PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions for Mathematics

If \[\tan x = \frac{a}{b},\] show that

If \[cosec x - \sin x = a^3 , \sec x - \cos x = b^3\], then prove that \[a^2 b^2 \left( a^2 + b^2 \right) = 1\]

If \[\cot x \left( 1 + \sin x \right) = 4 m \text{ and }\cot x \left( 1 - \sin x \right) = 4 n,\] \[\left( m^2 + n^2 \right)^2 = mn\]

If \[\sin x + \cos x = m\], then prove that \[\sin^6 x + \cos^6 x = \frac{4 - 3 \left( m^2 - 1 \right)^2}{4}\], where \[m^2 \leq 2\]

If \[a = \sec x - \tan x \text{ and }b = cosec x + \cot x\], then shown that  \[ab + a - b + 1 = 0\]

Prove the:
\[ \sqrt{\frac{1 - \sin x}{1 + \sin x}} + \sqrt{\frac{1 + \sin x}{1 - \sin x}} = - \frac{2}{\cos x},\text{ where }\frac{\pi}{2} < x < \pi\]

If \[T_n = \sin^n x + \cos^n x\], prove that \[\frac{T_3 - T_5}{T_1} = \frac{T_5 - T_7}{T_3}\]

If \[T_n = \sin^n x + \cos^n x\], prove that  \[2 T_6 - 3 T_4 + 1 = 0\]

If \[T_n = \sin^n x + \cos^n x\], prove that \[6 T_{10} - 15 T_8 + 10 T_6 - 1 = 0\]

Prove that:  tan 225° cot 405° + tan 765° cot 675° = 0

Prove that:

If a set contains n elements, then write the number of elements in its power set. 

Write the number of elements in the power set of null set. 

Let A = {x : x ∈ N, x is a multiple of 3} and B = {x : x ∈ N and x is a multiple of 5}. Write \[A \cap B\] 

Let A and B be two sets having 3 and 6 elements respectively. Write the minimum number of elements that \[A \cup B\] 

If A = {x ∈ C : x² = 1} and B = {x ∈ C : x⁴ = 1}, then write A − B and B − A. 

Prove that: cos 24° + cos 55° + cos 125° + cos 204° + cos 300° = \[\frac{1}{2}\]

If A and B are two sets such that \[A \subset B\], then write B' − A' in terms of A and B.

Let A and B be two sets having 4 and 7 elements respectively. Then write the maximum number of elements that \[A \cup B\] can have. 

If \[A = \left\{ \left( x, y \right) : y = \frac{1}{x}, 0 \neq x \in R \right\}\]and\[B = \left\{ \left( x, y \right) : y = - x, x \in R \right\}\] then write\[A \cap B\]