CBSE Class 10 2025 Mathematics Set-3 (430/6/3) Question Paper : Download Solution PDF With Answer Key

Question 1:

In two concentric circles centred at \( O \), a chord \( AB \) of the larger circle touches the smaller circle at \( C \). If \( OA = 3.5 \, cm \), \( OC = 2.1 \, cm \), then \( AB \) is equal to

• (A) 5.6 cm
• (B) 2.8 cm
• (C) 3.5 cm
• (D) 4.2 cm

Three coins are tossed together. The probability that at least one head comes up is

• (A) \(\dfrac{3}{8}\)
• (B) \(\dfrac{7}{8}\)
• (C) \(\dfrac{1}{8}\)
• (D) \(\dfrac{3}{4}\)

The volume of air in a hollow cylinder is \(450 \, cm^3\). A cone of same height and radius as that of cylinder is kept inside it. The volume of empty space in the cylinder is

• (A) \(225 \, cm^3\)
• (B) \(150 \, cm^3\)
• (C) \(250 \, cm^3\)
• (D) \(300 \, cm^3\)

If the length of the shadow of a tower is \(\sqrt{3}\) times its height, then the angle of elevation of the sun is

• (A) \(45^\circ\)
• (B) \(30^\circ\)
• (C) \(60^\circ\)
• (D) \(0^\circ\)

22nd term of the A.P.: \(\frac{3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2}, \ldots\) is

• (A) \(\dfrac{45}{2}\)
• (B) \(-9\)
• (C) \(-\dfrac{39}{2}\)
• (D) \(-21\)

In the given graph, the polynomial \(p(x)\) is shown. Number of zeroes of \(p(x)\) is

• (A) 3
• (B) 2
• (C) 1
• (D) 4

If probability of happening of an event is 57%, then probability of non-happening of the event is

• (A) 0.43
• (B) 0.57
• (C) 53%
• (D) \(\dfrac{1}{57}\)

OAB is a sector of a circle with centre O and radius 7 cm. If length of arc \(\overset{\frown}{AB} = \dfrac{22}{3} \, cm\), then \(\angle AOB\) is equal to

• (A) \(\left(\dfrac{120}{7}\right)^\circ\)
• (B) \(45^\circ\)
• (C) \(60^\circ\)
• (D) \(30^\circ\)

If the sum of first \(n\) terms of an A.P. is given by \(S_n = \dfrac{n}{2}(3n + 1)\), then the first term of the A.P. is

• (A) 2
• (B) \(\dfrac{3}{2}\)
• (C) 4
• (D) \(\dfrac{5}{2}\)

To calculate mean of grouped data, Rahul used assumed mean method. He used \(d = (x - A)\), where \(A\) is the assumed mean. Then \(\bar{x}\) is equal to

• (A) \(A + \bar{d}\)
• (B) \(A + h\bar{d}\)
• (C) \(h(A + \bar{d})\)
• (D) \(A - h\bar{d}\)

The point \((3, -5)\) lies on the line \(mx - y = 11\). The value of \(m\) is

• (A) 3
• (B) -2
• (C) -8
• (D) 2

If \(\sqrt{3} \sin \theta = \cos \theta\), then value of \(\theta\) is

• (A) \(\sqrt{3}\)
• (B) \(60^\circ\)
• (C) \(\dfrac{1}{\sqrt{3}}\)
• (D) \(30^\circ\)

ABCD is a rectangle with its vertices at \((2, -2), (8, 4), (4, 8), (-2, 2)\) taken in order. Length of its diagonal is

• (A) \(4\sqrt{2}\)
• (B) \(6\sqrt{2}\)
• (C) \(4\sqrt{26}\)
• (D) \(2\sqrt{26}\)

Two dice are rolled together. The probability of getting a sum more than 9 is

• (A) \(\dfrac{5}{6}\)
• (B) \(\dfrac{5}{18}\)
• (C) \(\dfrac{1}{6}\)
• (D) \(\dfrac{1}{2}\)

In \(\triangle ABC\), \(DE \parallel BC\). If \(AE = (2x + 1)\) cm, \(EC = 4\) cm, \(AD = (x + 1)\) cm and \(DB = 3\) cm, then value of \(x\) is

• (A) 1
• (B) 0
• (C) -1
• (D) \(\dfrac{1}{3}\)

The value of \(k\) for which the system of equations \(3x - 7y = 1\) and \(kx + 14y = 6\) is inconsistent, is

• (A) -6
• (B) \(\dfrac{2}{3}\)
• (C) 6
• (D) \(-\dfrac{3}{2}\)

In the given figure, \(PA\) is tangent to a circle with centre \(O\). If \(\angle APO = 30^\circ\) and \(OA = 2.5\, cm\), then \(OP\) is equal to

• (A) 2.5 cm
• (B) 5 cm
• (C) \(\dfrac{5}{\sqrt{3}}\, cm\)
• (D) 2 cm

Two identical cones are joined as shown in the figure. If radius of base is 4 cm and slant height of the cone is 6 cm, then height of the solid is

• (A) 8 cm
• (B) \(4\sqrt{5}\, cm\)
• (C) \(2\sqrt{5}\, cm\)
• (D) 12 cm

Assertion (A): \((a + \sqrt{b}) \cdot (a - \sqrt{b})\) is a rational number, where \(a\) and \(b\) are positive integers.

Reason (R): Product of two irrationals is always rational.

• (A) Both Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
• (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for Assertion (A).
• (C) Assertion (A) is true, but Reason (R) is false.
• (C) Assertion (A) is false, but Reason (R) is true.

Assertion (A): \(\triangle ABC \sim \triangle PQR\) such that \(\angle A = 65^\circ\), \(\angle C = 60^\circ\), \(\angle Q = 55^\circ\). Hence \(\angle Q = 55^\circ\).

Reason (R): Sum of all angles of a triangle is \(180^\circ\).

• (A) Both Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
• (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for Assertion (A).
• (C) Assertion (A) is true, but Reason (R) is false.
• (C) Assertion (A) is false, but Reason (R) is true.

A box contains 120 discs, which are numbered from 1 to 120. If one disc is drawn at random from the box, find the probability that

• (i) it bears a 2-digit number
• (ii) the number is a perfect square.

(a) Evaluate: \(\dfrac{\cos 45^\circ}{\tan 30^\circ + \sin 60^\circ}\)

(b) Verify that \(\sin 2A = \dfrac{2\tan A}{1 + \tan^2 A}\), for \(A = 30^\circ\)

(a) Solve the quadratic equation \(\sqrt{3}x^2 + 10x + 7\sqrt{3} = 0\) using the quadratic formula.

(b) Find the nature of roots of the equation \(4x^2 - 4a^2x + a^4 - b^4 = 0\), where \(b \ne 0\).

Using prime factorisation, find the HCF of 180, 140 and 210.

The perimeters of two similar triangles are 22 cm and 33 cm respectively. If one side of the first triangle is 9 cm, then find the length of the corresponding side of the second triangle.

Given that \(\sqrt{5}\) is an irrational number, prove that \(2 + 3\sqrt{5}\) is an irrational number.

(a) Find the A.P. whose third term is 16 and seventh term exceeds the fifth term by 12. Also, find the sum of first 29 terms of the A.P.

(b) Find the sum of first 20 terms of an A.P. whose \(n^th\) term is given by \(a_n = 5 + 2n\). Can 52 be a term of this A.P.?

Prove that \(\dfrac{\sin \theta}{1 + \cos \theta} + \dfrac{1 + \cos \theta}{\sin \theta} = 2\csc \theta\)

Find the length and breadth of a rectangular park whose perimeter is 100 m and area is \(600\, m^2\).

AB and CD are diameters of a circle with centre \(O\) and radius 7 cm. If \(\angle BOD = 30^\circ\), then find the area and perimeter of the shaded region.

(a) \(\alpha, \beta\) are zeroes of the polynomial \(3x^2 - 8x + k\). Find the value of \(k\), if \(\alpha^2 + \beta^2 = \dfrac{40}{9}\)

(b) Find the zeroes of the polynomial \(2x^2 + 7x + 5\) and verify the relationship between its zeroes and coefficients.

Find the ‘mean’ and ‘mode’ marks of the following data:

(a) Solve the following pair of linear equations by graphical method:
\[ 2x + y = 9 \quad and \quad x - 2y = 2 \]

(b) Nidhi received simple interest of ₹1200 when invested ₹\(x\) at 6% p.a. and ₹\(y\) at 5% p.a. for 1 year.

Had she invested ₹\(x\) at 3% p.a. and ₹\(y\) at 8% p.a. for that year, she would have received simple interest of ₹1260.

Find the values of \(x\) and \(y\).

(a) The given figure shows a circle with centre \(O\) and radius \(4\, cm\) inscribed in \(\triangle ABC\). \(BC\) touches the circle at \(D\), such that \(BD = 6\, cm\) and \(DC = 10\, cm\). Find the length of \(AE\), where \(E\) is the point of contact on \(AC\).

(b) PA and PB are tangents drawn to a circle with centre \(O\). If \(\angle AOB = 120^\circ\) and \(OA = 10\, cm\), then:

• (i) Find \(\angle OPA\)
• (ii) Find the perimeter of AOAP.
  (iii) Find the length of chord AB.

A drone is flying at a height of \(h\) metres. At an instant it observes the angle of elevation of the top of an industrial turbine as \(60^\circ\) and the angle of depression of the foot of the turbine as \(30^\circ\). If the height of the turbine is \(200\, metres\), find the value of \(h\) and the distance of the drone from the turbine.

(Use \(\sqrt{3} = 1.73\))

A triangular window of a building is shown above. Its diagram represents a \(\triangle ABC\) with \(\angle A = 90^\circ\) and \(AB = AC\). Points \(P\) and \(R\) trisect \(AB\), and \(PQ \parallel RS \parallel AC\).

Based on the figure, answer the following:

• (A) [(i)] Show that \(\triangle BPQ \sim \triangle BAC\)
• (B) [(ii)] Prove that \(PQ = \dfrac{1}{3} AC\)
• (C) [(iii)(a)] If \(AB = 3\, m\), find length \(BQ\) and \(BS\). Verify that \(BQ = \dfrac{1}{2} BS\)
  
  \textbf{OR}
• (D) [(iii)(b)] Prove that \(BR^2 + RS^2 = \dfrac{4}{9} BC^2\)

Gurveer and Arushi built a robot that can paint a path as it moves on a graph paper. Some co-ordinate of points are marked on it. It starts from (0, 0), moves to the points listed in order (in straight lines) and ends at (0, 0).



Arushi entered the points P(8, 6), Q(12, 2) and S(- 6, 6) in order. The path drawn by robot is shown in the figure.
Based on the above, answer the following:

• (A) [(i)] Determine the distance \(OP\)
• (B) [(ii)] \(QS\) is represented by the equation \(2x + 9y = 42\). Find the coordinates of the point where it intersects the y-axis.
• (C) [(iii)(a)] Point \(R(4.8, y)\) divides the line segment \(OP\) in a certain ratio. Find the value of \(y\) and hence the ratio.
  
  \textbf{OR}
• (D) [(iii)(b)] Using distance formula, show that \(\dfrac{PQ}{OS} = \dfrac{2}{3}\)

A hemispherical bowl is packed in a cuboidal box. The bowl just fits in the box. Inner radius of the bowl is \(10\, cm\). Outer radius of the bowl is \(10.5\, cm\).
Answer the following questions:

• (A) [(i)] Find the dimensions of the cuboidal box.
• (B) [(ii)] Find the total outer surface area of the box.
• (C) [(iii)(a)] Find the difference between the capacity of the bowl and the volume of the box. (Use \(\pi = 3.14\))
  \textbf{OR}
• (D) [(iii)(b)] The inner surface of the bowl and the thickness is to be painted. Find the area to be painted.