9th Standard Mathematics All Chapter Important Creative Questions-II-2020

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Questions:

Generic 2

24 x 2 = 48

What is the probability that a number selected from the numbers 1, 2, 3, ..., 25 is prime number when each of the given numbers is equally likely to be selected?

Find the value of cot 15°. cot 30°. cot 45°. cot 60°. cot 75°

Find the area of a quadrilateral whose sides are PQ = 15 cm, QR = 8 cm, RS = 25 m.

If the centroid of a triangle is at (10, -1) and two vertices are (3, 2) and (5, -11). Find the third vertex of a triangle.

The probability of guessing the correct answer to a certain question is \(\frac { x }{ 3 } \). If the probability of not guessing the correct answer is \(\frac { x }{ 5 } \), then find the value of x.

The dimensions of a cuboidal box are 6 m × 400 cm × 1.5 m. Find the cost of painting its entire outer surface at the rate of Rs. 22 per cm².

Find the value of the following:
\(\left( \frac { cos47° }{ sin43° } \right) +\left( \frac { sin72° }{ cos18° } \right) -2\cos^{ 2 }45°\)

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

We used to write \(\pi\) as \(\frac{22}{7}.\) Can we say \(\pi\) is a rational number?

10.

Find the value of \({ 729 }^{ \frac { -5 }{ 6 } }\)

11.

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?

12.

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

13.

In a week, temperature of a certain place is measured during winter are as follows 26^oC, 24^oC, 28^oC, 31^oC, 30^oC, 26^oC, 24^oC. Find the mean temperature of the week.

14.

Find the mode for the set of values 17, 18, 20, 20, 21, 21, 22, 22.

15.

Expand the following using identities: (2x - 3b)²

16.

Find the value of \(\left( 49 \right) ^{ \frac { 1 }{ 2 } }\)

17.

Twice the area of a right angled triangle 15x² + 19x + 6 sq. units. If its altitude is 5x + 3 units, find its base in terms of x.

18.

Express the following in the form \({p\over q},\) where p and q are integers and q \(\ne\) 0.
\(0.5\overline {7}\)

19.

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)'

20.

In a class there are 50 students who are offered either History, geography or both. 35 were offered History and 10 were offered History and Geography both. How many students were offered Geography?

21.

If U = {a, b, c, d, e, f, g, h}, A = {b, d, f, h} and B = {a, d, e, h}, find the following sets. (A′)′

22.

If U = {a, b, c, d, e, f, g, h}, A = {b, d, f, h} and B = {a, d, e, h}, find the following sets. (A\(\cap \)B)′

23.

Verify whether the following are zeros of the polynomial indicated against them, or not.
p(x) = x³ - 1, x = 1

24.

Find the value of x

Multiple Choice Question

40 x 1 = 40

25.

The total surface area of a cuboid with dimension 10 cm × 6 cm × 5 cm is _______.

(a)

280 cm²

(b)

300 cm²

(c)

360 cm²

(d)

600 cm²

26.

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

27.

The probability of an event cannot be _______.

(a)

Equal to zero

(b)

Greater than zero

(c)

Equal to one

(d)

Less than zero

28.

A number between 0 and 1 that is used to measure uncertainty is called _______.

(a)

Random variable

(b)

Trial

(c)

Simple event

(d)

Probability

29.

The value of \(\frac { sin{ 29 }^{ 0 }31' }{ cos{ 60 }^{ 0 }29' } \) is

(a)

(b)

(c)

(d)

-1

30.

The value of 3 sin 70⁰sec 20⁰ + 2 sin 49⁰sec 51⁰ is ________.

(a)

(b)

(c)

(d)

31.

If \(P(\frac{a}{3},\frac{b}{2})\)is the mid-point of the line segment joining A(−4, 3) and B(−2, 4) then (a, b) is ______.

(a)

(-9, 7)

(b)

\((-3, \frac{7}{2})\)

(c)

(9, -7)

(d)

\((3, -\frac{7}{2})\)

32.

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

(a)

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

(b)

\(P\cup Q\)

(c)

\({ P }^{ ' }\)

(d)

\({ Q }^{ ' }\)

33.

\(\left( A\cup B\cup C \right) \cup \left( A\cup BC \right) ^{ ' }\)=_______

(a)

\(\Phi \)

(b)

(c)

(d)

\(A\cap B\cap C\)

34.

In a town 30% like only coffee, 20% like only tea,10% like only like,15% like only any two of them, 5% only like all the three. What is the percentage of people who like none them.

(a)

15%

(b)

20%

(c)

10%

(d)

35.

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

36.

(a-b) (a²+ab+b²)=_________

(a)

a²+ b³+ c³- 3abc

(b)

a²- b²

(c)

a³+ b³

(d)

a³- b³

37.

Zero of (7+4x) is_______

(a)

\(\cfrac { 4 }{ 7 } \)

(b)

\(\cfrac { -7 }{ 4 } \)

(c)

(d)

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.

The angle subtend by a semicircle at the centre is_________

(a)

60^o

(b)

90°

(c)

120^o

(d)

180^o

40.

The angle subtend by equal chords of a circle at the centre is________

(a)

Complementary

(b)

Supplementary

(c)

equal

(d)

unequal

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.

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

(a)

Frequency

(b)

range

(c)

mode

(d)

median

44.

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)

45.

The mean of the first 10 prime numbers is ___________

(a)

12.6

(b)

12.7

(c)

12.8

(d)

12.9

46.

The mean of the square of first 11 natural numbers is _______.

(a)

(b)

(c)

(d)

47.

If there are 36 students of class 9 and 48 students of class 10, what is the minimum number of rows to arrange them in which each row consists of same class with same number of students ?

(a)

(b)

144

(c)

(d)

48.

The remainder when (x²-2x+7) is divided by (x + 4) is __________

(a)

(b)

(c)

(d)

49.

Which is the best example of a number written in scientific notation?

(a)

0.5 \(\times\) 10⁵

(b)

0.1254

(c)

5.367 \(\times\) 10^-6

(d)

12.5 \(\times\) 10²

50.

The degree of the polynomial \( \sqrt { 2 } { x }^{ 2 }-\frac { 7 }{ 2 } { x }^{ 4 }+x-5x^{ 3 }\) is _______________

(a)

(b)

(c)

(d)

51.

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

(a)

30°

(b)

60°

(c)

45°

(d)

90°

52.

The angle sum of a convex polygon with number of sides 7 is ________

(a)

900°

(b)

1080°

(c)

1444°

(d)

720°

53.

Which one of the following has terminating decimal expansion?

(a)

\(\frac { 7 }{ 9 } \)

(b)

\(\frac { 8 }{ 15 } \)

(c)

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

(d)

\(\frac { 5 }{ 32 } \)

54.

If a number has a non-terminating and non-recurring decimal expansion, then it is______________ .

(a)

a rational number

(b)

a natural number

(c)

an irrational number

(d)

an integer

55.

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)

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

56.

The distance between the points (4, -1) and the origin is___________

(a)

\(\sqrt{24}\)

(b)

\(\sqrt{37}\)

(c)

\(\sqrt{26}\)

(d)

\(\sqrt{17}\)

57.

A point which lies in the III quadrant is__________________

(a)

(5, 4)

(b)

(5, - 4)

(c)

(-5, - 4)

(d)

(-5,4)

58.

The point whose abscissa is 5 and lies on the x-axis is__________

(a)

(-5, 0)

(b)

(5,5)

(c)

(0,5)

(d)

(5,0)

59.

Which one of the following has a terminating decimal expansion?

(a)

\(\frac { 5 }{ 64 } \)

(b)

\(\frac { 8 }{ 9 } \)

(c)

\(\frac { 14 }{ 15 } \)

(d)

\(\frac { 1 }{ 12 } \)

60.

Point (–3, 5) lie in the ________ quadrant

(a)

(b)

(c)

III

(d)

61.

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)

62.

If U = {x | x ∈ N, x < 10} and A = {x | x ∈ N, 2 ≤ x < 6} then (A′)′ is –––––––––.

(a)

{1, 6, 7, 8, 9}

(b)

{1, 2, 3, 4}

(c)

{2, 3, 4, 5}

(d)

{ }

63.

If bisectors of ∠A and ∠B of a quadrilateral ABCD meet at O, then ∠AOB is ________.

(a)

∠C + ∠D

(b)

\(\frac { 1 }{ 2 } (\angle C+\angle D)\)

(c)

\(\frac { 1 }{ 2 } \angle C+\frac { 1 }{ 3 } \angle D\)

(d)

\(\frac { 1 }{ 3 } \angle C+\frac { 1 }{ 2 } \angle D\).

64.

ABCD is a square, diagonals AC and BD meet at O. The number of pairs of congruent triangles with vertex O are ________.

(a)

(b)

(c)

(d)

Generic 3

14 x 3 = 42

65.

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.

66.

A cube has the Total Surface Area of 486 cm². Find its lateral surface area.

67.

Find the value of k, for the following system of equation has infinitely many solutions. 2x − 3y = 7;(k + 2)x − (2k +1)y = 3(2k −1)

68.

If tan A = \(\frac { 2 }{ 3 } \) , then find all the other trigonometric ratios.

69.

If A = {2,5,6,7} and B = {3,5,7,8}, then verify the commutative property of union sets

70.

Show that (x-3) is a factor of x³+ 9x²- x - 105

71.

Multiply \(\sqrt [ 4 ]{ 400 } \) and \(\sqrt [ 4 ]{ 567 } \)

72.

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

73.

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

74.

In a college, 240 students play cricket, 180 students play football, 164 students play hockey, 42 play both cricket and football, 38 play both football and hockey, 40 play both cricket and hockey and 16 play all the three games. If each student participate in atleast one game, then find
(i) the number of students in the college
(ii) the number of students who play only one game.

75.

In the figure find x⁰ and y⁰.

76.

(i) Where do the following points lie?
P (-2, 0), Q (2, 0), and R(3, 0)
(ii) Find the distance between the following points using coordinates given in question
(a) P and R (b) Q and R.

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. \(2.\overline { 327 } \)

Generic 5

8 x 5 = 40

79.

Find the area of a quadrilateral ABCD whose sides are AB = 8cm, BC = 15 cm, CD = 12 cm, AD = 25 cm and = 90°.

80.

If P = {x : x\(\in \) N and 1\(\in \) N and n<6} and R = {4,6,8,9,1,12} then verify \(P-\left( Q\cap R \right) =\left( P-Q \right) \cup \left( P-R \right) \)

81.

Find the quotient and remainder when 4x³ + 6x² + 7x + 2 is divided by x - 2

82.

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

83.

The following are the scored by the students in the Summative Assessment exam

Class	0-10	10-20	20-30	30-40	40-50	50-60
No.of students	2	7	15	10	11	5

84.

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

85.

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

86.

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

Generic 8

3 x 8 = 24

87.

Construct the centroid of \(\triangle\)PQR such that PQ = 9 cm, PQ = 7cm, RP = 8 cm.

88.

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

89.

Draw a right triangle whose hypotenuse is 10 cm and one of the legs is 8 cm. Locate its incentre and also draw the incircle.

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Answers Key:

Generic 2

24 x 2 = 48

A = {2, 3, 5, 7, 11, 13, 17, 19, 23}
P(A) = \(\frac{9}{25}\).

cot (90° - 75) cot (90° - 60°) cot 45° cot 60° cot 75°
= tan 75° tan 6.0° (1) cot 60° cot 75° = 1

PR =\(\sqrt { 225+64 } =\sqrt { 289 } \) = 17 cm
s = \(\frac { 15+17+18 }{ 2 } \) = 20 cm
Area of Δ PQR = \(\\ \sqrt { 20\times 5\times 12\times 3 } =\sqrt { 5\times 4\times 5\times 4\times 3\times 3 } \)=60
ΔPRS
Area = \(\sqrt { 27(27-17)(27-25)(27-12) } =\sqrt { 27\times 10\times 2\times 15 } \)
=\(\sqrt { 3\times 3\times 3\times 5\times 2\times 2\times 5\times 3 } \)=9 x 5 x 2 =90
Total area = 90 + 60 =150 cm².

\((10,-1)=\left( \frac { 3+5+x }{ 3 } ,\frac { 2-11+y }{ 3 } \right) \)
\(10=\frac { 3+5+x }{ 3 } ,\)
30 = 3 + 5 + x
22 = x
\(-1=\frac { 2-11+y }{ 3 } \)
-3 = 2 - 11 + y
-3 + 9 = y
y = 6
(22, 6)

\(\frac { x }{ 3 } +\frac { x }{ 5 } \) = 1
\(\frac { 5x+3x }{ 15 } \) = 1
\(\frac { 8x }{ 15 } \) = 1
8x = 15
x = \(\frac { 15 }{ 8 } \)

l \(\times\) b \(\times\) h = 6 m \(\times\) 400 cm \(\times\) 1.5 m
l = 6m, b = 4 m, h = 1.5 m
Total surface area of the cuboid = Outer surface area
= 2 (lb + bh + hI)
= 2 ((6 \(\times\) 4) + (4 \(\times\) 1.5) + (1.5 \(\times\) 6))
= 2 (24 + 6 + 9) = 2 (39) m²
Cost of painting 1 m² = Rs. 22
Cost of painting 78 m²= 78 \(\times\) 22 = Rs. 1716

\(\left( \frac { cos47° }{ sin43° } \right) +\left( \frac { sin72° }{ cos18° } \right) -2cos^{ 2 }45°\)
= \(\left( \cfrac { cos\left( { 90 }^{ 0 }-{ 43 }^{ 0 } \right) }{ { sin43 }^{ 0 } } \right) ^{ 2 }+\left( \cfrac { sin\left( { 90 }^{ 0 }-{ 18 }^{ 0 } \right) }{ cos18 } \right) ^{ 2 }-2{ cos }^{ 0 }{ 45 }^{ 0 }\)
= \(\left( \cfrac { sin43 }{ sin43 } \right) ^{ 2 }+\left( \cfrac { cos18 }{ cos18 } \right) ^{ 2 }-2\left( \cfrac { 1 }{ \sqrt { 2 } } \right) ^{ 2 }\)

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

\(\frac{22}{7}\) = 3.142857142 ...... and \(\pi\) = 3.141592653289 ........
It is an irrational number.
We usually take \(\pi\) as \(\frac{22}{7}\) (a rational number only Butupto 2 decimal places).
But it is not exactly equal to \(\frac{22}{7},\) it is approximate value.

10.

\({ 729 }^{ \frac { -5 }{ 6 } }\) =\(\cfrac { 1 }{ { 729 }^{ \cfrac { 5 }{ 6 } } } =\cfrac { 1 }{ \left( \sqrt [ 6 ]{ 729 } \right) ^{ 5 } } =\cfrac { 1 }{ \left( \sqrt [ 6 ]{ { 3 }^{ 6 } } \right) } =\cfrac { 1 }{ { 3 }^{ 5 } } =\cfrac { 1 }{ 243 } \)

11.

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.

12.

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

13.

Mean \(\bar { x } =\cfrac { \Sigma x }{ n } \)
= \(\cfrac { 26+24+28+31+30+26+24 }{ 7 } =\cfrac { 189 }{ 7 } \)
Mean temperature of the week = 27⁰C

14.

In this example, three values 20, 21, 22 occur two times each. There are three modes for the given data!

15.

(2x - 3b)² [we have (a - b)²= a²- 2ab + b²]
(2a - 3b)²= (2a)²-2(2a)(3b)+(3b)² put [a = 2a, b = 3b]
= 4a²-12ab + 9b²

16.

\(\left( 49 \right) ^{ \frac { 1 }{ 2 } }=(7\times7)^{\frac{1}{2}}=7\)

17.

(3x + 2) units

18.

Let x = 0.57777 ......... \(\rightarrow\) (1)
10x = 5.77777 ............ \(\rightarrow\) (2)
100x = 57.7777 ........... \(\rightarrow\) (3)

(3) - (2) \(\Rightarrow\) 100x - 10x	= 57.7777 ..........
	= 57.7777 ........
99x	= 52.0000 .......

\(x={52\over 90}={26\over 45}\)
\(\therefore 0.5\overline{7}={26\over 45}\)

19.

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

20.

21.

A' = {a, c, e, g}
(A')' = U - A'
= {a, b, c, d, e, f, g, h} - {a, c, e, g}
= {b, d, f, h}

22.

(A\(\cap\)B) = {b, d, f, h} \(\cap\) {a, d, e, h}
= {d, h}
(A\(\cap\)B)' = U-(A\(\cap\)B)
= {a, b, c, d, e, f, g, h}-{d, h}
= {a, b, c, e, f, g}

23.

p(x) = x³ - 1 ; x = 1
p(1) = 1³ - 1
= 1 - 1
= 0
\(\therefore\) 1 is the zero of the polynomial

24.

\(\angle POS+\angle SOR=180^0\) (angle ofa straight line)
105° + x°= 180°
x°=180°-105°
=75°
The value of x = 75°.

Multiple Choice Question

40 x 1 = 40

25.

(a)
280 cm²

26.

(b)
6 cm²

27.

(d)
Less than zero

28.

(d)
Probability

29.

30.

31.

(a)
(-9, 7)

32.

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

33.

34.

(b)
20%

35.

36.

(d)
a³- b³

37.

(b)
\(\cfrac { -7 }{ 4 } \)

38.

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

39.

(d)
180^o

40.

41.

42.

(d)
12.9

43.

44.

(b)
100

45.

(d)
12.9

46.

(b)
46

47.

48.

(b)
31

49.

50.

51.

(b)
60°

52.

(a)
900°

53.

(d)
\(\frac { 5 }{ 32 } \)

54.

55.

56.

(d)
\(\sqrt{17}\)

57.

58.

(d)
(5,0)

59.

(a)
\(\frac { 5 }{ 64 } \)

60.

(b)
II

61.

(b)
X

62.

63.

(b)
\(\frac { 1 }{ 2 } (\angle C+\angle D)\)

64.

(a)
6

Generic 3

14 x 3 = 42

65.

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 \(=\frac { 15 }{ 42 } =\frac { 5 }{ 14 } \)

66.

Here, Total Surface Area of the cube = 486 cm²
6a² = 486 \(\Rightarrow\) a²= \(\frac{486}{6}\) and so, a² = 81 . This gives a = 9.
The side of the cube = 9 cm
Lateral Surface Area = 4a² = 4 × 9² = 4 × 81 = 324 cm²

67.

Given two linear equations are
2x - 3y = 7;
(k + 2)x - (2k + 1)y = 3(2k - 1)
\(\left[ \begin{matrix} { a }_{ 1 }x+{ b }_{ 1 }y+{ c }_{ 1 }=0 \\ { a }_{ 2 }x+{ b }_{ 2 }y+{ c }_{ 2 }=0 \end{matrix} \right] \)
Here a₁ = 2, b₁ = −3, a₂ = (k + 2), b₂ = −(2k +1), c₁ = 7, c₂ = 3(2k −1)
For infinite number of solution we consider \(\frac { { a }_{ 1 } }{ { a }_{ 2 } } =\frac { { b }_{ 1 } }{ { b }_{ 2 } } =\frac { { c }_{ 1 } }{ { c }_{ 2 } } \)
\(\frac { 2 }{ k+2 } =\frac { -3 }{ -(2k+1) } =\frac { 7 }{ 3(2k-1) } \)
\(\frac { 2 }{ k+2 } =\frac { -3 }{ -(2k+1) } \)
2(2k +1) = 3(k + 2)
4k + 2 = 3k + 6
k = 4
\(\frac { -3 }{ -(2k+1) } =\frac { 7 }{ 392k-1) } \)
9(2k −1) = 7(2k +1)
18k − 9 = 14k + 7
4k = 16
k = 4

68.

tan A = \(\frac { opposite\ side }{ adjacent\ side } =\frac { 2 }{ 3 } \)
By Pythagoras theorem,
\(AC=\sqrt { { AB }^{ 2 }+{ BC }^{ 2 } } \)
= \(\sqrt { { 3 }^{ 2 }+{ 2 }^{ 2 } } =\sqrt { 9+4 } =\sqrt { 13 } \)
AC = \(\sqrt { 13 } \)
\(sin\ A=\frac { opposite\ side }{ hypotenuse } =\frac { 2 }{ \sqrt { 13 } } \)
\(cosec\ A=\frac { hypotenuse }{ opposite\ side } =\frac { \sqrt { 13 } }{ 2 } \)
\(cos\ A=\frac { adjacent\ side }{ hypotenuse } =\frac { 3 }{ \sqrt { 13 } } \)
\(sec\ A=\frac { hypotenuse }{ adjacent\ side } =\frac { \sqrt { 13 } }{ 3 } \)
\(cot\ A=\frac { adjacent\ side }{ opposite\ side }= \frac { 3 }{ 2 } \)

69.

Given, A = {2,5,6,7} and B = {3,5,7,8}
\(A\cup B\) = {2,3,5,6,7,8} ...........(1)
\(B\cup A\) = {2,3,5,6,7,8} ..............(2)

70.

Let p(x)= x³+ 9x²- x - 105
By factor theorem,x-3 is a factor of p(x),if p(3) = 0
p(3) = 33 + 9(3)³ - 3 - 105
= 27 + 81 - 3 -105
= 108-108
p(3) = 0
Therefore,x-3 is a factor of x³+ 9x²- x - 105

71.

\(\sqrt [ 4 ]{ 400 } \times \sqrt [ 4 ]{ 567 } \) =\(\left( \sqrt [ 4 ]{ 2\times 2\times 2\times 2\times 5\times 5 } \right) \left( \sqrt [ 3 ]{ 3\times 3\times 3\times 3\times 7 } \right) \)
= \(\left( 2\times \sqrt [ 4 ]{ 25 } \right) \times \left( 3\times \sqrt [ 4 ]{ 7 } \right) =6\times \left( \sqrt [ 4 ]{ 25 } \times \sqrt [ 4 ]{ 7 } \right) \)
= \(6\times \left( \sqrt [ 4 ]{ 25 } \times \sqrt [ 4 ]{ 7 } \right) =6\times \sqrt [ 4 ]{ 25\times 7 } =6\sqrt [ 4 ]{ 175 } \)

72.

56.96 (or) 57 (approximately)

73.

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 \(\bar X={\sum^6_{i=1}(x_i+4)\over6} \)
\(={(x_1+4)+(x_2+4)+(x_3+4)+(x_4+4)+(x_5+4)+(x_6+4)\over6}\)
\(={\sum^6_{i=1}(x_i+24)\over6}={\sum_{i=1}^6x_i\over6}+4\)
\(\bar X=45+4=49\)

74.

Let C, F and H represent sets of students who play Cricket, Football and Hockey respectively.
Then , n(C) = 240, n(F) = 180, n(H) = 164, n(C ∩ F) = 42,
n(F ∩ H) = 38, n(C ∩ H) = 40, n(C ∩ F ∩ H) = 16.
Let us represent the given data in a Venn diagram.
(i) The number of students in the college = 174 + 26 + 116 + 22 + 102 + 24 + 16 = 480
(ii) The number of students who play only one game = 174 + 116 + 102 = 392

75.

ㄥACD = ㄥA + ㄥB
(An exterior angle of a triangle is sum of its interior opposite angles)
120° = 50° + x⁰
x⁰ = 120° - 50°
= 70°
In the the triangle ABC
ㄥA + ㄥB + ㄥACB = 180° (Sum of the angles of a Δ)
50° +x + ㄥACB = 180°
50° + 70° + ㄥACB = 180°
ㄥACB = 180° - 120°
y = 60° (OR)
ㄥACD + ㄥACB = 180⁰(Angles of a linear pair)
ㄥACB = 180° - 120°
= 60°
The value ofx = 70° andy = 60°.

76.

(i) The point P (-2, 0) , Q (2, 0), and R (3, 0) lies on the x-axis,
(ii) Distance =\(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
(a) PR =\(\sqrt{(3+2)^2+(0-0)^2}\)
\(=\sqrt{5^2+0}=\sqrt{25}=5units\)
(b) QR =\(\sqrt{(3-2)^2+(0-0)^2}\)
\(=\sqrt{1^2+0}=\sqrt{1}=1 \ units\)

77.

(3, 5)

78.

Let \(2.\overline { 327 } \) = 2.327327327 ..... \(\rightarrow\) (1)
Here period of decimal is 3, multiply equation (1) by 1000
1000 x = 2327.327327 ..... \(\rightarrow\) (2)
(2) - (1) \(\Rightarrow\) 1000x - x = 999x = 2325
\(x=\frac{2325}{999}\) (or) \(775\over 333\)

Generic 5

8 x 5 = 40

79.

In the quadrilateral ABCD, join one of the diagonals, say AC.
Area of \(\triangle\)ABC = \(\frac{1}{2}\)\(\times\) base \(\times\) height
=\(\frac{1}{2}\)\(\times\)8\(\times\)15\(\times\) 60 cm²
By Pythagoras theorem, in right angled triangle ABC,
AC² = AB² + BC²
= 8² +15² = 64 + 225 = 289 cm
Therefore, AC =\(\sqrt{289}\) =17cm
Now, for\(\triangle\)ACD, let us consider a = 17 cm, b =12 cm, c =25 cm
then, s = \(\frac{a+b+c}{2}=\frac{17+12+25}{2}=\frac{54}{2}\) = 27cm

Area of \(\triangle\)ACD =\(\sqrt { s(s-a)(s-b)(s-c) } \)
=\(\sqrt{27(27-17)(27-12)(27-25)}\)
=\(\sqrt{27\times10\times15\times2}\)
=\(\sqrt{3\times3\times3\times2\times5\times5\times3\times2}\)
= 3 × 3 × 2 × 5 = 90cm²
Therefore, Area of quadrilateral ABCD
=Area of \(\triangle\)ABC + Area of \(\triangle\)ACD
= 60 + 90 = 150 cm²

80.

The roster form of sets P,Q and R are P = {2,3,4,5,6,7,8,9,10}, Q = {2,4,6,8,10} and R = {4,6,8,9,1,12}
First,we find \(Q\cap R\) = {4,6,8,10}
Then, \(P-\left( Q\cap R \right) \) = {4, 6 ,8 ,10}
Next, P - Q = {3, 5, 7, ,9} ..........(1)
and P - R = {2, 3, 5, 7}
and so,\(\left( P-Q \right) \cup \left( P-Q \right) \)
= {2, 3, 5, 7, 9} ...............(2)
Hence from (1) and (2), it verified that \(P-\left( Q\cap R \right) =\left( P-Q \right) \cup \left( P-R \right) \)
Finding the elements of set Q

81.

P (x) = 4x³ + 6x² + 7x + 2
d(x) = x - 2
Standard form of p (x) 4x³+ 6x² + 7x + 2 and
d(x) x - 2

4x² + 6x²+ 7x + 2 = (x - 2)(4x³ + 14x + 35) + 72
Hence the quotient is 4x³ + 14x + 35 and remainder is 72

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.

Calculate the median

Class	Number of students	Cummulative frequency(cf)
0-10	2	2
10-20	7	9
20-30	15	24
30-40	10	34
40-50	11	45
50-60	5	50

N = 50
Median Class = \(\left(\cfrac{N}{2}\right)^{th}\)Value
=\(\left(\cfrac{50}{2}\right)^{th}\) Value = 25^th Value
Median value = 30 - 40
\(\cfrac{N}{2}\)=25, l = 30
m = 24, c =10, f =10
\(\therefore\) Median = \(\cfrac{l+\left(\cfrac{N}{2}-m\right)}{f}\times{c} \)
=\(\cfrac{30+25-24}{10}\times{10}=31\)

84.

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

85.

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 \(\frac { 1 }{ 24 } ,\frac { 2 }{ 24 } ,\frac { 3 }{ 24 } ,\frac { 4 }{ 24 } ,\frac { 5 }{ 24 } ,\frac { 6 }{ 24 } ,\frac { 7 }{ 24 } \)

86.

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

3 x 8 = 24

87.

In \(\triangle\)PQR, PQ = 5 cm, PR = 6 cm, \(\angle\)QPR = 60°

Construction:
Step 1: Draw \(\triangle\) PQR using the given measurements PQ = 9 cm, QR = 7 cm and RP = 8 cm and construct the perpendicular bisector of any two sides (PQ and QR) to find the mid-points M and N of PQ and QR respectively.
Step 2: Draw the medians PN and RM and let them meet at G. The point G is the centroid of the given \(\triangle\)PQR.

88.

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.

89.

Construction :
Step 1: Draw \(\triangle \)ABC with BC = 8 cm. AC = 10 cm with right angle at B.
Step 2: Construct angle bisectors of any two angles (B and C) and let them meet at I. I is the incentre.
Step 3: Draw perpendicular from I to any side of the triangle to meet BC at D.
Step 4: With I as centre, ID as radius draw the incircle, which touches all the three sides of the triangle internally. In radius = 1.9 cm.