Let \(\alpha_1\) and \(\alpha_2\) be the ordinates of two points \(A\) and \(B\) on a parabola \(y^2=4 a x\) and let \(\alpha_3\) be the ordinate of the point of intersection of its tangents at \(A\) and \(B\). Then, \(\alpha_3-\alpha_2=\)

If \(f(x)=\left\{\begin{array}{cl}1+6 x-3 x^2, & x \leq 1 \\ x+\log _2\left(b^2+7\right), & x>1\end{array}\right.\) is continuous at all real \(x\), then \(b=\)

The set of solutions of the equation \((\sqrt{3}-1) \sin \theta+(\sqrt{3}+1) \cos \theta=2\) is

If \(\int \sqrt{\frac{2}{1+\sin x}} d x=2 \log |A(x)-B(x)|+C\) and \(0 \leq x \leq \frac{\pi}{2}\) then \(B\left(\frac{\pi}{4}\right)=\)

Let \(\vec{a}\) and \(\vec{b}\) be two non-collinear vectors of unit modulus.
If \(\vec{u}=\vec{a}-(\vec{a} \cdot \vec{b}) \vec{b}\) and \(\vec{v}=\vec{a} \times \vec{b}\), then \(|\vec{v}|=\)

If a function \(f\) defined by \(f(x)=\left\{\begin{array}{cc}\frac{1-\cos 4 x}{x^2}, & x < 0 \\ a, & x=0 \\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x>0\end{array}\right.\) is continuous at \(\mathrm{x}=0\), then \(\mathrm{a}=\)

If a point \(\mathrm{C}\) divides the line segment joining the points with the position vectors \(2 \hat{i}-3 \hat{j}+2 \hat{k}\) and \(3 \hat{i}-\hat{j}-2 \hat{k}\) in the ratio \(2: 3\), then the distance of \(\mathrm{C}\) from the point with position vector \(2 \hat{i}-\hat{j}+\hat{k}\) is

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) \(3 \mathrm{Mg}_{(\mathrm{s})}+\mathrm{N}_{2(\mathrm{~g})} \rightarrow \mathrm{Mg}_3 \mathrm{~N}_{2(\mathrm{~s})}\)

The modulus of the complex number \(\left(\frac{2+i \sqrt{5}}{2-i \sqrt{5}}\right)^{10}+\left(\frac{2-i \sqrt{5}}{2+i \sqrt{5}}\right)^{10}\) is

Two sides of a triangle are given. If the area of the triangle is maximum, then the angle between the given sides is

One mole of a gas having \(\gamma=\frac{7}{5}\) is mixed with one mole of a gas having \(\gamma=\frac{4}{3}\) The value of \(\gamma\) for the mixture is ( \(\gamma\) is the ratio of the specific heats of the gas)

If \(\alpha\) is the common root of the quadratic equations \(x^2-5 x+4 a=0\), \(x^2-2 a x-8=0\), where \(a \in \mathbb{R}\), then the value of \(\alpha^4-\alpha^3+68\) is

The wavelength of the first line of Balmer series of hydrogen atom is \(\lambda\), what will be the wavelength of the same line in doubly ionized lithium?