CBSE Class 10 Maths Sample Paper 2026–27 | Board & Pre-Board Exams

1 Mark Three containers hold 84 mL, 108 mL and 132 mL of different liquids. The liquids are to be divided into equal-sized portions without mixing or leaving any quantity unused. What is the greatest possible capacity of each portion?

1 Mark The number of bacteria in a culture is represented by the polynomial \(p(t)=t^2-6t+8\). At which two values of \(t\) does the polynomial become zero?

1 Mark A business records its monthly profit (in thousand rupees) using \(P(x)=x^2-7x+12\). In which two months will the profit be zero?

1 Mark A plantation drive includes 100 neem and peepal saplings. Each neem tree absorbs 20 units of carbon dioxide annually, while each peepal tree absorbs 30 units. If the total annual absorption is 2400 units, how many saplings of each type are required?

1 Mark Two roads on a map are represented by \(x+y=8\) and \(2x-y=4\). At what point do they intersect?

1 Mark If the product of the zeroes of \(4x^2-2x+k-4=0\) is 1, find the value of \(k\).

1 Mark The monthly salary of an employee increases by ₹2,000 every year. If the starting monthly salary is ₹30,000, what will it be during the eighth year?

1 Mark An auditorium has 20 seats in its first row. Each successive row has two additional seats. Which row will contain 48 seats?

1 Mark If \(DE\parallel BC\) in triangle ABC, \(AD=3\) cm, \(DB=5\) cm and \(AE=4.5\) cm, then the length of EC is:

1 Mark A bus stop is to be placed exactly halfway between two locations A(2, 3) and B(8, 11) on a city map. Find its coordinates.

1 Mark A tower of height 15 m casts a shadow of 24 m. At the same time, a nearby pole casts a shadow of 16 m. What is the pole's height?

1 Mark A sound wave is represented by \(y=A\sin\theta\). If its amplitude is 8 units and \(\theta=30^\circ\), determine the value of \(y\).

1 Mark In a right-angled triangle ABC, \(\angle B=90^\circ\). What is the value of \(\cos(A+C)\)?

1 Mark From a point 30 m away from the base of a vertical tower, the angle of elevation of its top is \(60^\circ\). If the observer's eye level is 1.6 m above the ground, find the approximate height of the tower. Use \(\sqrt3=1.73\).

1 Mark A tangent is drawn from point T to a circle with centre O. The radius is 5 cm and the distance OT is 13 cm. Find the tangent length.

1 Mark A sprinkler covers a sector of \(90^\circ\) in a circular lawn of radius 14 m. What area of the lawn receives water? Use \(\pi=\frac{22}{7}\).

1 Mark A capsule consists of a cylinder with a hemisphere attached at each end. If the cylindrical portion has radius \(r\) and height \(h\), which expression gives the complete outer surface area?

1 Mark A toy is formed by joining a hemisphere and a cone of equal base radius 3 cm. The cone has height 4 cm. Find the exposed surface area of the toy, excluding the common circular base.

1 Mark Assertion (A): The distance between A(2, 3) and B(6, 3) is 4 units.
Reason (R): When two points have the same ordinate, their distance is the absolute difference between their abscissae.

1 Mark Assertion (A): A sphere has twice the volume of a hemisphere when both have the same radius.
Reason (R): The volume of a sphere is \(\frac43\pi r^3\), whereas the volume of a hemisphere is \(\frac23\pi r^3\).

2 Marks Find the LCM of the smallest two-digit composite number and the greatest one-digit composite number. Three walkers take steps of lengths 40 cm, 42 cm and 45 cm respectively. Find the minimum distance each must walk so that all three complete an exact number of steps.

2 Marks A drone travels in a straight line from A(−2, 3) to B(4, −5). A signal station is to be installed at the point that divides AB internally in the ratio 1:2. Determine its coordinates. On a map, a delivery vehicle travels from A(2, 3) to B(8, 11), with each coordinate unit representing 1 km. Calculate the straight-line distance between the two locations.

2 Marks A father is currently three times as old as his son. After five years, he will be two and a half times his son's age. Calculate the father's present age. Solve the following pair of linear equations and verify the values obtained: \[ 7x-2y=5,\qquad 8x+7y=15 \]

2 Marks Evaluate: \[ 2\sqrt2\cos45^\circ\sin30^\circ+ 2\sqrt3\cos30^\circ \] For \(A=60^\circ\) and \(B=30^\circ\), verify the identity: \[ \sin(A+B)=\sin A\cos B+\cos A\sin B \]

2 Marks If the quadratic equation \(ax^2+bx+c=0\) has equal real roots, express \(c\) in terms of \(a\) and \(b\).

3 Marks Prove that \(\sqrt3\) is an irrational number.

3 Marks A student solves 12 mathematics problems on the first day of preparation and increases the daily target by 4 problems every day. Find:

3 Marks A satellite is initially observed from a ground station at an angle of elevation of \(60^\circ\). It moves horizontally away from the station, maintaining a constant height of 300 km, until its angle of elevation becomes \(30^\circ\). Calculate the horizontal distance covered by the satellite. Use \(\sqrt3=1.73\). A researcher observes the top of a tree at an angle of elevation of \(30^\circ\). After walking 40 m towards the tree, the angle becomes \(60^\circ\). Draw a labelled diagram and calculate the tree's height.

3 Marks Prove that a rectangle which circumscribes a circle must be a square.

3 Marks A circular glass panel has a radius of 14 cm. A chord cuts off a minor segment and subtends an angle of \(60^\circ\) at the centre. Find the area of this minor segment. Use \(\sqrt3=1.73\). A sprinkler at a field corner rotates through \(60^\circ\) and sprays water up to a distance of 21 m. Calculate the area covered by the water.

3 Marks A decorative solid is made by placing a cone on a hemisphere. The cone is 4 cm high and the common base has a diameter of 6 cm. Find the total exposed surface area of the solid. Take \(\pi=3.14\).

5 Marks A ball is projected vertically upwards from the roof of a 40 m tall building with an initial velocity of 30 m/s. Its height above the ground after \(t\) seconds is: \[ h=40+30t-5t^2 \] Find: Use \(\sqrt{68}=8.25\).

5 Marks In triangle ABC, a beam DE is placed parallel to BC, meeting AB at D and AC at E. Given \(AD=3\) m, \(DB=5\) m and \(AE=4.5\) m:

5 Marks A person is hovering above the ground near a 60 m high tower. From the foot of the tower, the angle of elevation of the person is \(60^\circ\). From the tower's top, the angle of elevation is \(30^\circ\). Assume the person is vertically above a point on the same horizontal ground. Use \(\sqrt3=1.73\). A lighthouse is 75 m tall. Two ships are seen on the same side of the lighthouse, with angles of depression of \(30^\circ\) and \(45^\circ\). Find the distance between the two ships. Use \(\sqrt3=1.73\).

5 Marks A model rocket has a cylindrical body of radius 7 cm and height 40 cm. A cone of the same radius and height 24 cm is fixed at its top. The bottom of the cylindrical part is closed. A camping tent has a cylindrical lower part of radius 2.1 m and height 3 m, with a conical roof of the same radius and vertical height 2.8 m.

4 Marks Case Study: A School Cricket Match If the number of fours were doubled while the number of singles remained unchanged, the total score would be 56. Check whether this statement is possible using the original information. (2)

4 Marks Case Study: Tracking a Cricket Ball Verify whether triangle ABC is isosceles by calculating the appropriate side lengths. (2)

4 Marks Case Study: A Miniature Railway Track

Answer the following questions: