Class 10 Maths Sample Paper 1 (CBSE 2026-27 pattern)

If $p=2^4\times 3\times 5^2$ and $q=2^2\times 3^3\times 5$, then $\text{HCF}(p,q)$ is

1. (a) $30$
2. (b) $60$
3. (c) $120$
4. (d) $180$

If $3$ is a zero of the polynomial $x^2-kx+12$, then the value of $k$ is

1. (a) $5$
2. (b) $7$
3. (c) $-7$
4. (d) $12$

The pair of equations $3x+5y=7$ and $6x+ky=11$ has no solution when $k$ equals

1. (a) $5$
2. (b) $-10$
3. (c) $10$
4. (d) ${\displaystyle \frac{22}{7}}$

The roots of $2x^2-6x+c=0$ are real and equal. The value of $c$ is

1. (a) $3$
2. (b) ${\displaystyle \frac{9}{2}}$
3. (c) $9$
4. (d) $18$

The 11th term of the AP $-7,-3,1,5,\dots$ is

1. (a) $29$
2. (b) $33$
3. (c) $37$
4. (d) $40$

The distance between the points $(a,-b)$ and $(-a,b)$ is

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

The mid-point of the segment joining $(a,4)$ and $(6,b)$ is $(4,7)$. Then $(a,b)$ is

1. (a) $(2,10)$
2. (b) $(10,2)$
3. (c) $(2,3)$
4. (d) $(-2,10)$

In $\mathrm{△}PQR$, $S$ and $T$ are points on $PQ$ and $PR$ with $ST\parallel QR$. If $PS=3$ cm, $SQ=4.5$ cm and $PT=2.4$ cm, then $TR$ is

1. (a) $1.6$ cm
2. (b) $3.6$ cm
3. (c) $4.8$ cm
4. (d) $5.4$ cm

$PA$ and $PB$ are tangents from an external point $P$ to a circle with centre $O$. If $\mathrm{\angle }APB={64}^{\circ }$, then $\mathrm{\angle }OAB$ is

1. (a) ${26}^{\circ }$
2. (b) ${32}^{\circ }$
3. (c) ${58}^{\circ }$
4. (d) ${64}^{\circ }$

If $7\sin A=24\cos A$, where $A$ is acute, then $\sec A-\tan A$ equals

1. (a) ${\displaystyle \frac{1}{7}}$
2. (b) $7$
3. (c) ${\displaystyle \frac{25}{7}}$
4. (d) ${\displaystyle \frac{1}{25}}$

A taut guy-wire $40$ m long runs from the top of a vertical telecom mast to a peg on level ground and makes an angle of ${30}^{\circ }$ with the ground. The height of the mast is

1. (a) $20\sqrt{3}$ m
2. (b) $40\sqrt{3}$ m
3. (c) $20$ m
4. (d) ${\displaystyle \frac{40}{\sqrt{3}}}$ m

The area of a sector of a circle of radius $21$ cm with central angle ${40}^{\circ }$ is $(\pi ={\displaystyle \frac{22}{7}})$

1. (a) $77{\text{cm}}^2$
2. (b) $132{\text{cm}}^2$
3. (c) $154{\text{cm}}^2$
4. (d) $308{\text{cm}}^2$

The length of the arc of a circle of radius $14$ cm that subtends an angle of ${45}^{\circ }$ at the centre is $(\pi ={\displaystyle \frac{22}{7}})$

1. (a) $5.5$ cm
2. (b) $11$ cm
3. (c) $22$ cm
4. (d) $44$ cm

A toy is a cone mounted on a hemisphere of the same radius $3.5$ cm. The slant height of the cone is $5$ cm. The total surface area of the toy is $(\pi ={\displaystyle \frac{22}{7}})$

1. (a) $93.5{\text{cm}}^2$
2. (b) $115.5{\text{cm}}^2$
3. (c) $132{\text{cm}}^2$
4. (d) $154{\text{cm}}^2$

For a grouped distribution, the modal class is $40\text{–}50$, and the frequencies of the modal class, the class before it and the class after it are $16$, $10$ and $12$ respectively. The mode is

1. (a) $44$
2. (b) $45$
3. (c) $46$
4. (d) $48$

For a moderately skewed distribution, the mean is $24$ and the median is $26$. Using the empirical relation, the mode is

1. (a) $22$
2. (b) $25$
3. (c) $28$
4. (d) $30$

A bag has $5$ red, $8$ green and $7$ blue discs of the same size. One disc is drawn at random. The probability that it is not green is

1. (a) ${\displaystyle \frac{2}{5}}$
2. (b) ${\displaystyle \frac{3}{5}}$
3. (c) ${\displaystyle \frac{7}{20}}$
4. (d) ${\displaystyle \frac{1}{4}}$

A number is chosen at random from $1,2,3,\dots ,30$. The probability that it is a multiple of both $3$ and $4$ is

1. (a) ${\displaystyle \frac{1}{15}}$
2. (b) ${\displaystyle \frac{1}{10}}$
3. (c) ${\displaystyle \frac{1}{6}}$
4. (d) ${\displaystyle \frac{2}{5}}$

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

Assertion (A): The graph of $y=x^2+4x+5$ does not meet the $x$-axis.

Reason (R): A quadratic polynomial $a{x}^2+bx+c$ has no real zeroes when $b^2-4ac<0$.

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): $0$ is a term of the AP $31,28,25,\dots$

Reason (R): An AP with a negative common difference eventually has negative terms.

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.

Two numbers are $2^4\times 3^a\times 5$ and $2^b\times 3^2\times 7$, where $a,b$ are natural numbers. Their HCF is $36$ and their LCM is $5040$. Find $a$ and $b$.

Find the zeroes of $10x^2-9x-7$ and verify the relationship between the zeroes and the coefficients.

In $\mathrm{△}ABC$, $D$ and $E$ are points on $AB$ and $AC$ respectively such that $DE\parallel BC$. If $AD=4$ cm, $DB=(x-4)$ cm, $AE=8$ cm and $EC=(3x-19)$ cm, find $x$.

OR

Evaluate: $4{\sin }^2{60}^{\circ }+3{\tan }^2{30}^{\circ }-8\sin {45}^{\circ }\cos {45}^{\circ }$.

Two fair dice are thrown together. Find the probability that the product of the numbers on the top faces is a perfect square.

OR

Prove that $\sqrt{3}$ is irrational. Hence show that $(2+\sqrt{3}{)}^2$ is irrational.

At a school canteen in Lucknow, 3 samosas and 2 plates of idli cost ₹130, while 2 samosas and 5 plates of idli cost ₹215. Find the cost of one samosa and of one plate of idli.

In $\mathrm{△}ABC$, $D$ is a point on $AB$ and $E$ a point on $AC$ such that $DE\parallel BC$. Through $E$ a line is drawn parallel to $AB$, meeting $BC$ at $F$. Prove that ${\displaystyle \frac{AD}{DB}}={\displaystyle \frac{BF}{FC}}$. Hence, if $AD=3$ cm, $DB=5$ cm and $BC=16$ cm, find $FC$.

OR

A circle is inscribed in $\mathrm{△}ABC$, touching $BC$, $CA$ and $AB$ at $D$, $E$ and $F$ respectively. If $AB=13$ cm, $BC=15$ cm and $CA=14$ cm, find $AF$, $BD$ and $CE$.

Prove that $(\sin A+\cos A)(\tan A+\cot A)=\sec A+\mathrm{cosec}A$.

A solid wooden toy is a cylinder of radius $3.5$ cm and height $6$ cm, with a cone of the same radius and height $12$ cm fixed on top. Find the total surface area and the volume of the toy. $(\pi ={\displaystyle \frac{22}{7}})$

OR

An electrical shop in Indore buys a lot of identical LED bulbs for ₹4800. If each bulb had cost ₹40 less, the shop would have got $20$ more bulbs for the same amount. Form a quadratic equation and find the number of bulbs bought and the cost of each bulb.

OR

State and prove the Basic Proportionality Theorem. Using it, solve: in $\mathrm{△}ABC$, $D$ and $E$ lie on $AB$ and $AC$ with $DE\parallel BC$, $AD:DB=2:3$, $DE=6$ cm and $AE=4.8$ cm. Find $BC$ and $AC$.

OR

A survey drone rises vertically from a point $D$ on level ground at a constant speed. An observer stands at $O$ on the same ground, $90$ m from $D$. At one instant the angle of elevation of the drone from $O$ is ${30}^{\circ }$; $20$ seconds later it is ${60}^{\circ }$. (a) Find the heights of the drone at the two instants. (b) Find the speed of ascent. (c) How many seconds after the first observation was the angle of elevation ${45}^{\circ }$? (Take $\sqrt{3}=1.73$.)

The daily electricity consumption (in units) of $50$ households in a Jaipur colony is given below.

(a) Find the mean consumption using the step-deviation method. (b) Find the median consumption. (c) The colony newsletter claims that 'more than half the households use at most 30 units a day'. Is the claim supported? Justify.

Daily savings. Aarav starts a savings jar for a school trip. He puts ₹5 in the jar on day 1, ₹8 on day 2, ₹11 on day 3, and so on, adding ₹3 more each day than the day before.

1. How much does he put in the jar on day 10? [1 mark]

2. On which day does he put ₹71 in the jar? [1 mark]

3. Find the total amount in the jar at the end of 30 days. [2 marks]

   OR

   After how many days will the total in the jar first reach ₹670? [2 marks]

Village planning. On a map of a village near Nashik drawn on a grid (1 unit = 1 km), the school is at $S(2,3)$ and the primary health centre is at $H(8,11)$. A straight road joins $S$ and $H$.

1. Find the length of the road $SH$. [1 mark]

2. A bus stop is at the mid-point of $SH$. Find its coordinates. [1 mark]

3. A water tank $T$ is on the road with $ST:TH=3:2$. Find the coordinates of $T$. [2 marks]

   OR

   A well is to be dug on the $x$-axis at equal distances from $S$ and $H$. Find its coordinates. [2 marks]

Watering the lawn. A rectangular lawn in a housing society measures $20$ m by $15$ m. A sprinkler fixed at one corner sprays water up to $14$ m and turns through ${90}^{\circ }$, so it waters a quarter-circle of the lawn. $(\pi ={\displaystyle \frac{22}{7}})$

1. Find the area of the lawn watered by the sprinkler. [1 mark]

2. Find the area of the lawn that is not watered. [1 mark]

3. The curved edge of the watered region is to be lined with bricks at ₹25 per metre, and the watered region is to be re-turfed at ₹40 per m². Find the total cost. [2 marks]

   OR

   Instead, the sprinkler is placed at the mid-point of a $20$ m side and turns through ${180}^{\circ }$ with a spray radius of $10$ m. Find the area watered and the percentage of the lawn it covers. [2 marks]