CBSE Class 10 Maths Standard 2025 Question Paper 30/1/2 with Solutions

In the given figure, $PA$ is a tangent from an external point $P$ to a circle with centre $O$. If $\angle POB=115^{\circ }$, then $\angle APO$ is equal to :

1. (A)$25^{\circ }$
2. (B)$65^{\circ }$
3. (C)$90^{\circ }$
4. (D)$35^{\circ }$

A piece of wire $20$ cm long is bent into the form of an arc of a circle of radius $\frac{60}{\pi }$ cm. The angle subtended by the arc at the centre of the circle is :

1. (A)$30^{\circ }$
2. (B)$60^{\circ }$
3. (C)$90^{\circ }$
4. (D)$50^{\circ }$

Three numbers in AP have the sum $30$. What is its middle term ?

1. (A)$4$
2. (B)$10$
3. (C)$16$
4. (D)$8$

An arc of a circle is of length $5\pi$ cm and the sector it bounds has an area of $20\pi$ cm${}^2$. Its radius is :

1. (A)$10$ cm
2. (B)$1$ cm
3. (C)$5$ cm
4. (D)$8$ cm

If $x=1$ and $y=2$ is a solution of the pair of linear equations $2x-3y+a=0$ and $2x+3y-b=0$, then :

1. (A)$a=2b$
2. (B)$2a=b$
3. (C)$a+2b=0$
4. (D)$2a+b=0$

Two polynomials are shown in the graph below. The number of distinct zeroes of both the polynomials is :

1. (A)$3$
2. (B)$5$
3. (C)$2$
4. (D)$4$

If $\alpha +\beta =90^{\circ }$ and $\alpha =2\beta$, then ${\cos }^2\alpha +{\sin }^2\beta$ is equal to :

1. (A)$0$
2. (B)$\frac{1}{2}$
3. (C)$1$
4. (D)$2$

A card is selected at random from a deck of 52 playing cards. The probability of it being a red face card is :

1. (A)$\frac{3}{13}$
2. (B)$\frac{2}{13}$
3. (C)$\frac{1}{2}$
4. (D)$\frac{3}{26}$

If $\alpha$ and $\beta$ are the zeroes of polynomial $3x^2+6x+k$ such that $\alpha +\beta +\alpha \beta =-\frac{2}{3}$, then the value of $k$ is :

1. (A)$-8$
2. (B)$8$
3. (C)$-4$
4. (D)$4$

The value of ${\tan }^2\theta -\left(\frac{1}{\cos \theta }\times \sec \theta \right)$ is :

1. (A)$1$
2. (B)$0$
3. (C)$-1$
4. (D)$2$

Which of the following is a rational number between $\sqrt{3}$ and $\sqrt{5}$ ?

1. (A)$1.4142387954012\dots$
2. (B)$2.32\overline{6}$
3. (C)$\pi$
4. (D)$1.857142$

If $\text{HCF}(98,28)=m$ and $\text{LCM}(98,28)=n$, then the value of $n-7m$ is :

1. (A)$0$
2. (B)$28$
3. (C)$98$
4. (D)$198$

If the length of a chord of a circle is equal to its radius, then the angle subtended by chord at the centre is :

1. (A)$60^{\circ }$
2. (B)$30^{\circ }$
3. (C)$120^{\circ }$
4. (D)$90^{\circ }$

The greatest number which divides $70$ and $125$, leaving remainders $5$ and $8$ respectively, is :

1. (A)$13$
2. (B)$65$
3. (C)$875$
4. (D)$1750$

A ladder $14$ m long leans against a wall. If the foot of the ladder is $7$ m from the wall, then the angle of elevation of the top of the wall is :

1. (A)$15^{\circ }$
2. (B)$30^{\circ }$
3. (C)$45^{\circ }$
4. (D)$60^{\circ }$

In triangles $ABC$ and $DEF$, $\angle B=\angle E$, $\angle F=\angle C$ and $AB=3DE$. Then, the two triangles are :

1. (A)congruent but not similar
2. (B)congruent as well as similar
3. (C)neither congruent nor similar
4. (D)similar but not congruent

The mid-point of the line segment joining the points $P(-4,5)$ and $Q(4,6)$ lies on :

1. (A)x-axis
2. (B)y-axis
3. (C)origin
4. (D)neither x-axis nor y-axis

Mode and Mean of a data are $15x$ and $18x$, respectively. Then the median of the data is :

1. (A)$x$
2. (B)$11x$
3. (C)$17x$
4. (D)$34x$

Assertion (A) : If we join two hemispheres of same radius along their bases, then we get a sphere.
Reason (R) : Total Surface Area of a sphere of radius $r$ is $3\pi r^2$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

Assertion (A) : The probability of selecting a number at random from the numbers 1 to 20 is 1.
Reason (R) : For any event E, if P(E) = 1, then E is called a sure event.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the 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 the zeroes of the polynomial $x^2+ax+b$ are in the ratio $3:4$, then prove that $12a^2=49b$.

A person is standing at $P$ outside a circular ground at a distance of $26$ m from the centre of the ground. He found that his distances from the points $A$ and $B$ on the ground are $10$ m ($PA$ and $PB$ are tangents to the circle). Find the radius of the circular ground.

If $△ABC\sim △PQR$ in which $AB=6$ cm, $BC=4$ cm, $AC=8$ cm and $PR=6$ cm, then find the length of $(PQ+QR)$.

In the given figure, $\frac{QR}{QS}=\frac{QT}{PR}$ and $\angle 1=\angle 2$, show that $△PQS\sim △TQR$.

If $x\cos 60^{\circ }+y\cos 0^{\circ }+\sin 30^{\circ }-\cot 45^{\circ }=5$, then find the value of $x+2y$.

Evaluate : $\frac{{\tan }^{2}60^{\circ }}{{\sin }^{2}60^{\circ }+{\cos }^{2}30^{\circ }}$

The coordinates of the centre of a circle are $(2a,a-7)$. Find the value(s) of 'a' if the circle passes through the point $(11,-9)$ and has diameter $10\sqrt{2}$ units.

If the radii of the bases of a cylinder and a cone are in the ratio $3:4$ and their heights are in the ratio $2:3$, find the ratio of their volumes.

Three sets of Physics, Chemistry and Mathematics books have to be stacked in such a way that all the books are stored subject-wise and the height of each stack is the same. The number of Physics books is $144$, the number of Chemistry books is $180$ and the number of Mathematics books is $192$. Assuming that the books are of same thickness, determine the number of stacks of Physics, Chemistry and Mathematics books.

Two dice are thrown at the same time. Determine the probability that the difference of the numbers on the two dice is 2.

Prove that : $\frac{\tan \theta }{1-\cot \theta }+\frac{\cot \theta }{1-\tan \theta }=1+\sec \theta \mathrm{cosec}\theta$

Prove that : $\frac{\sin A+\cos A}{\sin A-\cos A}+\frac{\sin A-\cos A}{\sin A+\cos A}=\frac{2}{2{\sin }^{2}A-1}$

In the given figure, $O$ is the centre of the circle and $BCD$ is tangent to it at $C$. Prove that $\angle BAC+\angle ACD=90^{\circ }$.

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Find the ratio in which the y-axis divides the line segment joining the points $(5,-6)$ and $(-1,-4)$. Also find the point of intersection.

The diagonal $BD$ of a parallelogram $ABCD$ intersects the line segment $AE$ at the point $F$, where $E$ is any point on the side $BC$. Prove that $DF\times EF=FB\times FA$.

In $△ABC$, if $AD\perp BC$ and $A{D}^2=BD\times DC$, then prove that $\angle BAC=90^{\circ }$.

The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the mean and mode of the data :
Monthly Consumption (in units): 65 – 85, 85 – 105, 105 – 125, 125 – 145, 145 – 165, 165 – 185, 185 – 205
Number of Consumers: 4, 5, 13, 20, 14, 8, 4

Vijay invested certain amounts of money in two schemes A and B, which offer interest at the rate of $8\%$ per annum and $9\%$ per annum, respectively. He received ₹ 1,860 as the total annual interest. However, had he interchanged the amounts of investments in the two schemes, he would have received ₹ 20 more as annual interest. How much money did he invest in each scheme ?

A two-digit number is such that the product of its digits is $12$. When $36$ is added to this number, the digits interchange their places. Find the number.

A student scored a total of $32$ marks in class tests in Mathematics and Science. Had he scored $2$ marks less in Science and $4$ marks more in Mathematics, the product of his marks would have been $253$. Find his marks in the two subjects.

A brooch is a decorative piece often worn on clothing like jackets, blouses or dresses to add elegance. Made from precious metals and decorated with gemstones, brooches come in many shapes and designs.
One such brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in the figure.
Based on the above given information, answer the following questions :
(i) Find the central angle of each sector. (1)
(ii) Find the length of the arc ACB. (1)
(iii) (a) Find the area of each sector of the brooch. (2)
OR
(iii) (b) Find the total length of the silver wire used. (2)

Amrita stood near the base of a lighthouse, gazing up at its towering height. She measured the angle of elevation to the top and found it to be $60^{\circ }$. Then, she climbed a nearby observation deck, 40 metres higher than her original position and noticed the angle of elevation to the top of lighthouse to be $45^{\circ }$.
Based on the above given information, answer the following questions :
(i) If CD is h metres, find the distance BD in terms of 'h'. (1)
(ii) Find distance BC in terms of 'h'. (1)
(iii) (a) Find the height CE of the lighthouse [Use $\sqrt{3}=1.73$] (2)
OR
(iii) (b) Find distance AE, if AC = 100 m. (2)

A school is organizing a charity run to raise funds for a local hospital. The run is planned as a series of rounds around a track, with each round being 300 metres. To make the event more challenging and engaging, the organizers decide to increase the distance of each subsequent round by 50 metres. For example, the second round will be 350 metres, the third round will be 400 metres and so on. The total number of rounds planned is 10.
Based on the information given above, answer the following questions :
(i) Write the fourth, fifth and sixth term of the Arithmetic Progression so formed. (1)
(ii) Determine the distance of the $8^{\text{th}}$ round. (1)
(iii) (a) Find the total distance run after completing all 10 rounds. (2)
OR
(iii) (b) If a runner completes only the first 6 rounds, what is the total distance run by the runner ? (2)