CBSE / NCERT Class 9 Maths Practice Question Papers with Answers (2026-27)

Brain Grain · braingrain.in

Maths — Practice Paper · Set 1

Class: 9CBSE / NCERTMax Marks: 36

Name: ____________________Reg No: ____________

Part I — Short Answer Questions 18 × 2 = 36

Answer briefly. (Answer all questions.)

1.Find the following squares using one of the above identities. Determine which of these identities will make these calculations easier. (i) $117^2$ (ii) $78^2$ (iii) $198^2$ (iv) $214^2$ (v) $1104^2$ (vi) $1120^2$[2]

2.The ratio of the perimeters of two circles is $5:4$ . What is the ratio of their radii?[2]

3.When a single 6-sided die is rolled, what is the total number of possible outcomes in the sample space?[2]

4.Fig. 8.12 shows Stages 0 to 3 of the Sierpiński square carpet. Stage 0 of this fractal is a square sheet of paper. To construct Stage 1, each side of the square is trisected and the points of trisection of opposite sides are joined to obtain nine smaller squares. The centre square is then removed and the 8 smaller squares are retained, leaving a square hole in the centre. The same process is repeated on the eight smaller shaded squares to obtain Stage 2 and so on. (i) How many red squares are there in Stages 0 to 3? (ii) Can you predict the number of red squares in Stages 4 and 5? (iii) Can you find a rule for the number of red squares at the nth stage? Write the explicit formula as well as the recursive formula for the number of red squares at any stage. (iv) Suppose the area of the square in Stage 0 is 1 square unit. What is the area of the red region in Stages 1, 2 and 3? What will be the area of the red region in Stages 4 and 5? Find the explicit as well as the recursive formula for the area of the red region at the nth stage. What happens to this area as n, the number of stages, goes on increasing?[2]

5.Find 5 rational numbers between $\dfrac{1}{6}$ and $\dfrac{2}{5}$ .[2]

6.An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.[2]

7.In a circle, a chord is 5 cm away from the centre. If the radius of the circle is 13 cm, what is the length of the chord?[2]

8.Suppose the length of a rectangular box is 7 cm and breadth is 11 cm. Find the volume if the height is (i) 5 cm, (ii) 9 cm, (iii) 13 cm. Find the linear pattern representing the volume of the rectangular box.[2]

9.Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: $\dfrac{7}{20}$ , $\dfrac{4}{15}$ and $\dfrac{13}{250}$ . Then check your answers by explicitly performing the long divisions and expressing these rational numbers as decimals.[2]

10.Find a GP for which the sum of the first two terms is $-4$ and the fifth term is 4 times the third term.[2]

11.When two chords intersect, each of them is divided into two line segments. Show that if the intersecting chords are of equal length, then the line segments of one chord are equal to the corresponding line segments of the other chord.[2]

12.A traffic signal board, indicating 'SCHOOL AHEAD', is an equilateral triangle with side 'a'. Find the area of the signal board, using Heron's formula. If its perimeter is 180 cm, what will be the area of the signal board?[2]

13.Find the perimeter of a sector (i.e., the curved portion as well as the two straight portions) of a circle of radius 14 cm and sector angle 75°.[2]

14.The sum of the first three terms of a geometric progression is 26, and the sum of their squares is 364. Find the terms of the GP.[2]

15.Find the quotient: (i) $\dfrac{2}{3}\div\dfrac{3}{10}$ (ii) $\dfrac{7}{11}\div\dfrac{5}{8}$ (iii) $-\dfrac{4}{7}\div\dfrac{5}{14}$[2]

16.A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.[2]

17.If the wheel of a bicycle has a diameter of 60 cm, find how far a cyclist will have travelled after the wheel has rotated 100 times.[2]

18.Sarita is reading a book of 500 pages. She reads 20 pages every day. How many pages will be left after 15 days? Express this as a linear pattern.[2]

🔑 Show Answer Key — Set 1

1. (i) $13689$ (ii) $6084$ (iii) $39204$ (iv) $45796$ (v) $1218816$ (vi) $1254400$
2. The ratio of their radii is $5:4$ .
3. There are $6$ possible outcomes: $\{1,2,3,4,5,6\}$ .
4. (i) Stages 0 to 3 have $1,8,64,512$ red squares. (ii) Stages 4 and 5 have $4096$ and $32768$ red squares. (iii) If Stage 0 is counted as $n=0$ , the number of red squares is $R_n=8^n$ ; recursively, $R_0=1$ and $R_n=8R_{n-1}$ for $n\geq1$ . (iv) Red areas for Stages 1, 2, 3 are $\dfrac89,\dfrac{64}{81},\dfrac{512}{729}$ . For Stages 4 and 5 they are $\dfrac{4096}{6561}$ and $\dfrac{32768}{59049}$ . Explicitly, $A_n=\left(\dfrac89\right)^n$ ; recursively, $A_0=1$ and $A_n=\dfrac89A_{n-1}$ . As n increases, the area approaches 0.
5. Five examples are $\dfrac{6}{30}$ , $\dfrac{7}{30}$ , $\dfrac{8}{30}$ , $\dfrac{9}{30}$ and $\dfrac{10}{30}$ .
6. $9\sqrt{15}\text{ cm}^2$ , approximately $34.9\text{ cm}^2$ .
7. The chord length is $24$ cm.
8. (i) $385\text{ cm}^3$ (ii) $693\text{ cm}^3$ (iii) $1001\text{ cm}^3$ The linear pattern is $V=77h$ , where $h$ is the height in cm.
9. $\dfrac{7}{20}$ terminates: $0.35$ . $\dfrac{4}{15}$ is non-terminating repeating: $0.2\overline{6}$ . $\dfrac{13}{250}$ terminates: $0.052$ .
10. Two possible GPs are $-\dfrac{4}{3},-\dfrac{8}{3},-\dfrac{16}{3},\ldots$ and $4,-8,16,-32,64,\ldots$ .
11. Let equal chords AB and CD intersect at P. Then $AP+PB=CP+PD$ because $AB=CD$ . Also, by the intersecting chords theorem, $AP\cdot PB=CP\cdot PD$ . Two positive segment-pairs with the same sum and same product are the same pair of numbers. Therefore the two segments of one chord are equal to the corresponding two segments of the other chord.
12. The area is $\dfrac{\sqrt3}{4}a^2$ . If the perimeter is $180$ cm, the area is $900\sqrt3\text{ cm}^2$ .
13. $\dfrac{139}{3}$ cm, i.e. about $46.3$ cm.
14. The terms are $2,6,18$ or $18,6,2$ .
15. (i) $\dfrac{20}{9}$ (ii) $\dfrac{56}{55}$ (iii) $-\dfrac{8}{5}$
16. At a point on the minor arc: $150^\circ$ ; at a point on the major arc: $30^\circ$ .
17. $\dfrac{132000}{7}$ cm, i.e. about $188.6$ m.
18. After 15 days, $200$ pages will be left. The linear pattern is $P=500-20d$ , where $P$ is the number of pages left after $d$ days.

Brain Grain · braingrain.in

Maths — Practice Paper · Set 2

Class: 9CBSE / NCERTMax Marks: 36

Name: ____________________Reg No: ____________

Part I — Short Answer Questions 18 × 2 = 36

Answer briefly. (Answer all questions.)

1.If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD.[2]

2.Determine the AP whose third term is 16 and whose 7th term exceeds the 5th term by 12.[2]

3.Curved surface area of a cone is 308 cm² and its slant height is 14 cm. Find (i) radius of the base and (ii) total surface area of the cone.[2]

4.Factor completely: (i) $9x^2 + 24xy + 16y^2$ (ii) $4s^2 + 20st + 25t^2$ (iii) $49x^2 + 28xy + 4y^2$ (iv) $64p^2 + \dfrac{32}{3}pq + \dfrac{4}{9}q^2$ (v) $3a^2 + 4ab + \dfrac{4}{3}b^2$ (vi) $\dfrac{9}{5}s^2 + 6sv + 5v^2$[2]

5.Find the total surface area of a hemisphere of radius 10 cm. (Use π = 3.14)[2]

6.Plot point $Z(5,-6)$ on the Cartesian plane. Construct a right-angled triangle IZN and find the lengths of the three sides. (Comment: Answers may differ from person to person.)[2]

7.Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see Fig. 9.27). Prove that ∠ACP = ∠QCD.[2]

8.Simplify the following rational expressions assuming that the expressions in the denominators are not equal to zero: (i) $\dfrac{3p^2 - 3pq -18q^2}{p^2 + 3pq - 10q^2}$ (ii) $\dfrac{n^3 - 3n^2m + 3nm^2 - m^3}{5m^2 - 10mn + 5n^2}$ (iii) $\dfrac{w^3 - v^3 + x^3 + 3wvx}{w^2 + v^2 + x^2 - 2wv - 2vx + 2wx}$ (iv) $\dfrac{4y^2 - 20yz + 25z^2}{25z^2 - 4y^2}$ (v) $\dfrac{(x^2 + x - 6)(x^2 - 7x + 12)}{(x^2 - 6x + 8)(x^2 - 9)}$ (vi) $\dfrac{p^4 - 16}{p^2 - 4p + 4}$[2]

9.A spice trader takes a loan (debt) of `850. The next day, he makes a profit (fortune) of `1,200. The following week, he incurs a loss of `450. Write this sequence as an equation using integers and calculate his final financial standing.[2]

10.ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠CAD = ∠CBD.[2]

11.There are two fruit baskets A and B. Basket A has one apple and two oranges. Basket B has one banana and one mango. You randomly pick one fruit from each basket. (i) Draw a tree diagram showing all possible pairs of fruits. (ii) List the sample space. (iii) What is the probability of picking one apple and one banana?[2]

12.Calculate the length of the arc of a circle if: (i) the radius is 3.5 cm and the angle at the centre is 60°, and (ii) the radius is 6.3 m and the angle at the centre is 120°.[2]

13.If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.[2]

14.Can you explain why the converse to Theorem 4 is true, i.e., why does the perpendicular from the centre of a circle to a chord of the circle bisect the chord? (Hint: Use Fig. 5.12. You are told that $\angle CMA = \angle CMB = 90°$ . You need to show that $AM = BM$ .)[2]

15.Harish started work at an annual salary of `5,00,000 and received an increment of `20,000 each year. After how many years did his income reach `7,00,000?[2]

16.A rational number has a terminating decimal expansion whose last non-zero digit occurs in the 4th decimal place. Show that such a number can be written in the form $\dfrac{p}{10^4}$ , where p is an integer not divisible by 10. Is it necessary that the denominator of this rational number, when written in the lowest form, is divisible by $2^4$ or $5^4$ ? Give reasons.[2]

17.In a circle with centre O, the central angle AOB is 60°. If the radius of the circle is 12 cm, what is the length of the chord AB?[2]

18.Factor using suitable identities: (i) $16y^2 - 24y + 9$ (ii) $\dfrac{9}{4}s^2 + 6st + 4t^2$ (iii) $\dfrac{m^2}{9} + \dfrac{mk}{3} + \dfrac{k^2}{4} + 3nk + 2mn + 9n^2$ (iv) $\dfrac{p^2}{16} - 2 + \dfrac{16}{p^2}$ (v) $9a^2 + 4b^2 + c^2 - 12ab + 6ac - 4bc$[2]

🔑 Show Answer Key — Set 2

1. $AB=CD$ .
2. The AP is $4,10,16,22,28,\ldots$ .
3. (i) $7$ cm (ii) $462\text{ cm}^2$
4. (i) $(3x+4y)^2$ (ii) $(2s+5t)^2$ (iii) $(7x+2y)^2$ (iv) $\left(8p+\dfrac{2}{3}q\right)^2$ (v) $\dfrac{1}{3}(3a+2b)^2$ (vi) $\dfrac{1}{5}(3s+5v)^2$
5. $942\text{ cm}^2$ .
6. One possible triangle is obtained by taking $I(5,0)$ and $N(0,-6)$ . Then $ZI=6$ units, $ZN=5$ units and $IN=\sqrt{5^2+6^2}=\sqrt{61}$ units.
7. $\angle ACP=\angle QCD$ .
8. (i) $\dfrac{3(p-3q)(p+2q)}{(p+5q)(p-2q)}$ (ii) $\dfrac{n-m}{5}$ (iii) $\dfrac{w^2+v^2+x^2+wv-vx-wx}{w-v+x}$ (iv) $\dfrac{5z-2y}{2y+5z}$ (v) $1$ (vi) $\dfrac{(p+2)(p^2+4)}{p-2}$
9. The integer equation is $-850+1200-450=-100$ . His final standing is a debt of `100.
10. $\angle CAD=\angle CBD$ .
11. (i) The tree branches from Basket A to Apple or Orange, and from each of these to Banana or Mango. (ii) By fruit type, $S=\{(\text{Apple},\text{Banana}),(\text{Apple},\text{Mango}),(\text{Orange},\text{Banana}),(\text{Orange},\text{Mango})\}$ . (iii) $P(\text{Apple and Banana})=\dfrac{1}{3}\times\dfrac{1}{2}=\dfrac{1}{6}$ .
12. (i) $\dfrac{11}{3}$ cm, i.e. about $3.67$ cm (ii) $13.2$ m
13. The cyclic quadrilateral is a rectangle.
14. Let C be the centre and let CM be perpendicular to chord AB. Join CA and CB. Since CA and CB are radii, $CA=CB$ . Also $CM$ is common and $\angle CMA=\angle CMB=90°$ . Therefore $\triangle CMA \cong \triangle CMB$ by RHS congruence, so $AM=BM$ . Hence the perpendicular from the centre bisects the chord.
15. His salary reaches `7,00,000 after 10 increments, i.e. in the 11th year of work.
16. If the last non-zero digit is in the 4th decimal place, the number can be written as $\dfrac{p}{10^4}$ with $p$ not divisible by 10. In lowest form, its denominator must be divisible by at least one of $2^4$ or $5^4$ .
17. $AB=12$ cm.
18. (i) $(4y-3)^2$ (ii) $\left(\dfrac{3}{2}s+2t\right)^2$ (iii) $\left(\dfrac{m}{3}+\dfrac{k}{2}+3n\right)^2$ (iv) $\left(\dfrac{p}{4}-\dfrac{4}{p}\right)^2$ (v) $(3a-2b+c)^2$

Brain Grain · braingrain.in

Maths — Practice Paper · Set 3

Class: 9CBSE / NCERTMax Marks: 36

Name: ____________________Reg No: ____________

Part I — Short Answer Questions 18 × 2 = 36

Answer briefly. (Answer all questions.)

1.Find the length of the chord of a circle where the radius is 7 cm and perpendicular distance is 6 cm.[2]

2.Factor the following algebraic expressions: (i) $4y^2 + 1 + \dfrac{1}{16y^2}$ (ii) $9m^2 - \dfrac{1}{25n^2}$ (iii) $27b^3 - \dfrac{1}{64b^3}$ (iv) $x^2 + \dfrac{5x}{6} + \dfrac{1}{6}$ (v) $27u^3 - \dfrac{1}{125} - \dfrac{27u^2}{5} + \dfrac{9u}{25}$ (vi) $64y^3 + \dfrac{1}{125}z^3$ (vii) $p^3 + 27q^3 + r^3 - 9pqr$ (viii) $9m^2 - 12m + 4$ (ix) $9x^3 - \dfrac{8}{3}y^3 + \dfrac{z^3}{3} + 6xyz$ (x) $4x^2 + 9y^2 + 36z^2 + 12xz + 36yz + 24xy$ (xi) $27u^3 - \dfrac{1}{216} - \dfrac{9u^2}{2} + \dfrac{u}{4}$[2]

3.Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.[2]

4.A cyclic quadrilateral has sides 5, 5, 12, 12 units. Find its area.[2]

5.Curved surface area of a right circular cylinder is 4.4 m². If the radius of the base of the cylinder is 0.7 m, find its height.[2]

6.What is the coefficient of z in the polynomial $4z^3 + 5z^2 - 11$ ?[2]

7.Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.[2]

8.Represent the rational numbers $\dfrac{2}{3}$ , $-\dfrac{5}{4}$ and $\dfrac{11}{2}$ on a single number line.[2]

9.Find the radius of a sphere whose surface area is 154 cm².[2]

10.Find x in Fig. 5.26.[2]

11.A chord of a circle of radius 15 cm subtends an angle of 60° at the centre of the circle. Find the areas of the corresponding minor and major segments of the circle. (Use $\pi \approx 3.14$ and $\sqrt{3}\approx 1.73$ .)[2]

12.What is the constant term of the polynomial $9x^3 + 5x^2 - 8x -10$ ?[2]

13.The paint in a certain container is sufficient to paint an area equal to 9.375 m². How many bricks of dimensions 22.5 cm × 10 cm × 7.5 cm can be painted out of this container?[2]

14.Prove that the following rational numbers are equal: (i) $\dfrac{2}{3}$ and $\dfrac{4}{6}$ (ii) $\dfrac{5}{4}$ and $\dfrac{10}{8}$ (iii) $-\dfrac{3}{5}$ and $-\dfrac{6}{10}$ (iv) $\dfrac{9}{3}$ and $3$[2]

15.The difference between two positive integers is 63. The ratio of the two integers is $2:5$ . Find the two integers.[2]

16.Find the area of a quadrant of a circle whose circumference is 66 cm.[2]

17.One diagonal of a rhombus is twice as long as the other diagonal. If the rhombus has area $128\text{ cm}^2$ , find the length of the shorter diagonal.[2]

18.Fig. 1.3 shows Reiaan's room with points OABC marking its corners. The x- and y-axes are marked in the figure. Point O is the origin. Referring to Fig. 1.3, answer the following questions: (i) If $D_1R_1$ represents the door to Reiaan's room, how far is the door from the left wall (the y-axis) of the room? How far is the door from the x-axis? (ii) What are the coordinates of $D_1$ ? (iii) If $R_1$ is the point $(11.5, 0)$ , how wide is the door? Do you think this is a comfortable width for the room door? If a person in a wheelchair wants to enter the room, will he/she be able to do so easily? (iv) If $B_1(0, 1.5)$ and $B_2(0, 4)$ represent the ends of the bathroom door, is the bathroom door narrower or wider than the room door?[2]

🔑 Show Answer Key — Set 3

1. The chord length is $2\sqrt{13}$ cm.
2. (i) $\left(2y+\dfrac{1}{4y}\right)^2$ (ii) $\left(3m-\dfrac{1}{5n}\right)\left(3m+\dfrac{1}{5n}\right)$ (iii) $\left(3b-\dfrac{1}{4b}\right)\left(9b^2+\dfrac{3}{4}+\dfrac{1}{16b^2}\right)$ (iv) $\left(x+\dfrac{1}{2}\right)\left(x+\dfrac{1}{3}\right)$ (v) $\left(3u-\dfrac{1}{5}\right)^3$ (vi) $\left(4y+\dfrac{z}{5}\right)\left(16y^2-\dfrac{4}{5}yz+\dfrac{z^2}{25}\right)$ (vii) $(p+3q+r)(p^2+9q^2+r^2-3pq-3qr-pr)$ (viii) $(3m-2)^2$ (ix) $\dfrac{1}{3}(3x-2y+z)(9x^2+4y^2+z^2+6xy+2yz-3xz)$ (x) As printed, the $xy$ and $xz$ coefficients are interchanged from a perfect square. The intended expression $4x^2+9y^2+36z^2+12xy+36yz+24xz$ factors as $(2x+3y+6z)^2$ . (xi) $\left(3u-\dfrac{1}{6}\right)^3$
3. $21\sqrt{11}\text{ cm}^2$ , approximately $69.6\text{ cm}^2$ .
4. The area is $60$ square units.
5. $1$ m.
6. The coefficient of $z$ is $0$ .
7. Chords of congruent circles that subtend equal central angles are equal.
8. $\dfrac{2}{3}$ lies between 0 and 1, two-thirds of the way from 0 to 1. $-\dfrac{5}{4}=-1.25$ lies between $-2$ and $-1$ . $\dfrac{11}{2}=5.5$ lies halfway between 5 and 6.
9. $3.5$ cm.
10. $x=80°$ .
11. Minor segment area $\approx 20.44\text{ cm}^2$ ; major segment area $\approx 686.06\text{ cm}^2$ .
12. The constant term is $-10$ .
13. $100$ bricks.
14. (i) $\dfrac{4}{6}=\dfrac{2}{3}$ (ii) $\dfrac{10}{8}=\dfrac{5}{4}$ (iii) $-\dfrac{6}{10}=-\dfrac{3}{5}$ (iv) $\dfrac{9}{3}=3$
15. The two integers are $42$ and $105$ .
16. $86.625\text{ cm}^2$ .
17. $8\sqrt{2}$ cm.
18. (i) The door is $8$ units from the left wall (the y-axis) and $0$ units from the x-axis. (ii) $D_1 = (8, 0)$ . (iii) The width is $11.5 - 8 = 3.5$ units. A width of $3.5$ ft is comfortable for a room door and should allow a wheelchair to enter easily. (iv) The bathroom door width is $4 - 1.5 = 2.5$ units, so it is narrower than the room door.

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