List of top Questions asked in WBJEE JENPAS UG

If \( \cos(\theta+\phi)=\frac{3}{5} \) and \( \sin(\theta-\phi)=\frac{5}{13} \), \( 0<\theta,\phi<\frac{\pi}{4} \), then \( \cot(2\theta) \) equals:
The probability that a non-leap year selected at random will have 53 Sundays or 53 Saturdays is:
If \( f(x) \) and \( g(x) \) are 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 roots of \( x^2-3x+2=0 \). Then the number of points in \( [\alpha,\beta] \) at which \( f \) is not differentiable is:
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 \( R \) equals:
Let \( a_n \) denote the term independent of \( x \) in the expansion of \[ \left[x + \frac{\sin(1/n)}{x^2}\right]^{3n}, \] then \[ \lim_{n\to\infty} \frac{(a_n)n!}{\,{}^{3n}P_n} \] equals:
The maximum number of common normals of \( y^2 = 4ax \) and \( x^2 = 4by \) is:
If \( |z_1|=|z_2|=|z_3|=1 \) and \( z_1+z_2+z_3=0 \), then the area of the triangle whose vertices are \( z_1,z_2,z_3 \) is:
The number of solutions of \[ \sin^{-1} x + \sin^{-1}(1-x) = \cos^{-1} x \] is:
If \( a,b,c \) are in A.P. and the equations \[ (b-c)x^2 + (c-a)x + (a-b) = 0 \] \[ 2(c+a)x^2 + (b+c)x = 0 \] have a common root, then:
If \( f \) is the inverse function of \( g \) and \( g'(x) = \dfrac{1{1 + x^n} \), then the value of \( f'(x) \) is:}
Let \[ f_n(x) = \tan\frac{x}{2}(1+\sec x)(1+\sec 2x)\cdots(1+\sec 2^{n-1}x), \] then:
Evaluate \[ \lim_{x \to 0} \frac{\tan\!\left(\lfloor -\pi^2 \rfloor x^2\right) - x^2 \tan\!\left(\lfloor -\pi^2 \rfloor\right)}{\sin^2 x} \]
If \( x=-1 \) and \( x=2 \) are extreme points of \[ f(x) = \alpha \log|x| + \beta x^2 + x \quad (x \ne 0), \] then:
The line \( y - \sqrt{3}x + 3 = 0 \) cuts the parabola \( y^2 = x + 2 \) at the points \( P \) and \( Q \). If the coordinates of the point \( X \) are \( (\sqrt{3}, 0) \), then the value of \( XP \cdot XQ \) is:
If \( (1 + x - 2x^2)^6 = 1 + a_1 x + a_2 x^2 + \cdots + a_{12x^{12} \), then the value of \( a_2 + a_4 + a_6 + \cdots + a_{12} \) is:}
Let \( \omega (\ne 1) \) be a cube root of unity. Then the minimum value of the set \[ \left\{ |a + b\omega + c\omega^2|^2 : a,b,c \text{ are distinct non-zero integers} \right\} \] equals:
The expression \( 2^{4n} - 15n - 1 \), where \( n \in \mathbb{N} \), is divisible by:
Let \( \vec{a}, \vec{b}, \vec{c} \) be unit vectors. Suppose \( \vec{a}\cdot\vec{b} = \vec{a}\cdot\vec{c} = 0 \) and the angle between \( \vec{b} \) and \( \vec{c} \) is \( \frac{\pi}{6} \). Then \( \vec{a} \) is:
If \( \vec{a} = 3\hat{i} - \hat{k} \), \( |\vec{b}| = \sqrt{5} \) and \( \vec{a} \cdot \vec{b} = 3 \), then the area of the parallelogram for which \( \vec{a} \) and \( \vec{b} \) are adjacent sides is:
If \( \theta \) is the angle between two vectors \( \vec{a} \) and \( \vec{b} \) such that \( |\vec{a}| = 7, |\vec{b}| = 1 \) and \[ |\vec{a} \times \vec{b}|^2 = k^2 - (\vec{a} - \vec{b})^2, \] then the values of \( k \) and \( \theta \) are:
The line parallel to the x-axis passing through the intersection of the lines \[ ax + 2by + 3b = 0 \quad \text{and} \quad bx - 2ay - 3a = 0 \] where \( (a,b) \neq (0,0) \), is:
Consider three points \( P(\cos\alpha, \sin\beta) \), \( Q(\sin\alpha, \cos\beta) \) and \( R(0,0) \), where \( 0 < \alpha, \beta < \frac{\pi}{4} \). Then:
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:
Suppose \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + qx + r = 0 \) (with \( r \ne 0 \)) 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: