Class 10 Maths Sample Paper 3 (CBSE 2026-27 pattern)

The LCM of two numbers is $14$ times their HCF, and the sum of the HCF and the LCM is $600$. If one number is $280$, the other number is

1. (a) $40$
2. (b) $80$
3. (c) $120$
4. (d) $160$

If $\alpha ,\beta$ are the zeroes of $x^2-px+6$ and $\alpha -\beta =1$, then $p$ is

1. (a) $\pm 5$
2. (b) $5$ only
3. (c) $\pm 7$
4. (d) $25$

If the zeroes of $a{x}^2+bx+c$ $(a\ne 0,c\ne 0)$ are reciprocals of each other, then

1. (a) $b=0$
2. (b) $c=a$
3. (c) $a+c=0$
4. (d) $b=a$

The lines $kx-4y=3$ and $6x-12y=9$ intersect in exactly one point if

1. (a) $k=2$
2. (b) $k\ne 2$
3. (c) $k=-2$
4. (d) $k\ne 3$

For real $k$, the roots of $x^2-(k+1)x+k=0$ are

1. (a) never real
2. (b) real for every $k$
3. (c) real only when $k>0$
4. (d) equal for every $k$

The sum of the first $n$ terms of an AP is $3n^2+5n$. Its 10th term is

1. (a) $58$
2. (b) $62$
3. (c) $64$
4. (d) $350$

The 17th term from the end of the AP $7,10,13,\dots ,184$ is

1. (a) $133$
2. (b) $136$
3. (c) $139$
4. (d) $55$

The points $A(0,6)$, $B(-5,3)$ and $C(3,1)$ are the vertices of

1. (a) an equilateral triangle
2. (b) an isosceles right triangle
3. (c) a scalene triangle
4. (d) no triangle (they are collinear)

$A(1,2)$, $B(4,3)$, $C(6,6)$ and $D$ are the vertices of parallelogram $ABCD$, in order. The coordinates of $D$ are

1. (a) $(3,5)$
2. (b) $(9,7)$
3. (c) $(-1,1)$
4. (d) $(3,4)$

In trapezium $ABCD$, $AB\parallel DC$ and $AB=3DC$. The diagonals meet at $O$. Then $AO:OC$ is

1. (a) $1:3$
2. (b) $2:1$
3. (c) $3:1$
4. (d) $9:1$

Two tangents from an external point $P$ to a circle of radius $5$ cm are inclined to each other at ${60}^{\circ }$. The length of each tangent is

1. (a) $5$ cm
2. (b) ${\displaystyle \frac{5}{\sqrt{3}}}$ cm
3. (c) $5\sqrt{3}$ cm
4. (d) $10$ cm

$\mathrm{△}ABC\sim \mathrm{△}QRP$. If $\mathrm{\angle }A={50}^{\circ }$ and $\mathrm{\angle }B={60}^{\circ }$, then $\mathrm{\angle }P$ is

1. (a) ${50}^{\circ }$
2. (b) ${60}^{\circ }$
3. (c) ${70}^{\circ }$
4. (d) ${80}^{\circ }$

If $\sin A+{\sin }^{2}A=1$, then ${\cos }^{2}A+{\cos }^{4}A$ equals

1. (a) $0$
2. (b) ${\displaystyle \frac{1}{2}}$
3. (c) $1$
4. (d) $2$

If $x=a\cos \theta$ and $y=b\sin \theta$, then $b^2{x}^2+a^2{y}^2$ equals

1. (a) $ab$
2. (b) $a^2+b^2$
3. (c) $a^2{b}^2$
4. (d) $a^4{b}^4$

A cylinder, a cone and a hemisphere have the same base radius $r$, and the cylinder and cone have height $r$. The ratio of their volumes is

1. (a) $1:2:3$
2. (b) $3:1:2$
3. (c) $2:1:3$
4. (d) $3:2:1$

The area of a sector (in cm²) is numerically $3.5$ times its arc length (in cm). The radius of the circle is

1. (a) $1.75$ cm
2. (b) $3.5$ cm
3. (c) $7$ cm
4. (d) $14$ cm

The mean of $20$ observations was found to be $15$. Later it was found that the observation $25$ had been wrongly copied as $45$. The correct mean is

1. (a) $13$
2. (b) $14$
3. (c) $16$
4. (d) $14.5$

A letter is chosen at random from the letters of the word VISAKHAPATNAM. The probability that it is a vowel is

1. (a) ${\displaystyle \frac{4}{13}}$
2. (b) ${\displaystyle \frac{5}{13}}$
3. (c) ${\displaystyle \frac{2}{13}}$
4. (d) ${\displaystyle \frac{8}{13}}$

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): Any two triangles whose corresponding angles are equal are congruent.

Reason (R): Two equiangular triangles are similar.

1. (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. (b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. (c) Assertion (A) is true but Reason (R) is false.
4. (d) Assertion (A) is false but Reason (R) is true.

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The volume of a cone is one-third of the volume of a cylinder with the same base radius and height.

Reason (R): The curved surface area of a cone is one-third of the curved surface area of a cylinder with the same base radius and height.

1. (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. (b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. (c) Assertion (A) is true but Reason (R) is false.
4. (d) Assertion (A) is false but Reason (R) is true.

Find the least natural number which leaves a remainder of $7$ when divided by each of $12$, $16$ and $24$, and which is itself divisible by $11$.

The zeroes of $x^2+px+q$ are double the zeroes of $2x^2-5x-3$. Find $p$ and $q$.

OR

If $\sqrt{3}\tan \theta =3\sin \theta$, where $\theta$ is acute, find the value of ${\sin }^2\theta -{\cos }^2\theta$.

A chord of a circle of radius $14$ cm subtends a right angle at the centre. Find the difference between the areas of the major and the minor segments. $(\pi ={\displaystyle \frac{22}{7}})$

A bag contains only red and white balls, with three times as many red balls as white. When $5$ more white balls are added, the probability of drawing a white ball becomes ${\displaystyle \frac{1}{3}}$. How many balls of each colour were there at first?

OR

Prove that $\sqrt{2}+\sqrt{3}$ is irrational. (You must prove any irrationality result you use about $\sqrt{6}$.)

Meera walks at a steady speed of $a$ km/h and jogs at $b$ km/h. Walking for $2$ h and jogging for $1$ h, she covers $17$ km; walking for $1$ h and jogging for $2$ h, she covers $22$ km. Find $a$ and $b$. How long must she jog to cover $36$ km in all if she walks for $3$ h?

$ABCD$ is a trapezium with $AB\parallel DC$, $AB=16$ cm and $DC=10$ cm. Points $E$ on $AD$ and $F$ on $BC$ are such that $EF\parallel AB$ and $AE:ED=3:2$. Find $EF$.

The incircle of $\mathrm{△}ABC$ touches $BC$, $CA$, $AB$ at $D$, $E$, $F$. Prove that $BD-DC=AB-AC$. Hence find $BD$ if $AB=10$ cm, $AC=7$ cm and $BC=11$ cm.

For an acute angle $\theta$, $\sin \theta -\cos \theta ={\displaystyle \frac{7}{13}}$. Find the values of $\sin \theta \cos \theta$ and $\sin \theta +\cos \theta$, and hence find $\sec \theta +\mathrm{cosec}\theta$.

OR

A fair spinner with $8$ equal sectors numbered $1$ to $8$ is spun and a fair die is thrown. Find the probability that (i) the sum of the two numbers is $9$; (ii) the spinner shows a larger number than the die; (iii) the product of the two numbers is a multiple of $4$.

OR

Sneha repays an interest-free education loan of ₹1,18,000 in monthly instalments that form an AP. The first instalment is ₹1000 and each instalment is ₹100 more than the previous one. (a) How many instalments does she pay? (b) Find the last instalment. (c) How much is still due after the 25th instalment?

$ABCD$ is a quadrilateral. Points $P,Q,R,S$ lie on $AB,BC,CD,DA$ respectively, with ${\displaystyle \frac{AP}{PB}}={\displaystyle \frac{AS}{SD}}={\displaystyle \frac{1}{2}}$ and ${\displaystyle \frac{CQ}{QB}}={\displaystyle \frac{CR}{RD}}={\displaystyle \frac{1}{2}}$. (a) Prove that $PS\parallel BD$ and $PS={\displaystyle \frac{1}{3}}BD$. (b) Prove that $QR\parallel BD$, $QR={\displaystyle \frac{1}{3}}BD$, and hence that $PQRS$ is a parallelogram. (c) If $BD=18$ cm and $AC=24$ cm, find the perimeter of $PQRS$.

OR

A helicopter $H$ hovers vertically above a point $F$ on level ground. Two observers $A$ and $B$, on the same side of $F$ and in line with it, are $200$ m apart. The angles of elevation of $H$ from $A$ and $B$ are ${60}^{\circ }$ and ${30}^{\circ }$ respectively. Find (a) the height of the helicopter, (b) the distances $FA$ and $FB$, (c) the distance $AH$. (Take $\sqrt{3}=1.73$.)

OR

A school trophy is a solid cube of edge $10$ cm with a solid hemisphere of diameter $7$ cm fixed centrally on its top face. $(\pi ={\displaystyle \frac{22}{7}})$ (a) Find the total surface area of the trophy. (b) The hemisphere is gold-plated at ₹5 per cm² and the rest of the surface (including the base) is silver-plated at ₹2 per cm². Find the total cost. (c) Find the volume of the trophy.

Delivery times. A food-delivery company in Bengaluru recorded the delivery times of $100$ orders. Two frequencies were lost from the record, but the median delivery time is known to be $36$ minutes.

1. Write an equation connecting $x$ and $y$. [1 mark]

2. Which is the median class? Give a reason. [1 mark]

3. Find $x$ and $y$. [2 marks]

   OR

   Taking $x=24$ and $y=26$, find the modal delivery time. [2 marks]

Drone delivery corridor. A drone-delivery company models a town on a coordinate grid (1 unit = 1 km). Its two depots are at $A(-2,1)$ and $B(6,7)$, and drones fly along the straight corridor $AB$.

1. Find the length of the corridor $AB$. [1 mark]

2. A relay tower is at the mid-point of $AB$. Find its coordinates. [1 mark]

3. A charging station $C$ on the $x$-axis is equidistant from $A$ and $B$. Find $C$. [2 marks]

   OR

   Find the ratio in which the $y$-axis divides the corridor $AB$, and the point of division. [2 marks]

Fitness challenge. Neha joins a 'run more every day' challenge. She runs $1.5$ km on day 1 and increases her distance by $0.25$ km every day.

1. How far does she run on day 15? [1 mark]

2. On which day does she first run at least $8$ km? [1 mark]

3. Find the total distance she runs in the first $30$ days. [2 marks]

   OR

   After how many days will her total distance be $39$ km? [2 marks]