Class 10 Maths Sample Paper 2 (CBSE 2026-27 pattern)

If $\alpha$ and $\beta$ are the zeroes of $3x^2+5x-2$, then $\alpha +\beta +\alpha \beta$ equals

1. (a) $-{\displaystyle \frac{7}{3}}$
2. (b) $-1$
3. (c) $1$
4. (d) ${\displaystyle \frac{7}{3}}$

A quadratic polynomial whose zeroes are $-3$ and ${\displaystyle \frac{5}{2}}$ is

1. (a) $2x^2-x-15$
2. (b) $2x^2+x-15$
3. (c) $2x^2+11x+15$
4. (d) $x^2+x-15$

The pair of equations $x-2y=3$ and $3x-6y=9$ has

1. (a) a unique solution
2. (b) no solution
3. (c) infinitely many solutions
4. (d) exactly two solutions

The quadratic equation $k{x}^2-6x+1=0$ has two distinct real roots if

1. (a) $k<9$
2. (b) $k<9$ and $k\ne 0$
3. (c) $k>9$
4. (d) $k=9$

The sum of the first $10$ terms of the AP whose first term is $12$ and common difference is $-5$ is

1. (a) $105$
2. (b) $-93$
3. (c) $-105$
4. (d) $-225$

The number of terms of the AP $4,9,14,19,\dots$ that lie between $100$ and $200$ is

1. (a) $19$
2. (b) $20$
3. (c) $21$
4. (d) $40$

The point $(k,3)$ is at a distance of $5$ units from $(2,-1)$. The possible values of $k$ are

1. (a) $5$ or $-1$
2. (b) $-5$ or $1$
3. (c) $3$ only
4. (d) $5$ only

$M(2,3)$ is the mid-point of $AB$, where $A$ is $(-1,7)$. The coordinates of $B$ are

1. (a) $(5,-1)$
2. (b) $(3,-4)$
3. (c) $(\frac{1}{2},5)$
4. (d) $(5,1)$

$\mathrm{△}ABC\sim \mathrm{△}PQR$ with ${\displaystyle \frac{AB}{PQ}}={\displaystyle \frac{3}{4}}$. If the perimeter of $\mathrm{△}PQR$ is $48$ cm, the perimeter of $\mathrm{△}ABC$ is

1. (a) $27$ cm
2. (b) $36$ cm
3. (c) $64$ cm
4. (d) $44$ cm

From a point $P$ at a distance of $17$ cm from the centre of a circle of radius $8$ cm, a tangent is drawn. The length of the tangent is

1. (a) $9$ cm
2. (b) $12$ cm
3. (c) $15$ cm
4. (d) $\sqrt{353}$ cm

If $\sin \theta =\cos \theta$ with $0^{\circ }<\theta <{90}^{\circ }$, then $2{\tan }^2\theta +{\sin }^2\theta -1$ equals

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

$({\sec }^2\theta -1)(1-{\sin }^2\theta )$ is equal to

1. (a) ${\cos }^2\theta$
2. (b) ${\sin }^2\theta$
3. (c) ${\tan }^2\theta$
4. (d) $1$

The angle of elevation of the top of a $30$ m high tower from a point on level ground is ${60}^{\circ }$. The distance of the point from the foot of the tower is

1. (a) $10$ m
2. (b) $15$ m
3. (c) $10\sqrt{3}$ m
4. (d) $30\sqrt{3}$ m

The perimeter of a sector of a circle of radius $7$ cm with central angle ${90}^{\circ }$ is $(\pi ={\displaystyle \frac{22}{7}})$

1. (a) $11$ cm
2. (b) $18$ cm
3. (c) $25$ cm
4. (d) $36$ cm

The volume of a hemispherical bowl of internal radius $9$ cm is

1. (a) $243\pi {\text{cm}}^3$
2. (b) $486\pi {\text{cm}}^3$
3. (c) $729\pi {\text{cm}}^3$
4. (d) $972\pi {\text{cm}}^3$

The largest possible hemisphere is scooped out of one face of a solid cube of edge $7$ cm. The total surface area of the solid

1. (a) decreases by $38.5{\text{cm}}^2$
2. (b) increases by $77{\text{cm}}^2$
3. (c) increases by $38.5{\text{cm}}^2$
4. (d) does not change

The mean of the distribution below is $5$. The value of $p$ is

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

A card is drawn at random from a well-shuffled deck of $52$ playing cards. The probability that it is a red face card is

1. (a) ${\displaystyle \frac{1}{26}}$
2. (b) ${\displaystyle \frac{3}{26}}$
3. (c) ${\displaystyle \frac{3}{13}}$
4. (d) ${\displaystyle \frac{6}{13}}$

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

Assertion (A): For every natural number $n$, $\text{HCF}(n,n+1)=1$.

Reason (R): For any two natural numbers $a$ and $b$, $\text{HCF}(a,b)\times \text{LCM}(a,b)=a\times b$.

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 triangle with vertices $(0,0)$, $(3,0)$ and $(0,4)$ has perimeter $12$ units.

Reason (R): The distance between $(x_1,y_1)$ and $(x_2,y_2)$ is $\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$.

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.

Using prime factorisation, find the HCF and LCM of $144$ and $198$, and verify that HCF $\times$ LCM $=$ product of the two numbers.

Find the value of $k$ for which the pair of equations $2x+ky=5$ and $kx+8y=10$ has infinitely many solutions.

OR

In $\mathrm{△}ABC$, $P$ and $Q$ are points on $AB$ and $AC$ with $AP=3$ cm, $PB=6$ cm, $AQ=2.5$ cm and $QC=5$ cm. Is $PQ\parallel BC$? If $BC=12$ cm, find $PQ$.

If $5\cos \theta -12\sin \theta =0$, find the value of ${\displaystyle \frac{\sin \theta +\cos \theta }{2\cos \theta -\sin \theta }}$.

OR

The perimeter of a sector of a circle of radius $12$ cm is $(24+4\pi )$ cm. Find the central angle of the sector and its area (in terms of $\pi$).

Prove that $\sqrt{2}$ is irrational. Hence prove that ${\displaystyle \frac{1}{\sqrt{2}-1}}$ is irrational.

A rectangular vegetable plot has a perimeter of $46$ m and a diagonal of $17$ m. Find its length and breadth.

The mid-point of the segment joining $A(2k,1)$ and $B(4,k)$ lies on the line $x+2y=9$. Find $k$, the coordinates of $A$ and $B$, and the length $AB$.

OR

In $\mathrm{△}ABC$, $D$ and $E$ are points on $AB$ and $AC$ with $DE\parallel BC$. The segments $BE$ and $CD$ intersect at $O$. Prove that $\mathrm{△}ODE\sim \mathrm{△}OCB$. If $AD:DB=1:2$ and $DE=5$ cm, find $BC$ and $OD:OC$.

If $\sec A+\tan A=3$, find $\sec A-\tan A$, and hence find $\sin A$ and $\cos A$.

OR

Two circles with centres $A$ and $B$ and radii $9$ cm and $4$ cm touch each other externally. A common tangent (not through the point of contact) touches the circles at $P$ and $Q$ respectively. Find the length $PQ$.

An open-air auditorium in Kochi has $20$ seats in the first row, and each row after that has $2$ more seats than the row in front of it. The auditorium has $1100$ seats in all. (a) Find the number of rows. (b) How many seats are there in the last row? (c) How many seats are there in rows 11 to 20 (both inclusive)?

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Using this result, answer: $P$ is a point $29$ cm from the centre $O$ of a circle of radius $20$ cm, and $PA$, $PB$ are the two tangents from $P$. Find (a) the length of each tangent, (b) the area of quadrilateral $OAPB$.

OR

A storage shed has a cuboidal base $14$ m long, $7$ m wide and $4$ m high, surmounted by a half-cylinder of diameter $7$ m and length $14$ m (the curved roof). $(\pi ={\displaystyle \frac{22}{7}})$ (a) Find the volume of air inside the shed. (b) The inside walls, the two semicircular end-walls and the inside of the curved roof are to be painted at ₹20 per m². Find the cost (the floor is not painted).

OR

A survey of $60$ Class X students recorded their daily screen time (in minutes).

(a) Find the mean screen time by the assumed-mean method. (b) Find the modal screen time. (c) Which of the two better describes a 'typical' student here? Give a reason.

Fire-rescue ladder. A fire engine has a turntable ladder whose foot is mounted on the truck $2$ m above the ground. The fully extended ladder is $20$ m long. (Take $\sqrt{3}=1.73$, $\sqrt{2}=1.41$.)

1. The ladder is extended fully at ${60}^{\circ }$ to the horizontal. How high above the ground does its top reach? [1 mark]

2. In part (i), what is the horizontal distance between the foot of the ladder and the wall it touches? [1 mark]

3. The truck stays in place (foot of ladder $10$ m from the wall). To reach a window $12$ m above the ground, find the angle of the ladder with the horizontal and the length of ladder needed. [2 marks]

   OR

   Keeping the full $20$ m length, the angle is reduced from ${60}^{\circ }$ to ${30}^{\circ }$. By how much does the height of the top of the ladder fall? [2 marks]

Park pathway. A rectangular park in Bhopal measures $30$ m by $20$ m. A paved path of uniform width $x$ m is laid along the inside of its boundary, and the remaining lawn has an area of $336{\text{m}}^2$.

1. Write the length and breadth of the lawn in terms of $x$. [1 mark]

2. Form the quadratic equation in standard form satisfied by $x$. [1 mark]

3. Solve the equation and find the width of the path, justifying your choice of root. [2 marks]

   OR

   With the width found, paving costs ₹150 per m². Find the cost of paving the path. [2 marks]

Mela dice game. At a Diwali mela stall, a player throws two fair dice together. The stall-keeper announces different prizes for different outcomes.

1. Find the probability that the sum of the numbers is $7$. [1 mark]

2. Find the probability that the sum is at least $10$. [1 mark]

3. Find the probability that the two numbers differ by at least $3$. [2 marks]

   OR

   Find the probability that the product of the two numbers is (a) odd, (b) even. [2 marks]