If \(f\left( x \right) = {\left( {\frac{3}{5}} \right)^x} + {\left( {\frac{4}{5}} \right)^x} - 1\) , \(x \in R\) , then the equation \(f(x) = 0\) has

The line \(y=x+1\) meets the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{2}=1\) at two points \(P\) and \(Q\). If \(r\) is the radius of the circle with \(PQ\) as diameter then \((3 r )^{2}\) is equal to

If two tangents drawn from a point \((\alpha, \beta)\) lying on the ellipse \(25 x^{2}+4 y^{2}=1\) to the parabola \(y^{2}=4 x\) are such that the slope of one tangent is four times the other, then the value of \((10 \alpha+5)^{2}+\left(16 \beta^{2}+50\right)^{2}\) equals

Let \(\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})\) be a point in the first octant, whose projection in the xy-plane is the point \(\mathrm{Q}\). Let \(\mathrm{OP}=\gamma\); the angle between \(OQ\) and the positive \(\mathrm{x}\)-axis be \(\theta\); and the angle between \(\mathrm{OP}\) and the positive \(\mathrm{z}\)-axis be \(\phi\), where \(\mathrm{O}\) is the origin. Then the distance of \(\mathrm{P}\) from the \(\mathrm{x}\)-axis is :

Let the focal chord of the parabola \(P: y^{2}=4 x\) along the line \(L: y=m x+c, m>0\) meet the parabola at the points \(M\) and \(N\). Let the line \(L\) be a tangent to the hyperbola \(H : x ^{2}- y ^{2}=4\). If \(O\) is the vertex of \(P\) and \(F\) is the focus of \(H\) on the positive \(x\)-axis, then the area of the quadrilateral \(OMFN\) is.

Let \(x_1, x_2, \ldots, x_{10}\) be ten observations such that \(\sum_{i=1}^{10}\left(x_i-2\right)=30, \sum_{i=1}^{10}\left(x_i-\beta\right)^2=98, \beta\gt2\), and their variance is \(\frac{4}{5}\). If \(\mu\) and \(\sigma^2\) are respectively the mean and the variance of \(2\left(x_1-1\right)+4 \beta\), \(2\left(x_2-1\right)+4 \beta, \ldots ., 2\left(x_{10}-1\right)+4 \beta\), then \(\frac{\beta \mu}{\sigma^2}\) is equal to :

If \(\int {\frac{{dx}}{{x + {x^7}}}}  = p(x)\) then, \(\int {\frac{{{x^6}}}{{x + {x^7}}}} dx\) is equal to

Consider the system of linear equations \(x+y+z=5, x+2 y+\lambda^2 z=9\) \(x+3 y+\lambda z=\mu\), where \(\lambda, \mu \in R\). Then, which of the following statement is NOT correct?

Let \(x = x(y)\) be the solution of the differential equation \(2y^2 \dfrac{dx}{dy} - 2xy + x^2 = 0\), \(y > 1\), \(x(e) = e\). Then \(x(e^2)\) is equal to:

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero non-coplanar vectors. Let the position vectors of four points \(A, B, \quad C\) and \(D\) be \(\vec{a}-\vec{b}+\vec{c}, \lambda \vec{a}-3 \vec{b}+4 \vec{c}\), \(-\vec{a}+2 \vec{b}-3 \vec{c}\) and \(2 \vec{a}-4 \vec{b}+6 \vec{c}\) respectively. If \(\overrightarrow{A B}\), \(\overline{ AC }\) and \(\overline{ AD }\) are coplanar, then \(\lambda\) is :

Let \(\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{d}}=\vec{a} \times \overrightarrow{\mathrm{b}}\). If \(\overrightarrow{\mathrm{c}}\) is a vector such that \(\vec{a} \cdot \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{c}}|\), \(|\overrightarrow{\mathrm{c}}-2 \vec{a}|^2=8\) and the angle between \(\overrightarrow{\mathrm{d}}\) and \(\overrightarrow{\mathrm{c}}\) is \(\frac{\pi}{4}\), then \(|10-3 \overrightarrow{\mathrm{~b}} \cdot \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{d}} \times \overrightarrow{\mathrm{c}}|^2\) is equal to

If \(\left({ }^{40} C _{0}\right)+\left({ }^{41} C _{1}\right)+\left({ }^{42} C _{2}\right)+\ldots+\left({ }^{\infty} C _{20}\right)=\frac{ m }{ n }{ }^{60} C _{20}, m\) and \(n\) are coprime, then \(m+n\) is equal to

If \(\frac{{{}^{n + 2}{C_6}}}{{{}^{n - 2}{P_2}}} = 11\), then \(n\) satisfies the equation