CBSE Class 10 Maths Standard 2026 Question Paper 30/4/3 with Solutions

The distance between the points $(a\cos \theta +b\sin \theta ,0)$ and $(0,a\sin \theta -b\cos \theta )$ is

1. (A)$\sqrt{a^2+b^2}$
2. (B)$a^2-b^2$
3. (C)$\sqrt{a^2-b^2}$
4. (D)$a^2+b^2$

An ice-cream cone of radius r and height h is completely filled by two spherical scoopes of ice-cream. If radius of each spherical scoop is $\frac{r}{2}$, then h : 2r equals

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

The value of k for which the system of linear equations $kx-y-2=0$ and $6x-2y-3=0$ has infinitely many solutions, is (does)

1. (A)$\frac{1}{2}$
2. (B)3
3. (C)4
4. (D)Not exist

A circle is divided into 16 identical sectors. If radius of the circle is 7 cm, area of each sector is

1. (A)$\frac{77}{4}$ cm${}^2$
2. (B)77 cm${}^2$
3. (C)154 cm${}^2$
4. (D)$\frac{77}{8}$ cm${}^2$

Mean and Median of a frequency distribution are 43 and 40 respectively. The value of mode is

1. (A)34
2. (B)43
3. (C)38.5
4. (D)41.5

Three tennis balls are just packed in a cylindrical jar. If radius of each ball is r, volume of air inside the jar is

1. (A)$2\pi r^3$
2. (B)$3\pi r^3$
3. (C)$5\pi r^3$
4. (D)$4\pi r^3$

Observe the graph of polynomial $p(x)$. The zeroes of the polynomial are

1. (A)$-2$, 0, 2.5
2. (B)$-2$, 2.5
3. (C)0, 4
4. (D)$-2$, 0

From an external point P, a tangent PT has been drawn to a circle with centre at O and radius 3 cm, intersecting its concentric circle at A and B. If AB = 8 cm and OA = AP, the length PQ equals.

1. (A)8 cm
2. (B)10 cm
3. (C)9 cm
4. (D)12 cm

$\frac{{\sec }^{2}A-1}{{\sin }^{2}A}$ is same as

1. (A)${\cos }^{2}A$
2. (B)${\sec }^{2}A$
3. (C)$-{\sec }^{2}A$
4. (D)${\cot }^{2}A$

PQ is tangent to a circle with centre O. If $\angle POR=65^{\circ }$, then $m\angle PTR$ is

1. (A)$65^{\circ }$
2. (B)$58.5^{\circ }$
3. (C)$57.5^{\circ }$
4. (D)$45^{\circ }$

In the given figure, a circle is centred at (1, 2). The diameter of the circle is

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

In the given figure, $DE\parallel BC$. If $\frac{AD}{DB}=\frac{1}{3}$ and AC = 6 cm, then length AE is

1. (A)1.5 cm
2. (B)1 cm
3. (C)2 cm
4. (D)3 cm

It is given that $△ABC\sim △EDF$. Which of the following is not true ?

1. (A)$\frac{\text{Perimeter of }△ABC}{\text{Perimeter of }△EDF}=\frac{AB}{ED}$
2. (B)$\frac{AB}{ED}=\frac{AC}{EF}$
3. (C)$\angle A=\angle D,\angle C=\angle F$
4. (D)$\frac{AB+BC}{AC}=\frac{DE+DF}{EF}$

If the quadratic equation $9x^2+8kx+16=0$ has real and equal roots, then the value of k is

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

$\frac{1+{\tan }^{2}A}{1+{\cot }^{2}A}$ equals to :

1. (A)${\tan }^{2}A$
2. (B)$-1$
3. (C)$-{\tan }^{2}A$
4. (D)${\cot }^{2}A$

If sum of first ten terms of an A.P. is zero with a as the first term and d, the common difference, which of the following relation is true ?

1. (A)$10a+9d=0$
2. (B)$2a=9d$
3. (C)$a_{10}=-a$
4. (D)$a_{10}=a$

Two dice are rolled together. The probability that sum of the numbers obtained is atmost 12, is

1. (A)1
2. (B)0
3. (C)$\frac{1}{2}$
4. (D)$\frac{35}{36}$

Arc PQ subtends an angle $\theta$ at the centre of the circle with radius 6.3 cm. If $\stackrel{⌢}{\mathrm{PQ}}=11$ cm, then the value of $\theta$ is

1. (A)$10^{\circ }$
2. (B)$60^{\circ }$
3. (C)$45^{\circ }$
4. (D)$100^{\circ }$

Assertion (A) : If probability of happening of an event is 0.2p, $p>0$, then p can't be more than 5.
Reason (R) : $P(\overline{E})=1-P(E)$ for an event E.

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 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) : $(\sqrt{3}+\sqrt{5})$ is an irrational number.
Reason (R) : Sum of the any two irrational numbers is always irrational.

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 Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

Prove that : $\frac{\tan \theta }{1+{\tan }^2\theta }+\frac{\cot \theta }{1+{\cot }^2\theta }=2\sin \theta \cos \theta$.

Evaluate : $\frac{1-2{\tan }^{2}30^{\circ }-{\sec }^{2}45^{\circ }}{{\sin }^{2}60^{\circ }}$

$\alpha$ and $\beta$ are the zeroes of the polynomial $5x^2-16x-10$. Find the value of $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$.

In the given figure, DEFG is a square. $△ABC$ is right angle triangle with $\angle A=90^{\circ }$. Prove that $AG\times DG=AF\times DB$.

Verify that roots of the quadratic equation $(p-q)x^2+(q-r)x+(r-p)=0$ are equal when $q+r=2p$.

Prove that $2+3\sqrt{5}$ is an irrational number given that $\sqrt{5}$ is irrational number.

If the HCF of 210 and 55 is expressed as $210\times 5+55m$, then find the value of m.

To protect plants from heat, a shed of iron rods covered with green cloth is made. The lower part of the shed is a cuboid mounted by semi-cylinder as shown in the figure. Find the area of the cloth required to make this shed, if dimensions of the cuboid are 14 m $\times$ 25 m $\times$ 16 m

The internal and external radii of a hollow hemisphere are $5\sqrt{2}$ cm and 10 cm respectively. A cone of height $5\sqrt{7}$ cm and radius $5\sqrt{2}$ cm is surmounted on the hemisphere as shown in the figure. Find the total surface area of the object in terms of $\pi$. (Use $\sqrt{2}=1.4$)

A bag contains 30 balls out of which ‘m’ number of balls are blue in colour.
(i) Find the probability that a ball drawn at random from the bag is not blue.
(ii) If 6 more blue balls are added in the bag, then the probability of drawing a blue ball will be $\frac{5}{4}$ times the probability of drawing a blue ball in the first case. Find the value of m.

Find the greatest number which divides 764 and 1198, leaving remainders 8 and 10 respectively.

In a class test, Veer scored 6 more than twice as many marks as Kevin scored. If one of them had scored 4 more marks, their total score would have been 40. Find the marks obtained by Veer and Kevin.

Solve the linear equations $3x+y=14$ and $y=2$ graphically.

Chord AB subtends an angle of $120^{\circ }$ at the centre O of the circle with radius $\frac{21}{2}$ cm. Find the perimeter of shaded segment ACB. (Use $\sqrt{3}=1.7$)

If $\sec \theta +\tan \theta =m$, show that $\frac{m^2-1}{m^2+1}=\sin \theta$.

A kite is flying at a height of 60 m above the ground level. Ravi, standing at the roof of the house is holding the string straight and observes the angle of elevation of kite as $30^{\circ }$. From the bottom of the same building, the angle of elevation of kite is $45^{\circ }$. Find the length of the string and height of roof from the ground. (Use $\sqrt{3}=1.73$)

A person on tour has ₹ 5,400 for his expenses. If he extends his tour by 5 days, he has to cut down his daily expenses by ₹ 180. Find the original duration of the tour and daily expense.

The total cost of certain piece of cloth was ₹ 2,100. During special sale time, the shopkeeper offered 2 m extra cloth for free thus reducing the price of cloth per metre by ₹ 120. What was the original per metre price of cloth and its length ?

Find mean and mode of the following frequency distribution :

If the median of the distribution given below is 28.5, find the values of $x$ and y.

In the given figure, TP and TQ are tangents to a circle with centre M, touching another circle with centre N at A and B respectively. It is given that MQ = 13 cm, NB = 8 cm, BQ = 35 cm and TP = 80 cm.
(i) Name the quadrilateral MQBN. (1)
(ii) Is MN parallel to PA ? Justify your answer. (1)
(iii) Find length TB. (1)
(iv) Find length MN. (2)

‘Kolam’ is a decorative art which is made with rice flour in South Indian States. It is drawn on grid pattern of dots. One such art work is shown below.
Observe the given figure carefully. There are 4 dots in first square, 8 dots in second square, 12 dots in third square and so on.
Based on the above, answer the following questions :
(i) Show that number of dots given above form an A.P. Write the first term and common difference. (1)
(ii) Write $n^{th}$ term of the A.P. formed. (1)
(iii) (a) The pattern is expanded on a large ground. If total 220 dots are used, then find the number of squares formed. (2)
OR (b) Is it possible to complete n number of squares using 100 dots ? If yes, then find the value of n.

Observe the map of Jaipur city placed on a Cartesian plane. Taking Rambagh Palace as origin, the location of some places are given below :
Point A : $(-4,2)$ Rajasthan High Court
Point B : $(4,-4)$ Birla Mandir
Point C : $(4,3)$ Heera Bagh
Point D : $(-5,-2)$ Amar Jawan Jyoti
Based on the above, answer the following questions :
(i) Advocate Rehana stays at Heera Bagh. How much distance she has to cover daily to go to the court and coming back home ? (1)
(ii) There is a crossing on X-axis which divides AD in a certain ratio. Find the ratio. (1)
(iii) (a) Is Birla Mandir equidistant from Heera Bagh and Amar Jawan Jyoti ? Justify your answer. (2)
OR (b) Using section formula, show that points A, O and B are not collinear.

Carom board is a very popular game. The board is a square of side length 65 cm. It has circular pockets in each corner.
Ansh strikes a disc, kept at position P with a striker. The disc, hits the boundary of the board at R and goes straight to pocket at corner C. It is given that PS = 9 cm, PQ = 35 cm, BR = $x$, $\angle PRQ=\alpha$ and $\angle CRB=\theta$.
Based on the above information, answer the following questions :
(i) Using law of reflection i.e. $\angle PRT=\angle CRT$, prove that $\theta =\alpha$. (1)
(ii) Prove that $△PQR\sim △CBR$ given that PQ is perpendicular to AB. (1)
(iii) (a) Find the value of $x$ using similarity of triangles. (2)
OR (b) If $\frac{\text{Area }△PQR}{\text{Area }△CBR}=\frac{P{Q}^2}{C{B}^2}$, then find the value of $x$.