Science (English Medium) इयत्ता १२ - CBSE Question Bank Solutions for Mathematics

Prove that the curves x = y² and xy = k cut at right angles if 8k² = 1. [Hint: Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.]

Find the equations of the tangent and normal to the hyperbola `x^2/a^2 - y^2/b^2` at the point `(x_0, y_0)`

The slope of the normal to the curve y = 2x² + 3 sin x at x = 0 is

(A) 3

(B) 1/3

(C) −3

(D) `-1/3`

The line y = x + 1 is a tangent to the curve y² = 4x at the point

(A) (1, 2)

(B) (2, 1)

(C) (1, −2)

(D) (−1, 2)

Show that the normal at any point θ to the curve x = a cosθ + a θ sinθ, y = a sinθ – aθ cosθ is at a constant distance from the origin.

The slope of the tangent to the curve x = t² + 3t – 8, y = 2t² – 2t – 5 at the point (2,– 1) is

(A) `22/7`

(B) `6/7`

(C) `7/6`

(D) `(-6)/7`

The line y = mx + 1 is a tangent to the curve y² = 4x if the value of m is

(A) 1

(B) 2

(C) 3

(D) 1/2

Evaluate the definite integral:

`int_(-1)^1 (x + 1)dx`

Evaluate the definite integral:

`int_2^3 1/x dx`

Evaluate the definite integral:

`int_1^2 (4x^3 - 5x^2 + 6x + 9)  dx`

Evaluate the definite integral:

`int_0^(pi/4) sin2xdx`

Evaluate the definite integral:

`int_0^(pi/2) cos 2x  dx`

Evaluate the definite integral:

`int_4^5 e^x dx`

Evaluate the definite integral:

`int_0^(pi/4) tan x dx`

Evaluate the definite integral:

`int_(pi/6)^(pi/4) cosec x  dx`

Evaluate the definite integral:

`int_0^1 dx/sqrt(1-x^2)`

Evaluate the definite integral:

`int_0^1 dx/(1+x^2)`