List of top Mathematics Questions asked in MHT CET

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:

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:

Let \[ \vec a=\hat i+\hat j+\hat k \] \[ \vec b=\hat i-\hat j+2\hat k \] If a vector \( \vec c \) is coplanar with \( \vec a \) and \( \vec b \) such that \[ \vec c\cdot \vec a=1 \] and \[ \vec c\cdot \vec b=2 \] then \( \vec c \) is:

If \[ y=(x-1)(x-2)(x-3)\cdots(x-100) \] and the value of \( \dfrac{dy}{dx} \) at \(x=0\) is equal to \[ \lambda\left(\frac{100!}{^{100}C_5}\right) \] then \( \lambda \) is:

The value of \[ \int \frac{x^2-1}{(x^4+3x^2+1)\tan^{-1}\left(x+\frac1x\right)}\,dx \] is:

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

Let \[ f(x)=\sqrt{x^2+1}, \quad g(x)=\frac{x+1}{x^2+1}, \quad h(x)=2x-3 \] Then, \[ f'(h'(g'(x))) = ? \]

The statement \( \neg(p \leftrightarrow q) \) is logically equivalent to:

For the parabola \( y^2 = 16x \), find the distance between the focus and the directrix.

The surrounding temperature is \( 20^\circ\text{C} \). A body cools from \( 100^\circ\text{C} \) to \( 60^\circ\text{C} \) in 20 minutes. Find the time taken for the body to cool down to \( 30^\circ\text{C} \).

Given \( f(x) = x - 1 \), \( h(1) = 4 \), and \( h'(1) = -2 \). If \( g(x) = (f[4(h(x)) + 3])^2 \), find the value of \( g'(1) \).

If the area enclosed between the circle \(x^2+y^2=25\) and the parabola \(y^2=16x\) is \(A\), then \(A=\ ?\)

If \(A\) is a \(3\times3\) matrix satisfying \(2A=3A^2+kI\), and \(\det(A)=1\), then the value of \(k\) is:

\( \displaystyle \lim_{x \to 2} \left( \frac{1}{x-2} - \frac{2}{x^3-3x^2+2x} \right) = \)

A wire 64 m long is bent into the shape of a rectangle. What should be its dimensions so that the enclosed area is maximum?

The value of \[ \int \frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}\,dx \] is:

If \( \vec{a} = 2\hat{i} + \hat{j} + 3\hat{k} \) and \( \vec{b} = p\hat{i} + 2\hat{j} + 2\hat{k} \) are perpendicular to each other, find the value of \(p\).

If \(z\) be a complex number such that \( |z| + z = 2 + i \), then find the value of \( |z| \).