If \(a+x=b+y=c+z+1,\) where \(a, b, c, x, y, z\) are non-zero distinct real numbers, then \(\left|\begin{array}{lll}x & a+y & x+a \\ y & b+y & y+b \\ z & c+y & z+c\end{array}\right|\) is equal to  

The coefficient of \(x^{7}\) in the expression \((1+x)^{10}+x(1+x)^{9}+x^{2}(1+x)^{8}+\ldots+x^{10}\) is

Let \(P\) and \(Q\) be the points on the line \(\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2}\) which are at a distance of \(6\) units from the point \(R(1,2,3)\). If the centroid of the triangle \(PQR\) is \((\alpha, \beta, \gamma)\), then \(\alpha^2+\beta^2+\gamma^2\) is :

If \(a_n=\frac{-2}{4 n^2-16 n+15}\), then \(a_1+a_2+\ldots \ldots+a_{25}\) is equal to :

In a triangle \(ABC\), coordianates of \(A\) are \((1, 2)\) and the equations of the medians through \(B\) and \(C\) are \(x + y = 5\) and \(x = 4\) respectively. Then area of \(\Delta ABC\) (in sq. units) is

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three units vectors such that \(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0} .\) If \(\lambda=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}} \) and \(\overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}},\) then the ordered pair \((\lambda, {\mathrm{\vec d}})\) is equal to 

The coefficient of \(x^{50}\) in the binomial expansion of \({\left( {1 + x} \right)^{1000}} + x{\left( {1 + x} \right)^{999}} + {x^2}{\left( {1 + x} \right)^{998}} + ..... + {x^{1000}}\) is

The coefficient of  \(x^2\) in the expansion of the product \((2 -x^2)\). \(((1 + 2x + 3x^2)^6 +(1 -4x^2)^6)\)  is

Let \(a , b , c\) and \(d\) be positive real numbers such that \(a+b+c+d=11\). If the maximum value of \(a^5 b^3 c^2 d\) is \(3750 \beta\), then the value of \(\beta\) is

Let \(f(x)=2+|x|-|x-1|+|x+1|, x \in R\). Consider \((S1)\): \(f^{\prime}\left(-\frac{3}{2}\right)+f^{\prime}\left(-\frac{1}{2}\right)+f^{\prime}\left(\frac{1}{2}\right)+f^{\prime}\left(\frac{3}{2}\right)=2\) \(( S 2): \int_{-2}^{2} f ( x ) dx =12\)Then,

If \(\int {\frac{{dx}}{{{{\left( {{x^2} - 2x + 10} \right)}^2}}} = A\left( {{{\tan }^{ - 1}}\left( {\frac{{x - 1}}{3}} \right) + \frac{{f\left( x \right)}}{{{x^2} - 2x + 10}}} \right)}  + C\) Where \(C\) is a constant of integration, then

The radius of circle, having minimum area, which touches the cruve  \(y = 4 - {x^2}\) and the lines \(y = \left| x \right|\) is :

The sum of the squares of the roots of \(|\mathrm{x}+2|^2+|\mathrm{x}-2|-2=0\) and the squares of the roots of \(x^2-2|x-3|-5=0\), is