List of top Mathematics Questions asked in MHT CET

If \( \sin x \cos x = \frac{1}{4} \), then the general solution is:

Evaluate the integral: \(\int \frac{x}{x + 2} \, dx\).

In a certain culture of bacteria, the rate of increase is proportional to the number of bacteria present at that instant. It is found that there are 10,000 bacteria at the end of 3 hours and 40,000 bacteria at the end of 5 hours, then the number of bacteria present in the beginning are:

Evaluate the integral: \( \displaystyle \int \frac{x}{x+2}\,dx \)

Find the area of the region bounded by the curve \(y^2 = 4x\) and the line \(x = 3\).

If a polygon has 54 diagonals, find the number of sides of the polygon.

Let \(f(x)\) be defined by: \[ f(x)= \begin{cases} \displaystyle \int_x^6 (|t-2|+3)\,dt, & x>4 2x+8, & x\le4 \end{cases} \] Then at \(x=4\), \(f(x)\) is:

If \[ f(x)=\int \frac{x^2}{(1-x^2)(1+\sqrt{1-x^2})}\,dx \] and \( f(0)=2 \), then \( f\left(\frac12\right) \) is:

\[ \int \frac{1}{x^{1/2}+x^{1/3}}\,dx \] is equal to:

If \(n\) is an odd natural number and \[ I_n=\int_0^1 e^x(x-1)^n\,dx \] then \( I_n+nI_{n-1} \) is equal to:

If y = y(x) satisfies the differential equation \[ \left(\frac{2+\sin x}{1+y}\right)\frac{dy}{dx}=-\cos x \] and \( y(0)=2 \), then \( y\left(\frac{\pi}{2}\right) \) is equal to:

If the statement \( (p \land q) \rightarrow (r \lor \neg s) \) is False (F), what are the truth values of \(p, q, r,\) and \(s\) respectively?

If \(A, B, C\) are vertices of a triangle with position vectors \(\vec{a}, \vec{b}, \vec{c}\) respectively, then find the position vector of the point \(D\) where the angle bisector from vertex \(A\) meets \(BC\).

If \( n \in \mathbb{Z} \), then the expression \[ \frac{2^n}{(1-i)^{2n}} + \frac{(1+i)^{2n}}{2^n} \] is equal to:

For \(n\in\mathbb{N}\), if \[ y=ax^{n+1}+bx^{-n}, \] then \[ x^2\frac{d^2y}{dx^2} \] is equal to:

For \(n\in\mathbb{N}\), if \[ y=ax^{n+1}+bx^{-n}, \] then \[ x^2\frac{d^2y}{dx^2} \] is equal to:

Solve the differential equation \( \frac{dy}{dx} = \frac{1+y^2}{1+x^2} \) given \(y(0)=1\).

Let \[ f(x)=\int_{1}^{4}\log[x]\ dx, \] where \([x]\) denotes the greatest integer function. Then the value of \(f(x)\) is:

Evaluate: \(\tan^{-1}(1) + \tan^{-1}(4) + \tan^{-1}(5) + \tan^{-1}\left(\frac{1}{4}\right) = \pi + \tan^{-1}\left(\frac{\alpha}{2}\right)\). Find the value of \(\alpha\).

Solve for \(x\): \[ x+\log_{15}(5+3^x)=x\log_{15}5+\log_{15}24 \]

Solve for \(x\): \[ x+\log_{15}(5+3^x)=x\log_{15}5+\log_{15}24 \]

If \[ (2+\sin x)\frac{dy}{dx}+(y+1)\cos x=0 \] and \( y(0)=1 \), then \( y\left(\frac{\pi}{2}\right) \) is equal to:

If \[ \left(\frac{2+\sin x}{1+y}\right)\frac{dy}{dx}=-\cos x \] and \[ y(0)=2, \] then find the value of \[ y\left(\frac{\pi}{2}\right). \]

If \[ (2+\sin x)\frac{dy}{dx}+(y+1)\cos x=0 \] and \[ y(0)=1, \] then find the value of \[ y\left(\frac{\pi}{2}\right). \]

The ratio of areas bounded by the curves \[ y=\cos x \] and \[ y=\cos 2x \] between \[ x=0,\qquad x=\frac{\pi}{3} \] and the \(x\)-axis is: