CBSE Class 10 2025 Mathematics Set-1 (430/6/1) Question Paper : Download Solution PDF With Answer Key

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

• (A) 3
• (B) 2
• (C) 1
• (D) 4

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

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

The line \(2x - 3y = 6\) intersects x-axis at

• (A) (0, –2)
• (B) (0, 3)
• (C) (–2, 0)
• (D) (3, 0)

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

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) \(-\dfrac{3}{2}\)

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) \(\dfrac{1}{2}\)

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) \(2\sqrt{26}\)

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

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

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

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

• (A) \(1\)
• (B) \(\frac{1}{2}\)
• (C) \(-1\)
• (D) \(\frac{1}{3}\)

Three coins are tossed together. The probability that exactly one coin shows head, is

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

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

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

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

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

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

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

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

In \(\triangle ABC, \angle B = 90^\circ\). If \(\frac{AB}{AC} = \frac{1}{2}\), then \(\cos C\) is equal to

• (A) \(\frac{3}{2}\)
• (B) \(\frac{1}{2}\)
• (C) \(\frac{\sqrt{3}}{2}\)
• (D) \(\frac{1}{\sqrt{3}}\)

The volume of air in a hollow cylinder is \(450\ cm^3\). A cone of same height and radius as that of the 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) \(300\ cm^3\)

Assertion (A): \((a+\sqrt{b})(a-\sqrt{b})\) 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 the correct explanation of Assertion (A).
• (B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
• (C) Assertion (A) is true, but Reason (R) is false.
• (D) Assertion (A) is false, but Reason (R) is true.

Assertion (A): \(\triangle ABC \sim \triangle PQR\) such that \(\angle A = 65^\circ, \angle C = 60^\circ\). 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 the correct explanation of Assertion (A).
• (B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
• (C) Assertion (A) is true, but Reason (R) is false.
• (D) Assertion (A) is false, but Reason (R) is true.

(a) Solve the equation \( 4x^2 - 9x + 3 = 0 \) using the quadratic formula.

(b) Find the nature of roots of the equation \( 3x^2 - 4\sqrt{3}x + 4 = 0 \).

In a trapezium \(ABCD\), \(AB \parallel DC\) and its diagonals intersect at \(O\). Prove that \[ \frac{OA}{OC} = \frac{OB}{OD} \]

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

(i) it bears a 2-digit number

(ii) the number is a perfect square

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

OR
Question 24:
Verify that \[ \sin 2A = \frac{2 \tan A}{1 + \tan^2 A} \]
for \( A = 30^\circ \)

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

(a) If \(\alpha, \beta\) are zeroes of the polynomial \(8x^2 - 5x - 1\), then form a quadratic polynomial in \(x\) whose zeroes are \(\frac{2}{\alpha}\) and \(\frac{2}{\beta}\).

(b) Find the zeroes of the polynomial \(p(x) = 3x^2 + x - 10\) and verify the relationship between zeroes and its coefficients.

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

Three measuring rods are of lengths \(120 \, cm, 100 \, cm\) and \(150 \, cm\). Find the least length of a fence that can be measured an exact number of times using any of the rods. How many times each rod will be used to measure the length of the fence?

\(AB\) 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.

Prove that \[ \frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = \sec \theta \csc \theta + 1 \]

(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.

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.?

(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 circumscribed by \(\triangle ABC\). \(BC\) touches the circle at \(D\) such that \(BD = 6 \, cm, DC = 10 \, cm\). Find the length of \(AE\).

(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 \(\triangle OAP\).

(iii) Find the length of chord \(AB\).

The angles of depression of the top and the foot of a 9 m tall building from the top of a multi-storeyed building are \(30^\circ\) and \(60^\circ\) respectively. Find the height of the multi-storeyed building and the distance between the two buildings.

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

Find the \textbf{mean} and \textbf{mode} of the following data:

Class       15 – 20   20 – 25  25 – 30  30 – 35   35 – 40   40 – 45
Frequency   12          10           15          11            7             5
 

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

\smallskip
Based on the above, answer the following questions:

(i) Show that \(\triangle BQ \sim \triangle BAC\)

(ii) Prove that \(PQ = \frac{1}{3} AC\)

(iii) (a) If \(AB = 3 \, m\), find length \(BQ\) and \(BS\). Verify that \(BQ = \frac{1}{2} BS\).

OR

(iii) (b) Prove that \(BR^2 + RS^2 = \frac{4}{9} BC^2\)

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 is 10.5 cm.
Based on the above, answer the following questions:

(i) Find the dimensions of the cuboidal box.

(ii) Find the total outer surface area of the box.

(iii) (a) Find the difference between the capacity of the bowl and the volume of the box. (Use \(\pi = 3.14\))

OR

(iii) (b) The inner surface of the bowl and the thickness is to be painted. Find the area to be painted.

Gurveer and Arushi built a robot that can paint a path as it moves on a graph paper. Some co-ordinate 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).

\smallskip
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 questions:

(i) Determine the distance \(OP\).

(ii) \(QS\) is represented by equation \(2x + 9y = 42\). Find the co-ordinates of the point where it intersects the y-axis.

(iii) (a) Point \(R(4.8, y)\) divides the line segment \(OP\) in a certain ratio. Find the ratio. Hence, find the value of \(y\).

OR

(iii) (b) Using distance formula, show that \(\frac{PQ}{QS} = \frac{2}{3}\)