Let the roots of the equation \(E_1 \equiv x^3+x^2+l x+n=0\) be \(x_i,(i=1,2,3)\) and the roots of \(E_2 \equiv x^3+a x^2+b x+c=0\) be \(\frac{x_i-1}{2}\). If the equation \(E_2=0\) is a equation of class one, then the roots of these two equations excluding the common roots are

The mean of four observations is 3 . If the sum of the squares of these observations is 48 , then their standard deviation is

Suppose \(\alpha, \beta, \gamma \quad\) are roots of \(x^3+x^2+2 x+3=0\). If \(f(x)=0\) is a cubic polynomial equation whose roots are \(\alpha+\beta, \beta+\gamma, \gamma+\alpha\), then \(f(x)\) is equal to

If a circle with its centre at the focus of the parabola \(y^2=2 p x\) is such that it touches the directrix of the parabola, then a point of intersection of the circle and the parabola is

If \(\omega\) is a complex cube root of unity and \(x=\omega^2-\omega+2\) then

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\) Assertion (A) : The identity \(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for \(\vec{a}\). Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\), \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\) Which of the following is correct?

If \(y=\tan ^{-1}\left(\frac{\sqrt{1+a^2 x^2}-1}{a x}\right)\), then \(\left(1+a^2 x^2\right){y^{\prime}}^{\prime}+2 a^2 x y^{\prime}\) is equal to

If the amplitude of \(z-2-3 i\) is \(\pi / 4\), then the locus of \(z=x+i y\) is

Consider the following electrode processes of a cell, \[ \mathrm{Cl}^{-} \rightarrow \frac{1}{2} \mathrm{Cl}_2+e^{-} \quad\left[\mathrm{MCl}+e^{-} \rightarrow M+\mathrm{Cl}^{-}\right] \] If EMF of this cell is \(-1.140 \mathrm{~V}\) and \(E^{\circ}\) value of the cell is \(-0.55 \mathrm{~V}\) at \(298 \mathrm{~K}\), the value of the equilibrium constant of the sparingly soluble salt \(M \mathrm{Cl}\) is in the order of

Two cars \(A\) and \(B\) are moving with speeds \(v_A=120 \mathrm{~km} / \mathrm{h}\) and \(v_B=50 \mathrm{~km} / \mathrm{h}\) respectively in the directions as indicated by the arrow in the figure below. What is the relative speed of the car \(B\) with respect to car \(A\) ?

Number of solutions of the equation $\sin x-\sin 2x+\sin 3x=2{\cos }^{2}x-2\cos x$ in $\left(0,\pi \right)$ is

For reaction \(2 \mathrm{~A}+\mathrm{B} \longrightarrow \mathrm{D}+\mathrm{E}\), following mechanism has been proposed \(\mathrm{A}+\mathrm{B} \longrightarrow \mathrm{C}+\mathrm{D}\) (Is slow) and \(\mathrm{A}+\mathrm{C} \rightarrow \mathrm{E}\) (Is fast).

The area of the region bounded by the curves \(x=y^2-2\) and \(x=y\) is