List of top Mathematics Questions asked in MHT CET

$\int_{\pi/4}^{\pi/2} 2\sin^{-4} x dx = \_\_\_\_\_\_.$
Note: The initial OCR showed "$23.4 \frac{/2}{/4}$". The "4" was a misread coefficient. The mathematical evaluation of the options indicates a coefficient of 2 is present in the intended question.

The lines $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j})$ and $\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + 3\hat{k})$ are \dots}

The altitude through vertex A of $\triangle ABC$ with position vectors of points A, B, C as $\vec{a}, \vec{b}, \vec{c}$ respectively is ______.

If $\tan^{-1}(x + 1) + \tan^{-1} x + \tan^{-1}(x - 1) = \tan^{-1} 3$, then for $x < 0$ the value of $500x^4 + 270x^2 + 997 = \dots$

$\int \frac{x^4 \cos(\tan^{-1} x^5)}{1 + x^{10}} dx$ equals ______.

The length of the perpendicular drawn from the origin on the normal to the curve $x^2 + 2xy - 3y^2 = 0$ at the point $(2, 2)$ is ______.

There are 11 points in a plane of which 5 points are collinear. Then the total number of distinct quadrilaterals with vertices at these points is ______.

If $f(x) = \log(1 + x) - \frac{2x}{2 + x}$, then $f(x)$ is increasing in ______.

Let $f : \mathbb{R} - \{2\} \rightarrow \mathbb{R} - \{1\}$ defined by $f(x) = \frac{x-3}{x-2}$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ defined by $g(x) = 3x - 2$, then sum of all values of $x$ for which $f^{-1}(x) + g^{-1}(x) = 19/6$ is ______.

The angle $\theta$, at which the curves $y = 3^x$ and $y = 7^x$ intersect, is given by ______.

The circumradius of a triangle whose sides are 10 units, 8 units and 6 units is ______.

If $\sqrt{y} - \sqrt{y} - \dots = \sqrt{x} + \sqrt{x} + \dots$ then $dy/dx = \dots$

The function $f(x) = x^3 - 6x^2 + ax + b$ satisfies the conditions of Rolle's theorem in $[1, 3]$. Then the values of $a$ and $b$ are respectively \dots

Let $\vec{a} = \hat{i} + \hat{j} - \hat{k}$ and $\vec{c} = 5\hat{i} - 3\hat{j} + 2\hat{k}$ and if $\vec{b} \times \vec{c} = \vec{a}$ then $|\vec{b}|$ = ______.

If $x = \sin t$ and $y = \sin pt$, then the value of $(1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + p^2 y = \dots$

With usual notations in $\triangle ABC$, if $\angle B = \pi/2$, and $\tan A, \tan C$ are roots of equation $px^2 + qx + r = 0, p \neq 0$, then ______.

The straight line passing through $(-3, 6)$ and midpoint of the line segment joining the points $(4, -5)$ and $(-2, 9)$ have inclination ______.

The eccentricity of the hyperbola which passes through the points $(3, 0)$ and $(3\sqrt{2}, 2)$ is \dots

The general solution of differential equation $(y^2 - x^2)dx = xy dy$ ($x \neq 0$) is ______.

$\cos^4(\pi/8) + \cos^4(3\pi/8) + \cos^4(5\pi/8) + \cos^4(7\pi/8) = \dots$

The general solution of the differential equation $\frac{dy}{dx} = \cot x \cdot \cot y$ is ______.

If $\tan(\pi \cos \theta) = \cot(\pi \sin \theta)$, then $\sin\left(\frac{\pi}{4} + \theta\right) = $ ______.

The line MN whose equation is $x - y - 2 = 0$ cuts the X-axis at M and coordinates of N are (4, 2). The line MN is rotated about M through $45^{\circ}$ in anticlockwise direction. The equation of the line MN in the new position is ______.

The least distance of the point A(10, 7) from the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ is length of seg AM. If MM' is the diameter of the circle, then the lengths of AM and AM' are respectively ______, ______ units.

If the foot of the perpendicular drawn from the origin to a plane is P(-1, -1, 2), then the equation of the plane is ______.