List of top Questions asked in VITEEE- 2024

Let \( f(x) \) be a polynomial function satisfying \[ f(x) \cdot f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right). \] If \( f(4) = 65 \) and \( I_1, I_2, I_3 \) are in GP, then \( f'(I_1), f'(I_2), f'(I_3) \) are in:

Evaluate the limit: \[ L = \lim_{x \to 0} \frac{35^x - 7^x - 5^x + 1}{(e^x - e^{-x}) \ln(1 - 3x)} \]

If the roots of the quadratic equation $$ (a^2 + b^2) \, x^2 - 2 \, (bc + ad) \, x + (c^2 + d^2) = 0 $$ are equal, then: 

Let \( z \neq 1 \) be a complex number and let \( \omega = x + iy, y \neq 0 \). If \[ \frac{\omega -\overline{\omega}z}{1 -z} \] is purely real, then \( |z| \) is equal to

If A, B, C, D are the angles of a quadrilateral, then \[ \frac{\tan A + \tan B + \tan C + \tan D}{\cot A + \cot B + \cot C + \cot D} = \]

The circle touching the y axis at a distance 4 units from the origin and cutting off an intercept 6 from the x axis is: (A) \(x^2 + y^2 \pm 10x - 8y + 16 = 0\)

The coordinates of the foot of perpendicular from the point \( (2, 3) \) on the line \( y = 3x + 4 \) is given by:

If \( \frac{1}{q + r}, \frac{1}{r + p}, \frac{1}{p + q} \) are in A.P., then:

The coefficient of \( x^{50} \) in \( (1 + x)^{101} (1 - x + x^2)^{100} \) is:

The number of different permutations of all the letters of the word "PERMUTATION" such that any two consecutive letters in the arrangement are neither both vowels nor both identical is:

Negation of the statement \( (p \land r) \rightarrow (r \lor q) \) is:

The equation of a common tangent to the parabolas \( y = x^2 \) and \( y = -(x - 2)^2 \) is:

A and B are independent events of a random experiment if and only if:

The integral \( I = \int_{\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{x + \frac{\pi}{4}}{2 - \cos 2x} \, dx \) is equal to:

The range of \( 2 \left| \sin x + \cos x \right| - \sqrt{2} \) is:

The length of the perpendicular from the point \( (1, -2, 5) \) on the line passing through \( (1, 2, 4) \) and parallel to the line given by \( x + y - z = 0 \) and \( x - 2y + 3z - 5 = 0 \) is:

The probability distribution of a random variable is given below: \[\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline X = x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline P(X = x) & 0 & K & 2K & 2K & 3K & K^2 & 2K^2 & 7K^2 + K \\ \hline \end{array}\]

Find \( P(0<X<5) \).

If the solution of \[ \left( 1 + 2e^\frac{x}{y} \right) dx + 2e^\frac{x}{y} \left( 1 - \frac{x}{y} \right) dy = 0 \] is \[ x + \lambda y e^\frac{x}{y} = c \quad \text{(where \(c\) is an arbitrary constant), then \( \lambda \) is:} \]

The solution of the differential equation: \[ x^4 \frac{dy}{dx} + x^3 y + \csc(xy) = 0 \] is equal to:

Evaluate the integral: $\int_{-\pi}^{\pi} x^2 \sin(x) \, dx$