List of top Questions asked in MHT CET- 2020

Evaluate the sum of the series: \[ \frac{1^2}{2} + \frac{1^2 + 2^2}{3} + \frac{1^2 + 2^2 + 3^2}{4} + \frac{1^2 + 2^2 + 3^2 + 4^2}{5} + \cdots \quad \text{upto 8 terms} \]

Evaluate \[ \int_{0}^{a} (a-x)^{\frac{3}{2}} x^2 \, dx \]

In a \( \triangle ABC \), if \( 2 \cos C = \sin B \cdot \csc A \), then

The probability density function of a random variable \(X\) is given by \[ f(x) = \frac{k}{\sqrt{x}}, \quad 0 \le x \le 4 \] \[ f(x) = 0, \quad \text{otherwise} \] Find \(P(1<X<4)\).

The general solution of the differential equation \( (1 - x^2) \frac{dy{dx} + 2xy = x(1 - x^2)^{\frac{1}{2}} \) is}

The probability mass function of a random variable \(X\) is given by \[ P(X = x) = \frac{5cx}{2^5}, \quad x = 0,1,2,3,4,5 \] \[ P(X = x) = 0, \quad \text{otherwise} \] Then find \(P(X \le 2)\).

The dual of the statement pattern \( \sim p \land (q \lor t) \) is (where \( t \) is a tautology and \( c \) is a contradiction)

The area bounded by the curve \(y = x^3\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is

If \[ x \cos \theta + y \sin \theta = 5, \quad x \sin \theta - y \cos \theta = 3, \quad \text{then the value of} \, x^2 + y^2 \, \text{is} \]

The statement pattern \[ [(p \vee q) \wedge \neg p] \wedge (\neg q) \] is

If \[ A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -1 \end{pmatrix} \] then \( A^4 A^{-1} = \)

The differential equation obtained from the function \( y = a(x - a)^2 \) is

The rate of decay of a certain substance is directly proportional to the amount present at that instant. Initially, there are 27 grams of the substance and 3 hours later it is found that 8 grams are left, then the amount left after one more hour is

Evaluate the integral \[ \int_0^{\frac{\pi}{2}} \frac{\sqrt[3]{\sec x}}{\sqrt[3]{\sec x} + \sqrt[3]{\csc x}} \, dx \]

The equation of a plane passing through the intersection of the planes \(x + 2y - 3z + 2 = 0\) and \(6x + y + z + 1 = 0\) and parallel to the line \(x - 1 = y + 2 = 7 - z\) is

If \[ \int \frac{\sin \theta}{\sin^3 \theta} \, d\theta = \frac{1}{2k} \log \left| \frac{k + \tan \theta}{k - \tan \theta} \right| + c, \text{ then } k = \]

The matrix \( A = \begin{bmatrix} a & -1 & 4 \\ -3 & 0 & 1 \\ -1 & 1 & 2 \end{bmatrix} \) is not invertible only if \( a = \)

If the p.m.f. is given by \( P(X) = k \left( \frac{4}{x} \right) \) for \( x = 0, 1, 2, 3, 4 \) and \( k>0 \), then the value of \( k \) is

Radium decomposes at a rate proportional to the amount present. If half the original amount disappears in 1600 years, then the percentage loss in 100 years is

Evaluate the integral \[ \int \frac{dx}{\sqrt{(x - 1)(x - 2)}} = \text{?}

Evaluate \[ \int_{-\pi/2}^{\pi/2} \sin^2 x\,dx \]

The integral \[ \int_0^a \frac{x}{\sqrt{a - x}} \, dx = \]

Simplify the expression \( \sin \left( \frac{\pi}{3} + x \right) - \cos \left( \frac{\pi}{6} + x \right) \)

The displacement of a particle at time \(t\) is \[ s = t^3 - 4t^2 - 5t \] Then the velocity of the particle at \(t = 2\) sec is