PUC Science कक्षा ११ - Karnataka Board PUC Question Bank Solutions

\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]

e^x log a + e^a long x + e^a log a

(2x² + 1) (3x + 2) 

 log₃ x + 3 log_e x + 2 tan x

\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\] 

\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\] 

\[\frac{2 x^2 + 3x + 4}{x}\] 

\[\frac{( x^3 + 1)(x - 2)}{x^2}\] 

\[\frac{a \cos x + b \sin x + c}{\sin x}\]

2 sec x + 3 cot x − 4 tan x

a₀ xⁿ + a₁ xⁿ⁻¹ + a₂ xⁿ⁻² + ... + a_n−1 x + a_n. 

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

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

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

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.