9th Standard Mathematics English Medium All Chapter Book Back and Creative Two Marks Questions 2020

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Questions:

Generic 2

36 x 2 = 72

A bag contains 3 red and 2 blue marbles. A marble is drawn at random. What is the probability of drawing a blue marble?

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

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

The outer dimensions of a closed wooden box are 10 cm by 8 cm by 7 cm, Thickness of the wood is 1 cm³. Find the total cost of wood required to make box if 1 cm³ of wood costs Rs. 2.00?

Find the six trigo'nometric ratios of the angle 0 using the diagram

Using Heron's formula, find the area of a triangle whose sides are 41 m, 15 m, 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.

Find the coordinates of the point which divides, the line segment joining the point A (3,7) and B (-11, -2) in the ratio 5 : 1.

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

10.

A metallic cube with side 15 cm is melted and formed into a cuboid. If the length and height of the cuboid is 25 cm and 9 cm respectively then find the breadth of the cuboid

11.

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.

12.

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

13.

Find the area of a right triangle whose hypotenuse is 10cm and one of the acute angle is 24⁰24'

14.

The perimeter of a triangular plot is 600 m. If the sides are in the ratio 5:12:13, then find the area of the plot

15.

Verify the following equalities :
sin²60⁰ + cos²60⁰ = 1

16.

The vertices of a triangle are (1, 2), (h, −3) and (−4, k). If the centroid of the triangle is at the point (5, −1) then find the value of \(\sqrt { { (h+k) }^{ 2 }+{ (h+3k) }^{ 2 } } \)

17.

Can we write 0.25 as 0.250000 ...? Can a terminating decimal be written as a recurring decimal?

18.

Factorise the following : 25m^-2-16n²

19.

Express the surds in the simple form \(\sqrt [ 3 ]{ 128 } \)

20.

Find the value of X^o

21.

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

22.

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.

23.

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.

24.

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

25.

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

26.

In the given figure, AC is the diameter of the circle with centre O. If ㄥADE = 30°; ㄥDAC = 35° and ㄥCAB = 40°. Find
(i) ㄥACD
(ii) ㄥACB
(iii) ㄥDAE

27.

Expand the following using identities: (3x + 4y)²

28.

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

29.

Without actual division classify the decimal expansion of the following numbers as terminating or non-terminating and recurring.
\(-{11\over 75}\)

30.

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 UB)'
(ii) (A ∩ B)'

31.

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

32.

Find A∪B, A∩B, A–B and B–A for the following sets.
A = Set of all letters in the word “mathematics” and
B = Set of all letters in the word “geometry”

33.

Find the distance between the following pairs of points. (1, 2) and (4, 3)

34.

Identify which ones are trapeziums and which are not.

35.

Draw Venn diagram and shade the region representing the following sets
(A – B)′

36.

Write the coefficient of x²and x in each of the following polynomials \(\sqrt { 3 } { x }^{ 2 }+\sqrt { 2 } x+0.5\)

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Answers Key:

Generic 2

36 x 2 = 72

Probability = \(\frac{2}{5}\).

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

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

Volume of wood = 12 \(\times\) 10\(\times\) 9 - 10 \(\times\) 8 \(\times\) 7
= 1080 - 560 = 520 cm³ \(\times\)2.00 = Rs. 1040

Hypotenuse = \(\sqrt { { 12 }^{ 2 }+{ 5 }^{ 2 } } \)
= \(\sqrt { 144+25\quad } =\sqrt { 69 } \)
= 13
\(sin\theta =\cfrac { 5 }{ 13 } ;cos\theta =\cfrac { 12 }{ 13 } ;tan\theta =\cfrac { 5 }{ 12 } ; \)
\(cosec\theta =\cfrac { 13 }{ 5 } ;sec\theta =\cfrac { 13 }{ 12 } ;cot\theta =\cfrac { 12 }{ 3 } \)

s = \(\frac { 41+15+25 }{ 2 } \) = 40.5
Area = \(\sqrt { s(s-a)(s-b)(s-c) } =\sqrt { 40.5\times 0.5\times 25.5\times 15.5 } \)
=\(\sqrt { \frac { 405 }{ 10 } \times \frac { 5 }{ 10 } \times \frac { 255 }{ 10 } \times \frac { 155 }{ 10 } } =\sqrt { \frac { 5\times 9\times 9\times 5\times 5\times 3\times 17\times 5\times 31 }{ 10\times 10\times 10\times 10 } } \)
=\(\frac { 1 }{ 100 } \times 5\times 9\times 5\times \sqrt { 3\times 17\times 31 } \)

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

\((x,y)=\left( \frac { 3(1)+5(-11) }{ 5+1 } ,\frac { 1(7)+5(-2) }{ 6 } \right) \)
\(=\left( \frac { 3-55 }{ 6 } ,\frac { 7-10 }{ 6 } \right) =\left( \frac { -52 }{ 6 } ,\frac { -3 }{ 6 } \right) =\left( \frac { -26 }{ 3 } ,\frac { -1 }{ 2 } \right) \)

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

10.

The volume of the cuboid formed = The volume of the cube melted.
lbh = a³

height of the cuboid = 15 cm.

11.

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

12.

(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?

13.

Hypotenuse = 10 cm
One of the acute angle = 24° 24'
sin 24° 24' = 0.4131'
\(\cfrac { x }{ 10 } =0.4131\)
x = 0.4131 \(\times\) 10
x = 4.131
cos 24° 24' = 0.9107
\(\cfrac { y }{ 10 } =0.9107\)
y = 9.107
\(\therefore\) Area of the triangle = \(\cfrac { 1 }{ 2 } bh\)
= \(\cfrac { 1 }{ 2 } \times y\times x\)
\(=\cfrac { 1 }{ 2 } \times 9.107\times 4.131=18.81sq.cm\)

14.

s = 600 m
side s are in the ratio 5 : 12 : 13
5x + 12x + 13x = 30x
s = 600 ⇒ \(\frac { 30x }{ 2 } \) = 600
30x = 1200
x = 40
∴ sides are 200 m, 480 m, 520 m.
∴ Area =\(\sqrt { s(s-a)(s-b)(s-c) } \)
=\(\sqrt { 600(600-200)(600-480)(600-520) } \)
=\(\sqrt { 600\times 400\times 120\times 80 } =\sqrt { 2304000000 } \)
=\(\sqrt { 48\times 48\times 1000\times 1000 } \) = 48 \(\times\) 1000 =48000 sq.m

15.

\({ \sin }^{ 2 }{ \tan }^{ 2 }+{ \cos }^{ 2 }{ 60 }^{ o }=\left( \cfrac { \sqrt { 3 } }{ 2 } \right) +\left( \cfrac { 1 }{ 2 } \right) ^{ 2 }=\cfrac { 3 }{ 4 } +\cfrac { 1 }{ 4 } =\cfrac { 4 }{ 4 } =1\)

16.

Vertices of a triangle
(x₁,y₁) = (1,2)
(x₂,y₂) = (h,-3)
(x₃, y₃) = (-4, k)
Centroid G (x,y) = (5, -1)
\(G=\left( \frac { { x }_{ 1 }+{ x }_{ 2 }+{ x }_{ 3 } }{ 3 } ,\frac { { y }_{ 1 }+{ y }_{ 2 }+{ y }_{ 3 } }{ 3 } \right) =(5,-1)\)
\(\left( \frac { 1+h+(-4) }{ 3 } ,\frac { 2+(-3)+k }{ 3 } \right) =(5,-1)\)
\(\Rightarrow \frac { -3+h }{ 3 } =5\)
-3 + h = 15
h = 18
\(\frac { 2-3+k }{ 3 } =-1\)
\(\therefore \sqrt { { (h+k) }^{ 2 }+{ (h+3k) }^{ 2 } } \)
\(=\sqrt { (18+(-2))^{ 2 }+{ (18+3(-2)) }^{ 2 } } \)
\(=\sqrt { { 16 }^{ 2 }+{ 12 }^{ 2 } } =\sqrt { 256+144 } =\sqrt { 400 } =20\)

17.

Yes, it is possible to write.

18.

25m^-2- 16n²= (5m)² - (4n)² [ \(\because \) a²-b²= (a - b)(a + b)]
= (5m - 4n) (5m+ 4n)

19.

\(\sqrt [ 3 ]{ 128 } \) = \(\sqrt [ 3 ]{ 2\times 2\times 2\times 2\times 2\times 2\times 2 } =4\sqrt [ 3 ]{ 2 } \)

20.

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

21.

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

22.

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

23.

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

24.

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

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

25.

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

26.

(i) \(\angle\)ACD = 180⁰-(90⁰+ 35⁰)
= 18⁰-125⁰= 55⁰
(ii)\(\angle\)ACB = 180⁰- (90⁰+ 40⁰)
= 180⁰-130⁰= 50⁰
(iii)\(\angle\)ADC = 90⁰
\(\angle\)CAE = 180⁰-120⁰= 60⁰
\(\therefore\) \(\angle\)DAE = 60⁰- 35⁰= 25⁰

27.

(3x + 4y)² [we have (a+b)²= a²+ 2ab + b²]
(3x + 4y)² = (3x)²+2(3x)(4y)+(4y)² put [a = 3x, b = 4y]
= 9x²+ 24xy + 16y²

28.

\(\frac { \sqrt [ 9 ]{ 8 } }{ \sqrt [ 6 ]{ 6 } } =\frac { { 8 }^{ \frac { 1 }{ 9 } } }{ 6^{ \frac { 1 }{ 6 } } } \) (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?) \(=\left( \frac { 8\times }{ 6\times 6\times 6 } \right) ^{ \frac { 1 }{ 18 } }\)
=\(\left( \frac { 8 }{ 27 } \right) ^{ \frac { 1 }{ 18 } }=\left[ \left( \frac { 2 }{ 3 } \right)^3 \right] ^{ \frac { 1 }{ 18 } }=\left( \frac { 2 }{ 3 } \right) ^{ \frac { 1 }{ 6 } }=6\sqrt { \frac { 2 }{ 3 } } \).

29.

\(-{11\over 75}=-{11\over 3\times 5^2}\)
Since it is not in the form of \(P\over 2^m\times 5^n\)
\(\therefore -{11\over 75}\) has non-terminating and recurring decimal expansion.

30.

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

31.

32.

A { m, a, t, h, e, i, c, s}
B = {g, e, o, m, t, r, y}
A\(\cup\)B = {m, a, t, h, e, c, s} \(\cup\) {g, e, o, m, t, r, y}
= {a, c, e, g, h, i, m, o, r, s, t, y}
A\(\cap\)B = {m, a, t, h, e, i, c, s} \(\cap\) {g, e, o, m, t, r, y}
= {e, m, t}
A-B = {m, a, t, h, e, i, c, s} - {g, e, o, m, t, r, y}
= {a, c, h, i, s}
B-A = {g, e, o, m, t, r, y} - { m,a, t, h, e, i, c, s}
= {g, o, r, y}

33.

Distance between the points (1, 2) and (4, 3)
= \(\sqrt { ({ x }_{ 2 }-{ x }_{ 1 })^{ 2 }+(y_{ 2 }-{ y }_{ 1 })^{ 2 } } \)
= \(\sqrt { (4-1)^{ 2 }+(3-2)^{ 2 } } =\sqrt { { 3 }^{ 2 }+{ 1 }^{ 2 } } \)
= \(\sqrt { 9+1 } =\sqrt { 10 } \) = units

34.

It is a trapezium

35.

(A – B)′