List of top Questions asked in WBJEE

Evaluate: \[ \lim_{n \to \infty} \frac{1}{n^{k+1}} \left[ 2^k + 4^k + 6^k + \dots + (2n)^k \right]. \]

Let \( \Gamma \) be the curve \( y = b e^{-x/a} \) and \( L \) be the straight line:
\[ \frac{x}{a} + \frac{y}{b} = 1, \quad a, b \in \mathbb{R}. \]
Then:

The angle between two diagonals of a cube will be:

If two circles which pass through the points \( (0, a) \) and \( (0, -a) \) and touch the line \( y = mx + c \) cut orthogonally, then:

Let
\[ A = \begin{bmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{bmatrix}. \]
Which of the following is true?

If 1000! = 3ⁿ × m, where m is an integer not divisible by 3, then n =?

The locus of the midpoint of the system of parallel chords parallel to the line \( y = 2x \) to the hyperbola \( 9x^2 - 4y^2 = 36 \) is:

For the real numbers \( x \) and \( y \), we write \( x \, P \, y \) iff \( x - y + \sqrt{2} \) is an irrational number.
Then the relation \( P \) is:

If \( A \) and \( B \) are acute angles such that \( \sin A = \sin^2 B \) and \( 2\cos^2 A = 3\cos^2 B \), then \( (A, B) \) is:

In a plane, \(\vec{a}\) and \(\vec{b}\) are the position vectors of two points \(A\) and \(B\) respectively. A point \(P\) with position vector \(\vec{r}\) moves on that plane in such a way that
\[ |\vec{r} - \vec{a}| - |\vec{r} - \vec{b}| = c \quad (\text{real constant}). \]
The locus of \(P\) is a conic section whose eccentricity is:

Five balls of different colors are to be placed in three boxes of different sizes. The number of ways in which we can place the balls in the boxes so that no box remains empty is:

If \( a_1, a_2, \dots, a_n \) are in A.P. with common difference \( \theta \), then the sum of the series:
\[ \sec a_1 \sec a_2 + \sec a_2 \sec a_3 + \dots + \sec a_{n-1} \sec a_n = k (\tan a_n - \tan a_1), \]
where \( k = ? \)

Let
\[ I(R) = \int_0^R e^{-R \sin x} \, dx, \quad R > 0. \]
Which of the following is correct?

Let
\[ A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 1 & 1 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 2 \\ 1 \\ 7 \end{bmatrix}. \]
For the validity of the result \(AX = B\), \(X\) is:

Consider the function
\[ f(x) = x(x - 1)(x - 2) \cdots (x - 100). \]
Which one of the following is correct?

Consider the function f(x) = (x−2)logx. Then the equation xlogx = 2−x has:

If \(\int \frac{\log(x + \sqrt{1 + x^2})}{1 + x^2} \, dx = f(g(x)) + c\), then:

If \(f(x) = \frac{e^x}{1+e^x}, I_1 = \int_{-a}^a xg(x(1-x)) \, dx\) and \(I_2 = \int_{-a}^a g(x(1-x)) \, dx\), then the value of \(\frac{I_2}{I_1}\) is:

Let \(f : \mathbb{R} \to \mathbb{R}\) be a differentiable function and \(f(1) = 4\). Then the value of
\[ \lim_{x \to 1} \int_{4}^{f(x)} \frac{2t}{x - 1} \, dt \]

The equation \(2x^5 + 5x = 3x^3 + 4x^4\) has:

If \(\alpha, \beta\) are the roots of the equation \(ax^2 + bx + c = 0\), then:
\[ \lim_{x \to \beta} \frac{1 - \cos(ax^2 + bx + c)}{(x - \beta)^2} \]

If \(U_n (n = 1, 2)\) denotes the \(n\)-th derivative (\(n = 1, 2\)) of \(U(x) = \frac{Lx + M}{x^2 - 2Bx + C}\) (\(L, M, B, C\) are constants), then \(PU_2 + QU_1 + RU = 0\) holds for:

Chords AB CD of a circle intersect at right angle at the point P. If the lengths of AP, PB, CP, PD are 2, 6, 3, 4 units respectively, then the radius of the circle is:

\(\triangle OAB\) is an equilateral triangle inscribed in the parabola \(y^2 = 4ax, \, a>0\) with \(O\) as the vertex. Then the length of the side of \(\triangle OAB\) is: