If the rank of the matrix \(A=\left[\begin{array}{cccc}1 & 2 & 1 & -1 \\ -1 & 2 & 3 & 5 \\ 0 & 1 & k & k\end{array}\right]\)is 2 and \(\mathrm{k}\) is a real number, then \(\mathrm{k}\) is a root of the following quadratic equation

The equation of the tangent to the circle \(x^2+y^2-9=0\) making an angle \(60^{\circ}\) with the \(X\)-axis is

If $A=\frac{1}{7}\left[\begin{array}{ccc}3& -2& 6\\ -6& -3& 2\\ -2& 6& 3\end{array}\right]$, then

If \(f(x)=\left\{\begin{array}{cc}4 x-5, & x \leq 2 \\ x-k, & x>2\end{array}\right.\) then the value of ' \(k\) ' if \(\lim _{x \rightarrow 2} f(x)\) may exist is equal to

The equation of the circle passing through \((0,0)\) and which makes intercepts \(a\) and \(b\) on the coordinate axes is

The probability for a contractor to get a road contract is \(\frac{2}{9}\) and to get a building contract is \(\frac{5}{9}\), if the probability to get both the contract is \(\frac{1}{6}\), then what is the probability to get neither of these two contracts?

If \(\alpha, \beta, \gamma\) are the roots of \(2 x^3-2 x-1=0\), then \((\Sigma \alpha \beta)^2\) is equal to

Which of the following is an example for heterogeneous catalysis reaction?

Which of the following pair of solutions is isotonic?
A. \(18 \mathrm{~g} / \mathrm{L}\) of glucose of solution and \(6 \mathrm{~g} / \mathrm{L}\) of Urea solution
B. \(10 \mathrm{~g} / \mathrm{L}\) of glucose solution and \(10 \mathrm{~g} / \mathrm{L}\) of Urea solution
C. \(\quad 0.01 \mathrm{M} \mathrm{NaOH}\) solution and \(0.02 \mathrm{M}\) glucose solution
D. \(0.01 \mathrm{M} \mathrm{NaCl}\) solution and \(0.01 \mathrm{M}\) glucose solution
(Assume that \(\mathrm{NaCl}\) undergoes complete dissociation)

The vector equation of any plane passing through the line of intersection of the planes \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_1=\mathrm{q}_1\) and \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_2=\mathrm{q}_2\) is given by \(\overrightarrow{\mathrm{r}}\left(\overrightarrow{\mathrm{m}}_1+\lambda \overrightarrow{\mathrm{m}}_2\right)=\mathrm{q}_1+\lambda \mathrm{q}_2\) for \(\lambda \in \mathbb{R}\). The vector equation of a plane passing through the point \(2 \hat{i}-3 \hat{j}+\hat{k}\) and the line of intersection of the planes r. \((\hat{i}-2 \hat{j}+3 \hat{k})=5\) and \(\vec{r} \cdot(3 \hat{i}+\hat{j}-2 \hat{k})=7\)

The equation of the circle which passes through the origin and cuts orthogonally each of the circles \(x^2+y^2-6 x+8=0\) and \(x^2+y^2-2 x-2 y=7\) is

If $f^{\text{'}}\left(x\right)=a\sin x+b\cos x, f^{\text{'}}\left(0\right)=4, f\left(0\right)=3$ and $f\left(\frac{\mathrm{\pi }}{2}\right)=5,$ then $f\left(x\right)=$

\(\int x \operatorname{Cos}^{-1}\left(\frac{1-x^2}{1+x^2}\right) d x(x>0)=\)