List of top Questions asked in MHT CET- 2023

Identify the product 'B' in the following reaction:

\(CH_3MgBr \xrightarrow{\text{Dry ice, Dry ether}} A \xrightarrow{\text{dil. HCl}, H_2O} B\)

The probability mass function of a random variable $X$ is given by: $$ P(X = r) = \begin{cases} \frac{{}^nC_r}{32}, & r = 0,1,2,\dots,n \\ 0, & \text{otherwise} \end{cases} $$ Find: $$ P(X \le 2) $$ 

Evaluate: $$ \lim_{x \to \frac{\pi}{2}} \frac{\cot x - \cos x}{(\pi - 2x)^3} $$ 

Evaluate: $$ \int \frac{e^{\tan^{-1} x}}{1 + x^2} \left[ \left(\sec^{-1}\sqrt{1 + x^2}\right)^2 + \cos^{-1}\left(\frac{1 - x^2}{1 + x^2}\right) \right] dx, \quad x > 0 $$ 

If $x^{2}+y^{2}=t+\frac{1}{t}$ and $x^{4}+y^{4}=t^{2}+\frac{1}{t^{2}}$, then $\frac{dy}{dx}$ is equal to
A source of sound is moving towards a stationary observer with $\left(\frac{1}{10}\right)^{\text{th}}$ the speed of sound. The ratio of apparent to real frequency is
Equation of simple harmonic progressive wave is given by $y=\frac{1}{\sqrt{a}}\sin(\omega t)\pm \frac{1}{\sqrt{b}}\cos(\omega t)$. Then the resultant amplitude of the wave is ($\cos 90^{\circ}=0$):
The value of c of Lagrange's mean value theorem for $f(x)=\sqrt{25-x^{2}}$ on $[1,5]$ is
The power factor of an R-L circuit is $\frac{1}{\sqrt{2}}$. If the frequency of AC is doubled the power factor will now be

Given \[ 0 \le x \le \frac{1}{2}, \] find the value of \[ \tan \left( \sin^{-1}\left( \frac{x}{\sqrt{2}} + \frac{\sqrt{1 - x^2}}{\sqrt{2}} \right) - \sin^{-1} x \right). \] 

If $\int \frac{x^2}{\sqrt{1-x }}dx = p\sqrt{1-x} (3x^2 + 4x + 8) + c$ where $c$ is a constant of integration, then the value of $p$ is
The value of \[ \int e^{x} \left(\frac{x^{2} + 4x + 4}{(x+4)^{2}}\right)\, dx \] is:
If $\sum_{r=1}^{50} \tan^{-1}\frac{1}{2r^{2 = p}}$ then $\tan p$ is
If \[ x = \sqrt{e^{\sin^{-1} t}} \quad \text{and} \quad y = \sqrt{e^{\cos^{-1} t}}, \] then find \[ \frac{d^2 y}{dx^2}. \]
In an AC circuit, the current is $i=5 \sin(100t - \frac{\pi}{2})$ A and voltage is $e=200 \sin(100t)$ volt. Power consumption in the circuit is ($\cos 90^{\circ}=0$)
From a metallic surface photoelectric emission is observed for frequencies $v_{1}$ and $v_{2}$ ($v_{1}>v_{2}$) of the incident light. The maximum values of the kinetic energy of the photoelectrons emitted in the two cases are in the ratio $1:x$. Hence the threshold frequency of the metallic surface is
Two discs of same mass and same thickness (t) are made from two different materials of densities $d_{1}$ and $d_{2}$ respectively. The ratio of the moment of inertia $I_{1}$ to $I_{2}$ of two discs about an axis passing through the centre and perpendicular to the plane of disc is
A solid sphere rolls without slipping on an inclined plane at an angle $\theta$. The ratio of total kinetic energy to its rotational kinetic energy is
A closed organ pipe of length $L_{1}$ and an open organ pipe contain diatomic gases of densities $\rho_{1}$ and $\rho_{2}$ respectively. The compressibilities of the gases are same in both pipes, which are vibrating in their first overtone with same frequency. The length of the open organ pipe is (Neglect end correction)
When a string of length 'l' is divided into three segments of length $l_{1}$, $l_{2}$ and $l_{3}$, the fundamental frequencies of three segments are $n_{1}$, $n_{2}$ and $n_{3}$ respectively. The original fundamental frequency 'n' of the string is
A seconds pendulum is placed in a space laboratory orbiting round the earth at a height '3R' from the earth's surface. The time period of the pendulum will be (R = radius of earth)
A solid metallic sphere has a charge +3Q. Concentric with this sphere is a conducting spherical shell having charge -Q. The radius of the sphere is 'A' and that of the spherical shell is 'B' ($B>A$). The electric field at a distance 'R' ($A<R<B$) from the centre is ($\epsilon_{0} =$ permittivity of vacuum)
The potential energy of charged parallel plate capacitor is $v_{0}$. If a slab of dielectric constant K is inserted between the plates, then the new potential energy will be
The radii of two soap bubbles are $r_{1}$ and $r_{2}$. In isothermal condition they combine with each other to form a single bubble. The radius of resultant bubble is
In a thermodynamic process, there is no exchange of heat between the system and surroundings. Then the thermodynamic process is