Vardaan Learning Institute
Mock Board Paper 2025-26
Set - 3
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) with sub-parts.
- All Questions are compulsory. However, an internal choice has been provided.
- 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)
(Section A consists of 20 questions of 1 mark each)
1.
If \(p\) and \(q\) are two positive integers such that \(p = a^3b^2\) and \(q =
ab^3\), where \(a\) and \(b\) are prime numbers, then HCF\((p, q)\) is:
1
2.
The number of polynomials having zeroes as -2 and 5 is:
1
3.
The value of \(k\) for which the system of equations \(x + y - 4 = 0\) and \(2x + ky
- 3 = 0\) has no solution, is:
1
4.
The quadratic equation \(2x^2 - \sqrt{5}x + 1 = 0\) has:
1
5.
If the common difference of an AP is 5, then what is \(a_{18} - a_{13}\)?
1
6.
In \(\Delta ABC\), \(D\) and \(E\) are points on side \(AB\) and \(AC\) respectively
such that \(DE \parallel BC\). If \(AE = 2\) cm, \(AD = 3\) cm and \(BD = 4.5\) cm, then \(CE\) is:
1
7.
The midpoint of a line segment joining two points \(A(2, 4)\) and \(B(-2, -4)\) is:
1
8.
If \(\sin \theta = x\) and \(\sec \theta = y\), then \(\tan \theta\) is:
1
9.
The value of \((1 + \tan^2 \theta)(1 - \sin \theta)(1 + \sin \theta)\) is:
1
10.
From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q
from the centre is 25 cm. The radius of the circle is:
1
11.
If the area of a circle is \(154 \text{ cm}^2\), then its perimeter is:
1
12.
A sphere of diameter 18 cm is dropped into a cylindrical vessel of diameter 36 cm,
partly filled with water. If the sphere is completely submerged, then the water level rises (in cm)
by:
1
13.
Consider the following frequency distribution:
The upper limit of the median class is:
| Class | 0-5 | 6-11 | 12-17 | 18-23 | 24-29 |
| Freq | 13 | 10 | 15 | 8 | 11 |
1
14.
If a fair coin is tossed twice, the probability of getting at most one head is:
1
15.
The probability of getting a bad egg in a lot of 400 is 0.035. The number of bad
eggs in the lot is:
1
16.
The distance between the points \((a \cos \theta + b \sin \theta, 0)\) and \((0, a
\sin \theta - b \cos \theta)\) is:
1
17.
If LCM(x, 18) = 36 and HCF(x, 18) = 2, then x is:
1
18.
The angle of depression of a car parked on the road from the top of a 150 m high
tower is \(30^\circ\). The distance of the car from the tower (in meters) is:
1
19.
Assertion (A): The value of \(\sin \theta\) increases as \(\theta\)
increases from \(0^\circ\) to \(90^\circ\).
Reason (R): The value of \(\cos \theta\) increases as \(\theta\) increases from \(0^\circ\) to \(90^\circ\).
Reason (R): The value of \(\cos \theta\) increases as \(\theta\) increases from \(0^\circ\) to \(90^\circ\).
1
20.
Assertion (A): If the distance between \((4, k)\) and \((1, 0)\) is
5, then \(k = \pm 4\).
Reason (R): The point \((x, y)\) divides the line segment joining \((x_1, y_1)\) and \((x_2, y_2)\) in ratio \(m:n\) given by section formula.
Reason (R): The point \((x, y)\) divides the line segment joining \((x_1, y_1)\) and \((x_2, y_2)\) in ratio \(m:n\) given by section formula.
1
SECTION B
(Section B consists of 5 questions of 2 marks each)
(Section B consists of 5 questions of 2 marks each)
21.
Prove that \(7\sqrt{5}\) is an irrational number, given that \(\sqrt{5}\) is
irrational.
2
22.
ABCD is a trapezium in which \(AB \parallel DC\) and its diagonals intersect each
other at the point O. Show that \(\frac{AO}{BO} = \frac{CO}{DO}\).
2
23.
Evaluate: \(2\sec^2 30^\circ + x \sin^2 60^\circ - \frac{3}{4} \tan^2 30^\circ =
10\). Find the value of x.
OR
If \(\sin(A+B) = 1\) and \(\cos(A-B) =
\sqrt{3}/2\), \(0^\circ < A+B \le 90^\circ\), \(A> B\), find A and B.2
24.
The length of the minute hand of a clock is 14 cm. Find the area swept by the minute
hand in 15 minutes.
2
25.
Two dice are thrown at the same time. Find the probability of getting (i) the same
number on both dice (ii) a sum of 8.
2
SECTION C
(Section C consists of 6 questions of 3 marks each)
(Section C consists of 6 questions of 3 marks each)
26.
Find the HCF and LCM of 144, 180 and 192 by prime factorization method.
OR
Three bells ring at intervals of 12, 15 and 18 minutes respectively.
If they start ringing together at 9 a.m., when will they next ring together?3
27.
If \(\alpha\) and \(\beta\) are the zeroes of the polynomial \(f(x) = x^2 - 5x + k\)
such that \(\alpha - \beta = 1\), find the value of k.
3
28.
Solve for x and y: \(\frac{10}{x+y} + \frac{2}{x-y} = 4\); \(\frac{15}{x+y} -
\frac{5}{x-y} = -2\).
3
29.
Prove that: \((\sin A + \text{cosec } A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A +
\cot^2 A\).
3
30.
In the given figure, a quadrilateral ABCD is drawn to circumscribe a circle. Prove
that \(AB + CD = AD + BC\).
3
31.
The mean of the following frequency distribution is 25.2. Find the missing frequency
x.
| Class | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
| Frequency | 8 | x | 10 | 11 | 9 |
3
SECTION D
(Section D consists of 4 questions of 5 marks each)
(Section D consists of 4 questions of 5 marks each)
32.
The angles of elevation of the top of a tower from two points at a distance of 4m
and 9m from the base of the tower and in the same straight line with it are complementary. Prove
that the height of the tower is 6m.
5
33.
A motor boat whose speed is 24 km/h in still water takes 1 hour more to go 32 km
upstream than to return downstream to the same spot. Find the speed of the stream.
OR
Two pipes running together can fill a cistern in \(3 \frac{1}{13}\)
minutes. If one pipe takes 3 minutes more than the other to fill it, find the time in which each
pipe would fill the cistern.5
34.
From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity
of the same height and same diameter is hollowed out. Find the total surface area of the remaining
solid to the nearest \(cm^2\).
OR
A vessel is in the form of a hollow
hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total
height of the vessel is 13 cm. Find the inner surface area of the vessel.5
35.
If the median of the distribution given below is 32.5, find the values of \(f_1\)
and \(f_2\).
| Class Interval | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | Total |
| Frequency | \(f_1\) | 5 | 9 | 12 | \(f_2\) | 3 | 2 | 40 |
5
SECTION E
(Case Study Based Questions. Each carries 4 marks)
(Case Study Based Questions. Each carries 4 marks)
36.
Case Study - 1 (Arithmetic Progression)
A manufacturer of laptop produced 6000 units in 3rd year and 7000 units in 7th year. Assuming that production increases uniformly by a fixed number every year.
1. Find the production in the 1st year. (1)
2. Find the production in the 5th year. (1)
3. Find the total production in first 7 years. (2)
OR
In which year the production will be 10000? (2)
A manufacturer of laptop produced 6000 units in 3rd year and 7000 units in 7th year. Assuming that production increases uniformly by a fixed number every year.
1. Find the production in the 1st year. (1)
2. Find the production in the 5th year. (1)
3. Find the total production in first 7 years. (2)
OR
In which year the production will be 10000? (2)
4
37.
Case Study - 2 (Coordinate Geometry)
In a village, the Panchayat decided to construct a health centre. The Sarpanch marked the locations of three houses A(3, 3), B(6, 5) and C(4, 1) and the health centre P is equidistant from these three houses.
1. Find the distance between house A and house B. (1)
2. Find the midpoint of the line segment AB. (1)
3. If the health centre P(x, y) is equidistant from A and C, find the linear relation between x and y. (2)
OR
Find the coordinates of the point that divides BC in ratio 2:1. (2)
In a village, the Panchayat decided to construct a health centre. The Sarpanch marked the locations of three houses A(3, 3), B(6, 5) and C(4, 1) and the health centre P is equidistant from these three houses.
1. Find the distance between house A and house B. (1)
2. Find the midpoint of the line segment AB. (1)
3. If the health centre P(x, y) is equidistant from A and C, find the linear relation between x and y. (2)
OR
Find the coordinates of the point that divides BC in ratio 2:1. (2)
4
38.
Case Study - 3 (Applications of Trigonometry)
A boy is flying a kite at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is \(60^\circ\).
1. Draw a diagram to represent the situation. (1)
2. Find the length of the string, assuming that there is no slack in the string. (1)
3. If the angle of inclination changes to \(45^\circ\) while the height remains the same, what will be the new length of the string? (2)
OR
If the string length remains same but angle becomes \(30^\circ\), what is the new height of the kite? (2)
A boy is flying a kite at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is \(60^\circ\).
1. Draw a diagram to represent the situation. (1)
2. Find the length of the string, assuming that there is no slack in the string. (1)
3. If the angle of inclination changes to \(45^\circ\) while the height remains the same, what will be the new length of the string? (2)
OR
If the string length remains same but angle becomes \(30^\circ\), what is the new height of the kite? (2)
4
*** END OF PAPER ***