List of top Questions asked in MHT CET- 2020

For a sequence if \( S_n = \dfrac{5^n - 2^n}{2^n} \), then its fourth term is

If the radius of a circular blot of oil is increasing at the rate of $2$ cm/min, then the rate of change of its area when its radius is $3$ cm is

If \[ f(x) = \begin{cases} 6\beta - 3x, & \text{if } -4 \leq x<-2,
4x + 1, & \text{if } -2 \leq x \leq 2, \end{cases} \] is continuous on \( [-4, 2] \), then \( \alpha + \beta = \)

The equation of normal to the curve \( y = \sin \left( \frac{\pi x}{4} \right) \) at the point (2, 5) is

If \( f(x) = 2x^2 + bx + c \), \( f(0) = 3 \) and \( f(2) = 1 \), then \( (f \circ f)(1) = \)

Water at \(100^\circ\text{C}\) cools in 15 minutes to \(75^\circ\text{C}\) in a room temperature of \(25^\circ\text{C}\). Then the temperature of water after 30 minutes is

Evaluate \( \int_0^{\frac{\pi}{2}} \left( e^{\sin x} - e^{\cos x} \right) \, dx \)

If the lines \[ \frac{x - 1}{5} = \frac{y + 1}{3} = \frac{3 - z}{\lambda} \quad \text{and} \quad \frac{x + 1}{4} = \frac{1 - 3y}{15} = \frac{z + 1}{1} \] are perpendicular to each other, then \( \lambda = \)

Evaluate \[ \int \frac{5^x}{\sqrt{5^{-2x}} - 5^{2x}}\,dx \]

If the line \( 6x - y - 4 = 0 \) touches the curve \( y^2 = ax^3 + b \) at the point (1, 2), then \( a + b = \)

A fair coin is tossed 2 times. A person receives \( X^3 \) if he gets \( X \) number of heads. His expected gain is

If \( y = \tan^{-1} \left( \frac{\sin 2x}{1 + \cos 2x} \right) \), then \[ \frac{dy}{dx} = \]

If the equation \[ kxy + 5x + 3y + 2 = 0 \text{ represents a pair of lines, then } k = \]

If the elements of matrix \( A \) are the reciprocals of the elements of the matrix \( \begin{pmatrix} 1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \end{pmatrix} \), where \( \omega \) is a complex cube root of unity, then

The length of the perpendicular from the point \( P(a,b) \) to the line \( \dfrac{x}{a} + \dfrac{y}{b} = 1 \) is

The logical expression \( [p \wedge (q \vee r)] \vee [\neg r \wedge \neg q \wedge p] \) is equivalent to

The integrating factor of the differential equation \[ \frac{dy}{dx} + \frac{1}{x}y = x^3 - 3 \] is

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

The population \( P(t) \) of a certain mouse species at time \( t \) satisfies the differential equation \[ \frac{dP(t)}{dt} = 0.5P(t) - 450. \quad \text{If} \, P(0) = 850, \, \text{then the time at which the population becomes zero is} \]

The perimeter of a triangle is \(10\) cm. If one of its sides is \(4\) cm, then the remaining sides of the triangle, when the area of the triangle is maximum, are

The particular solution of the differential equation \[ y \frac{dx}{dy} = x \log x \quad \text{at} \quad x = e \text{ and } y = 1 \] is

The joint equation of bisectors of the angle between the lines represented by $3x^2+2xy-y^2=0$ is

Find the value of the following expression:
\[ 5^2 + 6^2 + 7^2 + \cdots + 20^2 = \]

If the points \( (1, 1, \lambda) \) and \( (-3, 0, 1) \) are equidistant from the plane \( 3x + 4y - 12z + 13 = 0 \), then the integer value of \( \lambda \) is

Evaluate \[ \int \frac{\log x - 1}{1 + (\log x)^2} \, dx \]