English Medium कक्षा १० - CBSE Question Bank Solutions for Mathematics

Evaluate 2 sec² θ + 3 cosec² θ – 2 sin θ cos θ if θ = 45°.

If sin θ – cos θ = 0, then find the value of sin⁴ θ + cos⁴ θ.

The distance of the point (3, 5) from x-axis (in units) is ______.

Assertion (A): The point (0, 4) lies on y-axis.

Reason (R): The x-coordinate of a point on y-axis is zero.

Find the roots of the quadratic equation x² – x – 2 = 0.

The boilers are used in thermal power plants to store water and then used to produce steam. One such boiler consists of a cylindrical part in middle and two hemispherical parts at its both ends.

Length of the cylindrical part is 7 m and radius of cylindrical part is `7/2` m.

Find the total surface area and the volume of the boiler. Also, find the ratio of the volume of cylindrical part to the volume of one hemispherical part.

(3 sin² 30° – 4 cos² 60°) is equal to ______.

In a right triangle PQR, right angled at Q. If tan P = `sqrt(3)`, then evaluate 2 sin P cos P.

In ΔBC, right angled at C, if tan A = `8/7`, then the value of cot B is ______.

Evaluate: 5 cosec² 45° – 3 sin² 90° + 5 cos 0°.

Distance of the point (6, 5) from the y-axis is ______.

Statement A (Assertion): Total Surface area of the top is the sum of the curved surface area of the hemisphere and the curved surface area of the cone.

Statement R( Reason): Top is obtained by joining the plane surfaces of the hemisphere and cone together.

A tent is in the shape of a cylinder surmounted by a conical top. If the height and radius of the cylindrical part are 3 m and 14 m respectively, and the total height of the tent is 13.5 m, find the area of the canvas required for making the tent, keeping a provision of 26 m² of canvas for stitching and wastage. Also, find the cost of the canvas to be purchased at the rate of ₹ 500 per m².

If the discriminant of the quadratic equation 3x² - 2x + c = 0 is 16, then the value of c is ______.

The ratio of total surface area of a solid hemisphere to the square of its radius is ______.

Tamper-proof tetra-packed milk guarantees both freshness and security. This milk ensures uncompromised quality, preserving the nutritional values within and making it a reliable choice for health-conscious individuals.

500 ml milk is packed in a cuboidal container of dimensions 15 cm × 8 cm × 5 cm. These milk packets are then packed in cuboidal cartons of dimensions 30 cm × 32 cm × 15 cm.

Based on the above-given information, answer the following questions:

i. Find the volume of the cuboidal carton. (1)

ii. a. Find the total surface area of the milk packet. (2)

OR

b. How many milk packets can be filled in a carton? (2)

iii. How much milk can the cup (as shown in the figure) hold? (1)

Ryan, from a very young age, was fascinated by the twinkling of stars and the vastness of space. He always dreamt of becoming an astronaut one day. So, he started to sketch his own rocket designs on the graph sheet. One such design is given below :

Based on the above, answer the following questions:

i. Find the mid-point of the segment joining F and G.    (1) 

ii. a. What is the distance between the points A and C?   (2)

OR

b. Find the coordinates of the points which divides the line segment joining the points A and B in the ratio 1 : 3 internally.    (2)

iii. What are the coordinates of the point D?    (1)

Assertion (A): Mid-point of a line segment divides the line segment in the ratio 1 : 1

Reason (R): The ratio in which the point (−3, k) divides the line segment joining the points (− 5, 4) and (− 2, 3) is 1 : 2.

If A(4, 3), B(-1, y) and C(3, 4) are the vertices of a right triangle ABC, right-angled at A, then find the value of y.

If the point P(x, y) is equidistant from the points A(a + b, b – a) and B(a – b, a + b). Prove that bx = ay.