List of top Questions asked in WBJEE

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
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
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
The value of $\int_0^{\pi/2} \frac{(\cos x)\sin x}{(\cos x)\sin x + (\sin x)\cos x} \, dx$ is
If $x \frac{dy}{dx} + y = x \frac{f(xy)}{f'(xy)}$, then $|f(xy)|$ is equal to
Let $\int \frac{x^{1/2}}{\sqrt{1 - x^3}} \, dx = \frac{2}{3} \, g(f(x)) + c$; then
$I = \int \cos(\ln x) \, dx$. Then $I =$
Let $f$ be derivable in $[0, 1]$, then
$\lim_{x \to 0} \left( \frac{1}{x} \ln \left( \frac{\sqrt{1 + x}}{\sqrt{1 - x}} \right) \right)$ is
Let $f:[a, b] \to \mathbb{R}$ be continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = 0 = f(b)$. Then (A) there exists at least one point $c \in (a, b)$ for which $f'(c) = f(c)$ (B) $f'(x) = f(x)$ does not hold at any point of $(a, b)$ (C) at every point of $(a, b)$, $f'(x)>f(x)$ (D) at every point of $(a, b)$, $f'(x)<f(x)$
Domain of $y = \sqrt{\log_{10} \left( \frac{3x - x^2}{2} \right)}$ is
If $y = e^{\tan^{-1} x}$, then
$AB$ 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 the sum of the distances of a point from two perpendicular lines in a plane is 1 unit, then its locus is
Let $P(3\sec\theta, 2\tan\theta)$ and $Q(3\sec\phi, 2\tan\phi)$ be two points on $\frac{x^2}{9} - \frac{y^2}{4} = 1$ such that $\theta + \phi = \frac{\pi}{2}$. Then the ordinate of the intersection of the normals at $P$ and $Q$ is
Let $P$ be a point on $(2, 0)$ and $Q$ be a variable point on $(y - 6)^2 = 2(x - 4)$. Then the locus of the midpoint of $PQ$ is
The line $x - 2y + 4z + 4 = 0$, $x + y + z - 8 = 0$ intersects the plane $x - y + 2z + 1 = 0$ at the point
$AB$ is a chord of a parabola $y^2 = 4ax$, $(a > 0)$ with vertex $A$, $BC$ is drawn perpendicular to $AB$ meeting the axis at $C$. The projection of $BC$ on the axis of the parabola is
The equation of the plane through the intersection of the planes $x + y + z = 1$ and $2x + 3y - z + 4 = 0$ and parallel to the $x$-axis is
If the algebraic sum of the distances from the points $(2, 0)$, $(0, 2)$, and $(1, 1)$ to a variable straight line is zero, then the line passes through the fixed point.
A determinant is chosen at random from the set of all determinants of order 2 with elements 0 and 1 only. The probability that the determinant chosen is non-zero is
For the mapping $f: \mathbb{R} \setminus \{1\} \to \mathbb{R} \setminus \{2\}$ given by $f(x) = \frac{2x}{x - 1}$, which of the following is correct?
The side $AB$ of $\triangle ABC$ is fixed and is of length $2a$ units. The vertex $C$ moves in the plane such that the vertical angle is always constant and is $\alpha$. Let the $x$-axis be along $AB$ and the origin be at $A$. Then the locus of the vertex is