List of top Questions asked in WBJEE

Let \( f(x) \) be continuous on \( [0, 5] \) and differentiable in \( (0, 5) \). If \( f(0) = 0 \) and \( |f'(x)| \leq \frac{1}{5} \) for all \( x \) in \( (0, 5) \), then for all \( x \) in \( [0, 5] \):
If the sum of \( n \) terms of an A.P. is \( 3n^2 + 5n \) and its \( m^{th} \) term is \( 164 \), then the value of \( m \) is:
Let \( f(x) = \max\{x + |x|, x - |x|, x - [x]\} \), where \( [x] \) stands for the greatest integer not greater than \( x \). Then \( \int_{-3}^{3} f(x) \, dx \) has the value:
If \( \cos(\theta + \phi) = \frac{3}{5} \) and \( \sin(\theta - \phi) = \frac{5}{13} \), \( 0<\theta, \phi<\frac{\pi}{4} \), then \( \cot(2\theta) \) has the value:
The value of the expression \( {}^{47} C_4 + \sum_{j=1}^{5} {}^{52-j} C_3 \) is:
If \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar vectors and \( \lambda \) is a real number, then the vectors \( \vec{a} + 2\vec{b} + 3\vec{c} \), \( \lambda \vec{b} + 4\vec{c} \), and \( (2\lambda - 1)\vec{c} \) are non-coplanar for:
If \( z_1, z_2 \) are complex numbers such that \( \frac{z_1}{3z_2} \) is a purely imaginary number, then the value of \( \left| \frac{z_1 - z_2}{z_1 + z_2} \right| \) is:
The value of the integral \( \int_{3}^{6} \frac{\sqrt{x}}{\sqrt{9 - x} + \sqrt{x}} \, dx \) is:
If \( \text{adj } B = \lambda I \) where \( |\lambda| = 1 \), then \( \text{adj} ((Q^{-1} B P^{-1})) \) =
Evaluate the limit \[ \lim_{x \to 0} \frac{\tan \left( -\pi^2 x^2 - x^2 \tan \left( -\pi^2 \right) \right)}{\sin^2 x} \] is:
Let \( \vec{a}, \vec{b}, \vec{c} \) be vectors of equal magnitude such that the angle between \( \vec{a} \) and \( \vec{b} \) is \( \alpha \), the angle between \( \vec{b} \) and \( \vec{c} \) is \( \beta \), and the angle between \( \vec{c} \) and \( \vec{a} \) is \( \gamma \). Then the minimum value of \( \cos \alpha + \cos \beta + \cos \gamma \) is:
If \( E \) and \( F \) are two independent events with \( P(E) = 0.3 \) and \( P(E \cup F) = 0.5 \), then \( P(E \mid F) - P(F \mid E) \) equals:
An \( n \times n \) matrix is formed using 0, 1 and -1 as its elements. The number of such matrices which are skew symmetric is:
The set of points of discontinuity of the function \( f(x) = x - [x], x \in \mathbb{R} \) is:
The function \( f(x) = 2x^3 - 3x^2 - 12x + 4 \), \( x \in \mathbb{R} \) has:
The expression \( 2^{4n} - 15n - 1 \), where \( n \in \mathbb{N} \) (the set of natural numbers), is divisible by:
The value of \( \displaystyle \int_0^{1.5} \left\lfloor x^2 \right\rfloor \, dx \) is:
If \( \cos^{-1} \alpha + \cos^{-1} \beta + \cos^{-1} \gamma = 3\pi \), then the value of \( \alpha(\beta+\gamma) + \beta(\gamma+\alpha) + \gamma(\alpha+\beta) \) is:
If \( \theta \) is the angle between two vectors \( \vec{a} \) and \( \vec{b} \) such that \( |\vec{a}| = 7 \), \( |\vec{b}| = 1 \) and \( |\vec{a} \times \vec{b}|^2 = k^2 - (\vec{a} \cdot \vec{b})^2 \), then the values of \( k \) and \( \theta \) are:
A function \( f : \mathbb{R} \to \mathbb{R} \), satisfies \[ \frac{f(x+y)}{3} = \frac{f(x) + f(y) + f(0)}{3} \quad \text{for all} \, x, y \in \mathbb{R}. \] If the function \( f \) is differentiable at \( x = 0 \), then \( f \) is:
If \( x = \int_0^y \frac{1}{\sqrt{1+9t^2}} \, dt \) and \( \frac{d^2y}{dx^2} = ay \), then the value of \( a \) is:
The sum of the first four terms of an arithmetic progression is 56. The sum of the last four terms is 112. If its first term is 11, then the number of terms is:
The value of the integral \( \int_{0}^{\pi/2} \log\left(\frac{4 + 3\sin x}{4 + 3\cos x}\right) dx \) is:
Let \( p(x) \) be a real polynomial of least degree which has a local maximum at \( x = 1 \) and a local minimum at \( x = 3 \). If \( p(1) = 6 \) and \( p(3) = 2 \), then \( p'(0) \) is equal to:
If \( (1 + x - 2x^2)^6 = 1 + a_1x + a_2x^2 + \ldots + a_{12}x^{12} \), then the value of \( a_2 + a_4 + a_6 + \ldots + a_{12} \) is: