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

Download Tamil Nadu 9th Standard Maths question papers, model tests, one-mark questions, important questions, and public exam papers in PDF format. Free study materials and answer keys for TN State Board students.

Questions:

Generic 2

24 x 2 = 48

Two unbiased coins are tossed simultaneously find the probability of getting
(i) two heads
(ii) one head
(iii) at least one head
(iv) at most one head

From the given figure, find all the trigonometric ratios of angle \(\theta\).

Find the area of an equilateral triangle whose perimeter is 150 m.

If A (10, 11) and B (2 ,3) are the coordinates of end points of diameter of circle. Then find the centre of the circle.

Solve \(\cfrac { 1 }{ 3 } x-\cfrac { 5 }{ 2 } =6\)

Two dice are rolled, find the probability that the sum is
(i) equal to 1
(ii) equal to 4
(iii) less than 13

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.

If sin \(\theta\) = \(\frac { a }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } \) then show that b sin \(\theta\) = a cos \(\theta\)

Which arrow best shows the position of \(\frac { 11 }{ 3 } \) on the number line?

10.

Express the following in the form 3ⁿ: 27

11.

Find the value of X^o

12.

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

13.

In a research laboratory scientists treated 6 mice with lung cancer using medicine. Ten days, they measured the volume of the tumor of the tumor in each mouse given the results in the table

Mouse marking	1	2	3	4	5	6
Tumor Volume(mm)³	145	148	142	141	139	140

Find the mean

14.

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

15.

Expand: (2a + 3b)³

16.

Evaluate: (0.1)^-4

17.

Express the following in the form of \(\frac { p }{ q } \) where p and q are integers and q \(\neq \)0.
(a)\(\overline { 0.6 } \)
(b) \(\overline { 0.47 } \)
(c) \(\overline { 0.001 } \)

18.

Represent U= {1,2,3,4,5,6,7,8,9,10} A = {1,2,4,6,8,10} and B = {3,6,1,10} in venn-diagram, then find
(i) (A U B)'
(ii) (A ∩ B)'

19.

In a class of 50 students, each of the students passed either in mathematics or in science or in both. 10 students passed in both and 28 passed in science. Find how many students passed in mathematics only

20.

Represent the following sets in set builder form.
E = The set of odd Whole numbers less than 9.

21.

By remainder theorem, find the remainder when, p(x) is divided by g(x) where
p(x) = x³- 3x²+ 4x + 50 g(x) = x - 3

22.

Which of the following are sets?
The Collection of rich people in India

23.

Identify which ones are trapeziums and which are not.

24.

Plot the following points in the coordinate plane. Join them in order. What type of geometrical shape is formed?
(–3, 3) (2, 3) (–6, –1) (5, –1)

Multiple Choice Question

40 x 1 = 40

25.

If the lateral surface area of a cube is 600 cm², then the total surface area is _______.

(a)

150 cm²

(b)

400 cm²

(c)

900 cm²

(d)

1350 cm²

26.

The six faces of the dice are called equally likely if the dice is _______.

(a)

Small

(b)

Fair

(c)

Six-faced

(d)

Round

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.

The perimeter of an equilateral triangle is 30 cm. The area is _______.

(a)

\(10\sqrt { 3 } \) cm²

(b)

\(12\sqrt { 3 } \) cm²

(c)

\(15\sqrt { 3 } \) cm²

(d)

\(25\sqrt { 3 } \) cm ²

29.

If cos A = \(\frac { 3 }{ 5 } \), then the value of tan A is

(a)

\(\frac { 4 }{ 5 } \)

(b)

\(\frac { 3 }{ 4 } \)

(c)

\(\frac { 5 }{ 3 } \)

(d)

\(\frac { 4 }{ 3 } \)

30.

If tan \(\theta\) cot 37⁰ , then the value of \(\theta\) is ________.

(a)

37⁰

(b)

53⁰

(c)

90⁰

(d)

1⁰

31.

Which of the following has a factor?

(a)

x²+ 2x

(b)

(x - 1)²

(c)

(x + 1)²

(d)

(x²- 2²)

32.

\(\cfrac { \sqrt [ 3 ]{ 27 } }{ \sqrt [ 5 ]{ 3 } } \) is equal to_______

(a)

\(5\sqrt { 9 } \)

(b)

\(5\sqrt { 6 } \)

(c)

\(5\sqrt { 24 } \)

(d)

\(\sqrt [ 5 ]{ 30 } \)

33.

Which of the following is not an irrational number?

(a)

\(\sqrt { 2 } \)

(b)

\(\sqrt { 5 } \)

(c)

\(\sqrt { 3 } \)

(d)

\(\sqrt { 25 } \)

34.

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°

35.

The mean of the first 10 whole numbers is

(a)

(b)

4.5

(c)

(d)

5.5

36.

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

(a)

Frequency

(b)

range

(c)

mode

(d)

median

37.

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

(a)

101

(b)

100

(c)

(d)

38.

Which one of the following is not a measure of central tendency?

(a)

Mean

(b)

Range

(c)

Median

(d)

Mode

39.

The mean of a 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)

\(\bar X\)

40.

The median of the first 10 whole numbers is ______________

(a)

(b)

4.5

(c)

(d)

5.5

41.

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

(a)

80°

(b)

100°

(c)

70°

(d)

90°

42.

In the figure, O is the centre of the circle and ㄥACB = 40° then ㄥAOB = ________.

(a)

80°

(b)

85°

(c)

70°

(d)

65°

43.

If x - 3 is a factor of p(x), then the remainder is _______.

(a)

(b)

-3

(c)

p(3)

(d)

p(-3)

44.

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

(a)

\(\sqrt{39}\)

(b)

5\(\sqrt{6}\)

(c)

5\(\sqrt{3}\)

(d)

3\(\sqrt{5}\)

45.

For any three sets A, B and C, (A−B)∩(B −C) is equal to ________.

(a)

A only

(b)

B only

(c)

C only

(d)

46.

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

(a)

(b)

(c)

(d)

47.

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

(a)

x = 0, 2

(b)

x = 1, 2

(c)

x = 1, -2

(d)

x = 0, -2

48.

Which of the following is trinomial?

(a)

-7z

(b)

\({ z }^{ 2 }-{ 4y }^{ 2 }\)

(c)

\({ x }^{ 2 }y-{ xy }^{ 2 }+y\)

(d)

\(12a=9ab+5b-3\)

49.

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

(a)

1, 7

(b)

2, 6

(c)

3, 5

(d)

4, 4

50.

Diagonal of which of the following quadrilaterals do not bisect it into two congruent triangles?

(a)

rhombus

(b)

trapezium

(c)

square

(d)

rectangle

51.

If all the four sides of a parallelogram are equal and the adjacent angles are of 120° and 60°, then the name of the quadrilateral is ______

(a)

rectangle

(b)

square

(c)

rhombus

(d)

kite

52.

The number of elements of the set {x : x ∈ Z, x² = 1} is ________

(a)

(b)

(c)

(d)

53.

Sets having the same number of elements are called ___________

(a)

overlapping sets

(b)

disjoint sets

(c)

equivalent sets

(d)

equal sets

54.

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

55.

Which of the following are irrational numbers?
\(\sqrt { 2+\sqrt { 3 } } \)
\(\sqrt [ 3 ]{ 5+\sqrt { 7 } } \)
\(\sqrt { 8-\sqrt [ 3 ]{ 8 } } \)
\(\sqrt { 4+\sqrt { 25 } } \)

(a)

(ii), (iii) and (iv)

(b)

(i), (ii) and (iv)

(c)

(i), (ii) and (iii)

(d)

(i), (iii) and (iv)

56.

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

(a)

(0, -5)

(b)

(-5,0)

(c)

(5,0)

(d)

(0,5)

57.

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)

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

58.

A point which lies in the III quadrant is__________________

(a)

(5, 4)

(b)

(5, -4)

(c)

(-5, -4)

(d)

(-5, 4)

59.

On which quadrant does the point (-4, 3) lie?

(a)

(b)

(c)

III

(d)

60.

The distance between the point ( 5, –1 ) and the origin is _______.

(a)

\(\sqrt { 24 } \)

(b)

\(\sqrt { 37 } \)

(c)

\(\sqrt { 26 } \)

(d)

\(\sqrt { 17 } \)

61.

The number \(0.\bar { 3 } \) in the form \(\frac { p }{ q } \) where p and q are integers and \(q\neq 0\)

(a)

\(\frac { 33 }{ 100 } \)

(b)

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

(c)

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

(d)

\(\frac { 3 }{ 100 } \)

62.

The zero of the polynomial 2x+5 is _______.

(a)

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

(b)

\(-\frac {5}{2}\)

(c)

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

(d)

\(-\frac {2}{5}\)

63.

The points (–5, 2) and (2, –5) lie in the ________.

(a)

same quadrant

(b)

II and III quadrant respectively

(c)

II and IV quadrant respectively

(d)

IV and II quadrant respectively

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}

Generic 3

13 x 3 = 39

65.

The length, breadth and height of a cuboid are in the ratio 7: 5: 2. Its volume is 35840 cm³. Find its dimensions.

66.

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?

67.

Find the value of cos19⁰59'

68.

Find the slopes of all the lines from the adjacent figure,

69.

Add \(\sqrt [ 5 ]{ 11 } \) and \(\sqrt [ 7 ]{ 11 } \). Check whether the sum is rational or irrational

70.

A researcher studying the behavior of mice has recorded the time (in seconds) taken by each mouse to locate its food by considering 13 different mice as 31, 33, 63, 33, 28, 29, 33, 27, 27, 34, 35, 28, 32. Find the median time that mice spent in searching its food.

71.

If A = {b,c,e,g,h}, B = {a,c,d,g,i} and C = {a,d,e,g,h}, then show that \(A-(B\cap C)=(A-B)\cup (A-C)\).

72.

The perimeter of a triangle is 8m² - 9m+ 7. If the two sides of the triangel be m² - m + 6 and 3m² - 2m - 4, find the third side.

73.

The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.

74.

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.

75.

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

76.

Express the following decimal expression into rational numbers \(3.1\overline { 7 } \)

77.

Let A(2, 3) and B(2, –4) be two points. If P lies on the x-axis, such that AP = \(\frac{3}{7}\) AB, find the coordinates of P.

Generic 5

9 x 5 = 45

78.

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.

79.

Draw Venn diagram for \(A\cap B\cap C\)

80.

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

81.

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

82.

Calculate the mean of the following distribution using Assumed Mean Method

Class Interval	0-10	10-20	20-30	30-40	40-50
Frequency	5	7	15	28	8

83.

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

84.

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

85.

Diagonal AC of a parallelogram ABCD bisects ∠A. Show that
(i) it bisects ∠C also
(ii) ABCD is a rhombus.

86.

Show that the point (3, -2), (3, 2), (-1, 2) and (-1, -2) taken in order are the vertices of a square.

Generic 8

2 x 8 = 16

87.

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

88.

Draw ΔABC given AB = 9 cm, ㄥCAB = 115° and ΔABC = 40°. Locate its incentre and also draw the incircle. (Note: You can check from the above examples that the incentre of any triangle is always in its interior).

Visit QB365-SLMS - https://www.qb365.in/slms/

Answers Key:

Generic 2

24 x 2 = 48

S = {HH, HT, TH, TT}
(i) probability of two heads = \(\frac { 1 }{ 4 } \)
(ii) probability of one head = \(\frac { 1 }{ 2 } \)
(iii) probability of at least one head = \(\frac { 3 }{ 4 } \)
(iv) probability of at most one head = \(\frac { 3 }{ 4 } \).

\(x=\sqrt { { 13 }^{ 2 }-{ 12 }^{ 2 } } \)
\(\sqrt { 169-144 } =\sqrt { 25 } =5\)
\(sin\theta =\cfrac { 5 }{ 13 } ;cos\theta =\cfrac { 12 }{ 13 } ;tan\cfrac { 5 }{ 12 } ;cosec\theta =\cfrac { 13 }{ 5 } ;\)
\(sec\theta =\cfrac { 13 }{ 12 } ;cot\theta =\cfrac { 12 }{ 5 } \)

3a = 150
a = \(\frac { 150 }{ 3 } \) = 50 m

Centre of the circle \(=\left( \frac { 10+2 }{ 2 } ,\frac { 11+3 }{ 2 } \right) =(6,7)\)

\(\cfrac { 1 }{ 3 } x-\cfrac { 5 }{ 2 } =6\)
\(\cfrac { 1 }{ 3 } x=6+\cfrac { 5 }{ 2 } =\cfrac { 12+5 }{ 2 } =\cfrac { 17 }{ 2 } \)
\(x=\cfrac { 17 }{ 2 } \times 3=\cfrac { 51 }{ 2 } \)

When two dice are rolled
Sample space
S = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4),(4,5),(4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
n(S) = 36
(i) Event of the sum is equal to 1 = 0
∴ Probability = \(\frac { 0 }{ n(S) } \) = 0
(ii) Event of the sum is equal to 4
B = {(1, 3), (2, 2), (3, 1)}
n(B) = 3
P(B)= \(\frac { n(B) }{ n(S) } =\frac { 3 }{ 6 } =\frac { 1 }{ 12 } \)
(iii) Event of the sum is equal to less than 13
C= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
n(C) = 36
P(C) = \(\frac { n(C) }{ n(S) } =\frac { 36 }{ 6 } \).

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

\(sin\ \theta =\cfrac { a }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } \)
\(cos\ \theta =\cfrac { b }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } \)
\(b\ sin\ \theta =b\times \cfrac { a }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } -\cfrac { ab }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } \) ... (1)
\(a\ cos\ \theta =a\times \cfrac { b }{ \sqrt { a^{ 2 }+{ b }^{ 2 } } } =\cfrac { ab }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } \) ...(2)
(1) = (2)\(\Rightarrow\) LHS = RHS
Hence proved

By By the Pythagoras theorem
BC² = AC² + AB²
= \(\left( \sqrt { { a }^{ 2 }+{ b }^{ 2 } } \right) ^{ 2 }-{ a }^{ 2 }\)
= a²+b²-a²
= b²
BC = b

\(\frac { 11 }{ 3 } \) = 3.66
Arrow D best shows the position of \(\frac { 11 }{ 3 } \) on the number line.

10.

27 = 3 \(\times\) 3 \(\times\) 3;
therefore 27 = 3³

11.

x^o=\(\cfrac { 1 }{ 2 } \angle BOC\)
=\(\cfrac { 1 }{ 2 } \) (50^o+30^o)
=\(\cfrac { 1 }{ 2 } \) \(\times\) 80 =40^o

12.

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 = 4 cm + 3 cm = 7 cm
Distance between the chord is 7 cm.

13.

\(\bar { x } =\frac { \Sigma x }{ n } =\frac { 145+148+142+141+139+140 }{ 6 } =\frac { 855 }{ 6 } \)
x = 142.5 mm²

14.

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

15.

We know that (a+b)³≡ a³+3a²b+3ab²+b³
(2a+3b)³= (2a)³+3(2a)²(3b)+3(2a)(3b)²+(3b)³
= 8a³+36a²b+54ab²+27b³

16.

(0.1)^-4 = \(\left( \frac { 1 }{ 10 } \right) ^{ -4 }=\frac { 1 }{ \left( \frac { 1 }{ 10 } \right) ^{ 4 } } =\frac { 1 }{ \left( \frac { 1 }{ 10000 } \right) } \)

17.

\((a)\frac { 2 }{ 3 } \)
\((b)\frac { 43 }{ 90 } \)
\((c)\frac { 1 }{ 999 } \)

18.

(i) {5,9}
(ii) {1,2,3,4,5,7,8,9}

19.

20.

E = {x : x is an odd whole number less than z < 9}

21.

p(x) = x³- 3x²+ 4x+ 50
g(x) = x - 3
p(3) = 33 - 3(3)² + 4(3) + 50
= 27 - 27 + 12 + 50 = 62
\(\therefore\) The Remainder 62

22.

not a set

23.

It is not a trapezium

24.

The shape of the geometrical figure Trapezium.

Multiple Choice Question

40 x 1 = 40

25.

26.

(b)
Fair

27.

(a)
Event

28.

(d)
\(25\sqrt { 3 } \) cm ²

29.

(d)
\(\frac { 4 }{ 3 } \)

30.

(b)
53⁰

31.

(a)
x²+ 2x

32.

(a)
\(5\sqrt { 9 } \)

33.

(d)
\(\sqrt { 25 } \)

34.

(a)
70°

35.

(b)
4.5

36.

37.

(b)
100

38.

(b)
Range

39.

40.

(b)
4.5

41.

(a)
80°

42.

(a)
80°

43.

44.

45.

(d)
ф

46.

47.

(d)
x = 0, -2

48.

49.

50.

(b)
trapezium

51.

52.

53.

54.

(d)
disjoint sets

55.

(d)
(i), (iii) and (iv)

56.

(a)
(0, -5)

57.

58.

59.

(b)
II

60.

61.

62.

(b)
\(-\frac {5}{2}\)

63.

64.

(b)
7 ∈ {1,2,3,4,5,6,7,8,9,10}

Generic 3

13 x 3 = 39

65.

Let the dimensions of the cuboid be
l = 7x, b = 5x and h = 2x.
Given that volume of cuboid = 35840 cm³
l × b × h = 35840
(7x)(5x)(2x) = 35840
70x³ = 35840
x³= \(\frac{35840}{70}\)
x²= 512
x =\(\sqrt [ 3 ]{ 8\times 8\times 8 } \)
x = 8 cm
Length of cuboid= 7x = 7 × 8 = 56cm
Breadth of cuboid= 5x = 5 × 8 = 40cm
Height of cuboid= 2x = 2 × 8 = 16 cm

66.

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

67.

12'

18'

24'

30'

36'

42'

48'

54'

Mean Difference

0.0⁰

0.1⁰

0.2⁰

0.3⁰

0.4⁰

0.5⁰

0.6⁰

0.7⁰

0.8⁰

0.9⁰

19⁰

0.9403

write 19⁰59' = 19⁰54' + 5'
From the table we have, cos19⁰54' = 0.9403

68.

Slope of AB = \(\frac { { y }_{ 2 }-{ y }_{ 1 } }{ { x }_{ 2 }-{ x }_{ 1 } } =\frac { 3 }{ 5 } \)
Slope of CD = \(\frac { change\ in\ y }{ change\ in\ x } =-\frac { 3 }{ 2 } \)
Slope of EF = \(\frac { 4 }{ 7 } \)
Slope of PQ = Undefined Slope of RS = 0.

69.

\(\sqrt [ 5 ]{ 11 } +\sqrt [ 7 ]{ 11 } =\left( 5+7 \right) \sqrt { 11 } =12\sqrt { 11 } \)

70.

Data in ascending order 27, 27, 28, 28, 29, 31, 32, 33, 33, 33, 34, 35, 63
Number of terms = 13
If n is odd then Median = \(\left[\frac{\mathrm{N}+1}{2}\right]^{t h}\) observation
\(=\left(\frac{13+1}{2}\right)^{t h}\)observation
= 7^th observation
7^th observation = 32

71.

A = {b,c,e,g,h}
B = {a,c,d,g,i}
C = {a,d,e,g,h}
B ∩ C = {a,d,g}
A - (B ∩ C) = {b,c,e,g,h} - {a,d,g} = {b,c,e,h}...(1)
A-B = {b,c,e,g,h} - {a,c,d,g,i} = {b,e,h}
A-C = {b,c,e,g,h} - {a,d,e,g,h} = {b,c}
(A - B)⋃(A - C) = {b,c,e,h}...(2)
From (1) and (2) it is verified that
A-(B ⋂ C) = (A - B) U (A - C)

72.

Let the side AC be x.

Perimeter of a ∆ABC = 8m²-9m+7
AB+BC+AC = 8m²-9m+7
m² - m + 6 + 3m² - 2m - 4 + x = 8m²-9m+7
4m² - 3m + 2 + x = 8m²-9m+7
x = 8m²-9m+7-(4m²-3m+2)
= 8m²-9m+7-4m²+3m-2
= 4m²-6m+5
The third side AC = 4m²-6m+5

73.

Let the angles of the quadrilateral be 3x, 5x, 9x and 13x.
Sum of all the angles of quadrilateral = 360°.
3x + 5x + 9x + 13x = 360°
30x = 360°
x = \(\frac { { 360 }^{ 0 } }{ 30 } \)
=12⁰
3x =3 \(\times\) 12 = 36°
5x = 5 \(\times\) 12 = 60°
9x = 9 \(\times\) 12 = 108°
13x = 13 \(\times\) 12 = 156°
The required angles of quadrilateral are 36°, 60° 108° and 156°.

74.

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 ∩ S)
40 = 26 + 24 - n (M ∩ S)
n (M ∩ S) = 26 + 24 - 40
= 50 - 40 = 10
∴ Number of students opted for Mathematics and Science = 10

75.

(3, 5)

76.

Let x = \(3.1\overline { 7 } \) = 3.1777 ....... (1)
Here period of decimal is 2, multiply equation (1) by 100
100x = 31.777 ......... (2)
(2)-(1)

\( =\frac{143}{45} \)

77.

Given points are A(2, 3) and B(2, -4)
The point P lies on the x-axis.
∴ The point P is (x, 0)
AP = \(\frac { 3 }{ 7 } \) AB
\(\frac { AP }{ AB } \) = \(\frac { 3 }{ 7 } \)
\(\frac { AP }{ PB } \) = \(\frac { 3 }{ 7 } \) ........(1)
Distance =\(\sqrt { ({ x }_{ 2 }-{ x }_{ 1 })^{ 2 }+(y_{ 2 }-y_{ 1 })^{ 2 } } \)
AP = \(\sqrt { (x-2)^{ 2 }+(0-3)^{ 2 } } \)
= \(\sqrt { { x }^{ 2 }-4x+4+9 } \)
= \(\sqrt { { x }^{ 2 }-4x+13 } \)
BP =\(\sqrt { (x-2)^{ 2 }+(0+4)^{ 2 } } \)
= \(\sqrt { x^{ 2 }-4x+4+16 } \)
= \(\sqrt { x^{ 2 }-4x+20 } \)
From (1) we get
\(\frac { AP }{ PB } \) = \(\frac { 3 }{4 } \)
\(\frac { \sqrt { { x }^{ 2 }-4x+13 } }{ \sqrt { { x }^{ 2 }-4x+20 } } =\frac { 3 }{ 4 } \) (Squaring on both sides)
\(\frac { { x }^{ 2 }-4x+13 }{ { x }^{ 2 }-4x+20 } =\frac { 9 }{ 16 } \)
16x²- 64x + 208 = 9x²- 36x + 180
7x²- 28x + 28 = 0
x²- 4x + 4 = 0
(x-2)²= 0
x-2 =0
x = 2
∴ The point P is (2, 0)

Generic 5

9 x 5 = 45

78.

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

79.

80.

Let p (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)

81.

\(\angle\)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

82.

Let Assumed mean be 25

Class Interval x	Frequency f	Mid x	A = 25 d = x -A	fd
0-10	5	5	-20	-100
10-20	7	15	-10	-70
20-30	15	25	0	0
30-40	28	35	10	280
40-50	8	45	20	160
	\(\Sigma f=63\)			\(\Sigma fd=270\)

Mean \(\bar{x}=A+\frac{\Sigma fd}{\Sigma f}\)
= \(25+\frac{270}{63}\)
= 25 + 4.29 = 29.29

83.

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 \(\left( -\frac { 9 }{ 11 } \right) \)

84.

Rational 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.

We have a parallelogram ABCD in which diagonals AC bisect ㄥA.
ㄥDAC = ㄥBAC
(i) To prove that AC bisects ∠C
∵ ABCD is a parallelogram
∴ AB II DC and AC is a transversal
∴ ㄥ1 = ㄥ3 (Alternate interior angle) .........(1)
.Also BC II AD and AC is a transversal.
∴ ㄥ2 = ㄥ4 (Alternate interior angle) ...........(2)
But AC bisects ㄥA
∴ ㄥ1 = ㄥ2
From (1), (2) and (3) we get
ㄥ3 = ㄥ4
∴ AC bisects ㄥC.
(ii) To prove that ABCD is a rhombus.
In ΔABC, we have ㄥ1 = ㄥ4 [ ∵ ㄥ1 = ㄥ2 = ㄥ4]
∴ BC = AB (side opposite to equal angles are equal) (4)
Similarly AD = DC ....... (5)
But ABCD is a parallelogram AB = DC (Opposite sides of a parallelogram) .......(6)
From (4), (5) and (6) we have AB = BC = CD = DA.
Thus ABCD is a rhombus.

86.

Distance = \(\sqrt{(x_2+x_1)^2+(y_2-y_1)^2}\)
AB =\(\sqrt{(3-3)^2+(2+2)^2}\)
=\(\sqrt{0+4^2}=\sqrt{16}=4\)
BC =\(\sqrt{(-1-3)^2+(2-2)^2}\)
\(\sqrt{(-4)^2+0}=\sqrt{16}=4\)
CD =\(\sqrt{(-1-1)^2+(2-2)^2}\)
\(\sqrt{0+(-4)^2}=\sqrt{16}=4\)
AD =\(\sqrt{(-1-3)^2+(-2+2)^2}\)
\(\sqrt{(-4)^2+0}=\sqrt{16}=4\)
AB = BC = CD = DA = 4. All the four sides are equal.

\(\therefore \)ABCD is a Rhombus .................(1)
Diagonal AC =\(\sqrt{(3+1)^2+(-2-2)^2}\)
\(=\sqrt{4^2+(-4)^2}=\sqrt{16+16}=\sqrt{32}\)
Diagonal BD =\(\sqrt{(3+1)^2+(2+2)^2}\)
\(=\sqrt{4^2+4^2}=\sqrt{16+16}=\sqrt{32}\)
Diagonal AC = Diagonal BD =\(\sqrt{32}\) ................ (2)
From (1) and (2) we get ABCD is a square.

Generic 8

2 x 8 = 16

87.

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 midpoints 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.

88.

Construction :
Step 1: Draw\(\triangle \)ABC with AB = 9 cm. ㄥA = 115°,ㄥB = 40°.
Step 2: Construct angle bisectors of any two angles (B and C). Let them meet at I. I is the incentre of \(\triangle \)ABC.
Step 3: Draw perpendicular from I to any side (BC) to meet BC at D.
Step 4: Draw incircle, with I as centre and ID a radius. Measure the inradius.