CBSE Class 10 Maths Standard 2025 Question Paper 30/1/3 with Solutions

What is the mode of a data if median and mean of the same data are $9.6$ and $10.5$, respectively ?

1. (A)$7.8$
2. (B)$12.3$
3. (C)$8.4$
4. (D)$7$

The value of $(\tan A\mathrm{cosec}A{)}^2-(\sin A\sec A{)}^2$ is :

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

A kite is flying at a height of $150$ m from the ground. It is attached to a string inclined at an angle of $30^{\circ }$ to the horizontal. The length of the string is :

1. (A)$100\sqrt{3}$ m
2. (B)$300$ m
3. (C)$150\sqrt{2}$ m
4. (D)$150\sqrt{3}$ m

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

If $\theta$ is an acute angle and $7+4\sin \theta =9$, then the value of $\theta$ is :

1. (A)$90^{\circ }$
2. (B)$30^{\circ }$
3. (C)$45^{\circ }$
4. (D)$60^{\circ }$

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)$4$

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)$50^{\circ }$

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)$198$

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)$1.857142$

The sum of the zeroes of the polynomial $p(x)=5x-7x^2+3$ is :

1. (A)$\frac{-7}{5}$
2. (B)$\frac{7}{5}$
3. (C)$\frac{5}{7}$
4. (D)$\frac{-5}{7}$

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)$2a+b=0$

If a sector of a circle has an area of $40\pi$ sq. units and a central angle of $72^{\circ }$, the radius of the circle is :

1. (A)$200$ units
2. (B)$100$ units
3. (C)$20$ units
4. (D)$10\sqrt{2}$ units

The tangents drawn at the extremities of the diameter of a circle are always :

1. (A)parallel
2. (B)perpendicular
3. (C)equal
4. (D)intersecting

If $(-1{)}^n+(-1{)}^8=0$, then $n$ is :

1. (A)any positive integer
2. (B)any negative integer
3. (C)any odd number
4. (D)any even number

The end points of a diameter of circle are $(2,4)$ and $(-3,-1)$. The length of its radius is :

1. (A)$\frac{5\sqrt{2}}{2}$ units
2. (B)$5\sqrt{2}$ units
3. (C)$3\sqrt{2}$ units
4. (D)$\pm \frac{5\sqrt{2}}{2}$ units

The $11^{\text{th}}$ and $13^{\text{th}}$ term of an AP are $39$ and $45$, respectively. What is the common difference of the AP ?

1. (A)$42$
2. (B)$21$
3. (C)$6$
4. (D)$3$

A card is drawn at random from a pack of $52$ cards. What is the probability that the card drawn is a spade or a king ?

1. (A)$\frac{1}{13}$
2. (B)$\frac{2}{13}$
3. (C)$\frac{4}{13}$
4. (D)$\frac{9}{13}$

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.

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.

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

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.

Find the zeroes of the polynomial $p(x)=x^2+\frac{4}{3}x-\frac{4}{3}$.

Find the length of the median through the vertex $B$ of $△ABC$ with vertices $A(9,-2)$, $B(-3,7)$ and $C(-1,10)$.

Prove that $\sqrt{5}$ is an irrational number.

Two dice are rolled together. Find the probability of getting :
(i) a multiple of 2 on one and a multiple of 3 on the other die.
(ii) the product of two numbers on the top of the two dice is a perfect square number.

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

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

A room is in the form of a cylinder surmounted by a hemispherical dome. The base radius of the hemisphere is half of the height of the cylindrical part. If the room contains $\frac{1408}{21}$ m${}^3$ of air, find the height of the cylindrical part. (Use $\pi =\frac{22}{7}$).

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 perimeter of a right triangle is $60$ cm and its hypotenuse is $25$ cm. Find the lengths of other two sides of the triangle.

A train travels a distance of $480$ km at a uniform speed. If the speed had been $8$ km/h less, then it would have taken $3$ hours more to cover the same distance. Find the speed of the train.

A bag contains some red and blue balls. Ten percent of the red balls, when added to twenty percent of the blue balls, give a total of $24$. If three times the number of red balls exceeds the number of blue balls by $20$, find the number of red and blue balls.

The lengths of 40 leaves of a plant are measured correct to the nearest millimetre, and the data obtained is represented in the following table :
Length (in mm): 118 – 126, 127 – 135, 136 – 144, 145 – 153, 154 – 162, 163 – 171, 172 – 180
Number of Leaves: 3, 5, 9, 12, 5, 4, 2
Find the median length of the leaves.

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 $\angle BAC=90^{\circ }$.

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)

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)