CBSE Class 10 Mathematics (Basic) Guess Paper 2027

The distance between the points (cos30°, sin30°) and (cos60°, −sin60°) is A) 0 unit B) √3 units C) 1 unit D) √2 units

The class mark of the median class of the following data is : (Class Interval / Frequency) 10–25 : 2, 25–40 : 3, 40–55 : 7, 55–70 : 6, 70–85 : 6, 85–100 : 6. (A) 40 (B) 55 (C) 47.5 (D) 62.5

[Case: A circular park has two gates at A(10, 20) and B(50, 50); two fountains at P and Q lie on AB such that AP = PQ = QB.] Find the coordinates of the centre C.

[Case: Saplings form a spiral of successive semicircles with centres alternately at A and B, starting at A, of radii 50 cm, 100 cm, 150 cm, ...; Spiral 1 has 10 flowers, Spiral 2 has 20, Spiral 3 has 30, and so on.] What is the radius of the 13th spiral ?

The value of (tan² A – 1/cos² A) is : (A) more than 1 (B) 1 (C) 0 (D) – 1

Find the nature of roots of the quadratic equation x² + 4x − 3√2 = 0.

Find a quadratic polynomial whose sum and product of zeroes are 0 and – 9, respectively. Also, find the zeroes of the polynomial so obtained.

Find the smallest number which is divisible by both 306 and 657.

(1 − tan² 30°)/(1 + tan² 30°) is equal to (A) cos 60° (B) sin 60° (C) 1 (D) tan² 60°

Find the value(s) of k for which the equation 2x² + kx + 3 = 0 has real and equal roots. Hence, find the roots of the equations so obtained.

In the given figure, PQ and PR are tangents to a circle centred at O. If ∠QPR = 35° then ∠QOR is equal to (A) 70° (B) 90° (C) 135° (D) 145°

From an external point Q, the length of tangent to a circle is 12 cm and the distance of Q from the centre of circle is 13 cm. The radius of circle (in cm) is A) 10 B) 5 C) 12 D) 7

If ΔABC ∼ ΔPQR (with AB = 3 cm, BC = 4 cm, AC = 5 cm and PR = 10 cm as shown), then perimeter of the triangle PQR (in cm) is A) 12 B) 24 C) 18 D) 20

The following distribution shows the number of runs scored by some batsmen in test matches : (Runs Scored / Number of Batsmen) 3000–4000 : 5, 4000–5000 : 10, 5000–6000 : 9, 6000–7000 : 8. The lower limit of the modal class is : (A) 3000 (B) 4000 (C) 5000 (D) 6000

Which of the following is not the criterion for similarity of triangles ? (A) AAA (B) SSS (C) SAS (D) RHS

If the HCF of two positive integers a and b is 1, then their LCM is : (A) a + b (B) a (C) b (D) ab

For which of the following solids is the lateral/curved surface area and total surface area the same ? (A) Cube (B) Cuboid (C) Hemisphere (D) Sphere

In the given figure, which of the following angles represents the angle of depression ? [Observer at top with horizontal line; line of sight to Object; marked angles x, y, z at observer/horizontal and a at object] (A) x (B) y (C) z (D) a

Assertion (A) : The value of p for which the system of equations 4x + py + 8 = 0 and 2x + 2y + 2 = 0 is consistent is 4. Reason (R) : The system of equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂ is consistent with infinitely many solutions, if a₁/a₂ = b₁/b₂ = c₁/c₂. (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): Two coins are tossed simultaneously. Possible outcomes are two heads, one head and one tail, two tails. Hence, the probability of getting two heads is 1/3. REASON (R): Probabilities of 'equally likely' outcomes of an experiment are always equal. 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.

The frequency distribution table of agriculture holdings in a village is given below — Area of land (in hectares): 1–3, 3–5, 5–7, 7–9, 9–11, 11–13 with No. of families: 20, 45, 80, 55, 40, 12 respectively. Find the modal agriculture holdings of the village.

[Case: Saplings form a spiral of successive semicircles with centres alternately at A and B, starting at A, of radii 50 cm, 100 cm, 150 cm, ...; Spiral 1 has 10 flowers, Spiral 2 has 20, Spiral 3 has 30, and so on.] Till which spiral, will there be a total of 450 saplings ?

Find the values of A and B (0 ≤ A < 90°, 0 ≤ B < 90°), if tan (A + B) = 1 and tan (A – B) = 1/√3.

[Case: Saplings form a spiral of successive semicircles with centres alternately at A and B, starting at A, of radii 50 cm, 100 cm, 150 cm, ...; Spiral 1 has 10 flowers, Spiral 2 has 20, Spiral 3 has 30, and so on.] Find the total number of saplings till the 11th spiral.

A chord of a circle of diameter 20 cm subtends an angle of 60° at the centre of the circle. Find the area of the corresponding minor segment of the circle. (Use π = 3.14 and √3 = 1.73)

Prove that : (1 + cot² A)/(1 + tan² A) = ((1 – cot A)/(1 – tan A))²

The following data gives the information on the observed lifetime (in hours) of 200 electrical components : (Lifetime in hours / Number of electrical components) 0–20 : 10, 20–40 : 35, 40–60 : 50, 60–80 : 60, 80–100 : 30, 100–120 : 15. Find the mean lifetime (in hours) of the electrical components.

Solve the following system of equations graphically : x + 3y = 6; 2x – 3y = 12

Prove that √3 is an irrational number.

Find the ratio in which the y-axis divides the line segment joining the points (4, −5) and (−1, 2). Also find the point of intersection.

[Case study: A jumbo pencil, when sharpened, is a combination of a cylinder and a cone. Length of the cylindrical portion is 21 cm, diameter of the base is 1 cm, and height of the conical portion is 1.2 cm.] Find the total surface area of the pencil (in terms of π).

Prove that a line drawn parallel to one side of a triangle intersecting other two sides in distinct points, divides the other two sides in the same ratio.

As observed from the top of a 75 m high lighthouse from the sea level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. (Use √3 = 1.732)

A train covers a distance of 300 km at a uniform speed. If the speed of the train is increased by 5 km/hr, it takes 2 hours less in the journey. Find the original speed of the train.

The angle of elevation of the top of a chimney from the foot of a tower is 60° and the angle of depression of the foot of the chimney from the top of the tower is 30°. If the height of the tower is 40 meters, find the height of the chimney. Also, find the length of the wire tied from the top of the chimney to the top of tower.

[Blood Donation Centre 2023 distribution of blood/Rhesus types (in %): O^- = x, O^+ = 30, A^- = 8, A^+ = 24, B^- = 6, B^+ = 18, {AB}^- = 1, {AB}^+ = 3.] Find the value of x.

[Water sprinkler shoots a stream of water a distance of 21 m and rotates through an angle equal to the complementary angle of 10°, watering a sector-shaped region.] What is the area of sector in terms of arc length?

[Case study: A survey of 200 students on preferred mode of transport to school — 120 preferred to walk, 25% preferred bicycles, 10% preferred the bus, and the remaining preferred to be dropped off by car.] What is the probability that a randomly selected student does not prefer to walk to school?