CBSE Class 10 Mathematics (Basic) Important Questions 2027 (Repeated in Board Exams)

Prove that √3 is an irrational number.

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

Show that the number 2 × 5 × 7 × 11 + 11 × 13 is a composite number.

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

Assertion (A) : For any two natural numbers a and b, the HCF of a and b is a factor of the LCM of a and b. Reason (R) : HCF of any two natural numbers divides both the numbers. (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 number 3 + √2 is : (A) a rational number (B) an irrational number (C) an integer (D) a natural number

The factor tree of a number x is shown below : x branches into 2 and y; y branches into 2 and 210; 210 branches into a and 70; 70 branches into 2 and 35; 35 branches into 5 and b. Find the values of x, y, a and b. Hence, write the product of the prime factors of the number x so obtained.

Prove that 6 - 4√5 is an irrational number, given that √5 is an irrational number.

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

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

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.

Solve the following system of equations for x and y : x/2 + 2y/3 = – 1 and x – y/3 = 3

x and y are complementary angles such that x : y = 1 : 2. Express the given information as a system of linear equations in two variables and hence solve it.

The greater of two supplementary angles exceeds the smaller by 18°. Find measures of these two angles.

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

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.

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.

The discriminant of the quadratic equation x² – 3x – 2 = 0 is : (A) 1 (B) 17 (C) √17 (D) – √17

The equation x + 1/x = 3 (x ≠ 0) is expressed as a quadratic equation in the form of ax² + bx + c = 0. The value of a – b + c is : (A) 5 (B) 2 (C) 1 (D) – 1

The difference of the squares of two positive numbers is 180. The square of the smaller number is 8 times the greater number. Find the two numbers.

[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 ?

[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 ?

[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.

[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.] If the radius of the nth spiral is 500 cm, find the value of n.

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.

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

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

From the figures given below, which of the following is true about the measure of ∠ P ? [Triangle ABC with ∠B = 60°, ∠A = 80°, AB = 3.8 cm, AC = 3√3 cm, BC = 6 cm; Triangle PQR with PR = 6√3 cm, RQ = 7.6 cm, PQ = 12 cm] (A) ∠ P = 60° (B) ∠ P = 80° (C) ∠ P = 40° (D) The measure of ∠ P cannot be determined

In the given figure, if PQ ∥ RS, then prove that Δ POQ ~ Δ SOR.

In the given figure, Δ OSR ~ Δ OQP, ∠ ROQ = 125° and ∠ ORS = 70°. Find the measures of ∠ OSR and ∠ OQP.

State “Basic Proportionality Theorem” and use it to prove the following : In a quadrilateral ABCD, diagonals AC and BD intersect each other at O such that AO/BO = CO/DO as shown in the given figure. Prove that ABCD is a trapezium.

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

[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: 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 point P.

[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 radius of the circular park.

[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 distance of the fountain at Q from gate A.

For a point (3, – 5), the value of (abscissa – ordinate) is : (A) – 8 (B) – 2 (C) 2 (D) 8

The mid-point of a line segment divides the line segment in the ratio : (A) 1 : 2 (B) 2 : 1 (C) 1 : 1 (D) 1/2 : 2

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

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

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

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

If 3 cot A = 4, where 0° < A < 90°, then sec A is equal to (A) 5/4 (B) 4/3 (C) 5/3 (D) 3/4

Prove that tan 45° = 1 geometrically.

Which of the following statements is false ? (A) tan 45° = cot 45° (B) sin 90° = tan 45° (C) sin 30° = cos 30° (D) sin 45° = cos 45°

[Case: An injured bird is on the roof of a 15 m high building; a fireman uses an adjustable ladder placed at 60° with the ground to reach the roof.] Find the length of the ladder used by the fireman to reach the roof.

[Case: An injured bird is on the roof of a 15 m high building; a fireman uses an adjustable ladder placed at 60° with the ground to reach the roof.] Find the distance of the point on the ground at which the ladder was fixed from the bottom of the building.

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)

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.

The angles of depression of the top and bottom of a 50m high building from the top of a tower are 45° and 60° respectively. Find the height of the tower and the horizontal distance between the tower and the building. (Use √3 = 1.73)

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

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

Prove that the lengths of tangents drawn from an external point to a circle are equal.

In the given figure, PA is a tangent to a circle with centre O. If OP = 10 cm, then the length of AP is : (A) 10√3 cm (B) 20 cm (C) 5 cm (D) 5√3 cm

Two concentric circles are of radii 6 cm and 10 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Prove that a rectangle circumscribing a circle is a square.

Assertion (A): If the PA and PB are tangents drawn to a circle with centre O from an external point P, then the quadrilateral OAPB is a cyclic quadrilateral. Reason (R): In a cyclic quadrilateral, opposite angles are equal. Options: (a) Both A and R are true and R explains A completely. (b) Both A and R are true but R does not explain A. (c) A is true but R is false. (d) A is false but R is true.

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)

The perimeter of the shaded region in the given figure is : [Circle with centre O; two radii r bounding a shaded sector, chord a subtending the sector, and arc l] (A) l (B) l + a (C) l + 2r (D) l + 2r + a

The ratio of the area of a quadrant of a circle to the area of the same circle is : (A) 1 : 2 (B) 2 : 1 (C) 1 : 4 (D) 4 : 1

[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).

A toy is in the form of a cone surmounted on a hemisphere. The cone and hemisphere have the same radii. The height of the conical part of the toy is equal to the diameter of its base. If the radius of the conical part is 5 cm, find the volume of the toy.

A cubical block is surmounted by a hemisphere of radius 3.5 cm. What is the smallest possible length of the edge of the cube so that the hemisphere can totally lie on the cube ? Find the total surface area of the solid so formed.

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

The radius of a sphere is 7/2 cm. The volume of the sphere is : (a) 231/3 cu cm (b) 539/12 cu cm (c) 539/3 cu cm (d) 154 cu cm

A juice glass is cylindrical in shape with hemi-spherical raised up portion at the bottom. The inner diameter of glass is 10 cm and its height is 14 cm. Find the capacity of the glass. (use π = 3.14)

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

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.

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.

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

[Case: Grouped frequency of leaf lengths (in mm): 70-80: 3, 80-90: 5, 90-100: 9, 100-110: 12, 110-120: 5, 120-130: 4, 130-140: 2.] Find median of the data.

A lot consists of 200 pens of which 180 are good and the rest are defective. A customer will buy a pen if it is not defective. The shopkeeper draws a pen at random and gives it to the customer. What is the probability that the customer will not buy it ? Another lot of 100 pens containing 80 good pens is mixed with the previous lot of 200 pens. The shopkeeper now draws one pen at random from the entire lot and gives it to the customer. What is the probability that the customer will buy the pen ?

In a random experiment of throwing a die, which of the following is a sure event ? (A) Getting a number between 1 and 6 (B) Getting an odd number < 7 (C) Getting an even number < 7 (D) Getting a natural number < 7

A card is drawn from a well shuffled deck of 52 playing cards. The probability that drawn card is a red queen, is : (a) 1/13 (b) 2/13 (c) 1/52 (d) 1/26