CBSE Class 10 Mathematics (Standard) Guess Paper 2027

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

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.

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

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

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

[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.

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

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

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.

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

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

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 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]

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

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

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

[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.

Assertion (A) : The surface area of the cuboid formed by joining two cubes of sides 4 cm each, end-to-end, is 160 cm². Reason (R) : The surface area of a cuboid of dimensions l x b x h is (lb + bh + hl). [Choose: both true R explains A / both true R does not explain A / A true R false / A false R true]

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

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

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.

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

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

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.

Prove that sqrt(5) is an irrational number.

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 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.

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.

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.

[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 ?

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

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.

[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: India Gate, 42m tall monument at New Delhi; student Shreya of height 1m visits it] What is the angle of elevation from Shreya's eye to the top of India Gate, if she is standing at a distance of 41m away from the India Gate?

[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 ?