List of practice Questions

Let $$ I_n = \int (\cos^n x + \sin^n x) \, dx,\quad \text{and} \quad \frac{n - 1}{n} I_{n - 2} = \frac{\sin x \cos x}{n} f(x) $$ Then, what is $ f(x) $?

In the given circuit, the values of $ I_1, I_2, I_3 $ are respectively:

The resistance of a wire at $ 0^\circ \text{C} $ is $ 20\, \Omega $. If the temperature coefficient of resistance is $ 5 \times 10^{-3}\, ^\circ \text{C}^{-1} $, at what temperature will the resistance double?

A particle of mass $ 2.2 \times 10^{-30}\, \text{kg} $ and charge $ 1.6 \times 10^{-19}\, \text{C} $ is moving at $ 10\, \text{km/s} $ in a circular path of radius $ 2.8\, \text{cm} $ inside a solenoid. The solenoid has $ 25\, \text{turns/cm} $, and the magnetic field is perpendicular to the plane of the particle’s path. The current in the solenoid is:
($ \mu_0 = 4\pi \times 10^{-7}\, \text{H/m} $)

All pairs $ (x, y) $ that satisfy the inequality \[ 2\sqrt{\sin^2x - 2\sin x + 5} \cdot \frac{1}{4^{\sin^2 y}} \leq 1 \] also satisfy the equation:

Given: \[ \lim_{x \to a} \left\lfloor \left( \frac{x^3}{a} - \left\lfloor \frac{x}{a} \right\rfloor^3 \right) \right\rfloor = k, \quad \lim_{x \to a} \left( \left\lfloor \frac{x^3}{a} \right\rfloor - \left\lfloor \frac{x}{a} \right\rfloor^3 \right) = l \] Then: \[ \boxed{?} \]

$[ \cdot ]$ represents the greatest integer function. If $ \int_{\sqrt{\frac{18}{3}}}^{[x]} dx = a + b\sqrt{2} + c\sqrt{3} $, then find the value of $ a + b + c $.

The length of the intercept on the line $4x - 3y - 10 = 0$ by the circle $x^2 + y^2 - 2x + 4y - 26 = 0$ is

If $(a, b, c)$ are the direction ratios of a line joining the points $(4, 3, -5)$ and $(-2, 1, -8)$, then the point $P = (a, 3b, 2c)$ lies on the plane

A ball of mass 0.5 kg moving horizontally at $ 10\, \text{ms}^{-1} $ strikes a vertical wall and rebounds with speed $ V $. The change in linear momentum is 8.0 kg·m/s. Find the magnitude of $ V $.

Evaluate: $ 1 + \cot^2 30^\circ - \sec^2 45^\circ $

The equation of the tangent to the circle $ x^2 + y^2 = 9 $ making an angle $ 60^\circ $ with the x-axis is:

If $ f(x) = \int \cos^{-1} \left( \frac{x}{\sqrt{a + x}} \right) dx + C $, then $ f'(a) = $

A gaseous mixture consists of 4 g of oxygen and 4 g of helium. The ratio $ \frac{C_p}{C_v} $ of the mixture is

A 100 $\mu F$ capacitor is connected to a 100 V, 50 Hz AC supply. The rms value of the current is

The least value of $n$ so that ${}^{n-1}C_3 + {}^{n-1}C_4>{}^nC_3$ is

Suppose, in triangle $\triangle ABC$, $x - y + 5 = 0$, $x + 2y = 0$ are respectively the equations of the perpendicular bisectors of the sides $AB$ and $AC$. If $A$ is $(1, -2)$, the equation of the line joining $B$ and $C$ is

By eliminating the arbitrary constants from \[ y = (a + b)\sin(x + c) - de^{x + te^t} \] the differential equation obtained is of order: