List of top Mathematics Questions asked in MHT CET

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

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

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

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

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

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

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

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 $\lambda =$

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.

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

The vector equation of the line whose Cartesian equations are $y = 2$ and $4x - 3z + 5 = 0$ is

$\int \frac{2 x^2-1}{x^4-x^2-20} d x=$

If $x = a(t + \sin t), y = a(1 - \cos t)$, then $\frac{dy}{dx} =$

Let $A=[a, b, c, d], B=[1,2,3]$. Relation $R_1, R_2, R_3, R_4$ are as follows :
$R_1=[(a, 1), (b, 2), (c, 1), (d, 2)]$
$R_2=[(a, 1), (b, 1), (c, 1), (d, 1)]$
$R_3=[(a, 2), (b, 3), (c, 2), (d, 2)]$
$R_4=[(a, 1), (b, 2), (a, 2), (d, 3)]$, then

If $\int_2^e \left[ \frac{1}{\log x} - \frac{1}{(\log x)^2} \right] dx = a + \frac{b}{\log 2}$, then

The population of a city increases at a rate proportional to the population at that time. If the population of the city increase from 20 lakhs to 40 lakhs in 30 years, then after another 15 years the population is

Two unbiased dice are thrown. Then the probability that neither a doublet nor a total of 10 will appear is

If $\cos x = \frac{24}{25}$ and $x$ lies in first quadrant, then $\sin \frac{x}{2} + \cos \frac{x}{2} =$

If $A^{-1}=\left[\begin{array}{ccc}3 & 2 & 6 \\ 1 & 1 & 2 \\ 2 & 5 & 5\end{array}\right]$, then $A=$

If $A^{-1} = \left[\begin{array}{cc}2 & -3 \\ -1 & 2\end{array}\right]$ and $B^{-1} = \left[\begin{array}{cc}1 & 0 \\ -3 & 1\end{array}\right]$, then $(AB)^{-1} =$

If $\vec{r} = -4\hat{i} - 6\hat{j} - 2\hat{k}$ is a linear combination of the vectors $\vec{a} = -\hat{i} + 4\hat{j} + 3\hat{k}$ and $\vec{b} = -8\hat{i} - \hat{j} + 3\hat{k}$, then

The joint equation of pair of lines through the origin and making an equilateral triangle with the line $y=3$ is

Negation of $(p \land q) \rightarrow (\sim p \lor r)$ is

The area bounded by the parabola $y = x^2$ and the line $y = x$ is