List of practice Questions

Evaluate the integral $ I = \int \frac{\sin 4x}{\sin 2x} \, dx $

Find the term independent of $ x $ in $ \left( 2x - \frac{5}{x^2} \right)^6 $

If $ \sum_{k=0}^{n+1} C_k^n = 512 $, find $ \sum_{k=0}^{n} C_k^n $.

Find the limit: $ \lim_{x \to 11} \frac{x - 11}{\sqrt{49 + x^2} - 13} $

Solve the differential equation: $ (1 + y) \, dx = (1 + x) \, dy $

Find the domain of the function: $ f(x) = \sqrt{7 - 11x} $

If $ a, a r, a r^2 $ are in a geometric progression (G.P.), then find the value of: $ \left| a \quad a r \right| \quad \left| a r^2 \quad a r^3 \right| = \left| a r^3 \quad a r^6 \right| $

Find the limit: $ \lim_{x \to 0^+} 2 \left\lfloor x \right\rfloor - \frac{x}{|x|} $

Find the limit: $ \lim_{x \to 0^+} 2 \left\lfloor x \right\rfloor - \frac{x}{|x|} $

Equation of parabola having focii (-3, 1) and (3, 1)

Solve the following differential equation and integrate: $ \frac{dy}{dx} + \frac{2x}{1 + x^2} \cdot y = x $

Find the value of $ \sin 60^\circ - \sin 80^\circ + \sin 100^\circ - \sin 120^\circ $

If $ f(x) = \frac{\sqrt{x^4}}{\sqrt{x^2}} $, find $ f'(27) $.

If $ f(x) = \frac{1}{x^2} $, $ u = f(x) $, and $ f'(x) $, then find $ \frac{du}{dx} $.

Given the system of equations: \[ Z + \bar{Z} = 4 \quad \text{and} \quad Z - \bar{Z} = 6 \] Find \( |Z| \).

Refer to Column I and Column II and mark the statement(s) that is/are “True” \[ \begin{array}{|c|c|} \hline \textbf{Column I: Class 8, Section A marks} & \textbf{Column II: Class 8, Section B marks} \\ \hline 12 & 18 \\ 12 & 18 \\ 11 & 15 \\ 10 & 15 \\ 10 & 10 \\ 9 & 5 \\ 8 & 2 \\ 8 & 2 \\ \hline \end{array} \]