Science (English Medium) Class 11 - CBSE Question Bank Solutions

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

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

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

\[\frac{2^x \cot x}{\sqrt{x}}\] 

x² sin x log x 

x⁵ e^x + x⁶ log x 

(x sin x + cos x) (x cos x − sin x) 

(x sin x + cos x ) (e^x + x² log x)