Let \(f(x)=(x+3)^2(x-2)^3, x \in[-4,4]\). If \(M\) and \(m\) are the maximum and minimum values of \(f\), respectively in \([-4,4]\), then the value of \(M-m\) is :

The area (in \(sq. units\)) of the smaller of the two circles that touch the parabola, \(y^2 = 4x\) at the points \((1, 2)\) and the axis is

The sum of the co-efficients of all odd degree terms in the expansion of  \({\left( {x + \sqrt {{x^3} - 1} } \right)^5} + {\left( {x - \sqrt {{x^3} - 1} } \right)^5},\left( {x > 1} \right)\) 

Let the function \(f(x)=\left(x^2+1\right)\left|x^2-a x+2\right|+\cos |x|\) be not differentiable at the two points \(x=\alpha=2\) and \(x=\beta\). Then the distance of the point \((\alpha, \beta)\) from the line \(12 x+5 y+10=0\) is equal to :

Let a circle \(C:(x-h)^{2}+(y-k)^{2}=r^{2}, k>0\), touch the \(x\)-axis at \((1,0)\). If the line \(x + y =0\) intersects the  circle \(C\) at \(P\) and \(Q\) such that the length of the chord  \(PQ\) is \(2\) , then the value of \(h + k + r\) is equal to

The circle passing through \(( 1, -2)\) and touching the axis of \(x\) at \((3, 0)\) also passes through the point

Let f and g be functions satisfying \( f(x+y)=f(x)f(y), f(1)=7 \) and \( g(x+y)=g(xy), g(1)=1 \) for all \( x, y\in\mathbb{N} \). If \( \sum_{x=1}^{n}(\frac{f(x)}{g(x)})=19607, \) then n is equal to:

The value of \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1+\sin ^{2} \mathrm{x}}{1+\pi^{\sin \mathrm{x}}}\right)\, \mathrm{dx}\) is

The area (in sq. units) of the parallelogram whose diagonals are along the vectors \(8\hat i - 6\hat j\) and \(3\hat i + 4\hat j - 12\hat k\) , is

If a circle \(C\) passing through the point \((4, 0)\) touches the circle \(x^2 + y^2 + 4x -6y = 12\) externally at the point \((1, -1)\), then the radius of \(C\) is

For two non-zero complex number \(z_1\) and \(z_2\), if \(\operatorname{Re}\left(z_1 z_2\right)=0\) and \(\operatorname{Re}\left(z_1+z_2\right)=0\), then which of the following are possible ? \((A)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((B)\) \(\operatorname{Im}\left(z_1\right) < 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((C)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) < 0\) \((D)\) \(\operatorname{Im}\left( z _1\right) < 0\) and \(\operatorname{Im}\left( z _2\right) < 0\) Choose the correct answer from the options given below :

If \(\int_{0}^{\pi}\left(\sin ^{3} x\right) e^{-\sin ^{2} x} d x=\alpha-\frac{\beta}{e} \int_{0}^{1} \sqrt{t} e^{t} d t\), then \(\alpha+\beta\) is equal to \(....\)

Let \(y=y(x)\) be the solution of the differential equation: \(\dfrac{dy}{dx}+\left(\dfrac{6x^2+(3x^2+2x^3+4)e^{-2x}}{(x^3+2)(2+e^{-2x})}\right)y=2+e^{-2x}\), \(x \in (-1,2)\), satisfying \(y(0)=\dfrac{3}{2}\). If \(y(1)=\alpha(2+e^{-2})\), then \(\alpha\) is equal to: