Let \( I(x)=\int\frac{3dx}{(4x+6)(\sqrt{4x^{2}+8x+3})} \) and \( I(0)=\frac{\sqrt{3}}{4}+20 \). If \( I(\frac{1}{2})=\frac{a\sqrt{2}}{b}+c \), where \( a, b, c \in N \) and \( gcd(a,b)=1 \), then \( a+b+c \) is equal to:

Let \(f (x) = a^x (a > 0)\) be written as \(f( x) = f_1( x) + f_2( x)\) , where \(f_1( x)\) is an even function and \(f_2( x)\) is an odd function. Then \(f_1( x + y) + f_1( x - y )\) equals

Let the product of \(\omega_1=(8+i) \sin \theta+(7+4 i) \cos \theta\) and \(\omega_2=(1+8 i) \sin \theta+(4+7 i) \cos \theta\) be \(\alpha+i \beta\), \(\mathrm{i}=\sqrt{-1}\). Let p and q be the maximum and the minimum values of \(\alpha+\beta\) respectively.

If the integral \(525 \int_0^{\frac{\pi}{2}} \sin 2 x \cos ^{\frac{11}{2}} x\left(1+\cos ^{\frac{5}{2}} x\right)^{\frac{1}{2}} d x\) is equal to \((n \sqrt{2}-64)\), then \(n\) is equal to

Let M denote the set of all real matrices of order \(3 \times 3\) and let \(\mathrm{S}=\{-3,-2,-1,1,2\}\). Let
\(\mathrm{S}_1=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_2=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=-\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_3=\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: a_{11}+a_{22}+a_{33}=0\) and \(a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\}\)
If \(n\left(\mathrm{~S}_1 \cup_2 \mathrm{US}_3\right)=125 \alpha\), then \(\alpha\) equals _______

If \(\int {\frac{{{x^2} - x + 1}}{{{x^2} + 1}}{e^{{{\cot }^{ - 1}}}}{}^x\,dx = A(x)} {e^{{{\cot }^{ - 1}}}}{}^x + C,\) then \(A(x)\) is equal to

Let a line passing through the point \((-1,2,3)\) intersect the lines \(L_1: \frac{x-1}{3}=\frac{y-2}{2}=\frac{z+1}{-2}\) at \(\mathrm{M}(\alpha, \beta, \gamma)\) and \(\mathrm{L}_2: \frac{\mathrm{x}+2}{-3}=\frac{\mathrm{y}-2}{-2}=\frac{\mathrm{z}-1}{4}\) at \(\mathrm{N}(\mathrm{a}, \mathrm{b}\), c). Then the value of \(\frac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2}\) equals

The number of natural numbers, between 212 and 999 , such that the sum of their digits is 15 , is

If \(\alpha \) and \( \beta \) are roots of the equation, \({x^2} - 4\sqrt 2\,kx + 2\,{e^{4\ln \,k}} - 1 = 0\) for some \(k\), and \({\alpha ^2} + {\beta ^2} = 66\), then \({\alpha ^3} + {\beta ^3}\) is equal to 

If \(I_{m, n}=\int_{0}^{1} x^{m-1}(1-x)^{n-1} d x,\) for \(m, n \geq 1\) and \(\int_{0}^{1} \frac{x^{m-1}+x^{n-1}}{(1+x)^{m+n}} d x=\alpha I_{m, n}, \alpha \in R,\) then \(\alpha\) equals .... .

The lengths of the sides of a triangle are \(10+ x ^{2}\), \(10+ x ^{2}\) and \(20-2 x ^{2}\). If for \(x = k\), the area of the triangle is maximum, then \(3 K ^{2}\) is equal to

Let \([t]\) denote the largest integer less than or equal to \(t\). If \(\int_0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right) d x=a+b \sqrt{2}-\sqrt{3}-\sqrt{5}+c \sqrt{6}-\sqrt{7},\) where \(a, b, c \in z\), then \(a+b+c\) is equal to ...........

Let \(a\) and \(b\) be any two numbers satisfying \(\frac{1}{{{a^2}}} + \frac{1}{{{b^2}}} = \frac{1}{4}\). Then, the foot of perpendicular from the origin on the variable line, \(\frac{x}{a} + \frac{y}{b} = 1\) , lies on