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

The LCM of 960 and 240 is : (A) 960 (B) 240 (C) 60 (D) 15

Prove that sqrt(5) is an irrational number.

For any natural number n, 5ⁿ ends with the digit : (A) 0 (B) 5 (C) 3 (D) 2

In a workshop, the number of teachers of English, Hindi and Science are 36, 60 and 84 respectively. Find the minimum number of rooms required, if in each room the same number of teachers are to be seated and all of them being of the same subject.

If alpha and beta are two zeroes of a polynomial f(x) = px² - 2x + 3p and alpha + beta = alpha*beta, then value of p is : (A) -2/3 (B) 2/3 (C) 1/3 (D) -1/3

The graph of y = f(x) is given. The number of distinct zeroes of y = f(x) is : (A) 0 (B) 1 (C) 2 (D) 3 [Figure: curve crossing x-axis at A and touching it at another point]

Find the zeroes of the quadratic polynomial 2x² - (1 + 2*sqrt(2))x + sqrt(2) and verify the relationship between the zeroes and coefficients of the polynomial.

A quadratic polynomial having zeroes -sqrt(5/2) and sqrt(5/2) is A) x² - 5*sqrt(2)x + 1 B) 8x² - 20 C) 15x² - 6 D) x² - 2*sqrt(5)x - 1

If alpha, beta are the zeroes of the quadratic polynomial px² + qx + r, then find the value of alpha³ * beta + beta³ * alpha.

Aarush bought 2 pencils and 3 chocolates for Rs 11 and Tanish bought 1 pencil and 2 chocolates for Rs 7 from the same shop. Represent this situation in the form of a pair of linear equations. Find the price of 1 pencil and 1 chocolate, graphically.

If the lines given by 3x + 2ky = 2 and 2x + 5y + 1 = 0 are not parallel, then k has to be (A) 15/4 (B) not equal to 15/4 (C) any rational number (D) any rational number having 4 as denominator

A person on a tour has Rs 4,200 for expenses. If he extends his tour for 3 days, he has to cut down his daily expenses by Rs 70. Find the original duration of the tour.

[case context: A rectangular floor can be tiled with 200 square tiles; if each tile's side is increased by 1 unit, only 128 tiles are needed.] Assuming the original length of each side of a tile is x units, make a quadratic equation from the above information.

[case context: A rectangular floor can be tiled with 200 square tiles; if each tile's side is increased by 1 unit, only 128 tiles are needed.] Write the corresponding quadratic equation in standard form.

[case context: elder brother repays a bank loan of Rs 1,18,000 in monthly instalments starting at Rs 1,000 and increasing by Rs 100 every month (an AP)] Find the amount paid by him in the 30th instalment.

[case context: Teacher asks students to write examples of A.P.; Aryan writes -5, -2, 1, 4, ... and Roshan writes 187, 184, 181, ...] Find the sum of first 10 terms of the progression written by Aryan.

If the sum of first m terms of an AP is 2m² + 3m, then its second term is : (A) 10 (B) 9 (C) 12 (D) 4

Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

If AP and DQ are medians of triangles ABC and DEF respectively, where triangle ABC ~ triangle DEF, then prove that AB/DE = AP/DQ

In the given figure, triangle AHK ~ triangle ABC. If AK = 10 cm, BC = 3.5 cm and HK = 7 cm, find the length of AC.

[case context: Class X students on picnic; positions of friends shown as points P(2,5), Q(4,4), R(8,3)] Let S be a point which divides the line joining PQ in ratio 2:3. Find the coordinates of S.

The distance of the point A(4a, 3a) from x-axis is : (A) 3a (B) -3a (C) 4a (D) -4a

Find the ratio in which the y-axis divides the line segment joining the points (5, -6) and (-1, -4). Also find the point of intersection.

[case context: Class X students on picnic; positions of friends Aman, Kirti, Chahat shown as points P(2,5), Q(4,4), R(8,3) on a coordinate grid] Find the distance between P and R.

Find the point(s) on the x-axis which is at a distance of sqrt(41) units from the point (8, -5).

Evaluate : tan² 60 deg / (sin² 60 deg + cos² 30 deg)

If 2sin(5x) = sqrt(3), 0 <= x <= 90 degrees, then x is equal to (A) 10 deg (B) 12 deg (C) 20 deg (D) 50 deg

If sec(theta) + tan(theta) = x, then sec(theta) - tan(theta) will be (A) x (B) x² (C) 2/x (D) 1/x

If sin(theta) + cos(theta) = sqrt(3), then prove that tan(theta) + cot(theta) = 1

Prove that (cosA - sinA + 1)/(cosA + sinA - 1) = cosecA + cotA

If tan(theta) = 5/2 then (4 sin(theta) + cos(theta)) / (4 sin(theta) - cos(theta)) is equal to A) 11/9 B) 3/2 C) 9/11 D) 4

Prove that : tan theta/(1 - cot theta) + cot theta/(1 - tan theta) = 1 + sec theta cosec theta

[case context: Tejas on top of a building observes a car approaching at uniform speed; angle of depression changes from 30 degrees to 60 degrees in 6 seconds, and at 60 degrees the car is 25 m from the building] What is the distance between the two positions of the car ?

[case context: Tejas on top of a building observes a car approaching at uniform speed; angle of depression changes from 30 degrees to 60 degrees in 6 seconds, and at 60 degrees the car is 25 m from the building] What is the distance of the observer from the car when it makes an angle of 60 degrees ?

From a point on the ground, which is 60 m away from the foot of a vertical tower, the angle of elevation of the top of the tower is found to be 45 degrees. The height (in metres) of the tower is : (A) 10 sqrt(3) (B) 30 sqrt(3) (C) 60 (D) 30

The angle of elevation of the top of a tower 24 m high from the foot of another tower in the same plane is 60 degrees. The angle of elevation of the top of second tower from the foot of the first tower is 30 degrees. Find the distance between two towers and the height of the other tower. Also, find the length of the wire attached to the tops of both the towers.

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

In Figure, XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X'Y' at B. Prove that angle AOB = 90 degrees. [For Visually Impaired candidates: Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that angle APB = 2(angle OAB)]

In the given figure, a tangent has been drawn at a point P on the circle centred at O. If angle TPQ = 110 deg then angle POQ is equal to A) 110 deg B) 70 deg C) 140 deg D) 55 deg

Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that angle PTQ = 2 angle OPQ.

The tangents drawn at the extremities of the diameter of a circle are always : (A) parallel (B) perpendicular (C) equal (D) intersecting

A circle with centre O and radius 8 cm is inscribed in a quadrilateral ABCD in which P, Q, R, S are the points of contact as shown. If AD is perpendicular to DC, BC = 30 cm and BS = 24 cm, then find the length DC.

Area of a segment of a circle of radius 'r' and central angle 60 degrees is : (A) (pi r²)/2 - (1/2) r² (B) (2 pi r)/4 - (sqrt(3)/4) r² (C) (pi r²)/6 - (sqrt(3)/4) r² (D) (2 pi r)/4 - r² sin 60

Find the area of the sector of a circle of radius 42 cm and of central angle 30 degrees. Also, find the area of the corresponding major sector. [Use pi = 22/7]

An empty cone of radius 3cm and height 12cm is filled with ice-cream such that the lower part of the cone which is (1/6)th of the volume of the cone is unfilled (empty) but a hemisphere is formed on the top. Find the volume of the ice-cream.

From a solid right circular cone, whose height is 6cm and radius of base is 12cm, a right circular cylindrical cavity of height 3cm and radius 4cm is hollowed out such that bases of cone and cylinder form concentric circles. Find the surface area of the remaining solid in terms of pi.

[case context] Silo = cylinder (r 1.5 m, h 7 m) + cone (r 1.5 m, h 2 m) on top. (ii) Find the curved surface area of the conical part of one silo.

A room is in the form of a cylinder surmounted by a hemispherical dome. The base radius of the hemisphere is half of the height of the cylindrical part. If the room contains 1408/21 m³ of air, find the height of the cylindrical part. (Use pi = 22/7).

If the mode of the following distribution is 55, then find the value of x. Hence, find the mean. Class Interval: 0-15, 15-30, 30-45, 45-60, 60-75, 75-90; Frequency: 10, 7, x, 15, 10, 12

The median of a set of 9 distinct observations is 20.5. If each of the observations of the set is increased by 2, then the median of the new set (A) is increased by 2 (B) is decreased by 2 (C) is two times the original number (D) Remains same as that of original observations

Which of the following gives the middle most observation of the data? A) Median B) Mean C) Range D) Mode

Consider the frequency distribution of 45 observations. Class: 0-10, 10-20, 20-30, 30-40, 40-50; Frequency: 5, 9, 15, 10, 6. The upper limit of median class is A) 20 B) 10 C) 30 D) 40

The mean and median of a frequency distribution are 43 and 43.4 respectively. The mode of the distribution is : (A) 43.4 (B) 42.4 (C) 44.2 (D) 49.3

[case context: BINGO game with 75 numbered balls; grouped frequency table of 48 announced numbers in classes 0-15(8), 15-30(9), 30-45(10), 45-60(12), 60-75(9).] Find median of the given data.

[case context: BINGO game with 75 numbered balls; grouped frequency table of 48 announced numbers in classes 0-15(8), 15-30(9), 30-45(10), 45-60(12), 60-75(9).] Find mode of the given data.

[case context: IMD table of sub-division wise monsoon rainfall (mm) in 2018 - classes 200-400 to 1600-1800 with frequencies 2,4,7,4,2,3,1,1] Find the mean rainfall in this season.

Two dice are thrown at the same time. Determine the probability that the (i) sum of the numbers on the two dice is 5, and (ii) difference of the numbers on the two dice is 3.

In a bag containing 24 balls, 4 are blue, 11 are green and the rest are white. One ball is drawn at random. The probability that drawn ball is white in colour is A) 1/6 B) 3/8 C) 11/24 D) 5/8

A card is selected at random from a deck of 52 playing cards. The probability of it being a red face card is : (A) 3/13 (B) 2/13 (C) 1/2 (D) 3/26

The probability for a randomly selected number out of 1, 2, 3, 4, ..., 25 to be a composite number is : (A) 15/25 (B) 10/25 (C) 11/25 (D) 9/25

If the probability of a player winning a game is 0.79, then the probability of his losing the same game is: (a) 1.79 (b) 0.31 (c) 0.21% (d) 0.21