List of top Questions asked in MHT CET- 2021

Let $f(x)$ be a function defined as: $$f(x) = \begin{cases} \frac{\sqrt{1 + px} - \sqrt{1 - px}}{x}, & \text{if } -1 \leq x < 0 \\ \frac{2x + 1}{x - 2}, & \text{if } 0 \leq x \leq 1 \end{cases}$$ If $f(x)$ is continuous in the interval $[-1, 1]$, then $p =$

The coordinates of the point $P \equiv (1,2,3)$ and $O \equiv (0,0,0)$ are given. The direction cosines of $\overline{OP}$ are

The equation of the plane containing the line $\frac{x+1}{-3} = \frac{y-3}{2} = \frac{z+2}{1}$ and the point $(0,7,-7)$ is

For $X \sim B(n, p)$, if $p = 0.6$ and $E(X) = 6$, then $\text{Var}(X) =$

If the lines $x^2 - 4xy + y^2 = 0$ make angles $\alpha$ and $\beta$ with the positive direction of the $X$-axis, then $\cot^2\alpha + \cot^2\beta =$

It is observed that 25% of the cases related to child labour reported to the police station are solved. If 6 new cases are reported, then the probability that at least 5 of them will be solved is

Evaluate the definite integral: $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\cos x}{1 + e^x}\ dx$$

An urn contains 9 balls of which 3 are red, 4 are blue and 2 are green. Three balls are drawn at random from the urn. The probability that the three balls have different colors is

Equation of the plane passing through the point $(2, 0, 5)$ and parallel to the vectors $\hat{i} - \hat{j} + \hat{k}$ and $3\hat{i} + 2\hat{j} - \hat{k}$ is

If $A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & a & 1 \end{bmatrix}$ and $A^{-1} = \frac{1}{2}\begin{bmatrix} 1 & -1 & 1 \\ -8 & 6 & 2c \\ 5 & -3 & 1 \end{bmatrix}$, then values of $a$ and $c$ are respectively

If $\frac{\pi}{2} < \theta < \pi$ and $|\bar{a}| = 5$, $|\bar{b}| = 13$, $|\bar{a} \times \bar{b}| = 25$, then the value of $\bar{a} \cdot \bar{b}$ is

The objective function $z = 4x + 5y$ subject to $2x + y \geq 7;\ 2x + 3y \leq 15;\ y \leq 3,\ x \geq 0;\ y \geq 0$ has its minimum value at a point located

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

Evaluate the indefinite integral:
$$\int \frac{1}{x^{\frac{1}{2}} + x^{\frac{1}{3}}}\ dx$$

Evaluate the definite integral: $$\int_{5}^{10} \frac{dx}{(x-1)(x-2)}$$

The direction cosines $\ell, m, n$ of the line $\frac{x+2}{2}=\frac{2y-5}{3} ; z=-1$ are

Evaluate the integral: $$\int [\sin(\log x) + \cos(\log x)]\ dx$$

If $A = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then $\text{adj}\ A =$

If $\bar{a} = \hat{i} + 2\hat{j} - 3\hat{k}$, $\bar{b} = 3\hat{i} - \hat{j} + 2\hat{k}$, $\bar{c} = \hat{i} + 3\hat{j} + \hat{k}$ and $\bar{a} + \lambda\bar{b}$ is perpendicular to $\bar{c}$, then $\lambda =$

The abscissa of the points, where the tangent to the curve $y = x^3 - 3x^2 - 9x + 5$ is parallel to $X$ axis are

If $p$ is the length of the perpendicular from origin to the line whose intercepts on the axes are $a$ and $b$, then $\frac{1}{a^2} + \frac{1}{b^2} =$

The differential equation of the family of circles touching $y$-axis at the origin is

The mean of five observations is 4 and their variance is 5.2. If three of these observations are 1, 2 and 6, then the other two are

If $(\text{m}+3\text{n})(3\text{m}+\text{n})=4\text{h}^2$, then the acute angle between the lines represented by $\text{m} \text{x}^2+2\text{h}\text{xy}+\text{n}\text{y}^2=0$ is

Consider the following differential equations:
I: $y' = \frac{y+x}{x}$
II: $y' = \frac{x^2+y}{x^3}$
III: $y' = \frac{2xy}{y^2-x^2}$
Which of the statements below is true regarding these equations?
S1: Differential equations given by I and II are homogeneous differential equations.
S2: Differential equations given by II and III are homogeneous differential equations.
S3: Differential equations given by I and III are homogeneous differential equations.