PUC Science Class 11 - Karnataka Board PUC Question Bank Solutions

If the major axis of an ellipse is three times the minor axis, then its eccentricity is equal to

The eccentricity of the ellipse 25x² + 16y² = 400 is

The eccentricity of the ellipse 5x² + 9y² = 1 is

The eccentricity of the ellipse 4x² + 9y² = 36 is

If p₁ and p₂ are the lengths of the perpendiculars from the origin upon the lines x sec θ + y cosec θ = a and x cos θ − y sin θ = a cos 2 θ respectively, then

If p be the length of the perpendicular from the origin on the line x/a + y/b = 1, then

What universal set (s) would you propose for the following:

The set of right triangles.

What universal set (s) would you propose for the following:

The set of isosceles triangles.

Given the sets, A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, the following may be considered as universal set (s) for all the three sets A, B and C?

{0, 1, 2, 3, 4, 5, 6}

Given the sets, A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, the following may be considered as universal set (s) for all the three sets A, B and C?

Φ

Given the sets, A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, the following may be considered as universal set (s) for all the three sets A, B and C?

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, the following may be considered as universal set (s) for all the three sets A, B and C?

{1, 2, 3, 4, 5, 6, 7, 8}

(i) If \[\left( \frac{a}{3} + 1, b - \frac{2}{3} \right) = \left( \frac{5}{3}, \frac{1}{3} \right)\] find the values of a and b. 

(ii) If (x + 1, 1) = (3, y − 2), find the values of x and y.

Prove the following identites

sec⁴x - sec²x = tan⁴x + tan²x

Prove the following identities
\[\sin^6 x + \cos^6 x = 1 - 3 \sin^2 x \cos^2 x\]

Prove the following identities
\[\left( cosec x - \sin x \right) \left( \sec x - \cos x \right) \left( \tan x + \cot x \right) = 1\]

Prove the following identities 
\[cosec x \left( \sec x - 1 \right) - \cot x \left( 1 - \cos x \right) = \tan x - \sin x\]

Prove the following identities
\[\frac{1 - \sin x \cos x}{\cos x \left( \sec x - cosec x \right)} \cdot \frac{\sin^2 x - \cos^2 x}{\sin^3 x + \cos^3 x} = \sin x\]

Prove the following identitie