Class 10 Maths Sample Paper 4 (CBSE 2026-27 pattern)

Which of the following is a rational number?

1. (a) $(3+\sqrt{2}{)}^2$
2. (b) $(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})$
3. (c) $\sqrt{2}\times \sqrt{8}\times \sqrt{3}$
4. (d) ${\displaystyle \frac{1}{\sqrt{2}+1}}$

If $\alpha ,\beta$ are the zeroes of $x^2-3x+1$, a quadratic polynomial whose zeroes are ${\alpha }^2$ and ${\beta }^2$ is

1. (a) $x^2-9x+1$
2. (b) $x^2-7x+1$
3. (c) $x^2+7x+1$
4. (d) $x^2-7x-1$

The graph of $y=a{x}^2+bx+c$ is a parabola opening downwards that cuts the $y$-axis above the origin. Then the zeroes of the polynomial

1. (a) are both positive
2. (b) are both negative
3. (c) have opposite signs
4. (d) are not real

If $x=a,y=b$ is the solution of $x+y=5$ and $2x-3y=0$, then $a-b$ equals

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

The equation $x^2-2(k-1)x+(k+5)=0$ has equal roots when $k$ is

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

If $a,b,c$ are in AP, then $(a-c{)}^2$ equals

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

The sum of all two-digit odd natural numbers is

1. (a) $2425$
2. (b) $2475$
3. (c) $2500$
4. (d) $2530$

The point $P(k,0)$ divides the join of $A(2,-2)$ and $B(-7,4)$ internally in the ratio $1:2$. The value of $k$ is

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

In $\mathrm{△}ABC$, $DE\parallel BC$ with $D$ on $AB$ and $E$ on $AC$. If ${\displaystyle \frac{AD}{DB}}={\displaystyle \frac{3}{5}}$, then ${\displaystyle \frac{DE}{BC}}$ is

1. (a) ${\displaystyle \frac{3}{5}}$
2. (b) ${\displaystyle \frac{3}{8}}$
3. (c) ${\displaystyle \frac{5}{8}}$
4. (d) ${\displaystyle \frac{9}{64}}$

The angle between two radii of a circle is ${130}^{\circ }$. The angle between the tangents drawn at the ends of these radii is

1. (a) ${40}^{\circ }$
2. (b) ${50}^{\circ }$
3. (c) ${65}^{\circ }$
4. (d) ${90}^{\circ }$

$AB$ is a chord of a circle with centre $O$ and $\mathrm{\angle }AOB={100}^{\circ }$. $AT$ is the tangent at $A$, on the side of $AB$ away from $O$. The angle $\mathrm{\angle }BAT$ is

1. (a) ${40}^{\circ }$
2. (b) ${50}^{\circ }$
3. (c) ${80}^{\circ }$
4. (d) ${100}^{\circ }$

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

1. (a) $0$
2. (b) $1$
3. (c) ${\sec }^2\theta$
4. (d) ${\cos }^2\theta$

The length of the shadow of a vertical tower on level ground is $\sqrt{3}$ times its height. The angle of elevation of the sun is

1. (a) ${30}^{\circ }$
2. (b) ${45}^{\circ }$
3. (c) ${60}^{\circ }$
4. (d) ${90}^{\circ }$

A sector of angle $\theta$ of a circle of radius $2r$ has the same area as a sector of angle ${60}^{\circ }$ of a circle of radius $r$. Then $\theta$ is

1. (a) ${15}^{\circ }$
2. (b) ${30}^{\circ }$
3. (c) ${120}^{\circ }$
4. (d) ${240}^{\circ }$

The total surface area of a solid hemisphere of radius $7$ cm is $(\pi ={\displaystyle \frac{22}{7}})$

1. (a) $308{\text{cm}}^2$
2. (b) $462{\text{cm}}^2$
3. (c) $616{\text{cm}}^2$
4. (d) $154{\text{cm}}^2$

A solid consists of a cylinder of radius $3$ cm and height $5$ cm with a cone of the same radius and height $4$ cm on top. Its volume is

1. (a) $45\pi {\text{cm}}^3$
2. (b) $51\pi {\text{cm}}^3$
3. (c) $57\pi {\text{cm}}^3$
4. (d) $60\pi {\text{cm}}^3$

For a grouped distribution with $N=60$, the median class is $30\text{–}40$, its frequency is $12$ and the cumulative frequency of the class before it is $22$. The median is (approximately)

1. (a) $32.5$
2. (b) $35$
3. (c) $36.67$
4. (d) $38.33$

Two fair coins are tossed together. The probability of getting at most one head is

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

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

Assertion (A): From a point outside a circle, exactly two tangents can be drawn to the circle.

Reason (R): A tangent to a circle meets it in exactly one point.

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 length of the arc of a sector of angle ${60}^{\circ }$ in a circle of radius $21$ cm is $22$ cm $(\pi ={\displaystyle \frac{22}{7}})$.

Reason (R): The length of the arc of a sector of angle $\theta$ in a circle of radius $r$ is ${\displaystyle \frac{\theta }{{360}^{\circ }}}\times \pi r^2$.

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.

If $2^x\times 3^y\times 5^z=10800$, find $x$, $y$, $z$. Hence find the HCF of $10800$ and $8100$.

Find all values of $p$ for which the quadratic equation $2x^2+px+8=0$ has real roots.

OR

Find the ratio in which the point $P(2,m)$ divides the line segment joining $A(-1,7)$ and $B(4,-3)$. Hence find $m$.

Find the acute angle $\theta$ for which ${\sec }^2\theta +{\tan }^2\theta =7$.

OR

A toy is a cone mounted on a hemisphere, both of radius $3$ cm. The total height of the toy is $10$ cm. Find the volume of the toy in terms of $\pi$.

Let $p$ be a prime. Using the Fundamental Theorem of Arithmetic, show that if $p$ divides $a^2$ ($a$ a positive integer), then $p$ divides $a$. Hence prove that $\sqrt{7}$ is irrational.

A jeweller in Jaipur has two gold alloys, one $60\mathrm{\%}$ gold and the other $85\mathrm{\%}$ gold by mass. How many grams of each must be melted together to obtain $50$ g of an alloy that is $70\mathrm{\%}$ gold?

OR

Find the point $P$ on the line $y=2x$ which is equidistant from $A(3,1)$ and $B(-1,5)$. Also find $PA$.

$ABCD$ is a square of side $12$ cm. $E$ is a point on $BC$ with $BE=8$ cm. $AE$ produced meets $DC$ produced at $F$. Prove that $\mathrm{△}ABE\sim \mathrm{△}FCE$ and find $CF$.

$TP$ and $TQ$ are tangents from an external point $T$ to a circle with centre $O$ and radius $6$ cm, and $\mathrm{\angle }PTQ={90}^{\circ }$. Prove that $OPTQ$ is a square, and find $OT$.

If $\tan \theta +\cot \theta ={\displaystyle \frac{4}{\sqrt{3}}}$, find the values of ${\tan }^2\theta +{\cot }^2\theta$ and ${\tan }^3\theta +{\cot }^3\theta$. Hence find the possible acute angles $\theta$.

OR

A school bus travels $120$ km to a picnic spot. On the return journey along the same road its average speed is $10$ km/h less, and the round trip takes $4$ hours $24$ minutes of driving time. Find the average speed on the onward journey. Justify your choice of root.

OR

In $\mathrm{△}ABC$, points $D$ and $E$ on $AB$ trisect it, with $AD=DE=EB$. Lines through $D$ and $E$ parallel to $BC$ meet $AC$ at $F$ and $G$ respectively. (a) Prove that $AF=FG=GC$. (b) If $BC=18$ cm, find $DF$ and $EG$. (c) Find the ratio of the perimeters of $\mathrm{△}ADF$, $\mathrm{△}AEG$ and $\mathrm{△}ABC$.

An escalator in a mall rises at ${30}^{\circ }$ to the horizontal from a point $B$ on the ground floor to a point $T$ on the first floor, $6$ m higher. Let $F$ be the point on the ground floor vertically below $T$. Standing at $T$, a shopper sees the foot $P$ of a pillar, which lies on the ground floor on segment $BF$, at an angle of depression of ${60}^{\circ }$. Find (a) the length of the escalator, (b) the horizontal distance $BF$, (c) the distance $FP$ and the distance $BP$. (Take $\sqrt{3}=1.73$.)

The ages of $80$ customers who visited a pharmacy in Guwahati one morning are given below.

(a) Find the median age. (b) Find the modal age. (c) The mean age is $39.75$ years. Use the empirical relation to estimate the mode from the mean and median, and compare it with your answer to (b).

OR

Community hall. A village community hall is a cylinder of radius $7$ m and height $10$ m, topped by a hemispherical dome of the same radius. $(\pi ={\displaystyle \frac{22}{7}})$

1. Find the curved surface area of the dome. [1 mark]

2. Find the volume of air inside the hall. [1 mark]

3. The inner curved wall and the inside of the dome are to be whitewashed at ₹15 per m². Find the cost. [2 marks]

   OR

   Safety norms require at least $11{\text{m}}^3$ of air per person. What is the greatest number of people the hall may hold? [2 marks]

Theatre pricing. A theatre in Chennai sells $300$ tickets a show at ₹200 each. A survey suggests that for every ₹10 cut in the price, $30$ more tickets would be sold. Let the price be cut by ₹$10x$.

1. Find the revenue per show at the present price. [1 mark]

2. Write the revenue $R$ as an expression in $x$. [1 mark]

3. Find the ticket price(s) that would give a revenue of ₹64,800. [2 marks]

   OR

   Show that a revenue of ₹70,000 per show cannot be reached. [2 marks]

Blood donation camp. At a blood donation camp in Chandigarh, $150$ donors registered: $60$ of group O, $36$ of group A, $42$ of group B and $12$ of group AB. One registration card is picked at random.

1. Find the probability that the donor is of group O. [1 mark]

2. Find the probability that the donor is not of group AB. [1 mark]

3. Later, $30$ more donors, all of group A, register. A card is now picked at random from all the cards. Find the probability that it belongs to group A. [2 marks]

   OR

   With the original $150$ donors, find the probability that the donor is (a) of group A or B, (b) of neither group O nor group B. [2 marks]