Let \((\lambda, 2,1)\) be a point on the plane which passes through the ponit \((4,-2,2) .\) If the plane is perpendicular to the line joining the points \((-2,-21,29)\) and \((-1,-16,23),\) then \(\left(\frac{\lambda}{11}\right)^{2}-\frac{4 \lambda}{11}-4\) is equal to

Let a unit vector which makes an angle of \(60^{\circ}\) with \(2 \hat{i}+2 \hat{j}-\hat{k}\) and an angle of \(45^{\circ}\) with \(\hat{i}-\hat{k}\) be \(\overrightarrow{\mathrm{C}}\). Then \(\overrightarrow{\mathrm{C}}+\left(-\frac{1}{2} \hat{\mathrm{i}}+\frac{1}{3 \sqrt{2}} \hat{\mathrm{j}}-\frac{\sqrt{2}}{3} \hat{\mathrm{k}}\right)\) is :

If \({\cos ^{ - 1}}\left( {\frac{2}{{3x}}} \right) + {\cos ^{ - 1}}\,\left( {\frac{3}{{4x}}} \right) = \frac{\pi }{2},\,x > \frac{3}{4}\) then \(x\) is equal to

Let \(A\) be the sum of the first \(20\) terms and \(B\) be the sum of the first \(40\) terms of the series  \({1^2} + 2 \cdot {2^2} + {3^2} + 2 \cdot {4^2} + {5^2} + .\;.\;.\;.\).If \(B - 2A = 100\lambda \) then \(\lambda \) is equal to : 

Let \(f(x)\) be a positive function such that the area bounded by \(y=f(x), y=0\) from \(x=0\) to \(x=a>0\) is \(\mathrm{e}^{-\mathrm{a}}+4 \mathrm{a}^2+\mathrm{a}-1\). Then the differential equation, whose general solution is \(y=c_1 f(x)+c_2\), where \(c_1\) and \(c_2\) are arbitrary constants, is :

The value of \(\sum_{n=1}^{100} \int_{n-1}^{n} e^{x-[x]} d x,\) where \([x]\) is the greatest integer \(\leq x ,\) is

Let \(f(x)=2 x+\tan ^{-1} x\) and \(g(x)=\log _e\left(\sqrt{1+x^2}+x\right)\), \(x \in[0,3]\). Then

Let \(\alpha, \beta(\alpha>\beta)\) be the roots of the quadratic equation \(x ^{2}- x -4=0\). If \(P _{ a }=\alpha^{ n }-\beta^{ n }, n \in N\), then \(\frac{ P _{15} P _{16}- P _{14} P _{16}- P _{15}^{2}+ P _{14} P _{15}}{ P _{13} P _{14}}\) is equal to\(......\)

\(\lim _{x \rightarrow 0} \operatorname{cosec} x\)\(\left(\sqrt{2 \cos ^2 x+3 \cos x}-\sqrt{\cos ^2 x+\sin x+4}\right)\) is:

If \(A> 0, B > 0\) and \(A + B = \frac{\pi }{6}\), then the minimum value of \(tan\,A + tan\,B\) is

Two poles standing on a horizontal ground are of heights \(5\,m\) and \(10\, m\) respectively. The line joining their tops makes an angle of \(15^o\) with ground. Then the distance (in \(m\)) between the poles, is

Let \(a\) be the sum of all coefficients in the expansion of \(\left(1-2 x+2 x^2\right)^{2023}\left(3-4 x^2+2 x^3\right)^{2024}\) and \(b=\lim _{x \rightarrow 0}\left(\frac{\int_0^x \frac{\log (1+t)}{t^{2024}+1} d t}{x^2}\right)\). If the equations \(\mathrm{cx}^2+\mathrm{dx}+\mathrm{e}=0\) and \(2 \mathrm{bx}^2+\mathrm{ax}+4=0\) have a common root, where \(c, d, e \in R\), then \(d: c: e\) equals

Let \(\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{c}}=\beta \hat{\mathrm{j}}-\hat{\mathrm{k}}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\) for which the area of the parallelogram of diagonals \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) is \(\frac{\sqrt{21}}{2}\), be \(\left(\alpha_1, \beta_1\right)\) and \(\left(\alpha_2, \beta_2\right)\). Then \(\alpha_1^2+\beta_1^2-\alpha_2 \beta_2\) is equal to