SAMPLE PAPER

MATHEMATICS

Name: Max Marks: 80

Class: IX Duration:3 Hrs
Sec: Date: / /2024
General Instructions:
1. This question paper has 5 sections A, B, C, D, and E.
2. Section A has 20 MCQs carrying 1 mark each
3. Section B has 5 questions carrying 2 marks each.
4. Section C has 6 questions carrying 3 marks each.
5. Section D has 4 questions carrying 5 marks each.
6. Section E has 3 case-based integrated units of assessment (4 marks each) with sub-parts
of the values of 1, 1 and 2 marks each respectively.
7. All questions are compulsory. However, an internal choice in 2 questions of 5 marks, 2 questions of
3 marks and 2 questions of 2 marks has been provided. An internal choice has been provided in the
2 marks questions of section E.
8. Draw neat figures wherever required. Take π =22/7 wherever required if not stated.

SECTION A

Section A consists of 20 questions of 1 mark each.

1 Any point on the y-axis is of the form 1

a) (𝑥, 𝑦) b) (0, 𝑦) c) (𝑥, 0) d) (𝑥, 𝑥)
2 Euclid’s fifth postulate is 1
(a) The whole is greater than the part.
(b) A circle may be described with any centre and any radius.
(c) All right angles are equal.
(d) If a straight line falling on two straight lines makes the interior angles on the same side of it
taken together less than two right angles, then the two straight lines if produced indefinitely,
meet on that side on which the sum of angles is less than two right angles.
3 2√3+ √3 is equal to 1
a) 2√6 b) 6 c) 3√3 d) 3√6
4 In the given figure, the ratio between ∠ABD and ∠ACD is: 1

a) 1:1 b) 2:1 c) 1:2 d) 3:1

D

B C

5 An angle is equal to one fourth of its supplement. Find its measure. 1

a) 60˚ b) 30˚ c) 18˚ d) 36˚

6 In the given figure, according to Euclid's fifth postulate, which are the two angles with the sum 1
less than two right angles?
a) ∠1 and ∠2 b) ∠2 and ∠4 c) ∠1 and ∠3 d) ∠2 and ∠3

7 The product of 5 - √3 and the other numbers is 5 +√3. Then the second number is _____ 1
14+5√3 14−5√3 28+5√3 14+10√3
a) b) c) d)
11 11 11 11

8 The area of a rectangle is 6𝑥 2 + 5x – 6. The possible dimensions of its length and breadth are 1

a) (2x-3), (3x-2) b) (2x+3), (3x-2) c) (2x-3), (3x+2) d) (2x+3), (3x+2)

9 In figure, if AB || CD || EF, PQ || RS, ∠RQD = 25° and ∠CQP = 60°, then ∠QRS is equal 1

(a) 85° (b) 135° (c) 145° (d) 110°

10 In figure 10.2, AB and CD are two equal chords of a circle with centre O. OP and OQ are 1
perpendiculars on chords AB and CD, respectively. If ∠POQ = 150º, then ∠APQ is equal to

(a) 30 ֯ (b) 75 ֯ (c) 15 ֯ (d) 60 ֯

11 The graph of line x – y = 0 passes through: 1

(a) (2, 3) (b) (3, 4) (c) (5, 6) (d) (0, 0)

2
12 An isosceles right triangle has area 8 𝑐𝑚 . The length of its hypotenuse is 1
(a ) √32 cm (b) 16 cm (C) √48 cm (D) √24 cm
𝑟
13 The total surface area of a cone whose radius is 2 and slant height 2l is 1
(a) 2πr (l + r) (b) πr (l + 4 r ) (c) πr (l + r) (d) 2πrl

14 Find the third side of a triangle whose other two sides are 3 cm and 4 cm and the semi 1
perimeter is 6 cm.
a) 10cm b) 12cm c) 5cm d) 16cm
15 If AB = 12 cm, BC = 16 cm and AB is perpendicular to BC, then the radius of the circle passing 1
through the points A, B and C is
a) 6 cm b) 8 cm c) 10 cm d) 12 cm

16 The cost of book (x) exceeds twice the cost of pen (y) by 10 rupees. This statement can be 1
expressed as linear equation as:

a) x -2y-10=0 b) 2x-y-10=0 c) 2x+y-10=0 d) x-2y+10=0

17 The length of each side of an equilateral triangle having an area of 9 √3 𝑐𝑚2 is 1

(a) 8 cm (b) 36 cm (c) 4 cm (d) 6 cm

18 The graph of line x – y = 0 passes through: 1

(a) (2, 3) (b) (3, 4) (c) (5, 6) (d) (0, 0)

 DIRECTION: In question numbers 19 and 20, a statement of assertion (A) is followed by a
statement of reason (R). Choose the correct option.
19 Assertion (A): In the given figure, ∠ABC = 70° and ∠ACB = 30°. Then, ∠BDC = 80°. 1

Reason (R): Angles in the same segment of a circle are equal.

a) Both A and R are correct and R is the correct explanation for A.
b) Both A and R are correct and R is not the correct explanation for A.
c) A is true but the R is false.
d) A is false but R is true.
20 Assertion (A): If f(x) =5𝑥 7 – 3𝑥 6 + 𝑥 + 2 is a polynomial, then its degree is 7. 1
Reason (R): The degree of a polynomial is the highest power of the variable in it.
a) Both A and R are correct and R is the correct explanation for A.
b) Both A and R are correct and R is not the correct explanation for A.
c) A is true but the R is false.
d) A is false but R is true.
SECTION B

Section B consists of 5 questions of 2 marks each.

21 If the point (3, 4) lies on the graph of 3y = ax + 7, then find the value of a. 2
OR
5
Determine the point on the graph of the equation 2x + 5y = 20 whose x-coordinate is 2 times its
ordinate.
22 In figure, PQ = PR and ∠Q = ∠R. Prove that ∆ PQS ≅ ∆ PRT. 2

23 In the given figure, EF || DQ and AB || CD. If FEB = 64°, PDC = 27°, then find PDQ, AED and DEF. 2
24 The sides of a triangular field are 41 m, 40 m and 9 m. Find the number of rose beds that can be 2
prepared in the field, if each rose bed, on average need 900 𝑐𝑚2 space.
25 If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the 2
quadrilateral, prove that it is a rectangle.
OR
ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠ CAD = ∠ CBD.

SECTION C

Section C consists of 6 questions of 3 marks each.

26 In the given figure, AP = BQ, PR = QS. ∠APS= ∠ BQR= 90°. Show that ΔAPS ≅ ΔBQR. 3
Give reason and also mention the criteria of congruence.

OR
Prove that “Angles opposite to equal sides of a triangle are equal”.

27 𝑝 3
Express 0.3 + 1. ̅24
̅̅̅ in the form , where p and q are integers and q ≠ 0 .
𝑞
28 Draw the graph of the linear equation 2x + 3y = 12. At what points, the graph cuts 3
the x-axis and the y-axis.

29 In the given figure, ∠1 = 55°, ∠2 = 20°, ∠3 = 35° and ∠4 = 145°. Prove that AB || CD. 3
30 Ashima donated a certain amount of money to a blind school. Her friend Manya wanted to 3
know the amount donated by her, but Ashima did not disclose the amount she donated, instead
1 1
she gave her a hint that if (𝑥 + 𝑥 ) = Rs. 7 then the amount donated by her is Rs. (𝑥 3 + 𝑥 3 ). Find
the amount donated by Ashima to the school.
31 If circles are drawn taking two sides of a triangle as diameters, prove that the point of 3
intersection of these circles lie on the third side.
SECTION D

Section D consists of 4 questions of 5 marks each.

32 Factorise 3𝑥 3 – 𝑥 2 – 3𝑥 + 1 5
OR
If the polynomials 𝑎𝑧 + 4𝑧 + 3𝑧 – 4 and 𝑧 3 – 4𝑧 + 𝑎 leave the same remainder when
3 2

divided by z – 3, find the value of a.

33 5+2√3 5
If = 𝑎 − 𝑏√3, find a and b where a and b are rational numbers.
7+√3

34 Draw a frequency polygon for the following data: 5

OR

Draw a histogram for the following data:

35 In a right triangle of given dimensions is revolved about one of its side as shown in the figure. 5
What is the volume of the solid figure so obtained? (Use 𝜋 = 22/ 7).
 SECTION E

Section E consists of 3 Case study-based question of 4 marks.

36 Cuboidal shapes are considered best in the category of rigid packaging as it is most suited across
the packaging supply chain both for performance & cost. So, Harry decided to pack his products
in a cuboidal box of breadth x centimetres, length 3 cm longer than the breadth and height 4 cm
shorter than length.

i. Express the volume of the cuboidal box as a function of x. 1

ii. How can you classify the given polynomial in terms of its degree? 1
iii. a) What would be the volume if breadth of the cuboid is 4 cm? 2
OR
b)When breadth is 5, surface area of the box is 16 sq.cm and the surface area is
represented by the equation 3𝑥 2 − ax − 4. Find the values of a and b.

37 A committee was appointed to estimate the prevalence and the socioeconomic and
demographic consequences of tobacco consumption in India. A survey was conducted by the
committe where they found that the number of people who die of lung cancer due to smoking
(Y) is three times the number of people who die of lung cancer due to other diseases (X).
i. Represent the above situation as a linear expression. 1
1
ii. Name the type of linear equation obtained in question (i)
iii. a) Find the number of people who die of lung cancer due to smoking, if the number of
2
people who die of lung cancer due to other diseases is 28.
OR
b)How many people die of lung cancer due to other diseases, if the number of people
who die of lung cancer due to smoking is 90.
38 3 groups of NSS students from Trinity School were on a field duty for helping people in a flood af
fected area.Group 1 had a canvas of area 551m2, which they used to make a conical tent of bas
e radius 7m. Approximately 1m2 of canvas was wasted in making the tent. 2nd group made a te
nt similar to the group 1 with same height and half the radius. 3rd group made the tent in such a
way that it could accommodate 10 persons,where each person occupied 4 m2on the base of th
e ground and 20 m3 of air to breathe.

i. What is the height of the tent made by group 1? 1

ii. Find the volume of tent made by group 1. 1
iii. a) Find the ratio of volume of tents made by group 1 and group 2.
2
OR
b) What would be the total surface area of tent made by group 2 if their tent has half the
radius and half the height of the tent created by group 1?