Vardaan Learning Institute
Previous Year Board Question Paper 2025
Set - 30-1
MATHEMATICS (STANDARD)
Class - X
Time: 3 Hours
Max. Marks: 80
General Instructions:
- This Question Paper has 5 Sections A, B, C, D and E.
- Section A has 20 MCQs carrying 1 mark each.
- Section B has 5 questions carrying 2 marks each.
- Section C has 6 questions carrying 3 marks each.
- Section D has 4 questions carrying 5 marks each.
- Section E has 3 case based integrated units of assessment (04 marks each).
SECTION A
1.
What is the mode of a data if median and mean of the same data are 9.6 and 10.5, respectively?
1
2.
The value of \( (\tan A \csc A)^2 - (\sin A \sec A)^2 \) is:
1
3.
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° to the horizontal. The length of the string is:
1
4.
In triangles ABC and DEF, \(\angle B = \angle E\), \(\angle F = \angle C\) and \(AB = 3 DE\). Then,
the two triangles are:
1
5.
If \(\theta\) is an acute angle and \(7 + 4 \sin^2 \theta = 9\), then the value of \(\theta\) is:
1
6.
Two polynomials are shown in the graph below. The number of distinct real zeroes of the two
polynomials is:
1
7.
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
8.
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
9.
If HCF(98, 28) = m and LCM(98, 28) = n, then the value of \(n - 7m\) is:
1
10.
Which of the following is a rational number between \(\sqrt{2}\) and \(\sqrt{3}\)?
1
11.
The sum of the zeroes of the polynomial \(p(x) = 5x - 7x^2 + 3\) is:
1
12.
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
13.
If a sector of a circle has an area of \(40\pi\) sq. units and a central angle of 72°, the radius of
the circle is:
1
14.
The tangents drawn at the extremities of the diameter of a circle are always:
1
15.
If \((-1)^n + (-1)^8 = 0\), then n is:
1
16.
The end points of a diameter of circle are (2, 4) and (-3, -1). The length of its radius is:
1
17.
The 11th and 13th term of an AP are 39 and 45, respectively. What is the common difference of the
AP?
1
18.
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
Questions number 19 and 20 are Assertion and Reason
based questions. Two statements are given, one labelled as Assertion (A) and the other is labelled as
Reason (R). Select the correct answer to these questions from the codes (A), (B), (C) and (D) as given
below:
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the
Assertion (A).
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(C) Assertion (A) is true, but Reason (R) is false.
(D) Assertion (A) is false, but Reason (R) is true.
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(C) Assertion (A) is true, but Reason (R) is false.
(D) Assertion (A) is false, but Reason (R) is true.
19.
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.
Reason (R): For any event E, if P(E) = 1, then E is called a sure event.
1
20.
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 \(3\pi r^2\).
Reason (R): Total Surface Area of a sphere of radius r is \(3\pi r^2\).
1
SECTION B
21.
(a) If \(\Delta ABC \sim \Delta PQR\) in which AB = 6 cm, BC = 4 cm, AC = 8 cm and PR = 6 cm, then
find the length of (PQ + QR).
OR
(b) In the given figure, \(\frac{QR}{QS} = \frac{QT}{PR}\) and \(\angle 1 = \angle 2\), show that \(\Delta PQS \sim \Delta TQR\).
OR
(b) In the given figure, \(\frac{QR}{QS} = \frac{QT}{PR}\) and \(\angle 1 = \angle 2\), show that \(\Delta PQS \sim \Delta TQR\).
2
22.
(a) If \(x \sin^2 60^\circ + y \cos 0^\circ + \sin 30^\circ - \cot 45^\circ = 5\), then find the
value of x + 2y.
OR
(b) Evaluate: \(\frac{\tan^2 60^\circ}{\sin^2 60^\circ + \cos^2 30^\circ}\)
OR
(b) Evaluate: \(\frac{\tan^2 60^\circ}{\sin^2 60^\circ + \cos^2 30^\circ}\)
2
23.
A person is standing at P outside a circular ground at a distance of 85 m from the centre of the
ground. He found that his distance from the points A and B on the ground are 84 m. PA and PB are
tangents to the circle. Find the radius of the circular ground.
2
24.
Find the zeros of the polynomial \(p(x) = x^2 - \frac{x}{6} - \frac{1}{6}\).
2
25.
Find the length of the median through the vertex B of \(\Delta ABC\) with vertices A(1, -1), B(0, 3)
and C(1, 10).
2
SECTION C
26.
Prove that \(\sqrt{7}\) is an irrational number.
3
27.
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.
(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.
3
28.
(a) Prove that: \(\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 +
\sec \theta \csc \theta\)
OR
(b) Prove that: \(\frac{\sin A - \cos A + 1}{\sin A + \cos A - 1} = \frac{1}{\sec A - \tan A}\)
OR
(b) Prove that: \(\frac{\sin A - \cos A + 1}{\sin A + \cos A - 1} = \frac{1}{\sec A - \tan A}\)
3
29.
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 \(1408/21\) m³ of
air, find the height of the cylindrical part. (Use \(\pi = 22/7\)).
3
30.
(a) In the given figure, O is the centre of the circle and BCD is a tangent to it at C. Prove that
\(\angle BAC + \angle ACD = 90^\circ\).
OR
(b) Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
OR
(b) Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
3
31.
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.
3
SECTION D
32.
(a) 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.
OR
(b) 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.
OR
(b) 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.
5
33.
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.
5
34.
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:
Find the median length of the leaves.
| Length (in mm) | Number of Leaves |
|---|---|
| 118 - 126 | 3 |
| 127 - 135 | 5 |
| 136 - 144 | 9 |
| 145 - 153 | 12 |
| 154 - 162 | 5 |
| 163 - 171 | 4 |
| 172 - 180 | 2 |
Find the median length of the leaves.
5
35.
(a) 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 \(DF \times EF = FB \times FA\).
OR
(b) In \(\Delta ABC\), if \(AD \perp BC\) and \(AD^2 = BD \times DC\), then prove that \(\angle BAC = 90^\circ\).
OR
(b) In \(\Delta ABC\), if \(AD \perp BC\) and \(AD^2 = BD \times DC\), then prove that \(\angle BAC = 90^\circ\).
5
SECTION E
36.
Case Study 1
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°. 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°.
Based on the above given information, answer the following questions:
(i) If CD is h metres, find the distance BD in terms of 'h'.
(ii) Find distance BC in terms of 'h'.
(iii) (a) Find the height CE of the lighthouse [Use \(\sqrt{3} = 1.73\)]
OR
(b) Find distance AE, if AC = 100 m.
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°. 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°.
Based on the above given information, answer the following questions:
(i) If CD is h metres, find the distance BD in terms of 'h'.
(ii) Find distance BC in terms of 'h'.
(iii) (a) Find the height CE of the lighthouse [Use \(\sqrt{3} = 1.73\)]
OR
(b) Find distance AE, if AC = 100 m.
4
37.
Case Study 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.
(ii) Determine the distance of the 8th round.
(iii) (a) Find the total distance run after completing all 10 rounds.
OR
(b) If a runner completes only the first 6 rounds, what is the total distance run by the runner?
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.
(ii) Determine the distance of the 8th round.
(iii) (a) Find the total distance run after completing all 10 rounds.
OR
(b) If a runner completes only the first 6 rounds, what is the total distance run by the runner?
4
38.
Case Study 3
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.
Based on the above given information, answer the following questions:
(i) Find the central angle of each sector.
(ii) Find the length of the arc ACB.
(iii) (a) Find the area of each sector of the brooch.
OR
(b) Find the total length of the silver wire used.
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.
Based on the above given information, answer the following questions:
(i) Find the central angle of each sector.
(ii) Find the length of the arc ACB.
(iii) (a) Find the area of each sector of the brooch.
OR
(b) Find the total length of the silver wire used.
4