Sample Paper 02
Class - 10th Exam - 2025-26
Mathematics - Basic
Time: 3 Hours | Max. Marks: 80
General Instructions:
- This question paper contains 38 questions.
- This Question Paper is divided into 5 Sections A, B, C, D and E.
- In Section A, Questions no. 1-18 are multiple choice questions (MCQs) and questions no. 19 and 20 are Assertion - Reason based questions of 1 mark each.
- In Section B, Questions no. 21-25 are very short answer (VSA) type questions, carrying 02 marks each.
- In Section C, Questions no. 26-31 are short answer (SA) type questions, carrying 03 marks each.
- In Section D, Questions no. 32-35 are long answer (LA) type questions, carrying 05 marks each.
- In Section E, Questions no. 36-38 are case study based questions carrying 4 marks each with sub parts of the values of 1, 1 and 2 marks each respectively.
- All Questions are compulsory. However, an internal choice in 2 Questions of Section B, 2 Questions of Section C and 2 Questions of Section D has been provided. An internal choice has been provided in all the 2 marks questions of Section E.
- Draw neat and clean figures wherever required.
- Take \( \pi = \frac{22}{7} \) wherever required if not stated.
- Use of calculators is not allowed.
Section - A
Section A consists of 20 questions of 1 mark each.
- If median is 137 and mean is 137.05, then the value of mode is
(a) 156.90
(b) 136.90
(c) 186.90
(d) 206.90 - The pair of equations \( x = a \) and \( y = b \) graphically represents lines which are
(a) Parallel
(b) Intersecting at (b, a)
(c) Coincident
(d) Intersecting at (a, b) - A tree casts a shadow 15 m long on the level of ground, when the angle of elevation of the sun is \( 45^\circ \). The height of a tree is
(a) 10 m
(b) 14 m
(c) 8 m
(d) 15 m - From a solid circular cylinder with height 10 cm and radius of the base 6 cm, a right circular cone of the same height and same base is removed, then the volume of remaining solid is
(a) \( 280\pi \text{ cm}^3 \)
(b) \( 330\pi \text{ cm}^3 \)
(c) \( 240\pi \text{ cm}^3 \)
(d) \( 440\pi \text{ cm}^3 \) - If the circumference of a circle increases from \( 4\pi \) to \( 8\pi \), then its area is
(a) halved
(b) doubled
(c) tripled
(d) quadrupled - A chord of a circle of radius 10 cm, subtends a right angle at its centre. The length of the chord (in cm) is
(a) \( 5\sqrt{2} \)
(b) \( 10\sqrt{2} \)
(c) \( 5\sqrt{2} \)
(d) \( 10\sqrt{3} \) - If a number \( x \) is chosen at random from the numbers -2, -1, 0, 1, 2. Then, the probability that \( x^2 < 2 \) is
(a) \( \frac{2}{5} \)
(b) \( \frac{4}{5} \)
(c) \( \frac{1}{5} \)
(d) \( \frac{3}{5} \) - \( (\cos^4 A - \sin^4 A) \) is equal to
(a) \( 1 - 2\cos^2 A \)
(b) \( 2\sin^2 A - 1 \)
(c) \( \sin^2 A - \cos^2 A \)
(d) \( 2\cos^2 A - 1 \) - The quadratic equation \( 4x^2 - 3x - 2 = 0 \) has
(a) Two distinct real roots
(b) Two equal real roots
(c) No real roots
(d) More than 2 real roots - If the square of difference of the zeroes of the quadratic polynomial \( x^2 + px + 45 \) is equal to 144, then the value of \( p \) is
(a) \( \pm 9 \)
(b) \( \pm 12 \)
(c) \( \pm 15 \)
(d) \( \pm 18 \) - Triangle ABC and Triangle BDE are two equilateral triangle such that D is the mid-point of BC. Ratio of the areas of triangles ABC and BDE is ................. .
(a) 1 : 4
(b) 4 : 1
(c) 1 : 3
(d) 3 : 1 - In the adjoining figure, TP and TQ are the two tangents to a circle with centre O. If \( \angle POQ = 110^\circ \) then \( \angle PTQ \) is
(a) \( 60^\circ \)[Insert Figure: Circle with tangents TP and TQ from point T. Angle POQ is marked as 110 degrees.]
(b) \( 70^\circ \)
(c) \( 80^\circ \)
(d) \( 90^\circ \) - A set of numbers consists of three 4’s, five 5’s, six 6’s, eight 8’s and seven 10’s. The mode of this set of numbers is
(a) 6
(b) 7
(c) 8
(d) 10 - Triangle ABC is an equilateral triangle with each side of length \( 2p \). If \( AD \perp BC \) then the value of AD is
(a) \( \sqrt{3} \)
(b) \( p\sqrt{3} \)
(c) \( 2p \)
(d) \( 4p \) - From an external point P, tangents PA and PB are drawn to a circle with centre O. If CD is the tangent to the circle at a point E and \( PA = 14 \) cm. The perimeter of Triangle PCD is
(a) 14 cm
(b) 21 cm
(c) 28 cm
(d) 35 cm - If \( x^2 + y^2 = 25 \), \( xy = 12 \), then x is
(a) (3, 4)
(b) (3, -3)
(c) (3, 4, -3, -4)
(d) (3, -3) - An observer, 1.5 m tall is 20.5 m away from a tower 22 m high, then the angle of elevation of the top of the tower from the eye of observer is
(a) \( 30^\circ \)
(b) \( 45^\circ \)
(c) \( 60^\circ \)
(d) \( 90^\circ \) - Which of the following relationship is the correct?
(a) \( P(E) + P(\bar{E}) = 1 \)
(b) \( P(\bar{E}) - P(E) = 1 \)
(c) \( P(E) = 1 + P(\bar{E}) \)
(d) None of these - Assertion: The value of \( \sin \theta = \frac{4}{3} \) is not possible.
Reason: Hypotenuse is the largest side in any right angled triangle.
(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: \( a_n - a_{n-1} \) is not independent of n then the given sequence is an AP.
Reason: Common difference \( d = a_n - a_{n-1} \) is constant or independent of n.
(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.
Section - B
Section B consists of 5 questions of 2 marks each.
- If \( \tan 2A = \cot(A - 18^\circ) \), where 2A is an acute angle, find the value of A.
- In Triangle ABC, \( AD \perp BC \) such that \( AD^2 = BD \times CD \). Prove that Triangle ABC is right angled at A.
- Find the mean the following distribution:
ORClass 3-5 5-7 7-9 9-11 11-13 Frequency 5 10 10 7 8
Find the mode of the following data:
Class 0-20 20-40 40-60 60-80 80-100 100-120 120-140 Frequency 6 8 10 12 6 5 3 - Explain why \( 7 \times 11 \times 13 + 13 \) and \( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 \) are composite numbers.
OR
Explain whether \( 3 \times 12 \times 101 + 4 \) is a prime number or a composite number. - In figure, a circle touches all the four sides of a quadrilateral ABCD. If \( AB = 6 \) cm, \( BC = 9 \) cm and \( CD = 8 \) cm, then find the length of AD.
[Insert Figure: Quadrilateral ABCD circumscribing a circle.]
Section - C
Section C consists of 6 questions of 3 marks each.
- Compute the mode for the following frequency distribution:
Size of items (in cm) 0-4 4-8 8-12 12-16 16-20 20-24 24-28 Frequency 5 7 9 17 12 10 6 - The sum of four consecutive number in AP is 32 and the ratio of the product of the first and last term to the product of two middle terms is 7 : 15. Find the numbers.
- A road which is 7 m wide surrounds a circular park whose circumference is 88 m. Find the area of the road.
OR
In Figure, PQ and AB are two arcs of concentric circles of radii 7 cm and 3.5 cm respectively, with centre O. If \( \angle POQ = 30^\circ \), then find the area of shaded region.[Insert Figure: Sector OQBP with arc AB and PQ. Shaded area between arcs.] - Find the ratio in which \( P(4, m) \) divides the segment joining the points \( A(2, 3) \) and \( B(6, -3) \). Hence find m.
OR
In the given figure Triangle ABC is an equilateral triangle of side 3 units. Find the co-ordinates of the other two vertices. - Given that \( \sqrt{5} \) is irrational, prove that \( 2\sqrt{5} - 3 \) is an irrational number.
- Prove that : \( \frac{\cot \theta - \csc \theta + 1}{\cot \theta + \csc \theta - 1} = \frac{1 + \cos \theta}{\sin \theta} \)
Section - D
Section D consists of 4 questions of 5 marks each.
- A solid is in the form of a cylinder with hemispherical end. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the solid. (Use \( \pi = \frac{22}{7} \))
- For what value of k, which the following pair of linear equations have infinitely many solutions: \( 2x + 3y = 7 \) and \( (k-1)x + (k+2)y = 3k \).
OR
The cost of 2 kg of apples and 1kg of grapes on a day was found to be Rs. 160. After a month, the cost of 4kg of apples and 2kg of grapes is Rs. 300. Represent the situations algebraically and geometrically. - Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
- The angles of depression of the top and bottom of an 8 m tall building from top of a multi-storeyed building are \( 30^\circ \) and \( 45^\circ \), respectively. Find the height of multi-storey building and distance between two buildings.
OR
Two poles of equal heights are standing opposite to each other on either side of a road, which is 80 m wide. From a point between them on the road, angles of elevation of their top are \( 30^\circ \) and \( 60^\circ \). Find the height of the poles and distance of point from poles.
Section - E
Section E consists of 3 case study based questions of 4 marks each.
- Case Study 1: Rohan and the Lamp Post
Rohan is very intelligent in maths. He always try to relate the concept of maths in daily life. One day he is walking away from the base of a lamp post at a speed of 1 m/s. Lamp is 4.5 m above the ground.
(i) If after 2 second, length of shadow is 1 meter, what is the height of Rohan?
(ii) What is the minimum time after which his shadow will become larger than his original height?
(iii) What is the distance of Rohan from pole at this point?
OR
(iv) What will be the length of his shadow after 4 seconds? - Case Study 2: Maximum Profit
An automobile manufacturer can produce up to 300 cars per day. The profit made from the sale of these vehicles can be modelled by the function \( P(x) = -x^2 + 350x - 6600 \) where \( P(x) \) is the profit in thousand Rupees and \( x \) is the number of automobiles made and sold. Answer the following questions based on this model:
(i) When no cars are produce what is a profit/loss?
(ii) What is the break even point? (Zero profit point is called break even)?
(iii) What is the profit/loss if 175 cars are produced?
OR
(iv) What is the profit if 400 cars are produced? - Case Study 3: Political Survey
Political survey questions are questions asked to gather the opinions and attitudes of potential voters. Political survey questions help you identify supporters and understand what the public needs. Using such questions, a political candidate or an organization can formulate policies to gain support from these people.
A survey of 100 voters was taken to gather information on critical issues and the demographic information collected is shown in the table. One out of the 100 voters is to be drawn at random to be interviewed on the India Today News on prime time.
(i) What is the probability the person is a woman or a Republican?Women Men Totals Republican 17 20 37 Democrat 22 17 39 Independent 8 7 15 Green Party 6 3 5 Totals 53 47 100
(ii) What is the probability the person is a Democrat?
(iii) What is the probability the person is a Independent men?
OR
(iv) What is the probability the person is a Independent men or green party men?