PUC Science Class 11 - Karnataka Board PUC Question Bank Solutions for Mathematics

\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\] 

\[\frac{(x + 5)(2 x^2 - 1)}{x}\]

\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\] 

cos (x + a)

\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]

\[\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}\]

Find the slope of the tangent to the curve f (x) = 2x⁶ + x⁴ − 1 at x = 1.

\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]  

Find the rate at which the function f (x) = x⁴ − 2x³ + 3x² + x + 5 changes with respect to x.

\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\] 

If for f (x) = λ x² + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ. 

For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]

x³ sin x 

x³ e^x 

x² e^x log x 

xⁿ tan x 

xⁿ log_a x 

(x³ + x² + 1) sin x 

sin x cos x

\[\frac{2^x \cot x}{\sqrt{x}}\]