If \({\left( {2 + \frac{x}{3}} \right)^{55}}\) is expanded in the ascending powers of \(x\) and the coefficients of powers of \(x\) in two consecutive terms of the expansion are equal, then these terms are

Let \(z_{1}\) and \(z_{2}\) be two complex numbers such that \(\arg \left(\mathrm{z}_{1}-\mathrm{z}_{2}\right)=\frac{\pi}{4}\) and \(\mathrm{z}_{1}, \mathrm{z}_{2}\) satisfy the equation \(|z-3|=\operatorname{Re}(z) .\) Then the imaginary part of \(z_{1}+z_{2}\) is equal to ..... .

The plane passing through the points \((1,2,1),(2,1,2)\) and parallel to the line, \(2 x=3 y, z=1\) also passes through the point 

Let \(\displaystyle\int_{-2}^{2} (|\sin x| + [x \sin x])\,dx = 2(3 - \cos 2) + \beta\), where \([\cdot]\) is the greatest integer function. Then \(\beta \sin\left(\dfrac{\beta}{2}\right)\) equals:

If \(a=\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{2 n}{n^{2}+k^{2}}\) and \(f(x)=\) \(\sqrt{\frac{1-\cos x}{1+\cos x}}, x \in(0,1)\), then.

The value of \(\lim\limits _{x \rightarrow 1} \frac{\left(x^{2}-1\right) \sin ^{2}(\pi x)}{x^{4}-2 x^{3}+2 x-1}\) is equal to

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous function satisfying \(f(0)=1\) and \(f(2 \mathrm{x})-f(\mathrm{x})=\mathrm{x}\) for all \(\mathrm{x} \in \mathbb{R}\). If \(\lim _{n \rightarrow \infty}\left\{f(x)-f\left(\frac{x}{2^n}\right)\right\}=G(x)\), then \(\sum_{r=1}^{10} G\left(r^2\right)\) is equal to

Number of integral terms in the expansion of  \(\left\{7^{\left(\frac{1}{2}\right)}+11^{\left(\frac{1}{6}\right)}\right\}^{824}\)  is equal to ...........

Let \(L\) be a tangent line to the parabola \(y^{2}=4 x-20\) at \((6,2)\) . If \(L\) is also a tangent to the ellipse \(\frac{ x ^{2}}{2}+\frac{ y ^{2}}{ b }=1,\) then the value of \(b\) is equal to ..... .

Let the lines \(L _{1}: \overrightarrow{ r }=\lambda(\hat{ i }+2 \hat{ j }+3 \hat{ k }), \lambda \in R\) \(L _{2}: \overrightarrow{ r }=(\hat{ i }+3 \hat{ j }+\hat{ k })+\mu(\hat{ i }+\hat{ j }+5 \hat{ k }) ; \mu \in R\) intersect at the point \(S\). If a plane \(ax + by - z\) \(+d=0\) passes through \(S\) and is parallel to both the lines \(L _{1}\) and \(L _{2}\), then the value of \(a + b +\) \(d\) is equal to

The least positive integral value of \(\alpha\), for which the angle between the vectors \(\alpha \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \mathrm{k}\) and \(\alpha \hat{\mathrm{i}}+2 \alpha \hat{\mathrm{j}}-2 \mathrm{k}\) is acute, is

Consider two vectors \(\overrightarrow{\mathrm{u}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}\) and \(\overrightarrow{\mathrm{v}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}}, \lambda \gt 0\). The angle between them is given by \(\cos ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)\). Let \(\overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{v}}_1+\overrightarrow{\mathrm{v}}_2\), where \(\overrightarrow{\mathrm{v}}_1\) is parallel to \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}_2\) is perpendicular to \(\overrightarrow{\mathrm{u}}\). Then the value \(\left|\overrightarrow{\mathrm{v}}_1\right|^2+\left|\overrightarrow{\mathrm{v}}_2\right|^2\) is equal to

If \(m\) is the \(A.M\) of two distinct real numbers \( l\)  and \(n (l,n>1) \) and  \(G_1, G_2\) and \(G_3\) are three geometric means between  \(l\) and \(n\) then \(G_1^4 + 2G_2^4 + G_3^4\) equals :