List of top Questions asked in MET- 2014

\( \int_{0}^{\pi/2} \frac{\sin x - \cos x}{1 + \sin x \cos x} \, dx \) is equal to

\( \lim_{x \to 0} \left(1^{\csc^2 x} + 2^{\csc^2 x} + \cdots + n^{\csc^2 x}\right)\sin^2 x \) is equal to

If \( x\sin(a+y) + \sin a \cos(a+y)=0 \), then \( \frac{dy}{dx} \) is equal to

The function \( f(x) = [x]\cos\left( \frac{2x-1}{2}\pi \right) \), where \( [\,\cdot\,] \) denotes the greatest integer function, is discontinuous at

The function \( f(x) = (x^2 - 1)|x^2 - 3x + 2| + \cos|x| \) is non-differentiable at

If $f(x+y)=f(x)f(y)$ for all $x,y$ and $f(15)=2,\ f'(0)=3$, then $f'(5)$ will be

The approximate value of $f(5.001)$, where $f(x)=x^3 - 7x^2 + 15$, is

What are the values of \( c \) for which Rolle’s theorem for the function \( f(x) = x^3 - 3x^2 + 2x \) in the interval \( [0,2] \) is verified?

The value of \( \lim_{x \to 0} \left( \frac{a^x + b^x + c^x}{3} \right)^{\frac{2}{x}} \), \( (a,b,c>0) \) is

The range of \( f(x) = \sec\left( \frac{\pi}{4} \cos^2 x \right), \; -\infty<x<\infty \) is

The value of $\lambda$ and $\mu$ for which the system of equations $x+y+z=6$, $x+2y+3z=10$ and $x+2y+\lambda z=\mu$ have no solution, are

In a trial, the probability of success is twice the probability of failure. In six trials, the probability of at least four successes will be

If \( \sin^{-1} a + \sin^{-1} b + \sin^{-1} c = \pi \), then the value of \( a\sqrt{1-a^2} + b\sqrt{1-b^2} + c\sqrt{1-c^2} \) will be

If the integers $m$ and $n$ are chosen at random between 1 and 100, then the probability that a number of the form $7^m + 7^n$ is divisible by 5, equals

If \( a, b, c \) are \( p^{th}, q^{th} \) and \( r^{th} \) terms of a G.P, then the vectors \( \log a \,\hat{i} + \log b \,\hat{j} + \log c \,\hat{k} \) and \( (q-r)\hat{i} + (r-p)\hat{j} + (p-q)\hat{k} \) are

The distance of the point \( (1,-5,9) \), from the plane \( \vec{r}\cdot(\hat{i}-\hat{j}+\hat{k}) = 5 \) measured along the line \( \vec{r} = \hat{i} + \hat{j} + \hat{k} \) is

If $|a|=1, |b|=4, a\cdot b = 2$ and $c = 2a \times b - 3b$, then the angle between $b$ and $c$ is

The line joining the points \( (1,1,2) \) and \( (3,-2,1) \) meets the plane \( 3x + 2y + z = 6 \) at the point

The image of the point with position vector \( \hat{i} + 3\hat{k} \) in the plane \( \vec{r}\cdot(\hat{i} + \hat{j} + \hat{k}) = 1 \) is

The equation of the curve for which the square of the ordinate is twice the rectangle contained by the abscissa and the intercept of the normal on the $x$-axis passing through $(2,1)$, is

The area bounded by \( y = x|\sin x| \) and X-axis between \( x = 0 \) and \( x = 2\pi \) is

\( \int \frac{dx}{\sin x - \cos x + \sqrt{2}} \) is equal to

Evaluate \( \int_{0}^{\frac{3\pi}{2}} \sin\left( \left\lfloor \frac{2x}{\pi} \right\rfloor \right) dx \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function.

The real value of \( m \) for which the substitution \( y = u^m \) will transform the differential equation \( 2x^4 y \frac{dy}{dx} + y^4 = 4x^6 \) into a homogeneous equation is

The area enclosed by the curves \( |y + x| \le 1 \), \( |y - x| \le 1 \) and \( 2x^2 + 2y^2 = 1 \) is