Saraswati Vidyamandir Logo
Book Free Orientation
Home Our Courses CBSE Resources
NCERT Solutions
Science — Class 9 Science — Class 10 About Us Contact Us Book Free Orientation
Previous Year Questions

CBSE Class 9 Mathematics
Previous Year Questions

Chapter-wise questions from 2018–2024 Annual Exams  ·  All question types  ·  Year & marks labelled

Questions are grouped by chapter and labelled with the exam year and marks. Covers CBSE annual and half-yearly exam papers from 2018 to 2024 across all 12 chapters of the current NCERT syllabus.

01 The World of Numbers 10 questions
1
Express 0.6̄ in the form p/q, where p and q are integers and q ≠ 0.
2019 Annual1 Mark
2
Find three rational numbers between 1/3 and 1/2.
2020 Annual2 Marks
3
Represent √3 on the number line.
2021 Annual2 Marks
4
Rationalize the denominator: 1 / (√7 − √6).
2022 Annual2 Marks
5
Simplify: (√5 + √2)2.
2018 Annual1 Mark
6
Evaluate: (81)3/4.
2023 Annual1 Mark
7
If a = 7 + 4√3, find the value of √a + 1/√a.
2020 Annual3 Marks
8
Represent √9.3 on the number line.
2024 Annual3 Marks
9
Simplify: (21/3 × 41/5) / 2−2/15.
2022 Annual2 Marks
10
If x = 2 − √3, find the value of x − 1/x.
2019 Annual3 Marks
02 Introduction to Linear Polynomials 10 questions
1
Find the value of the polynomial p(x) = 5x − 4x2 + 3 at x = −1.
2019 Annual1 Mark
2
Find the remainder when x3 + 3x2 + 3x + 1 is divided by (x + π).
2018 Annual1 Mark
3
Using factor theorem, show that (x − 5) is a factor of x3 − 3x2 − 4x − 30.
2022 Annual2 Marks
4
Expand using suitable identity: (x + 2y + 4z)2.
2020 Annual2 Marks
5
Evaluate using a suitable identity: 1023.
2023 Annual2 Marks
6
If x + y + z = 0, show that x3 + y3 + z3 = 3xyz.
2019 Annual3 Marks
7
Factorise: 8a3 + b3 + 12a2b + 6ab2.
2021 Annual2 Marks
8
Factorise: x3 − 23x2 + 142x − 120.
2020 Annual3 Marks
9
Factorise: 2y3 + y2 − 2y − 1.
2024 Annual3 Marks
10
Without actual division, prove that 2x4 − 5x3 + 2x2 − x + 2 is exactly divisible by x2 − 3x + 2.
2021 Annual3 Marks
03 Orienting Yourself: The Use of Coordinates 8 questions
1
In which quadrant or on which axis do each of the following points lie: (−2, 4), (3, −1), (−1, 0), (1, 2), (−3, −5)?
2019 Annual2 Marks
2
Write the coordinates of a point that lies on the x-axis at a distance of 7 units to the right of the y-axis.
2020 Annual1 Mark
3
Plot the points A(0, 2), B(3, 0), C(0, −2) and D(−3, 0) on a graph. Join them in order. What shape is formed?
2018 Annual3 Marks
4
The base of an equilateral triangle with side 2a lies along the y-axis such that the mid-point of the base is at the origin. Find the vertices of the triangle.
2021 Annual3 Marks
5
What are the coordinates of the point where the x-axis and y-axis intersect? What is this point called?
2022 Annual1 Mark
6
Write the coordinates of a point on the y-axis which is equidistant from the points (6, 5) and (−4, 3).
2023 Annual2 Marks
7
Find the mirror image of the point (2, 3) in the x-axis. Also find the mirror image of (−4, 1) in the y-axis.
2024 Annual1 Mark
8
In a Cartesian plane, plot the points P(1, 0), Q(4, 0), R(4, 3) and S(1, 3). Name the figure PQRS and find its area.
2019 Annual3 Marks
04 Exploring Algebraic Identities 8 questions
1
Find four different solutions of the equation x + 2y = 6.
2019 Annual2 Marks
2
If the point (3, k) lies on the line represented by 3x − y = 6, find the value of k.
2020 Annual1 Mark
3
Express the equation 3y − 4 = 0 in the form ax + by + c = 0 and state the values of a, b and c.
2018 Annual1 Mark
4
The taxi fare in a city is as follows: for the first kilometre the fare is &rupee;8 and for the subsequent distance it is &rupee;5 per km. If the distance covered is x km and the total fare is &rupee;y, write a linear equation for this and draw its graph.
2022 Annual3 Marks
5
Draw the graphs of y = 3x and y = 3 on the same pair of axes. Write the coordinates of the point where the two graphs intersect.
2021 Annual3 Marks
6
If x = 2, y = 1 is a solution of the equation 2x + 3y = k, find the value of k.
2023 Annual1 Mark
7
Draw the graph of 2x + y = 6 and 2x − y + 2 = 0. Shade the region bounded by these lines and the x-axis. Find the area of the shaded region.
2024 Annual4 Marks
8
Give the geometric interpretation of the equation y = 3: (i) as an equation in one variable, (ii) as an equation in two variables.
2019 Annual2 Marks
05 Introduction to Euclid's Geometry 6 questions
1
State Euclid's fifth postulate. How would you rewrite it so that it is easier to understand? Does Euclid's fifth postulate imply the existence of parallel lines? Explain.
2019 Annual3 Marks
2
In the figure, if AC = BD, then prove that AB = CD. State the Euclid's axiom used.
2020 Annual2 Marks
3
What is the difference between an axiom and a postulate? Give one example of each.
2018 Annual2 Marks
4
If AB = PQ and PQ = XY, then AB = XY. Name the Euclid's axiom that best supports this statement.
2021 Annual1 Mark
5
"Two distinct intersecting lines cannot be parallel to the same line." State whether this is true or false and justify your answer using Euclid's postulates.
2022 Annual2 Marks
6
How many lines can pass through two given distinct points? State the postulate that supports your answer. How many points does a unique line determine?
2023 Annual2 Marks
06 I'm Up and Down, and Round and Round 30 questions
1
In the figure, lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE.
2019 Annual3 Marks
2
Prove that if two lines intersect each other, then the vertically opposite angles are equal.
2020 Annual3 Marks
3
If the complement of an angle is 79°, what is its supplement?
2018 Annual1 Mark
4
In the figure, if AB ∥ CD, EF ⊥ CD and ∠GED = 126°, find ∠AGE, ∠GEF and ∠FGE.
2021 Annual3 Marks
5
Two lines AB and CD intersect at point O. If ∠AOC = 4x and ∠COB = 3x + 20°, find x and hence find all four angles.
2022 Annual3 Marks
6
In the figure, if PQ ∥ ST, ∠PQR = 110° and ∠RST = 130°, find ∠QRS.
2018 Annual3 Marks
7
If the angles of a triangle are in the ratio 5:3:7, find all three angles. Are any of these angles obtuse?
2023 Annual2 Marks
8
Prove that the sum of the angles of a triangle is 180°.
2024 Annual3 Marks
9
An exterior angle of a triangle is 110°. If one of its opposite interior angles is 45°, find the other two interior angles of the triangle.
2021 Annual2 Marks
10
In the figure, the side QR of ▵PQR is produced to a point S. If the bisectors of ∠PQR and ∠PRS meet at point T, then prove that ∠QTR = ½∠QPR.
2020 Annual3 Marks
11
In ▵ABC, AB = AC and ∠ACD = 130°. Find ∠A.
2019 Annual2 Marks
12
Prove that angles opposite to equal sides of an isosceles triangle are equal.
2022 Annual3 Marks
13
In ▵ABC, AB = BC and AP ⊥ BC, CQ ⊥ AB. Prove that AP = CQ.
2020 Annual3 Marks
14
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 B. Show that: (i) ▵AMC ≅ ▵BMD, (ii) ∠DBC is a right angle, (iii) ▵DBC ≅ ▵ACB.
2019 Annual5 Marks
15
In the figure, AD is the altitude of ▵ABC in which AB = AC. Show that: (i) AD bisects BC, (ii) AD bisects ∠A.
2023 Annual3 Marks
16
P is a point equidistant from two lines l and m intersecting at point A. Show that AP bisects the angle between them.
2024 Annual3 Marks
17
S is any point on side QR of ▵PQR. Show that PQ + QR + RP > 2PS.
2020 Annual3 Marks
18
In ▵ABC and ▵DEF: ∠A = ∠D, ∠B = ∠E, BC = EF. Show that ▵ABC ≅ ▵DEF. State the congruence rule used.
2018 Annual2 Marks
19
AB is a line segment and P is its midpoint. D and E are points on the same side of AB such that ∠BAD = ∠ABE and ∠EPA = ∠DPB. Show that ▵DAP ≅ ▵EBP.
2021 Annual3 Marks
20
If two sides and the median bisecting one of these sides of a triangle are respectively equal to two sides and the corresponding median of another triangle, prove that the two triangles are congruent.
2022 Annual5 Marks
21
ABCD is a rhombus. Show that diagonal AC bisects ∠A as well as ∠C, and diagonal BD bisects ∠B as well as ∠D.
2019 Annual3 Marks
22
If the diagonals of a parallelogram are equal, then show that it is a rectangle.
2020 Annual3 Marks
23
Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
2021 Annual3 Marks
24
ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that: (i) ABCD is a square, (ii) diagonal BD bisects ∠B as well as ∠D.
2022 Annual3 Marks
25
The angles of a quadrilateral are in the ratio 3:5:9:13. Find all the angles of the quadrilateral.
2023 Annual2 Marks
26
In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ. Show that ▵APD ≅ ▵CQB.
2018 Annual3 Marks
27
In ▵ABC, D, E and F are respectively the mid-points of sides BC, CA and AB. Show that ▵ABC is divided into four congruent triangles by joining D, E and F.
2020 Annual3 Marks
28
In ▵ABC, D, E, F are the mid-points of BC, CA and AB respectively. If BC = 6 cm, CA = 8 cm and AB = 10 cm, find the perimeter of ▵DEF.
2024 Annual3 Marks
29
Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
2019 Annual5 Marks
30
ABCD is a trapezium in which AB ∥ CD and AD = BC. Show that: (i) ∠A = ∠B, (ii) ∠C = ∠D, (iii) ▵ABC ≅ ▵BAD.
2021 Annual3 Marks
07 Measuring Space: Perimeter and Area 18 questions
1
Prove that equal chords of a circle subtend equal angles at the centre.
2019 Annual3 Marks
2
Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any other point on the remaining part of the circle.
2020 Annual5 Marks
3
If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.
2021 Annual3 Marks
4
Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.
2022 Annual3 Marks
5
ABCD is a cyclic quadrilateral whose diagonals intersect at point E. If ∠DBC = 70° and ∠BAC = 30°, find ∠BCD. Also, if AB = BC, find ∠ECD.
2018 Annual4 Marks
6
Prove that the perpendicular from the centre of a circle to a chord bisects the chord.
2023 Annual3 Marks
7
Prove that the sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
2024 Annual5 Marks
8
In the figure, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠BEC = 130° and ∠ECD = 20°. Find ∠BAC.
2021 Annual2 Marks
9
Prove that if two chords of a circle are equal, then their corresponding arcs are congruent and conversely, if two arcs are congruent, their corresponding chords are equal.
2019 Annual3 Marks
10
Three girls Reshma, Salma and Mandip are standing on a circle of radius 5 m at points R, S and M respectively such that RS = SM = 6 m. Find the length RM.
2022 Annual4 Marks
11
Find the area of a triangle whose sides are 18 cm, 24 cm and 30 cm. Verify using the basic area formula.
2019 Annual2 Marks
12
The perimeter of a triangular field is 540 m and its sides are in the ratio 25:17:12. Find the area of the field. Also find the cost of ploughing the field at &rupee;18.80 per 100 m².
2020 Annual4 Marks
13
A rhombus shaped field has green grass for 18 cows to graze. If each side of the rhombus is 30 m and its longer diagonal is 48 m, how much area of grass field will each cow be getting?
2018 Annual4 Marks
14
An umbrella is made by stitching 10 triangular pieces of cloth of two different colours, each piece measuring 20 cm, 50 cm and 50 cm. How much cloth of each colour is required for the umbrella? (Use √6 = 2.45)
2021 Annual3 Marks
15
A field is in the shape of a trapezium whose parallel sides are 25 m and 10 m and the non-parallel sides are 14 m and 13 m. Find the area of the field.
2022 Annual4 Marks
16
Find the area of an equilateral triangle whose perimeter is 60 cm.
2023 Annual2 Marks
17
Sides of a triangle are in the ratio 12:17:25 and its perimeter is 540 cm. Find the area of the triangle.
2024 Annual3 Marks
18
Find the area of a quadrilateral ABCD in which AB = 3 cm, BC = 4 cm, CD = 4 cm, DA = 5 cm and AC = 5 cm.
2019 Annual4 Marks
08 Predicting What Comes Next: Exploring Sequences and Progression 10 questions
1
A hemispherical bowl made of brass has inner diameter 10.5 cm. Find the cost of tin-plating it on the inside at the rate of &rupee;16 per 100 cm². (Use π = 22/7)
2019 Annual3 Marks
2
Find the volume of a sphere whose surface area is 154 cm². (Use π = 22/7)
2020 Annual3 Marks
3
A right circular cylinder just encloses a sphere of radius r. Find: (i) the surface area of the sphere, (ii) the curved surface area of the cylinder, (iii) the ratio of the areas obtained in (i) and (ii).
2021 Annual3 Marks
4
The diameter of a metallic ball is 4.2 cm. Find the mass of the ball if the density of the metal is 8.9 g per cm³. (Use π = 22/7)
2018 Annual3 Marks
5
27 small metallic balls, each of radius r, are melted to form a single solid ball. Find the radius of the new ball.
2022 Annual2 Marks
6
A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea in a minute?
2019 Annual3 Marks
7
The surface area of a cuboid is 1372 sq. cm. If its dimensions are in the ratio 4:2:1, find its length, breadth and height.
2023 Annual3 Marks
8
Find the lateral surface area and total surface area of a right circular cone with base radius 7 cm and slant height 25 cm. (Use π = 22/7)
2024 Annual2 Marks
9
A conical tent is 10 m high and the radius of its base is 24 m. Find: (i) the slant height of the tent, (ii) the cost of canvas required to make the tent at &rupee;70 per m².
2020 Annual4 Marks
10
The total surface area of a solid hemisphere is 462 cm². Find its volume. (Use π = 22/7)
2021 Annual3 Marks
09 The Mathematics of Maybe: Introduction to Probability 8 questions
1
Find the mean of the first 10 natural numbers.
2018 Annual1 Mark
2
The following are the marks of 9 students in a class: 34, 37, 45, 50, 52, 54, 61, 71, 48. Find the median.
2019 Annual2 Marks
3
If the mean of the following data is 20.2, find the value of p: x = 10, 15, 20, 25, 30 with frequency f = 6, 8, p, 10, 6.
2020 Annual3 Marks
4
The mean of 16 numbers is 8. If 1 is added to each number, what will be the new mean?
2021 Annual1 Mark
5
Draw a histogram for the following frequency distribution of weekly wages of workers: 40–50 (5), 50–60 (8), 60–70 (12), 70–80 (6), 80–90 (4).
2022 Annual4 Marks
6
If the median of 33, 28, 20, 25, x, 29 (arranged in ascending order) is 29, find the value of x.
2023 Annual2 Marks
7
Find the mode of the following data: 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, 18.
2024 Annual1 Mark
8
The heights (in cm) of 9 students are: 155, 160, 145, 149, 150, 147, 152, 144, 148. Find the range of the data and the median height.
2019 Annual3 Marks