frac { 17 }{ 40 } \right) \)
=\(\frac { 1 }{ 2 } \left( \frac { 20+17 }{ 40 } \right) \)
=\(\frac { 37 }{ 80 } \)
Thus the rational numbers are \(\frac { 7 }{ 20 } ,\frac { 17 }{ 40 } and\frac { 37 }{ 80 } \)

86.

Let us convert the given rational numbers having the same denominators.
L.C.M of 8 and 6 is 24.
\(\frac { 5 }{ 8 } \times \frac { 5\times 3 }{ 8\times 3 } =\frac { 15 }{ 24 } \)
\(\frac { 5 }{ 8 } \times \frac { 5\times 3 }{ 8\times 3 } =\frac { 15 }{ 24 } \)
Now the rational numbers between \(-\frac { 202 }{ 24 } and\frac { 15 }{ 24 } are-\frac { 19 }{ 24 } ,-\frac { 18 }{ 24 } ,-\frac { 17 }{ 24 } ,....,\frac { 0 }{ 24 } ,\frac { 1 }{ 24 } ,\frac { 22 }{ 24 } ,...,\frac { 14 }{ 24 } \)
we can take any seven of them \(\frac { 1 }{ 24 } ,\frac { 2 }{ 24 } ,\frac { 3 }{ 24 } ,\frac { 4 }{ 24 } ,\frac { 5 }{ 24 } ,\frac { 6 }{ 24 } ,\frac { 7 }{ 24 } \)

87.

Let the point A (-1, -1), (1, 1) and (\(-\sqrt{13},\sqrt{13}\))
Distance = \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
AB =\(\sqrt{(1+1)^2+(1+1)^2}\)
\(\sqrt{2^2+2^2}=\sqrt{4+4}=\sqrt{8}\)
BC =\(\sqrt{(-\sqrt{3}+1)^2+(\sqrt{3}-1)^2}\)
\(\sqrt{(\sqrt{3}+1)^2+(\sqrt{3}-1)^2}\)
\(\sqrt{3+1+2\sqrt{3}+3+1-2\sqrt{3}}\)
\(\sqrt{3+1+3+1}=\sqrt{8}\)
AC =\(\sqrt{(-\sqrt{3}+1)^2+(\sqrt{3}+1)^2}\)
\(=\sqrt{3+1-2\sqrt{3}+3+1+2\sqrt{3}}\)
\(=\sqrt{4+4}=\sqrt{8}\)

AB= BC=AC= \(\sqrt{8}\)
\(\therefore\)ABC is an equilateral triangle.

Generic 8

2 x 8 = 16

88.

In \(\triangle\)ABC, AB = BC = 8 cm, \(\angle\)B = 70°.

Construction :
Step 1: Draw \(\triangle\)ABC with BC = 8 cm, \(\angle\)B = 70°. AB = 8.
Step 2: Construct the angle bisectors of any two angles (B and C) and let them meet at I. Then I is the incentre of \(\triangle\)ADC. Draw perpendicular from I to anyone of the side (BC) to meet BC at D.
Step 3: With I as centre and ID as radius draw a circle. This circle touches all the sides of the triangle internally.

89.

Construction :
Step 1: Draw \(\triangle\)ABC with AB = BC = CA = 8 cm
Step 2: Construct angle bisectors of any two angles (A and B) and let them meet at I. I is the incentre of \(\triangle\)ABC.
Step 3: Draw perpendicular from I to any one of the side (AB) to meet AB at D.
Step 4: With I as centre, ID as radius draw the circle. This circle touches all the sides of triangle internally.