If \(a x^2-34 x y-5 y^2+2 x+26 y-5=0\) represents a pair of straight lines, then the value of \(a\) is

If \(\left|z_1\right|=1,\left|z_2\right|=2,\left|z_3\right|=3\) and \(\left|9 z_1 z_2+4 z_1 z_3+z_2 z_3\right|=12\), then the value of \(\left|z_1+z_2+z_3\right|\) is

If \(k \in R\) is such that the equation \(2 \cosh ^2 x=3 \sinh x+k\) has no real solution, then which of the following is correct?

Let $A=\left[\begin{array}{ccc}-2& x& 1\\ x& 1& 1\\ 2& 3& -1\end{array}\right]$. If the roots of the equation $detA=0$ are $l,m$, then $l^3-m^3=$

If \(\vec{a}=2 \hat{i}-\hat{j}+3 \hat{k}, \vec{b}=-3 \hat{i}+5 \hat{j}-4 \hat{k}\) and \(\vec{c}=6 \hat{i}-4 \hat{j}+5 \hat{k}\), then \((\vec{a} \times \vec{b}) \cdot(\vec{b} \times \vec{c})=\)

The number of different ways of preparing a garland by using 6 distinct white roses and 6 distinct red roses such that no two red roses come together is

The equation of the circle concentric with the circle \(x^2+y^2-6 x+12 y+15=0\) and of double its area is

From the following, identify the reactions which are responsible for depletion of ozone layer.
\(\begin{aligned}
& \mathrm{Cl} \dot{\mathrm{O}}(g)+\mathrm{NO}_2(g) \rightarrow \mathrm{ClONO}_2(g) \\
& \mathrm{CF}_2 \mathrm{Cl}_2(g) \xrightarrow{\text { uv }} \mathrm{Cl}(g)+\dot{\mathrm{CF}}_2 \mathrm{Cl}(g) \\
& \dot{\mathrm{Cl}}(g)+\mathrm{CH}_4(g) \rightarrow \mathrm{HCl}(g)+\dot{\mathrm{C}} \mathrm{H}_3(g) \\
& \dot{\mathrm{C}} \mathrm{l}(\mathrm{g})+\mathrm{O}_3(g) \rightarrow \mathrm{Cl} \dot{\mathrm{O}}(g)+\mathrm{O}_2(g) \\
& \mathrm{Cl} \dot{\mathrm{O}}(g)+\mathrm{O}(g) \rightarrow \dot{\mathrm{C}} \mathrm{l}(g)+\mathrm{O}_2(g)
\end{aligned}\)

If \(A\) and \(B\) are both \(3 \times 3\) matrices, then which of the following statements are true?
(i) \(\mathrm{AB}=0 \Rightarrow \mathrm{~A}=0\) or \(\mathrm{B}=0\)
(ii) \(\mathrm{AB}=\mathrm{I}_3 \Rightarrow \mathrm{~A}^{-1}=\mathrm{B}\)
(iii) \((A-B)^2=A^2-2 A B+B^2\)

If the lines represented by the equation \(4 x y+6 x-8 y+c=0\) form a rectangle with the coordinate axis then the area of the rectangle (in sq. units) is

If \(\omega\) is a complex cube root of unity, the \(\left[\frac{51+73 \omega+87 \omega^2}{73+87 \omega+51 \omega^2}+\frac{51+73 \omega+87 \omega^2}{87+51 \omega+73 \omega^2}\right]=\)

If \(y=x^x+x^7+7^x+7^7\), then \(\frac{d y}{d x}=\)

$7.8 \mathrm{g}$ of a compound having molecular formula ${\mathrm{C}}_6{\mathrm{H}}_6$, on reacting with ${\mathrm{CH}}_3\mathrm{COCl}/{\mathrm{AlCl}}_3$ gives $8.4 \mathrm{g}$ of a product which has molecular formula ${\mathrm{C}}_8{\mathrm{H}}_8\mathrm{O}$. Calculate the percentage yield of the product ${\mathrm{C}}_8{\mathrm{H}}_8\mathrm{O}$. (Given atomic weights of $\mathrm{H}, \mathrm{C}$ and $\mathrm{O}$ respectively are $1,12$ and $16$)