List of top Mathematics Questions asked in MHT CET

For all real $x$, the minimum value of the function $f(x) = \frac{1 - x + x^2}{1 + x + x^2}$ is

The area of the triangle ABC is $10\sqrt{3}\ \text{cm}^2$, angle B is $60^\circ$ and its perimeter is $20\ \text{cm}$, then $\ell(\text{AC}) =$

With usual notations in $\triangle \text{ABC}$, if $\frac{\sin \text{A}}{\sin \text{C}} = \frac{\sin(\text{A} - \text{B})}{\sin(\text{B} - \text{C})}$, then $a^2, b^2, c^2$ are in

The equation of the tangent to the curve $y = \sqrt{2} \sin\left(2x + \frac{\pi}{4}\right)$ at $x = \frac{\pi}{4}$ is

If $\vec{a} = 2\hat{\text{i}} - \hat{\text{j}} + \hat{\text{k}}$, $\vec{b} = \hat{\text{i}} + 2\hat{\text{j}} - 3\hat{\text{k}}$ and $\vec{c} = 3\hat{\text{i}} + \lambda\hat{\text{j}} + 5\hat{\text{k}}$ are coplanar, then $\lambda$ is the root of the equation

The domain of the function $\log_{10}(x^2 - 5x + 6)$ is

The general solution of the differential equation $\cos(x + y) \frac{dy}{dx} = 1$ is

The derivative of the function $\cot^{-1}\left[(\cos 2x)^{1/3}\right]$ at $x = \frac{\pi}{6}$ is

The order and degree of the differential equation $$\frac{d^2y}{dx^2} = \sqrt{\frac{dy}{dx}}$$ are respectively

Evaluate the definite integral $$\int_{1}^{3} \left[ \tan^{-1}\left(\frac{x}{x^2-1}\right) + \tan^{-1}\left(\frac{x^2-1}{x}\right) \right] dx$$

Let two cards be drawn at random from a pack of 52 playing cards. Let X be the number of aces obtained. Then the value of E(X) is

If $\int \frac{\sqrt{x}}{x(x+1)} \, dx = k \tan^{-1} m + c$ (where $c$ is a constant of integration), then

If the amplitude of $(z - 2 - 3i)$ is $\frac{3\pi}{4}$, then the locus of $z$ is (where $z = x + iy$)

A fair coin is tossed 100 times. The probability of getting a head for an even number of times is

The negation of the inverse of $\sim p \rightarrow q$ is

If $\sin(y + z - x)$, $\sin(z + x - y)$ and $\sin(x + y - z)$ are in AP, then

Let $\vec{a} = 2\hat{\text{i}} + \hat{\text{j}} - 2\hat{\text{k}}$ and $\vec{b} = \hat{\text{i}} + \hat{\text{j}}$. If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c} = |\vec{c}|$, $|\vec{c} - \vec{a}| = 2\sqrt{2}$ and the angle between $\vec{a} \times \vec{b}$ and $\vec{c}$ is $60^\circ$, then $|(\vec{a} \times \vec{b}) \times \vec{c}| =$

The projection of $\vec{a} = \hat{\text{i}} - 2\hat{\text{j}} + \hat{\text{k}}$ on $\vec{b} = 2\hat{\text{i}} - \hat{\text{j}} + \hat{\text{k}}$ is

If the function given by
$f(x) = -2 \sin x$ for $-\pi \le x < -\frac{\pi}{2}$
$f(x) = a \sin x + b$ for $-\frac{\pi}{2} < x < \frac{\pi}{2}$
$f(x) = \cos x$ for $\frac{\pi}{2} \le x \le \pi$
is continuous in $[-\pi, \pi]$, then the value of $(3a + 2b)^3$ is

If $\tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}$, where $x > 0$, then $x =$

If $x = a(\theta + \sin \theta)$ and $y = a(1 - \cos \theta)$, then $\left(\frac{d^2y}{dx^2}\right)_{\text{at } \theta = \pi / 2} =$

If $^{11}\text{C}_4 + {^{11}\text{C}_5} + {^{12}\text{C}_6} + {^{13}\text{C}_7} = {^{14}\text{C}_r}$, then the value of $r$ is

Which of the following matrices are invertible?
$$\text{A} = \left[\begin{array}{cc} 2 & 3 \\ 10 & 15 \end{array}\right], \quad \text{B} = \left[\begin{array}{ccc} 1 & 2 & 3 \\ 2 & -1 & 3 \\ 1 & 2 & 3 \end{array}\right], \quad \text{C} = \left[\begin{array}{ccc} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 4 & 6 & 8 \end{array}\right], \quad \text{D} = \left[\begin{array}{ccc} 2 & 4 & 2 \\ 1 & 1 & 0 \\ 1 & 4 & 5 \end{array}\right]$$

The co-ordinates of the foot of the perpendicular drawn from the point $2\hat{\text{i}} - \hat{\text{j}} + 5\hat{\text{k}}$ to the line $\vec{\text{r}} = (11\hat{\text{i}} - 2\hat{\text{j}} - 8\hat{\text{k}}) + \lambda(10\hat{\text{i}} - 4\hat{\text{j}} - 11\hat{\text{k}})$ are

Evaluate the indefinite integral $\int \sec^{-1}x \, dx$