List of practice Questions

If $P_1P_2$ and $P_3P_4$ are two focal chords of the parabola $y^2 = 4ax$, then the chords $P_1P_3$ and $P_2P_4$ intersect on the
$f: X \to \mathbb{R}, X = \{x | 0<x<1\}$ is defined as $f(x) = \frac{2x - 1}{1 - |2x - 1|}$. Then
If $\mathbf{a} = \hat{i} + \hat{j} - \hat{k}$, $\mathbf{b} = \hat{i} - \hat{j} + \hat{k}$, and $\mathbf{c}$ is a unit vector perpendicular to $\mathbf{a}$ and coplanar with $\mathbf{a}$ and $\mathbf{b}$, then the unit vector $\mathbf{d}$ perpendicular to both $\mathbf{a}$ and $\mathbf{c}$ is
If the equation of one tangent to the circle with center at $(2, -1)$ from the origin is $3x + y = 0$, then the equation of the other tangent through the origin is
If $a$, $b$, and $c$ are in GP, and $\log a - \log 2b$, $\log 2b - \log 3c$, $\log 3c - \log a$ are in A.P., then $a$, $b$, and $c$ are the lengths of the sides of a triangle which is
Let $a_n = (1^2 + 2^2 + \cdots + n^2)$ and $b_n = n^n (n!)$. Then
If $z = x - iy$ and $z^{1/3} = p + iq$ ($x, y, p, q \in \mathbb{R}$), then $\frac{\left( \frac{x}{p} + \frac{y}{q} \right)}{p^2 + q^2}$ is equal to
If $a$, $b$ are odd integers, then the roots of the equation $2ax^2 + (2a + b)x + b = 0$, where $a \neq 0$, are
Let $f(x) = \int \cos x \sin x \, e^{-t^2} dt$. Then $f' \left( \frac{\pi}{4} \right)$ equals
The point of contact of the tangent to the parabola $y^2 = 9x$ which passes through the point $(4, 10)$ and makes an angle $\theta$ with the positive side of the axis of the parabola, where $\tan \theta>2$, is
Let $f(x) = (x - 2)^{17} (x + 5)^{24}$. Then
The solution of $\cos y \frac{dy}{dx} = e^x + \sin y + x^2 e^{\sin y}$ is $f(x) + e^{-\sin y} = C$ (where $C$ is an arbitrary real constant), where $f(x)$ is equal to
The area of the figure bounded by the parabola $y^2 + 8x = 16$ and $y^2 - 24x = 48$ is
A particle moving in a straight line starts from rest, and the acceleration at any time $t$ is $a - kt^2$, where $a$ and $k$ are positive constants. The maximum velocity attained by the particle is
$I = \int \cos(\ln x) \, dx$. Then $I =$
Let $f$ be derivable in $[0, 1]$, then
The value of $\int_0^{\pi/2} \frac{(\cos x)\sin x}{(\cos x)\sin x + (\sin x)\cos x} \, dx$ is
Let $\lim_{\varepsilon \to 0^+} \int_\varepsilon^x \frac{b t \cos 4t - a \sin 4t}{t^2} \, dt = \frac{a \sin 4x}{x} - 1,\quad (0<x<\frac{\pi}{4})$. Then $a$ and $b$ are given by
Let $\int \frac{x^{1/2}}{\sqrt{1 - x^3}} \, dx = \frac{2}{3} \, g(f(x)) + c$; then
If $x \frac{dy}{dx} + y = x \frac{f(xy)}{f'(xy)}$, then $|f(xy)|$ is equal to
A curve passes through the point $(3, 2)$ for which the segment of the tangent line contained between the coordinate axes is bisected at the point of contact. The equation of the curve is
$AB$ is a variable chord of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. If $AB$ subtends a right angle at the origin $O$, then $\frac{1}{OA^2} + \frac{1}{OB^2}$ equals to
The values of $a, b, c$ for which the function $f(x) = \begin{cases} \sin((a + 1)x) + \sin x, & x<0 \\ c, & x = 0 \\ \frac{(\sqrt{x + bx^2}) - \sqrt{x}}{bx^{1/2}}, & x > 0 \end{cases}$ is continuous at $x = 0$, are
Let $f(x) = a_0 + a_1|x| + a_2|x^2| + a_3|x^3|$, where $a_0, a_1, a_2, a_3$ are real constants. Then $f(x)$ is differentiable at $x = 0$
If $y = e^{\tan^{-1} x}$, then