9th Standard Mathematics All Chapter Important Creative Questions-I- 2019-2020

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Questions:

Generic 2

25 x 2 = 50

One card is drawn from a pack of 52 cards, each of the 52 cards being equally likely to be drawn. Find the probability that the card drawn is
(i) an ace,
(ii) either red card or king.

Find the value of \(\cfrac { cos{ 63 }^{ 0 }20' }{ sin{ 26 }^{ 0 }40' } \)

Find the surface area of a cube whose edge is
(i) 27 cm
(ii) 3 cm
(iii) 6 cm
(iv) 2.1 cm

Find the coordinates of the point which divides the line segment joining the points (3, 1) and (5, 13) internally in the ratio 3 : 5.

A company manufactures 10000 Laptops in 6 months. In that 25 of them are found to be defective. When you choose one Laptop from the manufactured, what is the probability that selected Laptop is a good one.

The parallel sides of a trapezium are 15 m and 10 m long and its non-parallel sides are 8 m and 7 m long. Find the area of the trapezium.

Akshaya has 2 rupee coins and 5 rupee coins in her purse. If in all she has 80 coins totalling Rs. 220, how many coins of each kind does she have.

Evaluate:
(i) sin 30⁰ +cos 30⁰
(ii) tan 60⁰cot 60⁰
(iii) \(\frac { tan45° }{ tan30°+tan60° } \)
(iv) sin²45⁰ +cos²45⁰

Using section formula, show that the points A(7, −5), B(9, −3) and C(13, 1) are collinear.

10.

Represent the following as decimal form
(i) \(\frac { -4 }{ 11 } \)
(ii) \(\frac { 11 }{ 75 } \)

11.

Factorise the following : 64m³+ 27n³

12.

Express the surds in the simple form \(\sqrt { 27 } \)

13.

Evaluate : \(\left( \cfrac { 1 }{ 9 } \right) ^{ -3 }\)

14.

In a circle, AB and CD are two parallel chords with centre 0 and radius 5 em such that AB = 8 cm and CD = 6 cm determine the distance between the chords?

15.

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

16.

The mean weight of 4 members of a family is 60 kg. Three of them have the weight 56kg, 68kg and 72 kg respectively. Find the weight of the fourth member.

17.

In a rice mill, seven labours are receiving the daily wages of Rs. 500, Rs. 600, Rs. 600, Rs. 800, Rs. 800, Rs. 800 and Rs. 1000, find the modal wage

18.

Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord

19.

Use a fractional index to write: \(\sqrt{5}\)

20.

If A =\(\left\{ -\frac { 1 }{ 2 } ,0,\frac { 1 }{ 4 } ,\frac { 3 }{ 4 } ,2 \right\} \), B =\(\left\{ 0,\frac { 1 }{ 4 } ,\frac { 3 }{ 4 } ,2,\frac { 5 }{ 2 } \right\} \) and C =\(\left\{ -\frac { 1 }{ 2 } .\frac { 1 }{ 4 } ,1,2,\frac { 5 }{ 2 } \right\} \), then verify \(A\cap (B\cap C)=(A\cap B)\cap C\) .

21.

Let U= {x : -3 : < x < 4} A = {-1,2,3} B = {0,1,2,3} C = {-3;-2,-1,0,1,2}. Find (i) A' UB' (ii) (A ∩ B)' (iii) (A ⋂ C)'

22.

IfA = {1,2,3} B = {3,5,6} C = {3,4,7}. Find
(i) A-(BUC)
(ii) BΔC
(iii) AΔB

23.

Verify whether the following are zeros of the polynomial indicated against them, or not.
p(x) = (x+3) (x-4), x = -3, x = 4

24.

Identify which ones are trapeziums and which are not.

25.

From the given Venn diagram, write the elements of

(i) A
(ii) B
(iii) A – B
(iv) B – A
(v) A′
(vi) B′
(vii) U

Multiple Choice Question

40 x 1 = 40

26.

The number of bricks each measuring 50 cm × 30 cm × 20 cm that will be required to build a wall whose dimensions are 5 m × 3 m × 2 m is _______.

(a)

1000

(b)

2000

(c)

3000

(d)

5000

27.

A collection of one or more outcomes of an experiment is called _______.

(a)

Event

(b)

Outcome

(c)

Sample point

(d)

None of the above

28.

If the sides of a triangle are 3 cm, 4 cm and 5 cm, then the area is _______.

(a)

3 cm²

(b)

6 cm²

(c)

9 cm²

(d)

12 cm²

29.

The probability of all possible outcomes of a random experiment is always equal to _______.

(a)

One

(b)

Zero

(c)

Infinity

(d)

Less than one

30.

The value of tan 1° tan 2° tan 3°...tan 89° is ________.

(a)

(b)

(c)

(d)

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

31.

The value of \(\frac { tan15° }{ cot75° } \) is

(a)

cos 90⁰

(b)

sin 30⁰

(c)

tan 45⁰

(d)

cos 30⁰

32.

The mid-point of the line joining (−a, 2b) and (−3a,−4b) is ______.

(a)

(2a, 3b)

(b)

(−2a, −b)

(c)

(2a, b)

(d)

(−2a, −3b)

33.

For any three sets, P,Q,R\(\left( P\cap Q \right) ^{ ' }\)

(a)

\({ P }^{ ' }\cup { Q }^{ ' }\)

(b)

\(P\cup Q\)

(c)

\({ P }^{ ' }\)

(d)

\({ Q }^{ ' }\)

34.

If U = {x : x\(\in \)W and x < 20}, A = {2,4,6,8}, B = {6,8,12,14} then\(\left[ n\left( A\cup B^{ ' } \right) \right] \)

(a)

15%

(b)

20%

(c)

10%

(d)

35.

(a-b-c)² is equal to________

(a)

(a+b+c)²

(b)

(a+b+c)²

(c)

(-a + b + c)²

(d)

(a+b+c)²

36.

If x - 2 is a factor of q(x), then the remainder is___________

(a)

q(-2)

(b)

x - 2

(c)

(d)

-2

37.

Which of the following has a factor?

(a)

x²+ 2x

(b)

(x - 1)²

(c)

(x + 1)²

(d)

(x²- 2²)

38.

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

(a)

(b)

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

(c)

\(\sqrt { 3 } \)

(d)

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

39.

\(\sqrt [ 5 ]{ 21 } \times 6\sqrt { 10 } \)

(a)

\(30\sqrt { 210 } \)

(b)

(c)

\(\sqrt { 210 } \)

(d)

\(210\sqrt { 30 } \)

40.

If sum of two opposite angles of a cyclic quadrilateral is_________

(a)

45°

(b)

90°

(c)

180°

(d)

360°

41.

Find the mean of the prime factors of 165 _____________

(a)

(b)

(c)

(d)

42.

The mean of the first 10 prime number is _______________

(a)

12.6

(b)

12.7

(c)

12.8

(d)

12.9

43.

If the mean of five observations x, x+2, x+4, x+6, x+8, is 11, then the mean of first three observations is __________

(a)

(b)

(c)

(d)

44.

For which set of number do the mean,median and mode all have the same valuas?

(a)

2,2,2,4

(b)

1,3,3,3,5

(c)

1,1,2,5,6

(d)

1,1,2,1,5

45.

For which set of numbers do the mean, median and mode all have the same values?

(a)

2, 2, 2, 4

(b)

1, 3, 3, 3, 5

(c)

1, 1, 2, 5, 6

(d)

1, 1, 2, 1, 5

46.

A particular observation which occurs maximum number of times in a given data is called its _______.

(a)

Frequency

(b)

range

(c)

mode

(d)

Median

47.

In the figure, PQRS and PTVS are two cyclic quadrilaterals, If ㄥQRS = 100°, then ㄥTVS = ______.

(a)

80°

(b)

100°

(c)

70°

(d)

90°

48.

The GCD of x⁴-y⁴ and x²-y² is _______.

(a)

x⁴-y⁴

(b)

x²-y²

(c)

(x+y)²

(d)

(x+y)⁴

49.

When written with a rational denominator, the expression \(\frac { 2\sqrt { 3 } }{ 3\sqrt { 2 } } \) can be simplified as ________.

(a)

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

(b)

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

(c)

\(\frac { \sqrt { 6 } }{ 3 } \)

(d)

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

50.

\(\sqrt{27}\)+\(\sqrt{12}\) = ________.

(a)

\(\sqrt{39}\)

(b)

5\(\sqrt{6}\)

(c)

5\(\sqrt{3}\)

(d)

3\(\sqrt{5}\)

51.

If n(A U B U C) = 100, n(A) = 4x, n(B) = 6x, n(C) = 5x, n(A ∩ B) = 20, n(B ⋂C) = 15, n(A C) = 25 and n(A ⋂ B ⋂ C) = 10 , then the value of x is ________.

(a)

(b)

(c)

(d)

52.

The coefficient of \({ x }^{ 2 }\ and\ x\ in\ 2{ x }^{ 3 }-5{ x }^{ 2 }+6x-3\) are respectively ____________

(a)

2, -5

(b)

2, 6

(c)

-5, 6

(d)

-5, -3

53.

Which of the following is a formula to find the sum of interior angles of a quadrilateral of n-sides?

(a)

\(\frac { n }{ 2 } \) \(\times\) 180

(b)

\(\left( \frac { n+1 }{ 2 } \right) \)180⁰

(c)

\(\left( \frac { n-1 }{ 2 } \right) \)180^o

(d)

(n-2)180^o

54.

In a parallelogram \(\angle{A}:\angle{B}=1:2\) Then ㄥA ............

(a)

30°

(b)

60°

(c)

45°

(d)

90°

55.

What is the name of a regular polygon of six sides?

(a)

Square

(b)

Equilateral triangle

(c)

Regular hexagon

(d)

Regular octagon

56.

The set {x: x ∈ A, x ∈ B, x ∉ A ∩ B} is___________.

(a)

A ∩ B

(b)

A U B

(c)

A - B

(d)

A Δ B

57.

The value of \(\bar { 0.03 } \) +\(\bar { 0.03 } \) is ___________.

(a)

\(\bar { 0.09 } \)

(b)

\(\bar { 0.09 } \)

(c)

\(\bar { 0.09 } \)

(d)

58.

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)

59.

The point which is on y-axis with ordinate - 5 is _____________

(a)

(0, - 5)

(b)

(-5,0)

(c)

(5,0)

(d)

(0,5)

60.

The point (0, -3) lies on _______________

(a)

+ ve x-axis

(b)

+ ve y-axis

(c)

- ve x-axis

(d)

- ve y-axis

61.

The centre of a circle is (0, 0). One end point of a diameter is (5, -1), then ______________

(a)

\(\sqrt{24}\)

(b)

\(\sqrt{37}\)

(c)

\(\sqrt{26}\)

(d)

\(\sqrt{17}\)

62.

If P( –1, 1), Q( 3, -4), R( 1, -1), S(-2, -3) and T( -4, 4) are plotted on a graph paper, then the points in the fourth quadrant are ______.

(a)

P and T

(b)

Q and R

(c)

only S

(d)

P and Q

63.

x³ – x² is a …………..

(a)

monomial

(b)

binomial

(c)

trinomial

(d)

constant polynomial

64.

Which of the following is correct?

(a)

{7} ∈ {1,2,3,4,5,6,7,8,9,10}

(b)

7 ∈ {1,2,3,4,5,6,7,8,9,10}

(c)

7 ∉ {1,2,3,4,5,6,7,8,9,10}

(d)

{7} \(\nsubseteq \) {1,2,3,4,5,6,7,8,9,10}

65.

If the diagonal of a rhombus are equal, then the rhombus is a ________.

(a)

Parallelogram but not a rectangle

(b)

Rectangle but not a square

(c)

Square

(d)

Parallelogram but not a square

Generic 3

14 x 3 = 42

66.

In an office, where 42 staff members work, 7 staff members use cars, 20 staff members use two-wheelers and the remaining 15 staff members use cycles. Find the relative frequencies.

67.

Find the Total Surface Area and Lateral Surface Area of the cube, whose side is 5 cm.

68.

Evaluate:
(i) \(\frac { sin\ 49° }{ cos\ 41° } \)
(ii) \(\frac { sec\ 63° }{ cosec\ 27° } \)

69.

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.

70.

Sho that \(\sqrt [ 3 ]{ 2 } >\sqrt [ 5 ]{ 3 } \)

71.

The following are scores obtained by 11 players in a cricket match 7, 21, 45, 12, 56, 35, 25, 0, 58, 66, 29. Find the median score.

72.

The arithmetic mean of 6 values is 45 and if each value is increased by 4, then find the arithmetic mean of new set of values

73.

(i) Prove that (x - 1) is a factor of x³- 7x²+ 13x - 7
(ii) Prove that (x + 1) is a factor of x³+ 7x²+ 13x + 7

74.

What must be subtracted from \({ y }^{ 4 }+{ 2y }^{ 3 }-3y+8\quad to\quad get\quad { y }^{ 4 }-{ 2y }^{ 3 }+6?\)

75.

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

76.

In a class there are 40 students. 26 have opted for Mathematics and 24 have opted for Science. How many student have opted for Mathematics and Science.

77.

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

78.

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

79.

The abscissa of a point A is equal to its ordinate, and its distance from the point B(1, 3) is 10 units, What are the coordinates of A?

Generic 5

9 x 5 = 45

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.

Factorise 2x³- x² - 12x - 9 into linear factors

82.

Find the angle of the given cyclic quadrilateral ABCD in the figure.

83.

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

84.

In the class, weight of students is measured for the class records. Caculate mean weight of the students using direct method.

Weight in kg	15-25	25-35	35-45	45-55	55-65	56-75
No.of students	4	11	19	14	0	2

85.

Represent \(-\frac { 2 }{ 11 } ,-\frac { 5 }{ 11 } and-\frac { 9 }{ 11 } \)on the number line.

86.

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

87.

Fill in the blanks with appropriate cardinal numbers

S.No	n(A)	n(B)	n(AUB)	n(A∩B)
1	30	45	65
2	20		55	10
3	50	65		25
4	30	43	70

88.

Show that the given points (1, 1), (5, 4), (-2, 5) are the vertices of an isosceles right angled triangle.

Generic 8

2 x 8 = 16

89.

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

90.

Draw and locate the centroid of the triangle ABC where right angle at A, AB = 4cm and AC = 3cm

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Answers Key:

Generic 2

25 x 2 = 50

\(\cfrac { cos{ 63 }^{ 0 }20' }{ sin{ 26 }^{ 0 }40' } =\cfrac { cos63^{ 0 }20' }{ cos{ 63 }^{ 0 }20' } =1\)

(i) TSA = 6a² = 6 \(\times\) 27² = 6 \(\times\) 729 = 4374 cm²
(ii) TSA = 6a² = 6 \(\times\)3² = 54 cm²
(iii) TSA = 6a² = 6 \(\times\) 6²= 6 \(\times\) 36 = 216 cm²
(iv) TSA = 6a² = 6 \(\times\) 2.1² = 6 \(\times\) 4.41 = 26.46 cm²

\(\left( \frac { 5(3)+3(5) }{ 5+3 } ,\frac { 5(1)+3(13) }{ 5+3 } \right) =\left( \frac { 15+15 }{ 8 } ,\frac { 5+39 }{ 8 } \right) =\left( \frac { 30 }{ 8 } ,\frac { 44 }{ 8 } \right) =\left( \frac { 15 }{ 4 } ,\frac { 11 }{ 2 } \right) \)

Total n(S) = 10,000
Defective n(A) = 25
Number of defective laptop is 25
Number of good laptop is = 10000 - 25 = 9975
Let A be the event of choosing a good laptop. n(A) = 9975
\(\mathrm{P}(\mathrm{A})=\frac{\mathrm{n}(\mathrm{A})}{\mathrm{n}(\mathrm{S})}=\frac{9975}{10000}\)
P(A) = 0.9975
Probability of getting a good laptop is 0.9975

In the figure AP 丄 CD and BQ 丄 CD
⇒ PQ = AB = 10, AP = BQ = h
Assume DP = x ⇒ QC = 15 - (10 + x)
= 15 - 10 - x = 5 - x
In Δ ADP, AP² = 7²- x²= 49 - x²
In Δ BQC, BQ² = 8²- (5 - x)² = 64 - (25 - 10x + x²) .
= 64 - 25 + 10x - x²
=39 - x²+ 10x
Equating AP² = BQ²
49 - x² = 39 - x² + 10x

⇒ AP=\(\sqrt { { 7 }^{ 2 }-{ x }^{ 2 } } =\sqrt { 49-1^{ 2 } } =\sqrt { 48 } \)
∴ Area of the trapezium = \(\frac{1}{2}\) h(a+b) sq.units
=\(\frac { 1 }{ 2 } \times \sqrt { 48 } \times (10+15)=\frac { 1 }{ 2 } \times \sqrt { 3\times 16 } \times 25\)

= 2 \(\times\) 25 \(\times\) 1.732 = 86.6 m²

Let the number of 2 rupee coins be x
Let the number of 5 rupee coins be y
x+y = 80
2x+ 5y = 220
x +y- 80 = 0
2x + 5y - 220 = 0
For cross multiplication method, we write the co-efficients as

\(\cfrac { x }{ -220 } =\cfrac { y }{ -160+220 } =\cfrac { 1 }{ 5-2 } \)
\(\cfrac { x }{ 180 } =\cfrac { y }{ 60 } =\cfrac { 1 }{ 3 } \)
\(\therefore \ \cfrac { x }{ 180 } =\cfrac { 1 }{ 3 } \ \ \cfrac { y }{ 60 } =\cfrac { 1 }{ 3 } \)

x = 60 y = 20
\(\therefore\) No. of 2 rupee coins = 60
No. of 5 rupee coins = 20

(i)sin 30⁰ + cos 30⁰ = \(\frac { 1 }{ 2 } +\frac { \sqrt { 3 } }{ 2 } =\frac { 1+\sqrt { 3 } }{ 2 } \)
(ii) tan 60⁰cot 60⁰ = \(\sqrt { 3 } \times \frac { 1 }{ \sqrt { 3 } } =1\)
(iii) \(\frac { tan45° }{ tan30°+tan60° } \) = \(\frac { 1 }{ \frac { 1 }{ \sqrt { 3 } } +\frac { \sqrt { 3 } }{ 1 } } =\frac { 1 }{ \frac { 1+\left( \sqrt { 3 } \right) ^{ 2 } }{ \sqrt { 3 } } } =\frac { 1 }{ \frac { 1+3 }{ \sqrt { 3 } } } = \frac { \sqrt { 3 } }{ 4 } \)
(iv) sin²45⁰ + cos²45⁰ = \(\left( \frac { 1 }{ \sqrt { 2 } } \right) ^{ 2 }+\left( \frac { 1 }{ \sqrt { 2 } } \right) ^{ 2 }=\frac { 1^{ 2 } }{ \left( \sqrt { 2 } \right) ^{ 2 } } +\frac { 1^{ 2 } }{ \left( \sqrt { 2 } \right) ^{ 2 } } =\frac { 1 }{ 2 } +\frac { 1 }{ 2 } =1\)

x₁ y₁ x₂ y₂
A(7, -5) B(9, -3)
\(\bar { AB } =\sqrt { { ({ x }_{ 2 }-{ x }_{ 1 }) }^{ 2 }+{ ({ y }_{ 2 }-{ y }_{ 1 }) }^{ 2 } } \)
\(=\sqrt { { (9-7) }^{ 2 }+{ (-3-(-5)) }^{ 2 } } =\sqrt { { 2 }^{ 2 }+{ (-3+5) }^{ 2 } } =\sqrt { 4+4 } \)
\(=\sqrt { 8 } =\sqrt { 4\times 2 } =2\sqrt { 2 } \)
\(\bar { BC } =\sqrt { { (13-7 })^{ 2 }+{ (1-(-5)) }^{ 2 } } =\sqrt { { 4 }^{ 2 }+{ 4 }^{ 2 } } =\sqrt { 16+16 } \)
\(=\sqrt { 32 } =\sqrt { 16\times 2 } =4\sqrt { 2 } \)
\(\bar { AC } =\sqrt { { (13-7) }^{ 2 }+{ (1-(-5)) }^{ 2 } } \)
\(=\sqrt { { 6 }^{ 2 }+{ 6 }^{ 2 } } =\sqrt { 36+36 } \)
\(=\sqrt { { 6 }^{ 2 }+{ 6 }^{ 2 } } =\sqrt { 36+36 } \)
\(2\sqrt { 2 } +4\sqrt { 2 } =6\sqrt { 2 } \)
∴ AB + BC = AC, Here B is the common Point
∴ A, B, C are collinear

10.

(i) \(\frac { -4 }{ 11 } \)

Thus we see that \(\frac { -4 }{ 11 } =-0.\overline { 36 } \)
(ii) \(\frac { 11 }{ 75 } \)

Thus we see that \(\frac { 11 }{ 75 } =0.14\overline { 6 } \)
A rational number can be expressed by
(i) either a terminating
(ii) or a non-terminating and recurring (repeating) decimal expansion.
The converse of this statement is also true. That is, if the decimal expansion of a number is terminating or non-terminating and recurring, then the number is a rational number.

11.

64m³+ 27n³= (4m)³+ (3n)³
= (4m + 3n) ((4m)² - (4m) (3n) + (3n)²) [\(\because \) a³+ b³= (a+b) (a²- ab + b²)]
= (4m + 3n)(16m²- 12mn + 9n²)

12.

\(\sqrt { 27 } \) =\(\sqrt { 3\times 3\times 3 } =\sqrt [ 3 ]{ 3 } \)

13.

\(\left( \cfrac { 1 }{ 9 } \right) ^{ -3 }\)=\(\cfrac { 1 }{ \left( \cfrac { 1 }{ 9 } \right) ^{ 3 } } =\cfrac { 1 }{ \left( \cfrac { 1 }{ 729 } \right) } =729\)

14.

The distance between the two chord
FE = OF + OE
OE = \(\sqrt { { 5 }^{ 2 }-3^{ 2 } } =\sqrt { 25-9 } \)
= \(\sqrt { 16 } \) = 4 cm
OF = \(\sqrt { { 5 }^{ 2 }-{ 4 }^{ 2 } } =\sqrt { 25-16 } =\sqrt { 9 } \) = 3 cm
\(\therefore\) DE = 4cm + 3cm = 7cm
Distance between the chord is 7 cm.

15.

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

16.

\(\bar { x } =60kg\)
\(\bar { x } =\cfrac { \Sigma x }{ n } =\cfrac { 56+68+72+x }{ 4 } =60\)
196 + x = 260
x = 240-196
\(\therefore \) The weight of the fourth number = 44 kg

17.

In the given data Rs. 800 occurs thrice. Hence the mode is Rs. 800

18.

OD = DP =\(\cfrac { 4cm }{ 2 } \) = 2 cm
AD = BD =\(\sqrt { { 5 }^{ 2 }-{ 4 }^{ 2 } } \)
= \(\sqrt { 25-19 } \)
= \(\sqrt { 9 } \)
= 3cm
\(\therefore\) The length of the common chrod AB = AD + BD = (3 + 3) cm = 6 cm

19.

\({ 5 }^{ \frac { 1 }{ 2 } }\)

20.

Now, \((B\cap C)=\left\{ \frac { 1 }{ 4 } ,2,\frac { 5 }{ 2 } \right\} \)
\(A\cap (B\cap C)=\left\{ \frac { 1 }{ 4 } ,2 \right\} \) .......(1)
Then, \(A\cap B=\left\{ 0,\frac { 1 }{ 4 } ,\frac { 3 }{ 4 } ,2 \right\} \)
\((A\cap B)\cap C=\left\{ \frac { 1 }{ 4 } ,2 \right\} \) .......(2)
From (1) and (2), it is verified that
\((A\cap B)\cap C=A\cap (B\cap C)\).

21.

(i) {-3,-2,-1,0,1,4} (ii) {-3,-2,-1,0,1,4} (iii) {-3,-2,0,1,3,4}

22.

(i) {1 ,2}
(ii) {4,5,6,7}
(iii) {1,2,5,6}

23.

p(x) = (x + 3)(x - 4); x = -3, x = 4
p( -3) = (- 3 + 3)(- 3 - 4)
= (0) (-7) = 0
p(4) = (4+3) (4-4)
= (7) (0) = 0
\(\therefore\) -3 and 4 are the zeros of the polynomial.

24.

It is not a trapezium

25.

(i) A = {a, e, i, o, u}
(ii) B = {b, c, e, o}
(iii) A – B = {a, i, u}
(iv) B – A = {b, c}
(v) A′ = {b, c, d, g}
(vi) B′ = {a, d, g, i, u}
(vii) U = {a, b, c, d, e, g, i, o, u}.

Multiple Choice Question

40 x 1 = 40

26.

(a)
1000

27.

(a)
Event

28.

(b)
6 cm²

29.

(a)
One

30.

(b)
1

31.

32.

(b)
(−2a, −b)

33.

(a)
\({ P }^{ ' }\cup { Q }^{ ' }\)

34.

(b)
20%

35.

(d)
(a+b+c)²

36.

37.

(a)
x²+ 2x

38.

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

39.

(a)
\(30\sqrt { 210 } \)

40.

41.

42.

(d)
12.9

43.

(a)
9

44.

(b)
1,3,3,3,5

45.

(b)
1, 3, 3, 3, 5

46.

47.

(a)
80°

48.

(b)
x²-y²

49.

50.

51.

(a)
10

52.

53.

(d)
(n-2)180^o

54.

(b)
60°

55.

56.

(d)
A Δ B

57.

58.

(a)
2

59.

(a)
(0, - 5)

60.

(d)
- ve y-axis

61.

62.

(b)
Q and R

63.

(b)
binomial

64.

(b)
7 ∈ {1,2,3,4,5,6,7,8,9,10}

65.

Generic 3

14 x 3 = 42

66.

Total number of staff members = 42
The relative frequencies:
Car users \(=\frac { 7 }{ 42 } =\frac { 1 }{ 6 } \)
Two-wheeler users \(=\frac { 20 }{ 42 } =\frac { 10 }{ 21 } \)
Cycle users \(=\frac { 15 }{ 42 } =\frac { 5 }{ 14 } \)

67.

The side of the cube (a) = 5 cm
Total Surface Area = 6a² = 6(5²) = 150 sq. cm
Lateral Surface Area = 4a² = 4(5²) = 100 sq. cm

68.

(i) \(\frac { sin\ 49° }{ cos\ 41° } \)
sin 49⁰ = sin(90⁰ - 41⁰) = cos 41⁰, since 49⁰ + 41⁰ = 90⁰ (complementary),
Hence on substituting sin 49^o = cos41^o we get, \( \frac { cos\ 41° }{ cos\ 41° } \)= 1
(ii) \(\frac { sec\ 63° }{ cosec\ 27° } \)
sec63^o = sec (90^o- 27^o) = cosec27^o, here, 63^o and 27^o are complementary angles
we have \(\frac { sec\ 63° }{ cosec\ 27° } =\frac { cosec\ 27° }{ cosec\ 27° } =1\)

69.

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 \(\left( A\cap B \right) '=A'\cup B'\)

70.

\(\sqrt [ 3 ]{ 2 } =\sqrt [ 15 ]{ { 2 }^{ 5 } } =\sqrt [ 15 ]{ 32 } \)
\(\sqrt [ 5 ]{ 3 } =\sqrt [ 15 ]{ 3^{ 3 } } =\sqrt [ 15 ]{ 27 } \)
\(\sqrt [ 15 ]{ 32 } >\sqrt [ 5 ]{ 3 } \)
Therefore, \(\sqrt [ 3 ]{ 2 } >\sqrt [ 5 ]{ 3 } \)

71.

Let us arrange the values in ascending order.
0, 7, 12, 21, 25, 29, 35, 45, 56, 58, 66
The number of values = 11 which is odd
Median \(=\left(11+1\over2\right)^{th}\)value
\(=\left(12\over2\right)^{th}\) value = 6^th value = 29

72.

Let x₁, x₂, x₃, x₄, x₅, x₆ be the given set of values then \({\sum^6_{i=1}\over6}=45\)
If each value is increased by 4, then the mean of new set of values is
New A.M \(\bar X={\sum^6_{i=1}(x_i+4)\over6} \)
\(={(x_1+4)+(x_2+4)+(x_3+4)+(x_4+4)+(x_5+4)+(x_6+4)\over6}\)
\(={\sum^6_{i=1}(x_i+24)\over6}={\sum_{i=1}^6x_i\over6}+4\)
\(\bar X=45+4=49\)

73.

(i) Let p(x) = x³- 7x + 13x - 7
Sum of coefficients = 1 − 7 + 13 − 7 = 0
Thus (x -1) is a factor of p (x)
(ii) Let q(x) = x³+ 7x²+ 13x + 7
Sum of coefficients of even powers of x with constant = 7 + 7 = 14
Sum of coefficients of odd powers of x = 1 + 13 = 14
Hence, (x + 1) is a factor of q (x)

74.

Let A be the required number to be subtracted
(y⁴ + 2y³ - 3y² + 8) - A = y⁴ - 2y³ + 6
y⁴ + 2y³ - 3y² + 8 - (y⁴ - 2y³ + 6) = A
y⁴ + 2y³ - 3y² + 8 - y⁴ + 2y³ - 6 = A
4y³-3y²+2 = A
Hence 4y³-3y²+2 must be subtracted.

75.

(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.

76.

Let M be the set of students opting for Mathematics.
Let S be the set of students opting for Science.
n (M U S) = 40, n (M) = 26, n (S) = 24
n (M U S) = n (M) + n (S) - n (M n S)
40 = 26 + 24 - n (M n S)
n (M n S) = 26 + 24 - 40
= 50 - 40 = 10
∴ Number of students opted for Mathematics and Science = 10

77.

(3, 5)

78.

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
\( x=\frac{17197.936}{999} \\ x=\frac{17197936}{999000} \\ x=\frac{5681}{330} \)

79.

Let the point A be (a, a), B is (1, 3)
Distance AB = 10 (Given)
By distance formula \(\sqrt { (a-1)^{ 2 }+(a-3)^{ 2 } } =10\)
Simplifying
2a²- 8a + 10 = 100
a²- 4a - 45 = 0
(a - 9)(a + 5) = 0
⇒ a = -5 ; A = (-5,-5)
a = 9 ; A = (9,9)

Generic 5

9 x 5 = 45

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.

Let p (x) 2x³ - x² - 12x - 9
Sum of the co-efficients = 2 - 1- 12- 9 = -20 \(\neq \) 0
Hence x-1 is not a factor
Sum of co-efficients of even powers with constant = -1 - 9 = -10
Sum of co-efficients of odd powers = 2 - 12= -10
Hence x + 1 is a factor of x.
Now we use synthetic division to find the other factors.

Then p (x) = (x + 1)(2x² - 3x - 9)
Now 2x² - 3x - 9 = 2x² - 6x + 3x - 9 = 2x (x - 3) + 3 (x - 3)
= (x - 3)(2x + 3)
Hence 2x³ - x²- 12x - 9 (x + 1) (x - 3) (2x + 3)

82.

\(\angle\)In the cyclic quadrilateral \(\angle\) A+\(\angle\)C = 180^o
y + 4^o + 3y^o+ 8^o = 180^o
4y^o + 12^o = 80^o
4y^o = 180^o -12^o = 168
y =\(\cfrac { 168 }{ 4 } \) = 42
\(\angle\)B + \(\angle\)D = 180^o
8x + 12 = 180^o
8x = 180^o- 12^o = 168
x = \(\cfrac { 168 }{ 8 } \) = 21^o
\(\therefore\) \(\angle\)A = y + 4^o = 42^o + 4^o = 46^o
\(\angle\)C = 3y + 8 = 3 \(\times\) 42 + 8 = 126 + 8 = 134^o
\(\angle\)B = 3x + 6 = 3 \(\times\) 21 + 6 = 63 + 6 = 69^o
\(\angle\)D = 5x + 6 = 5 \(\times\) 21 + 6
= 105 + 6 = 111^o

83.

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

84.

Weight in kgs(x)	Number of students(f)	Mid value of x	fx
15-25	4	20	80
25-35	11	30	33
35-45	19	40	760
45-55	14	50	700
55-65	0	60	0
65-75	2	70	140
	\(\Sigma{f}=50\)		2010

85.

To represent \(-\frac { 2 }{ 11 } ,-\frac { 5 }{ 11 } and-\frac { 9 }{ 11 } \)on the number line we make 11 markings each being equal distance \(\frac { 1 }{ 11 } \) on the left of 0.
The point A represents \(\left( -\frac { 2 }{ 11 } \right) \) , the point B represents\(\left( -\frac { 2 }{ 11 } \right) \) and the point C represents \(\left( -\frac { 9 }{ 11 } \right) \)

86.

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 \(\frac { 7 }{ 20 } ,\frac { 17 }{ 40 } and\frac { 37 }{ 80 } \)

87.

S.No	n(A)	n(B)	n(AUB)	n(A∩B)	n(A-B)	n(B-A)
1	30	45	65	10	20	35
2	20	45	55	10	10	35
3	50	65	90	25	25	40
4	30	43	70	3	27	40

88.

Let A(1, 1),B(5, 4) and C(-2, 5)
Distance =\(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
AB =\(\sqrt{(5-1)^2+(4-1)^2}\)
\(=\sqrt{4^2+3^2}=\sqrt{16+9}=\sqrt{25}=5\)
BC =\(\sqrt{(-2-5)^2+(5-4)^2}\)
\(=\sqrt{(-7)^2+1^2}=\sqrt{49+1}=\sqrt{50}\)
AC =\(\sqrt{(-2-1)^2+(5-1)^2}\)
\(=\sqrt{(-3)^2+4^2}=\sqrt{9+16}=\sqrt{25}=5\)

AB = 5, AC = 5,
\(\therefore\) ABC is an isosceles triangle .................(1)
BC² = AB² + AC²
50 = 25 + 25\(\Rightarrow\) 50 = 50
\(\therefore\) \(\angle\)A = 90° ...................(2)
From (1) and (2) we get ABC is an isosceles right angle triangle.

Generic 8

2 x 8 = 16

89.

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.

90.

In \(\triangle \)ABC,
AB = 4 cm, AC = 3 cm, \(\angle\)A = 90°

Construction :
Step 1: Draw \(\triangle \)ABC with AB = 4 cm, AC = 3 cm, \(\angle\)A = 90°
Step 2: Draw perpendicular bisectors of any two sides (AB and AC) to find the mid points of AB and AC.
Step 3: Draw the medians CD and BE. Let them meet at G.
Step 4: G is the centroid of the given triangle.