Class 9 Question Paper 2024 Maths (with Answers)

ANNUAL EXAMINATION

Question Paper
2024

NCERT BASED SYLLABUS
FOR CBSE AND STATE BOARD
FOLLOWING NCERT

Annual Exam Question Paper 2024

SESSION ENDING EXAMINATION 2023-24
Class – IX Subject – Mathematics
Time: 3 Hours Max. Marks: 80
General Instructions:
1. This Question Paper has 5 Sections A, B, C, D, and E.
2. Section A has 20 Multiple Choice Questions (MCQs) carrying 1 mark each.
3. Section B has 5 Short Answer-I (SA-I) type questions carrying 2 marks each.
4. Section C has 6 Short Answer-II (SA-II) type questions carrying 3 marks each.
5. Section D has 4 Long Answer (LA) type 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 Qs of 5 marks, 2 Qs of 3
marks and 2 Questions of 2 marks has been provided. An internal cho ice has been
provided in the 2marks questions of Section E
8. Draw neat figures wherever required. Take π =22/7 wherever required if not stated.

Section A
1 Factors of 6x2 + 5x -6 are 1
a. (x – 6) (x+5) b. (2x – 2) (x – 3) c. (2x -3) (3x +2) d. (2x +3) (3x – 2)
2 The bisectors of any two adjacent angles of a parallelogram intersect at: 1
(a) 30 degree (b) 45 degree (c) 60 degree (d) 90 degree
3 Which is not the criteria of congruency? 1
(a) SSS (b) AAS (C) AAA (d) SAS
4 The linear equation x + 0y + 9 = 0 in two variables is 1
a) parallel to x axis b) parallel to y axis c) passing through origin d) none of these
5 Which of the following is not a solution of linear equation 2x – 3y = 12 1
(a) (0,-4) (b) (2, 3) (c) (6, 0) (d) (3, -2)
6 An angle is 18 degree less than its complementary angle. The measure of this angle is 1
(a) 36 degree (b) 48 degree (c) 83 degree (d) 81 degree
7 For drawing a frequency polygon of a continuous frequency distribution, we plot the points 1
whose ordinates are the frequencies of the respective classes and abscissa are respectively:
(a) upper limits of the classes (b) lower limits of the classes
(c) class marks of the classes (d) upper limits of preceding classes
8 The value of semi-perimeter of an equilateral triangle having area 4√3 sq cm is 1
(a) 8 cm (b) 36 cm (c) 16 cm (d) 6 cm
9 Euclid stated that all right angles are equal to each other in the form of 1
(a) A Postulate (b) A Proof (c)An Axiom (d)A Definition

10 0.01233333….. can be expressed in rational form 1
as
(d)111/900
(a)900/111 (b) 111/9000 (c)123/100

11 The expanded form of (3x – 5)3 is: 1
3 2
(a) 27x + 135x + 225x – 125
(b) 27x3 + 135x2 –225x + 125
(c ) 27x3 – 135x2 + 225x – 125
(d)None of the above
12 How many linear equations can be satisfied by x = 2 and y = 3? 1
(a) Only one (b) many (c) two (d) none of these
13 Abscissa of a point is negative in a
(a)Quadrant IV only (b) Quadrant II and III(c)Quadrant I and IV (d) Quadrant I only

14 The ordinate of any point on x – axis 1
(a) 0 (b) -1 (c) 1 (d) any
15 If √5 = 2.236, then 1/√5= ? 1
(a) 44.72 (b) 0.4472 (c) 0.04472 (d) 4.472
16 D and E are the mid points of sides AB and AC of a ∆ ABC. If BC = 5.6 cm, find DE. 1
(a) 2.8 cm (b) 3 cm (c) 2.9 cm (d) 2.5 cm
17 A polynomial of two terms is called 1
(a) quadratic (b) binomial (c) monomial (d) cubic
18 Find the value of k , if x=1 , y= -3 is a solution of equation 2x – 3y=k 1
(a) 32 (b) 10 (c) 11 (d) 22
3 2
19 Assertion (A): If (x – 1) is a factor of 4x + 3x - 4x +k , then k = -3 1
Reason (R): (x – a) is a factor of the polynomial p(x) if p(a)=0.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion
(A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of
assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
20 Assertion (A): 2 + √6 is an irrational number. 1
Reason (R): Sum of a rational number and an irrational number is always an irrational
number.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion
(A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of
assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Section B
21 A shot-putt is a metallic sphere of radius 4.9 cm. if the density of the metal is 7.8g per cm cube , 2
find the mass of the shot-putt.

22 3
Factorise 8𝑥 - (2𝑥 − 𝑦)
3 2

23 Write the coordinates of the point: 2
i) Which lie on x and y axes both.
ii) Whose ordinate is -4 and which lies on y axis.
24 If point C lies between two points A and B such that AC = BC, then prove that AC = 1/2𝐴𝐵. 2
Explain by drawing the figure.
OR
Why is Axiom 5, The whole is greater than a part, considered a “universal truth
25 The surface areas of two spheres are in the ratio 1:4, find the ratio of their volumes. 2
OR
Find the capacity of a conical vessel in litres with height 12 cm and slant height 13 cm

Section C

26 If 𝑥 −
1
= 5, find the value of 3
𝑥

3 1
𝑥 − 3
𝑥

OR
3 2
Factorise: 𝑥 − 23𝑥 + 142𝑥 − 120
27 Represent √9.3 on number line. 3
28 Prove that the angle subtended by an arc at the centre is double the angle subtended by any point on 3
the remaining part of the circle.
29 Triangle ABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that 3
AD=AB . So that angle BCD is a Right angle.
OR
In right triangle ABC, right angled at C, M is
the mid-point of hypotenuse AB. C is joined
to M and produced to a point D such that
DM = CM. Point D is joined to point B
(see Figure). Show that:
(i) Δ AMC ≅ Δ BMD
(ii) ∠ DBC is a right angle.
(iii) Δ DBC ≅ Δ ACB

30 A triangular park ABC is inside the campus of green valley apartments. All age group peoples 3
used to visit the park. This park has sides 120m, 80m and 50m. A gardener dhania has to put a
fence all around it and also plant grass inside. Based on the above information answer the
following questions.
i) How much area does he need to plant grass?
ii) If the shape of the park is similar to equilateral triangle of perimeter 180 m, then what will

be the area of the park?

31 3
Find three different solutions of the equation 2x + y = 6.

Section D
32 i) In the figure, ray OS stands on a line POQ. Ray OR and OT are angle bisectors of ∠𝑃OS and 2
∠SOQ, respectively. If ∠𝑃OS= 2a , find ∠ROT

ii) In given figure, PQ and RS are two mirrors placed
parallel to each other. An incident ray AB strikes
the mirror PQ at B, the reflected ray moves along 3
the path BC and strikes the mirror RS at C and
again reflects back along CD. Prove that
AB || CD.

33 Monica has a piece of canvas whose area is 551 m2 . She uses it to have a conical tent made, with a 5
base radius of 7 m. assuming that all the stitching margins and the wastage incurred while cutting,
amounts to approximately 1 m2, Find the volume of the tent that can be made with it.

OR
A right triangle ABC with sides 5 cm, 12 cm and 13 cm.
(a) If the triangle ABC is revolved about the side 12 cm, then find the volume of the solid so
obtained.
(b) If the triangle ABC is revolved about the side 5 cm, then find the volume of the solid so
obtained.
(c) Find the ratio of the volumes of the two solids obtained

34 In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig.). 5
Show that:

(i) Δ APD ≅ Δ CQB
(ii) AP = CQ
(iii) Δ AQB ≅ Δ CPD
(iv) AQ = CP
(v) APCQ is a parallelogram
OR
ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel

to BC intersects AC at D. Show that:
(i) D is the mid-point of AC
(ii)MD ⊥AC
(iii) CM = MA = 1/ 2 AB

35 .Draw a Frequency Polygon to represent the following grouped frequency distribution : 5

Age in 20-24 25-29 30-34 35-39 40-44 45-49 50-54
years
No.of 10 28 32 48 50 35 12
teachers

Section E
36 Two partners start a business together. They decided to share their capitals depending upon the
variable expenditure. The capital of two partners together is given 4x² – 8x – 5 which is the product
of their individual share factors.
a. Write the degree of given polynomial and also write the name of polynomial on the basis of 1
degree. 1
b. Name of the polynomial of amount invested by each partner. 2
c. What are the shares of two partners invested individually?
OR
What is the total amount invested by both, if x = 100.
37 Sanjay and his mother visited in a mall. He
observes that three shops are situated at P, Q and
R as shown in the figure from where they have
to purchase things according to their need.
Distance between shop P and Q is 8m and
distance between shop P and R is 6m.
Considering O as centre of the circle.
Answer the following questions
(a) Find the measure of ∠ QPR 1
(b) Find the radius of the circle. 2
OR
Find the perimeter of the circle. 1
(c) Find the measure of ∠ QSR
38 To judge the preparation of students of class 9th on the topic rationalisation of irrational numbers,
Mathematics teacher defined rationalisation saying that when the denominator of an expression
contains the term with the square root, the procedure of converting it to an equivalent expression
whose denominator is rational number is called rationalising that denominator
Based on the above information answer the following questions
1
a) Rationalize 1
3+ √2
b) Classify the following as Rational or irrational number ( 6 – √27) – ( –√27 -8) 1
2
c) Simplify (√14 − 2√2) 2
OR
7
Rationalize the denominators of
11− √5 4

*****************************************

SESSION ENDING EXAMINATION 2023-24

Class – IX Marking Scheme Subject - Mathematics
S. Section A Marks
No. Section A consists of 20 questions of 1 mark each
1 D 1
2 D 1
3 C 1
4 B 1
5 B 1
6 A 1
7 C 1
8 D 1
9 A 1
10 B 1
11 C 1
12 B 1
13 B 1
14 A 1
15 B 1
16 A 1
17 B 1
18 C 1
19 A 1
20 A 1
Section B consists of 5 questions of 2 marks each
21 493 cm cube ( nearly) ( correct 1+1
formula, correct calculation,
correct answer)

3845.44 g(nearly )
22 1+1
⌊(2𝑥 − (2𝑥 − 𝑦)⌋ {(2𝑥)2 + 2𝑥(2𝑥 − 𝑦) + (2𝑥 − 𝑦)2 }
𝑦(12𝑥 2 − 6𝑥𝑦 + 𝑦 2 )

23 (0,0) (0,-4) 1+1
24 1
1

2

Or
Correct
explanation
25 1+1

Section C consists of 6 questions of 3 marks each

26

1

1

1

or
P(x) = x3– 23x2 + 142x – 120
P(1) = 0 , (x-1) is factor of p(x) 1
P(x) = x2(x-1) -22x(x-1) + 120 (x-1)
= (x-1) (x2 - 22x + 120) 1

=(x-1)(x-10)(x-12) 1
27 For correct representation + construction 1+2
28 For correct diagram 1
For correct proof 2

29 For correct figure 1
For correct proof 2
OR
In ΔAMC and ΔBMD 1+1+1
AM=BM (M is midpoint of AB)
∠AMC=∠BMD(vertically opposite angles)CM=DM (given)
∴ΔAMC≅ΔBMD (by SAS congruence rule)
∴ AC=BD (by CPCT)
⇒∠DBC+∠ACB=180∘(co−interior angles)
⇒ ∠DBC+90∘=180∘(∠ACB=90∘)⇒∠DBC=180∘−90∘⇒∠DBC=90∘
⇒DB=AC (By CPCT)…(i)
In ΔDBC and ΔACB
DB=AC (From (i))BC=BC(Common)∠DBC=∠ACB=90∘
∴ΔDBC≅ΔACB
by SAS
congruence
30 i) 375√15m² (Using correct formula + correct calculation + correct answer) ½
ii) 900√3m² (Using correct formula + correct calculation + correct answer) +1/2+1/2
½+1/2+
1/2
31 For 3 correct solutions 1+1+1
Section D

Section D consists of 4 questions of 5 marks each
32 1
i)∠𝑃OS + ∠SOQ = 1800 1
∠ROT=900
Ii) Since BE and FC are normal to PQ and RS respectively, therefore, BE||FC
Let, ∠ABE=∠EBC=x[PQ is a mirror, so angle of incidence is equal to angle of reflection]
∠FCD=∠BCF=y[RS is a mirror, so angle of incidence is equal to angle of reflection] .5
Now considering BE and FC, taking BC as transversal, .5

∠EBC=∠BCF........(i) [alternate interior angle]
i.e. x=y .5
i.e. ∠ABE=∠FCD......(ii)
Adding equation (i) and (ii) .5
∠EBC+∠ABE=∠BCF+∠FCD
∠ABC=∠BCD
Now if we take line AB and CD in consideration, alternate interior angles that .5
are ∠ABC and ∠BCD are equal.
.5
Therefore, AB∥CD

33 find Slant height = 25 m (Using correct formula + correct calculation + correct answer) 1+1
find height = 24 m (Using correct formula + correct calculation + correct answer)
1
3
volume of the tent 1232m (Using correct formula + correct calculation + correct answer)
2
or

(a) Since the triangle is revolved about the side 12 cm, a
solid cone is formed with a height of 12 cm and radius of the base of
5 cm as shown below. 2
Volume of a cone having radius 'r', and height 'h', = 1/3πr²h
Radius of the cone, 'r' = 5 cm
Height of the cone, 'h' = 12 cm
Volume of the cone = 1/3πr²h 2
= 1/3 × π × 5 cm × 5 cm × 12 cm
= 100π cm³
Volume of the cone is 100π cm³.
1
(b) Since the triangle is revolved about the side 5 cm, a
solid cone if formed with a height of 5 cm and radius of the
base of 12 cm.
Volume of a cone having radius 'r' and height 'h' =
1/3πr²h
Radius of the cone, r =12cm
Height of the cone, h = 5cm
Volume of the cone = 1/3πr²h
= (1/3) × π × 12cm × 12cm × 5cm
= 240π cm³
( c) Ratio = Volume of the cone in (a)/ Volume of the cone in (b)
= 100π : 240π
= 5 :12
The volume
of the cone is
240π cm³ and
the
required ratio
is 5 :12

1 mark for each part
Or
34 For correct figure
(i) In ΔACB,
M is the midpoint of AB and MD || BC 1

D is the midpoint of AC (Converse of mid point theorem) 1
(ii) ∠ACB = ∠ADM (Corresponding angles) 1
also, ∠ACB = 90° 1
∠ADM = 90° and MD ⊥ AC 1
(iii) In ΔAMD and ΔCMD,
AD = CD (D is the midpoint of side AC)
∠ADM = ∠CDM (Each 90°)
DM = DM (common)
ΔAMD ≅ ΔCMD [SAS congruency] AM =
CM [CPCT]
also, AM = ½ AB (M is midpoint of AB) Hence,
CM = MA = ½ AB
35 1
For making class interval continuous
For class mark+ proper scaling 1+1/2
frequency polygon 2 1/2

Section E
Section E consists of 3 questions of 4 marks each
36
a) degree is 2 , Quadratic polynomial ½ +1/2
b) Linear 1
c) (2x+1), (2x-5) 2
or
39195

37 (a) Angle QPR = 90 degree 1
(b) Triangle QPR is a right angled
triangle 4r2 = 64 +36 2
r = 5 cm
OR
Perimeter of circle = 2𝜋𝑟
= 2 ×22/7 ×5 = 31.4 cm 1
(c) Angle QSR = Angle QPR = 90 degree ( Angle in the same segment )

38 i) √3 − √2 1
ii) Rational
iii) 22- 4√28 1
Or 2
√77 − √35
-----------------
6

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