List of top Mathematics Questions asked in WBJEE

Let \( f(\theta) = \begin{vmatrix} 1 & \cos \theta & -1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1 \end{vmatrix} \). Suppose \( A \) and \( B \) are respectively the maximum and minimum values of \( f(\theta) \). Then \( (A, B) \) is equal to:
If \( a, b, c \) are in A.P. and if the equations \( (b - c)x^2 + (c - a)x + (a - b) = 0 \) and \( 2(c + a)x^2 + (b + c)x = 0 \) have a common root, then
If \( g(f(x)) = |\sin x| \) and \( f(g(x)) = (\sin \sqrt{x})^2 \), then:
If for a matrix \( A \), \( |A| = 6 \) and \( \text{adj } A = \begin{bmatrix} 1 & -2 & 4 \\ 4 & 1 & 1 \\ -1 & k & 0 \end{bmatrix} \), then \( k \) is equal to:
If the matrix \[ \begin{pmatrix} 0 & a & a\\ 2b & b & -b\\ c & -c & c \end{pmatrix} \] is orthogonal, then the values of \( a, b, c \) are:
If \( a, b, c \) are positive real numbers each distinct from unity, then the value of the determinant \[ \left| \begin{matrix} 1 & \log_a b & \log_a c \\ \log_b a & 1 & \log_b c \\ \log_c a & \log_c b & 1 \end{matrix} \right| \] is:
Let \( A = \begin{pmatrix} 5 & 5\alpha & \alpha \\ 0 & \alpha & 5\alpha \\ 0 & 0 & 5 \end{pmatrix} \). If \( |A|^2 = 25 \), then \( |\alpha| \) equals to
Suppose \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + qx + r = 0 \) (with \( r \neq 0 \)) and they are in A.P. Then the rank of the matrix \( \begin{pmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{pmatrix} \) is:

If \(f(x) = \begin{cases} x^2 + 3x + a, & x \leq 1  bx + 2, & x>1 \end{cases}\), \(x \in \mathbb{R}\), is everywhere differentiable, then

Let \( f: [0, 1] \to \mathbb{R} \) and \( g: [0, 1] \to \mathbb{R} \) be defined as follows:
 
The function \( f(x) \) is defined as:
\[ f(x) = \begin{cases} 1 & \text{if } x \in \mathbb{Q} \cap [0, 1] \\ 0 & \text{if } x \in (\mathbb{R} \setminus \mathbb{Q}) \cap [0, 1] \end{cases} \]
The function \( g(x) \) is defined as:
\[ g(x) = \begin{cases} 0 & \text{if } x \in \mathbb{Q} \cap [0, 1] \\ 1 & \text{if } x \in (\mathbb{R} \setminus \mathbb{Q}) \cap [0, 1] \end{cases} \]
Then:
If \( f(x) = \int_{0}^{\sin^2 x} \sin^{-1} \sqrt{t} \, dt \) and \( g(x) = \int_{0}^{\cos^2 x} \sin^{-1} \sqrt{t} \, dt \), then the value of \( f(x) + g(x) \) is:
If \( P \) is a non-singular matrix of order \( 5 \times 5 \) and the sum of the elements of each row is 1, then the sum of the elements of each row in \( P^{-1} \) is:
The population \( p(t) \) at time \( t \) of a certain mouse species follows the differential equation \( \frac{dp(t)}{dt} = 0.5p(t) - 450 \). If \( p(0) = 850 \), then the time at which the population becomes zero is:
Let \( f(x) = x^2 \), \( x \in [-1, 1] \). Then which of the following are correct?
The solution set of the equation \( \tan(\pi \tan x) = \cot(\pi \cot x) \), \( x \in \left(0, \frac{\pi}{2}\right) \), is:
If \( 0 \leq a, b \leq 3 \) and the equation \( x^2 + 4 + 3\cos(ax + b) = 2x \) has real solutions, then the value of \( (a + b) \) is:
If the equation \( \sin^4 x - (p + 2)\sin^2 x - (p + 3) = 0 \) has a solution, then \( p \) must lie in the interval:
Three numbers are chosen at random without replacement from \( \{1, 2, \dots, 10\} \). The probability that the minimum of the chosen numbers is 3 or their maximum is 7, is:
The value of \( \int_{-100}^{100} \frac{x + x^3 + x^5}{1 + x^2 + x^4 + x^6} dx \) is:
If \( f(x) \) and \( g(x) \) are two polynomials such that \( \phi(x) = f(x^3) + xg(x^3) \) is divisible by \( x^2 + x + 1 \), then:
Let \( f(x) = |x - \alpha| + |x - \beta| \), where \( \alpha, \beta \) are the roots of the equation \( x^2 - 3x + 2 = 0 \). Then the number of points in \( [\alpha, \beta] \) at which \( f \) is not differentiable is:
If \( f(x) = \frac{3x - 4}{2x - 3} \), then \( f(f(f(x))) \) will be:
Let \( x - y = 0 \) and \( x + y = 1 \) be two perpendicular diameters of a circle of radius \( R \). The circle will pass through the origin if \( R \) is equal to:
Let \( u + v + w = 3 \), \( u, v, w \in \mathbb{R} \) and \( f(x) = ux^2 + vx + w \) be such that \( f(x + y) = f(x) + f(y) + xy \) for all \( x, y \in \mathbb{R} \). Then \( f(1) \) is equal to:
Let \( a_n \) denote the term independent of \( x \) in the expansion of \( \left[ x + \frac{\sin(1/n)}{x^2} \right]^{3n} \). Then \( \lim_{n \to \infty} \frac{(a_n) n!}{3^n n^n} \) equals: