If the pairs of straight lines \(x^2-2 p x y-y^2=0\) and \(x^2-2 q x y-y^2=0\) be such that each pair bisects the angle between the other pair, then

If the letters of the word "CAPITAL" are permuted in all possible ways and the words are arranged in a dictionary order, then the rank of the word "CAPITAL" is

If \(p=a_1 x+b_1 y+k_1=0, q=a_2 x+b_2 y+k_2=0\) and \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{k_1}{k_2}\), then the curve \(p+c q\) is

In \(\triangle A B C\), the sum of the lengths of two sides is \(x\) and the product of those lengths is \(y\). If \(c\) is the length of its third side and \(x^2-c^2=y\), then the circumradius of that triangle is

\(\mathbf{a}, \mathbf{b}, \mathbf{c}\) are non-coplanar vectors. If \(\begin{aligned} & \mathbf{a}+3 \mathbf{b}+4 \mathbf{c}=x(\mathbf{a}-2 \mathbf{b}+3 \mathbf{c})+y(\mathbf{a}+5 \mathbf{b}-2 \mathbf{c}) \\ & +z(6 \mathbf{a}+14 \mathbf{b}+4 \mathbf{c}), \text { then } x+y+z=\end{aligned}\)

Area of the triangle formed by the complex numbers \(\mathrm{z}, \mathrm{iz}\) and \(\mathrm{z}+\mathrm{iz}\) in the Argand diagram as vertices is

The origin is shifted to the point \((2,3)\) by translation of axes and then the coordinate axes are rotated about the origin through an angle \(\theta\) in the counter-clockwise sense. Due to this if the equation \(3 x^2+2 x y+3 y^2-18 x-22 y+\) \(50=0\) is transformed to \(4 x^2+2 y^2-1=0\), then the angle \(\theta=\)

Glyscosidic linkage in carbohydrates is called

At \(1000 \mathrm{~K}\), the equilibrium constant. \(K_C\) for the reaction \(2 \mathrm{NOCl}(g) \rightleftharpoons 2 \mathrm{NO}(g)+\mathrm{Cl}_2(g)\) is \(4.0 \times 10^{-6} \mathrm{~mol} \mathrm{~L}^{-1}\). The \(K_P\) (in bar) at the same temperature is \(\left(R=0.083 \mathrm{~L} \mathrm{bar} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right)\)

Which of the following are disproportionation reactions?
\((\mathrm{g}=\sigma, l=(ద, \mathrm{aq}=\text { జe, } \mathrm{s}=\text { ఘ) }\)
(A) \(2 \mathrm{NO}_{2(\mathrm{~g})}+2 \mathrm{NaOH}_{(\mathrm{aq})} \rightarrow \mathrm{NaNO}_{2(\mathrm{aq})}+\mathrm{NaNO}_{3(\mathrm{aq})}+\mathrm{H}_2 \mathrm{O}_{(l)}\)
(B) \(\mathrm{Cl}_{2(\mathrm{~g})}+2 \mathrm{NaOH}_{(\mathrm{aq})} \rightarrow \mathrm{NaClO}_{(\mathrm{aq})}+\mathrm{NaCl}_{(\mathrm{aq})}+\mathrm{H}_2 \mathrm{O}_{(\mathrm{l})}\)
(C) \(3 \mathrm{ClO}^{-} \rightarrow 2 \mathrm{Cl}^{-}+\mathrm{ClO}_3^{-}\)
(D) \(3 \mathrm{Mg}_{(\mathrm{s})}+\mathrm{N}_{2(\mathrm{~g})} \rightarrow \mathrm{Mg}_3 \mathrm{~N}_{2(\mathrm{~s})}\)

If \(\left|\begin{array}{cc}x^3+2 x^2+3 x-2 & x^2+2 x+4 \\ x^3-x^2-2 x-1 & 3 x^3-2 x^2+4 x-2\end{array}\right|\) \(=a x^6+b x^5+c x^4+d x^3+e x^2+f x+g\), then \(a+b+c+d+e+f\) is equal to

A body executes simple harmonic motion under the action of a force \(F_1\) with a time period \(\frac{4}{5} \mathrm{~s}\). If the force is changed to \(F_2\) it executes SHM with time period \(\frac{3}{5} \mathrm{~s}\). If both the forces \(F_1\) and \(F_2\) act simultaneously in the same direction on the body, its time period in seconds is