\(1+3+5^2+7+9^2+\ldots\) upto 40 terms is equal to

If the tangent to the curve \(y=x^{3}-x^{2}+x\) at the point \((a, b)\) is also tangent to the curve \(y=5 x^{2}+\) \(2 x -25\) at the point \((2,-1)\), then \(|2 a +9 b |\) is equal to \(........\)

Let \(a , b , c\) be three distinct positive real numbers such that \((2 a)^{\log _{\varepsilon} a}=(b c)^{\log _e b}\) and \(b^{\log _e 2}=a^{\log _e c}\). Then \(6 a+5 b c\) is equal to \(........\).

If the function \(f(x)=\left\{\begin{array}{c}\frac{\log _{e}\left(1-x+x^{2}\right)+\log _{e}\left(1+x+x^{2}\right)}{\sec x-\cos x}, x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)-\{0\} \\ k \end{array}\right.\) is continuous at \(x =0\), then \(k\) is equal to.

The angle of elevation of the summit of a mountain from a point on the ground is \(45^{\circ}\). After climding up one \(km\) towards the summit at an inclination of \(30^{\circ}\) from the ground, the angle of elevation of the summit is found to be \(60^{\circ} .\) Then the height (in \(km\) ) of the summit from the ground is

If the coefficients of \(x^2\) and \(x^3\) are both zero, in the expansion of the expression \((1 + ax + bx^2) (1 -3x)^{t5}\) in powers of \(x\), then the ordered pair \((a, b)\) is equal to

The set of all values of \(\lambda\) for which the system of linear  \(2{x_1} - 2{x_2} + {x_3} = \lambda {x_1}\;,\;2{x_1} - 3{x_2} + 2{x_3} = \lambda {x_2}\;\;,\)\(\;\; - {x_1} + 2{x_2} = \lambda {x_3}\) has a non-trivial solution

The total number of matrices \(A = \left[ {\begin{array}{*{20}{c}}
0&{2x}&{2x}\\
{2y}&y&{ - y}\\
1&{ - 1}&1
\end{array}} \right];\,\left( {x,y \in R,\,x \ne y} \right)\) for which \({A^T}A = 3{I_3}\)

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

The shortest distance between the line \(x-y=1\) and the curve \(x^{2}=2 y\) is .... .

Let \(L\) be a common tangent line to the curves \(4 x^{2}+9 y^{2}=36\) and \((2 x)^{2}+(2 y)^{2}=31\). Then the square of the slope of the line \(L\) is ..... .

The shortest distance between the lines \(\frac{x-1}{0}=\frac{y+1}{-1}=\frac{z}{1}\) and \(x+y+z+1=0\), \(2 x-y+z+3=0\) is

Let \(\mathrm{y}=\mathrm{y}(\mathrm{x})\) be the solution curve of the differential equation, \(\quad\left(y^{2}-x\right) \frac{d y}{d x}=1\) satisfying \(\mathrm{y}(0)=1 .\) This curve intersects the \(\mathrm{x}\) -axis at a point whose abscissa is