Class IX - Mathematics - Paper - 1 - Practice for Everyone

Kindly share you feedback about the website – Click here

Mathematics

Class – IX

Time Allowed: 3 hours Maximum Marks: 80

General Instructions:

1. This question paper contains two parts A and B

2. Both Part A and Part B have internal choices

Part – A:

1. It consists of two section I and II

2. Section I has 22 questions. Internal choices provided in 7 questions.

3. Section II has Q.23 and 24 two case study-based questions. Each case study has 5 case –based sub parts. An examinee is to attempt any 4 out of 5 sub-parts.

Part – B:

1. Question No 25 to 31 are Very Short Answer type questions of 2 marks each

2. Question No 32 to 38 are Short Answer type questions of 3 marks each

3. Question No 39 to 41 are Long Answer type questions of 5 marks each.

4. Internal choice is provided in 2 questions of 2 marks, 2 questions of 3 marks and 1 question of 5 marks.

Part – A

Section – I

1. Write a rational number between √2 and √3

View Answer

Ans. We know that,

√2 = 1.4142135….. and √3 = 1.732050807…

We see that 1.5 is a rational number which lies between 1.4142135….. and 1.732050807…

How many rational numbers are there between 13 and 17?

View Answer

Ans. There are infinite rational number between 13 and 17.

Eg. 13.4, 13.5……., 14.5, 14.6…….

2. Which of the following is irrational?

√4/7, √7

View Answer

Ans. √4/7 = 2/7 is a rational number

√7 is an irrational number

3. Which one of the following is a polynomial?

View Answer

Ans.

How many zeroes are there in cubic equation?

View Answer

Ans. 3 zeroes are there in a cubic equation

4. If (2, 0) is a solution of the linear equation 2x + 4y = k, then find the value of k

View Answer

Ans. Put x = 2, y = 0 in the equation

2x + 3y = k

2(2) + 3(0) = k

4 + 0 = k

Hence, k = 4

5. What is the ordinate of the points that lie on the x-axis?

View Answer

Ans. Because ordinate or y-coordinate of a point is perpendicular distance of this point from the x-axis measured along the y-axis. If a point lies on x-axis, then the perpendicular distance of a point from x-axis will be zero, so ordinate will be zero.

In which quadrant, point (-3, -4) lies?

View Answer

Ans. 3^rd quadrant

6. The angles of a triangle are in the ratio 5: 3: 7. What is the type of the triangle?

View Answer

Ans. Let the angles of triangle be 5x, 3x, 7x

According to question

5x + 3x + 7x = 180 (angle sum property)

15x = 180

x = 12

1^st angle = 5 x 12 = 60

2^nd angle = 3 x 12 = 36

3^rd angle = 7 x 12 = 84

All the angles of the triangle are acute

The triangle is an acute angled triangle

7. If bisectors of ∠A and ∠B of a quadrilateral ABCD intersect each other at P, of ∠B and ∠C at Q, of ∠C and ∠D at R and of ∠D and ∠A at S, then PQRS is ___________ whose opposite angles are ___________.

View Answer

Ans. PQRS is a quadrilateral whose opposite angles are supplementary.

8. ABCD is a cyclic quadrilateral such that AB is a diameter of the circle circumscribing it and ∠ADC = 140°, then find ∠BAC.

View Answer

Ans. Given that,

∠ADC + ∠ABC = 180

140 + ∠ABC = 180

∠ABC = 180 – 140

= 40

ABCD is a cyclic quadrilateral such that AB is the diameter of the circle circumscribing it.

Now, join AC. ∠C = 90 (Angles in a semi-circle is a right angle)

In ΔABC, we have

∠BAC = 180 – (90 + 40) = 50

9. The radius of a hemispherical balloon increases from 6 cm to 12 cm as air is being pumped into it. What is the ratios of the surface areas of the balloon in the two cases?

View Answer

Ans. Radius r = 6 cm

Surface area = 3 πr² = 3 π x 6² = 3 π x 6 x 6

Radius R = 12 cm

Surface area = 3 πR² = 3 π x 12² = 3 π x 12 x 12

Ratio = 3 π x 6 x 6 : 3 π x 12 x 12

= 1 : 4

10. Factorize: x² – 3x

View Answer

Ans. x(x – 3)

11. ∠A + ∠B + ∠C is in ΔABC = __________

View Answer

Ans. Sum of measures of all the angles of a triangle is 180

A ray has __________ end point.

View Answer

Ans. A ray has one end point

It is also called fixed point

12. What are the three steps from solids to points?

View Answer

Ans. The three steps are: solids – surface – lines – points

13. Bisector of an angle divides it in _________ equal parts.

View Answer

Ans. Two

14. The angles of a quadrilateral are in the ratio 2: 3: 6: 7. The largest angle of the quadrilateral is ________

View Answer

Ans. Let the angle of the quadrilateral be 2x, 3x, 6x and 7x

2x + 3x + 6x + 7x = 360° (Angle sum property of quadrilateral)

Or 18x = 360°

X = 20°

Largest angle = 7x = 7 x 20° = 140°

15. Write an example of a constant polynomial

View Answer

Ans. Constant polynomial is 7

Name of the polynomial on the basis of degree: -3x + 2

View Answer

Ans. Linear polynomial

16. In the given figure, PQ|| RS and EF || QS. If ∠PQS = 60°, then what will be the measure of ∠RFE?

View Answer

Ans. ∠PQS + ∠QSF = 180 (Consecutive Interior angle)

Or ∠PQS + ∠RFE = 180, as ∠QSF = ∠EFR

Or 60 + ∠RFE = 180 (corresponding angles)

∠RFE = 120

17. During maths lab activity, teacher give four sticks of length 6cm, 6 cm, 4 cm and 4 cm to each student to make different types of quadrilateral. Write the name of quadrilateral that can be formed with these sticks.

View Answer

Ans. Kite, rectangle, parallelogram

Opposite sides of a parallelogram and a rectangle are equal. Also, adjacent sides of kite are equal. Thus kite, rectangle and parallelogram can be formed with the sticks of lengths 6 cm, 6 cm , 4 cm and 4 cm

18. Is trapezium is a parallelogram?

View Answer

Ans. A trapezium has only one pair of parallel sides. Thus, it cannot be a parallelogram

19. Find any point on the line y = x

View Answer

Ans. Every point on the line y = x has same value of x and y-coordinates, i.e., x = a and y = a. Hence (a, a) is a point on the line y = x

In 5x – 8y = 17, express y in terms of x

View Answer

Ans. 5x – 8y = 17

5x – 17 = 8y

y = (5/8)x – (17/8)

20. What is a straight line?

View Answer

Ans. Two planes intersect each other to form a straight line

Who was the teacher of Pythagoras?

View Answer

Ans. Thales was the teacher of Pythagoras.

21. Can the angles 110, 80, 70 and 95 be the angles of a quadrilateral? Why or why not?

View Answer

Ans. Here, sum of angles = 110 + 80 + 70 + 95 = 355

But we, know that the sum of angles of a quadrilateral is always 360°. Hence 110°, 80°, 70° and 95° cannot be the angles of a quadrilateral.

22. If x⁵¹ + 51 is divided by x + 1, then find remainder.

View Answer

Ans. Let p(x) = x⁵¹ – 51 ……… (i)

i.e.,

p(-1) = (-1)⁵¹+ 51

= -1 + 51

= 50

Hence, remainder is 50

Section – II

23. Nikita has to make her project on ‘Monument in India’. She decided to make her project on Gol Gumbaz monument. She already known following things about it:

It is located in a small town in Northern Karnataka
It reaches up to 51 metres in height while the giant dome has an external diameter of 44 meters, making it one of the largest domes ever built.
At each of the four corners of the cube is a dome shaped octagonal tower seven stories high with a staircase inside.

Help her in making project by answering the following questions:

A) What is the total area of a hemispherical dome?

i) 1/3 πr²

ii) 3 πr²

iii) 1/2 πr²

iv) 2 πr²

View Answer

Ans. ii) 3 πr²

B) What is the curved surface area of hemispherical dome?

i) 908 πm²

ii) 968 πm²

iii) 340 πm²

iv) 780 πm²

View Answer

Ans. ii) 968 πm²

Diameter = 44 m

Radius = 22 m

Curved surface area of hemispherical dome

= 2πr²

= 2π (22)²

= 968π m²

C) What is the volume of the sphere?

i) 4/3 πr³

ii) 2/3 πr³

iii) 1/3 πr³

iv) 1/2 πr³

View Answer

Ans. i) 4/3 πr³

D) What is the circumference of the base of the dome?

i) 34 π

ii) 22 π

iii) 44 π

iv) 55 π

View Answer

Ans. iii) 44 π

Circumference of the base of the dome = 2πr

= 2π(22)

= 44 π

E) Find the cost of painting the dome, given the cost of painting is Rs 100 per cm²

i) ₹980π

ii) ₹9690π

iii) ₹968000π

iv) ₹9700π

View Answer

Ans. iii) ₹968000π

Curved surface area of hemispherical dome

= 968 π m²

Cost of painting 100 cm²= ₹10

Cost of painting 1 m² = ₹1000

Thus, cost of painting the dome = ₹1000 x 968 π cm²

= ₹968000 π

24. Prime Minister’s National Relief Fund (also called PMNRF in short) is the fund raised to provide support for people affected by natural and man-made disasters. Natural disasters that are covered under this include flood, cyclone, earthquake etc. Man-made disasters that are included are major accidents, acid attacks, riots, etc.

Two friends Sita and Gita, together contributed Rs 200 towards Prime Minister’s Relief Fund. Answer the following:

A) Which out of the following is not the equation in two variables?

i) 2x = 3

ii) 4 = 5x – 4y

iii) x² + x = 1

iv) x – √2y = 3

View Answer

Ans. iii) x² + x = 1

x² + x = 1 is not linear as highest power of 2. Also it is an equation in one variable. Thus, it is not a linear equation in two variables.

B) How to represent the above situation in linear equations in two variables?

i) 2x + y = 200

ii) x + y = 200

iii) 200x = y

iv) 200 + x = y

View Answer

Ans. ii) x + y = 200

Here, x represents Sita’s contribution and y repreents Gita’s contribution.

C) If Sita contributed ₹76, then how much was contributed by Gita?

i) ₹120

ii) ₹123

iii) ₹124

iv) ₹125

View Answer

Ans. iii) ₹124

If x = 76 then 76 + y = 200

y = 200 – 76

y = 124

D) If both contributed equally, then how much is contributed by each?

i) ₹50, ₹150

ii) ₹100, ₹100

iii) ₹50, ₹50

iv) ₹120, ₹120

View Answer

Ans. ii) ₹100, ₹100

If x = y then x + x = 200

2x = 200

X = 100

Thus each contribution is ₹100

E) Which is the standard form of linear equations x = – 5?

i) x + 5 = 0

ii) 1.x – 5 = 0

iii) 1.x + 0.y + 5 = 0

iv) 1.x + 0.y = 5

View Answer

Ans. iii) 1.x + 0.y + 5 = 0

Since, x= -5

X + 5 = 0

Thus, standard form of x = -5 is 1.x + 0.y + 5 = 0

Part – B

25. If 7x = 1, then find the decimal expansion of x

View Answer

Ans.

26. If (3x – 15°) and (x + 5°) are complementary angles, find the angles

View Answer

Ans. (3x – 15°) + (x + 5°) = 90

4x = 90 + 10 = 100

X = 100/4 = 25

Angles are 60 and 30

In the given figure, find the angles a and b and then show that LM || XY

View Answer

Ans. ∠a = 180 – 50

= 130 (Linear pair)

∠b = 130 (Since vertically opposite angles are equal)

This shows that ∠a = ∠b

But they are alternate interior angles.

LM || XY

27. In the figure, ΔABC and ΔDBC are two isosceles triangles on the same base BC. Prove that ∠ABD = ∠ACD

View Answer

Ans. Join AD

In ΔABD and ΔACD,

AB = AC (Given)

BD = CD (Given)

AD = AD (Common)

By using SSS Congruency Rule

ΔABD ≅ ΔACD

∠ABD = ∠ACD (CPCT)

28. In the given figure, find the value of x.

View Answer

Ans. In a cyclic quadrilateral,

∠A + ∠C = 180 (opposite angles of cyclic quadrilateral are supplementary)

2x + 4 + 4x – 64 = 180

6x – 60 = 180

6x = 180 + 60

x = 240/6

x = 40

29. In the given figure, BC is a diameter of the circle and ∠BAO = 60°, then find ∠ADC

View Answer

Ans. In ΔOAB,

OA – OB (Radii of the same circle)

∠ABO = ∠BAO (Angles opposite to equal sides are equal)

∠ABO = ∠BAO= 60 (Angles in the same segment of a circle are equal)

Therefore ∠ADC = 60

In the figure, if ∠DAB = 65°, ∠ABD = 45°, then find ∠ACB.

View Answer

Ans. In ΔADB, using angle sum property

∠ABD + ∠ADB + ∠BAD = 180

45 + ∠ADB + 65 = 180

∠ADB = 180 – (45 + 65)

= 70

Thus, ACD = AOB = 70 (Angles in the same segment of a circle are equal)

30. Find the area of a triangle, two sides of which are 8 cm and 11 cm and the perimeter is 32 cm.

View Answer

Ans. Let the side AB(c) = 8 cm

Side AC(b) = 11 cm

Perimeter of ΔABC = 32 cm

a + b + c = 32 cm

a + 8 + 11 = 32

a = 32 – 19 = 13 cm

ar(ΔABC) = |√s(s – a)(s – b)(s – c)|

ar(ΔABC) = |√16(16 – 13)(16 – 8)(16 – 11)|

= |√16 x 3 x 8 x 5|

= |√4 x 4 x 2 x 2 x 2 x 3 x 5|

= |4 x 2√2 x 3 x 5|

ar(ΔABC) = 8√30 cm²

31. Coronavirus disease (COVID-19) is an infectious disease. It is caused by a newly discovered coronavirus. Most people who fall sick with COVID-19 experience mild to moderate symptoms and recover without special treatment

Coronavirus is mainly transmitted through droplets generated when an infected person coughs, sneezes or exhales. Look at the bar graph and answer the following:

i) What is horizontal line representing?

View Answer

Ans. Number of cases

ii) How many confirmed cases were more than recovered cases?

View Answer

Ans. Required number = confirmed cases – recovered cases

= 8089593 – 7371898

= 717695

32. Find three rational numbers between 5/7 and 9/11.

View Answer

Ans.

View Answer

Ans.

33. Point A(4, 2), B(-1, 2) and D(4, -5) are three vertices of a rectangle ABCD. Plot these points and hence find the vertex C.

View Answer

Ans.

34. If f(x) = x³ – 3x² + 3x – 4, find f(2) + f(-2) + f(0)

View Answer

Ans. Given f(x) = x³ – 3x² + 3x – 4

f(2)= (2)³ – 3(2)² + 3(2) – 4

= 8 – 12 + 6 – 4

f(2) = -2

and f(-2) = (-2)3 – 3(-2)2 + 3(-2) – 4

= -8 – 12 – 6 – 4

f(-2) = -30

And f(0) = -4

Therefore f(2) + f(-2) +f(0) = -2 – 30 – 4 = -36

Draw the quadrilateral ABCD whose vertices are A(0, 0), B(5, 0), C(3, 2) and D(0, 2)

View Answer

Ans.

35. Prove that if two lines intersect, vertically opposite angles are equal.

View Answer

Ans. Given Two lines AB and CD intersect at O

To prove

i) ∠AOC = ∠BOD

ii) ∠AOD = ∠BOC

Proof: Ray OA stands on line CD

Therefore ∠AOC + ∠AOD = 180 (linear pair) ——(i)

Again ray OD stands on line AB

Therefore ∠AOD + ∠BOD = 180 (linear pair) ——(ii)

From eqn (i) and (ii)

∠AOC + ∠AOD = ∠AOD + ∠BOD (Each equal to 180)

Therefore ∠AOC = ∠BOD

Similarly ∠AOD = ∠BOC

36. Find the area of a triangular field of sides 18 m, 24 m and 30 m. Also find the altitude corresponding to the shortest side.

View Answer

Ans. s = (18 + 24 + 30)/2 = 36 m

Area = √s(s – a)(s – b)(s –c)

= √36(36 – 18)(36 – 24)(36 – 30)

= √36 x 18 x 12 x 6

= 216 m²

Altitude (height) corresponding to the shortest side

= 2 x Area/Base = 2 x 216/18 = 24 m

37. How many litres of milk can a hemispherical bowl of diameter 10.5 cm hold? Use π = 3.14)

View Answer

Ans. Given, diameter of hemispherical bowl = 10.5 cm

Radius of hemispherical bowl r = 10.5/2 = 5.25 cm

Volume of hemispherical bowl = 2/3 πr³

= 2/3 x 3.14 x (5.25)³

= 302.91

= 303 cm³

Therefore, Amount of milk that the hemispherical bowl can hold = 0.303 litres

38. ABCD is a rhombus. Show that the diagonal AC bisect ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D.

View Answer

Ans. Given ABCD is a rhombus

To prove:

∠BAC = ∠DAC and ∠DCA = ∠BCA

So, AB = BC = CD = AD

In ΔADC and ΔABC

AB = AD (given)

CD = CB (given)

AC = AC (common)

So, ΔADC ≅ ΔABC (By SSS Congruency Rule)

So, ∠DAC = ∠BAC (CPCT)

Similarly,

ΔDCA ≅ ΔBCA

∠DCA = ∠BCA (CPCT) Hence proved

39 a) The blood groups of 30 students of Class IX are recorded as follows:

A, B, O, O, AB, O, A, O, B, A, O, B, A. O, O

A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O

i) Represent this data in the form of a frequency distribution table

ii) Which is the most common and which the rarest blood group among these students?

View Answer

Ans. i) Frequency Distribution Table:

ii) Blood group ‘O’ is most common as it has highest frequency i.e., 12. Blood group AB is rarest.

b) Facts or information collected with a definite purpose are called.

View Answer

Ans. Data

40. In the given figure, ABCD and BPQ are straight lines. If BP = BC and DQ is parallel to CP prove that:

i) CP = CD

ii) DP bisects CDQ

View Answer

Ans. BP = BC (Given)

∠BCP = ∠BPC

= y

∠1 = y (corresponding angles)

∠2 = x (Alternate angles)

∠3 = ∠1 + ∠2 (external angle)

= x + y

4x = 2y (external angle)

∠2 = x (Alternate angles)

Y = x + ∠3

Or x = ∠3

Or CP = CD

Also, we have ∠2 = x, x = ∠3

Or DP bisects ∠CDQ

Hence proved

A student Amit of class IX is unable to write in his examination, due to fracture in his arm. Akhil a student of class VI writes for him. The sum of their ages is 25 years.

i) Write a linear equation for the above situation and represent it graphically.

ii) Find the age of Akhil from the graph, when age of Amit is 14 years

View Answer

Ans. Let age of Amit = x years

Age of Akhil = y years

i) According to the question the linear equation for the above situation is

x + y = 25

or y = 25 – x

ii) From the graph when Amit’s age = 14 years, then Akhil’s age = 11 years

41. It is known that if a + b = 10, then a + b – c = 10 – c. Write the Euclid’s axiom that best illustrates this statement. Also give two more axioms other than the axiom used in the above situation.

View Answer

Ans. Axiom: If equals are subtracted from equals, the remainders are equal.

Two more axioms:

i) Things which are halves of the same thing are equal to one other

ii) The whole is greater than the part

Class IX – Mathematics – Paper – 1