List of top Questions asked in MHT CET

Evaluate the integral \[ \int \frac{x^2+1}{x^4+x^2+1}\,dx \]

If \( a, b, c \) are lengths of the sides \( BC, CA, AB \) respectively of \( \triangle ABC \) and \( a\vec{AH} + b\vec{BH} + c\vec{CH} = \vec{0} \), then point \( H \) is the

Which of the following statement patterns is a tautology?
\[ \begin{aligned} S_1 &: (\sim q \land p) \land q
S_2 &: [p \land (p \rightarrow q)] \rightarrow q
S_3 &: (p \land q) \land (\sim p \lor \sim q)
S_4 &: (p \land q) \rightarrow r \end{aligned} \]

If \( \vec{AB} = 3\hat{i} + 5\hat{j} + 4\hat{k} \), \( \vec{AC} = 5\hat{i} - 5\hat{j} + 2\hat{k} \) represent the sides of triangle \( ABC \), then the length of the median through \( A \) is

A plane \( E_1 \) makes intercepts \( 1, -3, 4 \) on the coordinate axes. The equation of a plane parallel to \( E_1 \) and passing through \( (2,6,-8) \) is

If a line in octant OXYZ makes equal angles with the coordinate axes, then

The radius of a circle is increasing at the rate of \( 2 \,\text{cm/sec} \). Find the rate at which its area is increasing when the radius of the circle is \( 5 \) decimeters.

Evaluate the integral \[ \int_{0}^{1} \tan^{-1}\!\left( \frac{2x - 1}{1 + x - x^2} \right) dx \]

In a certain culture of bacteria, the rate of increase is proportional to the number present. It is found that the number doubles in 4 hours. Then the number of times the bacteria are increased in 12 hours is

If one of the lines given by the equation \( x^2 + kxy + 2y^2 = 0 \) is \( x + 2y = 0 \), then the value of \( k \) is

The area bounded by the parabola \( x^2 = 4y \), the lines \( y = 2 \), \( y = 4 \) and the Y-axis is

Evaluate the integral \[ \int \frac{\sin 2x}{\sin^2 x \cos^2 x}\, dx \]

If \( y = e^{4x}\cos 5x \), then find \( \dfrac{d^2y}{dx^2} \) at \( x = 0 \).

The joint equation of the pair of lines passing through the point of intersection of the lines \[ 2x^2 - xy - 15y^2 - 7x + 32y - 9 = 0 \] and parallel to the coordinate axes is

A tangent to the curve \( x = at^2,\; y = 2at \) is perpendicular to the X-axis. Then the point of contact is

If \( P(A') = 0.6 \), \( P(B) = 0.8 \) and \( P(B|A) = 0.3 \), then find \( P(A|B) \).

The coordinates of the foci of the ellipse \( 16x^2 + 9y^2 = 144 \) are

Evaluate the integral \[ \int \frac{\sec x}{\sqrt{\log(\sec x + \tan x)}}\, dx \]

The L.P.P. to maximize \( z = x + y \), subject to \[ x + y \le 30,\; x \le 15,\; y \le 20,\; x + y \ge 15,\; x \ge 0,\; y \ge 0 \] has

With usual notations, in \( \triangle ABC \), if \( b\cos^2\frac{C}{2} + c\cos^2\frac{B}{2} = \frac{3a}{2} \), then

Evaluate: \[ \cos x \cos 7x - \cos 5x \cos 13x \]

If \( f : \mathbb{R} \rightarrow \mathbb{R} \) such that \[ f(x) = \frac{e^{x} + e^{-x}}{e^{x} - e^{-x}}, \] then \( f \) is

Evaluate the integral \[ \int_{0}^{\pi/2} \frac{\sin^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x}\, dx \]

Given below is the probability distribution of a discrete random variable \( X \):
\[ \begin{array}{c|cccccc} X=x & 1 & 2 & 3 & 4 & 5 & 6
\hline P(X=x) & k & 0 & 2k & 5k & k & 3k \end{array} \] Then find \( P(X \ge 4) \).

If \( \tan \theta = \dfrac{1}{3} \), then find \( \cos 2\theta \).