Vardaan Learning Institute

Mock Board Paper 2025-26

Set - 1
MATHEMATICS (STANDARD) Class - X
Time: 3 Hours
Max. Marks: 80
General Instructions:
  1. This Question Paper has 5 Sections A, B, C, D and E.
  2. Section A has 20 MCQs carrying 1 mark each.
  3. Section B has 5 questions carrying 2 marks each.
  4. Section C has 6 questions carrying 3 marks each.
  5. Section D has 4 questions carrying 5 marks each.
  6. Section E has 3 case based integrated units of assessment (04 marks each) with sub-parts.
  7. All Questions are compulsory. However, an internal choice in 2 Qs of 5 marks, 2 Qs of 3 marks and 2 Questions of 2 marks has been provided.
  8. Draw neat figures wherever required. Take \(\pi = 22/7\) wherever required if not stated.
SECTION A
(Section A consists of 20 questions of 1 mark each)
1.
If HCF(336, 54) = 6, find LCM(336, 54).
(a) 3000
(b) 3024
(c) 3014
(d) 3034
1
2.
The roots of the quadratic equation \(x^2 - 0.04 = 0\) are:
(a) \(\pm 0.2\)
(b) \(\pm 0.02\)
(c) 0.4
(d) 2
1
3.
For what value of \(k\) will the system of linear equations \(x + 2y = 3\) and \(5x + ky + 7 = 0\) have no solution?
(a) 10
(b) 5
(c) -7
(d) 2
1
4.
If the distance between the points (4, p) and (1, 0) is 5, then the value of p is:
(a) 4 only
(b) \(\pm 4\)
(c) -4 only
(d) 0
1
5.
In \(\Delta ABC\), \(DE \parallel BC\). If \(AD = x\), \(DB = x-2\), \(AE = x+2\) and \(EC = x-1\), find the value of x.
(a) 3
(b) 4
(c) 5
(d) 2.5
1
6.
The value of \(\frac{2 \tan 30^\circ}{1 + \tan^2 30^\circ}\) is:
(a) \(\sin 60^\circ\)
(b) \(\cos 60^\circ\)
(c) \(\tan 60^\circ\)
(d) \(\sin 30^\circ\)
1
7.
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of \(80^\circ\), then \(\angle POA\) is equal to:
(a) \(50^\circ\)
(b) \(60^\circ\)
(c) \(70^\circ\)
(d) \(80^\circ\)
1
8.
The length of the shadow of a tower on the plane ground is \(\sqrt{3}\) times the height of the tower. The angle of elevation of the sun is:
(a) \(45^\circ\)
(b) \(30^\circ\)
(c) \(60^\circ\)
(d) \(90^\circ\)
1
9.
The area of a quadrant of a circle with circumference 22 cm is:
(a) \(77/8 \text{ cm}^2\)
(b) \(77/2 \text{ cm}^2\)
(c) \(77/4 \text{ cm}^2\)
(d) \(77 \text{ cm}^2\)
1
10.
Two cubes each of volume \(64 \text{ cm}^3\) are joined end to end. The surface area of the resulting cuboid is:
(a) \(140 \text{ cm}^2\)
(b) \(150 \text{ cm}^2\)
(c) \(160 \text{ cm}^2\)
(d) \(180 \text{ cm}^2\)
1
11.
The mean of the following distribution is:
Class 0-10 10-20 20-30 30-40 40-50
Frequency 3 5 9 5 3
(a) 24
(b) 25
(c) 26
(d) 27
1
12.
A card is selected at random from a well shuffled deck of 52 cards. The probability of its being a red face card is:
(a) 3/26
(b) 3/13
(c) 2/13
(d) 1/2
1
13.
If the perimeter of a circle is equal to that of a square, then the ratio of their areas is:
(a) 22:7
(b) 14:11
(c) 7:22
(d) 11:14
1
14.
The sum of the first 20 odd natural numbers is:
(a) 100
(b) 210
(c) 400
(d) 420
1
15.
If \(\sin A = 1/2\) and \(\cos B = 1/2\), then \(A + B =\) ?
(a) \(0^\circ\)
(b) \(60^\circ\)
(c) \(90^\circ\)
(d) \(30^\circ\)
1
16.
The probability that a non-leap year selected at random will contain 53 Sundays is:
(a) 1/7
(b) 2/7
(c) 3/7
(d) 5/7
1
17.
The coordinates of the point which divides the line segment joining the points (4, -3) and (8, 5) in the ratio 3:1 internally is:
(a) (7, 3)
(b) (3, 7)
(c) (7, -3)
(d) (4, 3)
1
18.
In a formula for mean, \(d_i = x_i - a\), then \(a\) is:
(a) Class Limit
(b) Class Mark
(c) Frequency
(d) Assumed Mean
1
Directions for Q19 and Q20: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Choose the correct option as:
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
19.
Assertion (A): The value of y is 6, for which the distance between P(2, -3) and Q(10, y) is 10.
Reason (R): Distance between two given points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).
1
20.
Assertion (A): The HCF of two numbers is 5 and their product is 150, then their LCM is 30.
Reason (R): For any two positive integers a and b, \(HCF(a, b) \times LCM(a, b) = a \times b\).
1
SECTION B
(Section B consists of 5 questions of 2 marks each)
21.
Prove that \(3 + 2\sqrt{5}\) is an irrational number, given that \(\sqrt{5}\) is irrational.
2
22.
In the given figure, if \(LM \parallel CB\) and \(LN \parallel CD\), prove that \(\frac{AM}{AB} = \frac{AN}{AD}\). Triangle Diagram
2
23.
Find the area of a quadrant of a circle whose circumference is 22 cm.
OR
The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.
2
24.
Evaluate: \(\frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}\)
2
25.
If \(\alpha\) and \(\beta\) are zeroes of the polynomial \(x^2 - 6x + k\) and \(3\alpha + 2\beta = 20\), find the value of \(k\).
2
SECTION C
(Section C consists of 6 questions of 3 marks each)
26.
The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.
OR
2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to finish the work, and also that taken by 1 man alone.
3
27.
Prove that: \(\frac{\sin \theta - \cos \theta + 1}{\sin \theta + \cos \theta - 1} = \frac{1}{\sec \theta - \tan \theta}\)
3
28.
Prove that the parallelogram circumscribing a circle is a rhombus. Parallelogram Circle Diagram
3
29.
Find the mean of the following data:
Class Interval 0-20 20-40 40-60 60-80 80-100 100-120
Frequency 5 8 x 12 7 8
Given that the mean is 62.8, find the value of x.
3
30.
Two dice are thrown at the same time. What is the probability that the sum of the two numbers appearing on the top of the dice is:
(i) 8?
(ii) 13?
(iii) less than or equal to 12?
3
31.
If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.
OR
Find the 20th term from the last term of the AP: 3, 8, 13, ..., 253.
3
SECTION D
(Section D consists of 4 questions of 5 marks each)
32.
State and prove Basic Proportionality Theorem (Thales Theorem). Using the theorem, solve: In \(\Delta ABC\), if line DE intersects sides AB and AC at D and E respectively and is parallel to BC, prove \(\frac{AD}{AB} = \frac{AE}{AC}\).
5
33.
A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of Rs 500 per \(m^2\). (Note that the base of the tent will not be covered with canvas.)
OR
A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that \(1 \text{ cm}^3\) of iron has approximately 8g mass. (Use \(\pi = 3.14\))
5
34.
As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are \(30^\circ\) and \(45^\circ\). If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
5
35.
A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
OR
Two water taps together can fill a tank in \(9 \frac{3}{8}\) hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
5
SECTION E
(Case Study Based Questions. Each carries 4 marks)
36.
Case Study - 1 (Arithmetic Progression)
India is competitive manufacturing location due to the low cost of manpower and strong technical and engineering capabilities contributing to higher quality production runs. The production of TV sets in a factory increases uniformly by a fixed number every year. It produced 16000 sets in the 6th year and 22600 in the 9th year.
4

Based on the above information, answer the following questions:

  1. Find the production in the 1st year. (1)
  2. Find the production in the 8th year. (1)
  3. Find the total production in first 3 years. (2)
    OR
    In which year the production will be 29200? (2)
37.
Case Study - 2 (Coordinate Geometry)
In order to conduct Sports Day activities, rectangular shaped school ground ABCD has been marked with lines of chalk powder at a distance of 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD. Niharika runs 1/4th the distance AD on the 2nd line and posts a green flag. Preet runs 1/5th the distance AD on the eighth line and posts a red flag. Sports Day Grid Diagram
4

Based on the above information, answer the following questions:

  1. Find the coordinates of the Green Flag. (1)
  2. Find the distance between the two flags. (1)
  3. If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag? (2)
    OR
    If Joy wants to post a flag at 1/4 distance from Green Flag to Red Flag, what are the coordinates? (2)
38.
Case Study - 3 (Applications of Trigonometry)
A Satellite flying at height h is watching the top of the two tallest mountains in Uttarakhand and Karnataka, them being Nanda Devi (height 7,816m) and Mullayanagiri (height 1,930m). The angles of depression from the satellite to the top of Nanda Devi and Mullayanagiri are \(30^\circ\) and \(60^\circ\) respectively. The distance between the peaks of the two mountains is 1937 km.
4

Based on the above information, answer the following questions:

  1. Make a labeled diagram representing the situation. (1)
  2. Find the distance of the satellite from the top of Nanda Devi. (1)
  3. Find the vertical height of the satellite from the ground (approximate). (2)
    OR
    What is the angle of elevation if a man standing at a distance of 7816m from Nanda Devi looks at the peak? (2)
*** END OF PAPER ***

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