The  real number \(k\) for which  the equation, \(2{x^2} + 3x + k = 0\) has two distinct real roots in \([0, 1]\) 

Consider the following two statements : Statement \(I\) : For any two non-zero complex numbers \(\mathrm{z}_1, \mathrm{z}_2\) \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)\) and Statement \(II\) : If \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are three distinct complex numbers and a, b, c are three positive real numbers such that \(\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}\), then \(\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1\) Between the above two statements,

Let \(f: R \rightarrow R\) satisfy the equation \(f(x+y)=f(x) \cdot f(y)\) for all \(x, y \in R\) and
\(f ( x ) \neq 0\) for any \(x \in R .\) If Ihe function \(f\) is differentiable at \(x =0\) and \(f^{\prime}(0)=3,\) then \(\lim _{h \rightarrow 0} \frac{1}{h}(f(h)-1)\) is equal to ....... .

Statement \(- 1:\) The function \(x^2 (e^x + e^{-x})\) is increasing for all \(x > 0.\) Statement \(-2:\) The functions \(x^2e^x\) and \(x^2e^{-x}\) are increasing for all \(x > 0\) and the sum of two increasing functions in any interval \((a, b)\) is an increasing function in \((a, b).\)

If \(\alpha = \displaystyle\int_0^{2\sqrt{3}} \log_2(x^2 + 4)\,dx + \displaystyle\int_2^4 \sqrt{2^x - 4}\,dx\), then \(\alpha^2\) is equal to _______.

Let S be the set of the first 11 natural numbers. Then the number of elements in \( A=\{B\subseteq S:n(B)\ge2 \) and the product of all elements of B is even} is ___ .

A ray of light through \((2,1)\) is reflected at a point \(P\) on the \(y\) - axis and then passes through the point \((5,3)\). If this reflected ray is the directrix of an ellipse with eccentrieity \(\frac{1}{3}\) and the distance of the nearer focus from this directrix is \(\frac{8}{\sqrt{53}}\), then the equation of the other directrix can be :

For \(p\,>\,0\), a vector \(\vec{v}_{2}=2 \hat{i}+(p+1) \hat{j}\) is obtained by rotating the vector \(\vec{v}_{1}=\sqrt{3} p \hat{i}+\hat{j}\) by an angle \(\theta\) about origin in counter clockwise direction. If \(\tan \theta=\frac{(\alpha \sqrt{3}-2)}{4 \sqrt{3}+3}\), then the value of \(\alpha\) is equal to \(....\)

Consider the three planes  \(P_{1}: 3 x+15 y+21 z=9\)  ;  \(P _{2}: x -3 y - z =5,\) and ;  \(P_{3}: 2 x+10 y+14 z=5\). Then, which one of the following is true ?

Let \(N\) be the set of natural numbers and two functions \(f\) and \(g\) be defined as \(f\), \(g : N \to N\) such that \(f\left( n \right) = \left\{ \begin{gathered}
  \frac{{n + 1}}{2}\,\,\,\,\,\,\,\,\,\,\,{\text{if n is odd}} \hfill \\
  \frac{n}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{if n is even}} \hfill \\ 
\end{gathered}  \right.\) and \(g(n) = n - (-1)^n\). Then \(fog\) is

Let the system of linear equations  \(x+y+k z=2\) ;  \(2 x+3 y-z=1\) ; \(3 x+4 y+2 z=k\) , have infinitely many solutions. Then the system \(( k +1) x +(2 k -1) y =7\) ; \((2 k +1) x +( k +5) y =10 \text { has : }\)

If \(P ( h , k )\) be point on the parabola \(x =4 y ^2\), which is nearest to the point \(Q(0,33)\), then the distance of \(P\) from the directrix of the parabola \(y ^2=4( x + y )\) is equal to :

Let \(f: \mathbf{R}-\{0\} \rightarrow(-\infty, 1)\) be a polynomial of degree 2, satisfying \(f(x) f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)\). If \(f(K)=-2 K\), then the sum of squares of all possible values of \(K\) is :