If \(a _{1}(>0), a _{2}, a _{3}, a _{4}, a _{5}\) are in a G.P., \(a _{2}+ a _{4}=2 a _{3}+1\) and \(3 a _{2}+ a _{3}=2 a _{4}\), then \(a _{2}+ a _{4}+2 a _{5}\) is equal to

For two non-zero complex number \(z_1\) and \(z_2\), if \(\operatorname{Re}\left(z_1 z_2\right)=0\) and \(\operatorname{Re}\left(z_1+z_2\right)=0\), then which of the following are possible ? \((A)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((B)\) \(\operatorname{Im}\left(z_1\right) < 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((C)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) < 0\) \((D)\) \(\operatorname{Im}\left( z _1\right) < 0\) and \(\operatorname{Im}\left( z _2\right) < 0\) Choose the correct answer from the options given below :

Let \(\alpha\) and \(\beta\) be the roots of the equation \(\mathrm{x}^{2}-\mathrm{x}-1=0 .\) If \(\mathrm{p}_{\mathrm{k}}=(\alpha)^{\mathrm{k}}+(\beta)^{\mathrm{k}}, \mathrm{k} \geq 1,\) then which one of the following statements is not true?

If \(\dfrac{\pi}{4} + \displaystyle\sum_{p=1}^{11} \tan^{-1}\left(\dfrac{2^{p-1}}{1 + 2^{2p-1}}\right) = \alpha\), then \(\tan\alpha\) is equal to __________.

The number of terms in the expansion of \((1 +x)^{101}  (1 +x^2 - x)^{100}\) in powers of \(x\) is

Suppose a differentiable function \(f ( x )\) satisfies the identity \(f(x+y)=f(x)+f(y)+x y^{2}+x^{2} y\) for all real \(x\) and \(y .\) If \(\lim \limits_{x \rightarrow 0} \frac{f(x)}{x}=1,\) then \(f^{\prime}(3)\) is equal to 

Let the line \(\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}\) intersect the lines \(\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}\) and \(\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}\) at the points \(A\) and \(B\) respectively. Then the distance of the mid-point of the line segment \(A B\) from the plane \(2 x-2 y+z=14\) is

If the value of the integral \(\int \limits_{0}^{\frac{1}{2}} \frac{x^{2}}{\left(1-x^{2}\right)^{3 / 2}} d x\) is \(\frac{ k }{6},\) then \(k\) is equal to

The sum of all values of \( \alpha \), for which the shortest distance between the lines \( \frac{x+1}{\alpha}=\frac{y-2}{-1}=\frac{z-4}{-\alpha} \) and \( \frac{x}{\alpha}=\frac{y-1}{2}=\frac{z-1}{2\alpha} \) is \( \sqrt{2} \), is

If tangents are drawn to the ellipse \(x^2 + 2y^2 = 2\) at all points on the ellipse other than its four vertices than the mid points of the tangents intercepted between the coordinate axes lie on the curve

If for some \(\alpha\) and \(\beta\) in \(R,\) the intersection of the following three planes  \(x+4 y-2 z=1\) ; \(x+7 y-5 z=\beta\) ; \(x+5 y+\alpha z=5\) is a line in \(\mathrm{R}^{3},\) then \(\alpha+\beta\) is equal to

Let \(A\) and \(B\) be two invertible matrices of order \(3 \times 3\). If det \((ABA^T) = 8\) and \(det\,(AB^{-1}) = 8\), then \(det\, (BA^{-1} B^T)\) is equal to

The distance of the point \((-1,2,3)\) from the plane \(\vec{r} .(\hat{i}-2 \hat{j}+3 \hat{k})=10\) parallel to the line of the shortest distance between the lines \(\overrightarrow{ r }=(\hat{ i }-\hat{ j })+\lambda(2 \hat{i}+\hat{ k })\) and \(\overrightarrow{ r }=(2 \hat{ i }-\hat{ j })+\mu(\hat{i}-\hat{j}+\hat{ k })\) is :