List of top Questions asked in MHT CET- 2014

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are unit vectors and $\theta$ is the angle between $\vec{a}$ and $\vec{c}$ and $\vec{a} + 2\vec{b} + 2\vec{c} = \vec{0}$, then $|\vec{a} \times \vec{c}|$ is equal to

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are three vectors with magnitudes $\sqrt{3}$, $1$, $2$ respectively, such that $\vec{a} \times (\vec{a} \times \vec{c}) + 3\vec{b} = \vec{0}$, if $\theta$ is the angle between $\vec{a}$ and $\vec{c}$, then $\sec^2 \theta$ is

The principal solutions of the equation $\sec x+\tan x=2\cos x$ are

If $a>0$ and $z = \frac{(1+i)^2}{a - i}$ ($i = \sqrt{-1}$) has magnitude $\frac{2}{\sqrt{5}}$, then $\overline{z}$ is

If $f'(x) = \tan^{-1}(\sec x + \tan x)$, $-\frac{\pi}{2}<x<\frac{\pi}{2}$ and $f(0) = 0$, then $f(1)$ is

$\int \frac{\sin 2x \left(1 - \frac{3}{2}\cos x\right)}{e^{\sin^2 x + \cos^3 x}} \, dx =$

If $\int \frac{\cos \theta}{5 + 7\sin \theta - 2\cos^2 \theta} \, d\theta = A \log_e |f(\theta)| + c$ (where $c$ is a constant of integration), then $\frac{f(\theta)}{A}$ can be

In $\triangle ABC$, with usual notations, $2ac \sin\left(\frac{A-B+C}{2}\right)$ is equal to

Let \( A=\begin{bmatrix}2 & -1 \\ 0 & 2\end{bmatrix} \). If \( B=I-{}^{3}C_{1}(\mathrm{adj}\,A)+{}^{3}C_{2}(\mathrm{adj}\,A)^{2}-{}^{3}C_{3}(\mathrm{adj}\,A)^{3} \), then the sum of all elements of the matrix \( B \) is

The angle between the lines, whose direction cosines \( l, m, n \) satisfy the equations \( l + m + n = 0 \) and \( 2l^{2} + 2m^{2} - n^{2} = 0 \), is

Let \( f:\mathbb{R}\rightarrow \mathbb{R} \) be a function such that \( f(x)=x^{3} + x^{2}f^{\prime}(1) + x f^{\prime\prime}(2) + 6 \) for \( x \in \mathbb{R} \), then \( f(2) \) equals

If $y = (\sin^{-1}x)^2 + (\cos^{-1}x)^2$, then $(1 - x^2)\,y'' - x\,y' = $

If $\triangle ABC$ is right angled at A, where $A\equiv(4,2,x)$, $B\equiv(3,1,8)$ and $C\equiv(2,-1,2)$, then the value of $x$ is

If $\sin(\theta-\alpha)$, $\sin~\theta$ and $\sin(\theta+\alpha)$ are in H.P., then the value of $\cos~2\theta$ is

The equation \( x^{3} + x - 1 = 0 \) has

If \( \vec{a}, \vec{b}, \vec{c} \) are three vectors such that \( |\vec{a}+\vec{b}+\vec{c}|=1 \), \( \vec{c}=\lambda(\vec{a}\times\vec{b}) \) and \( |\vec{a}|=\frac{1}{\sqrt{2}} \), \( |\vec{b}|=\frac{1}{\sqrt{3}} \), \( |\vec{c}|=\frac{1}{\sqrt{6}} \), then the angle between \( \vec{a} \) and \( \vec{b} \) is

Let \( \vec{a}, \vec{b}, \vec{c} \) be three vectors such that \( |\vec{a}|=\sqrt{3} \), \( |\vec{b}|=5 \), \( \vec{b}\cdot\vec{c}=10 \) and the angle between \( \vec{b} \) and \( \vec{c} \) is \( \frac{\pi}{3} \). If \( \vec{a} \) is perpendicular to the vector \( \vec{b}\times\vec{c} \), then \( |\vec{a}\times(\vec{b}\times\vec{c})| \) is equal to

The variance of 20 observations is 5. If each observation is multiplied by 2, then variance of resulting observations is

General solution of the differential equation $\cos x(1+\cos y)dx-\sin y(1+\sin x)dy=0$ is

If the line \( x - 2y = m \, (m \in \mathbb{Z}) \) intersects the circle \( x^{2} + y^{2} = 2x + 4y \) at two distinct points, then the number of possible values of \( m \) are

The solution set of the inequalities $4x+3y\le60$, $y\ge2x$, $x\ge3$, $x, y\ge0$ is represented by region

The statement $[(p\rightarrow q)\wedge\sim q]\rightarrow r$ is tautology, when $r$ is equivalent to

The negation of the statement "The number is an odd number if and only if it is divisible by 3."

Two cards are drawn successively with replacement from well shuffled pack of 52 cards, then the probability distribution of number of queens is

For an initial screening of an entrance exam, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is \( \frac{4}{5} \), then the probability that he is unable to solve less than two problems is