Let [ ] denote the greatest integer function and \( f(x)=lim_{n\rightarrow\infty}\frac{1}{n^{3}}\sum_{k=1}^{n}[\frac{k^{2}}{3^{x}}] \) Then \( 12\sum_{j=1}^{x}f(j) \) is equal to ........... .

If \( f(x) \) satisfies the relation \( f(x)=e^{x}+\int_{0}^{1}(y+xe^{x})f(y)dy, \) then \( e + f(0) \) is equal to ___ .

The urns \(A, B\) and \(C\) contain \(4\) red, \(6\) black;\(5\) red,\(5\) black and \(\lambda\) red,\(4\) black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn \(C\) is \(0.4\) then the square of the length of the side of the largest equilateral triangle, inscribed in the parabola \(y^2=\lambda x\) with one vertex at the vertex of the parabola is

If some three consecutive in the binomial expansion of \({\left( {x + 1} \right)^n}\) in powers of \(x\) are in the ratio \(2 : 15 : 70\), then the average of these three coefficient is

The term independent of \(x\) in the expansion of \(\left( {\frac{1}{{60}} - \frac{{{x^8}}}{{81}}} \right).{\left( {2{x^2} - \frac{3}{{{x^2}}}} \right)^6}\) is equal to

Let \(\overrightarrow{ a }=2 \hat{ i }+\hat{ j }+\hat{ k }\), and \(\overrightarrow{ b }\) and \(\overrightarrow{ c }\) be two nonzero vectors such that \(|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}| \quad\) and \(\vec{b} \cdot \vec{c}=0\). Consider the following two statement: \((A)\) \(|\overrightarrow{ a }+\lambda \overrightarrow{ c }| \geq|\overrightarrow{ a }|\) for all \(\lambda \in R\). \((B)\) \(\overrightarrow{ a }\) and \(\overrightarrow{ c }\) are always parallel

Let a smooth curve \(y=f(x)\) be such that the slope of the tangent at any point \((x, y)\) on it is directly proportional to \(\left(\frac{-y}{x}\right)\). If the curve passes through the point \((1,2)\) and \((8,1)\), then \(\left| y \left(\frac{1}{8}\right)\right|\) is equal to

A ray of light passing through the point \(P (2,3)\) reflects on the \(x-\)axis at point \(A\) and the reflected ray passes through the point \(Q(5,4)\). Let \(R\) be the point that divides the line segment \(AQ\) internally into the ratio \(2: 1\). Let the co-ordinates of the foot of the perpendicular \(M\) from \(R\) on the bisector of the angle \(PAQ\) be \((\alpha, \beta)\). Then, the value of \(7 \alpha+3 \beta\) is equal to.......

If the mean deviation about the mean of the numbers \(1,2,3, \ldots ., n\), where \(n\) is odd, is \(\frac{5(n+1)}{n}\), then \(n\) is equal to

Statement \(-1\) : The system of linear equations \(x + \left( {\sin \,\alpha } \right)y + \left( {\cos \,\alpha } \right)z = 0\) \(x + \left( {\cos \,\alpha } \right)y + \left( {\sin \alpha } \right)z = 0\) \(x - \left( {\sin \,\alpha } \right)y - \left( {\cos \alpha } \right)z = 0\) has a non-trivial solution for only one value of \(\alpha \) lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)  Statement \(-2\) : The equation in \(\alpha \) \(\left| {\begin{array}{*{20}{c}}
  {\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha } \\ 
  {\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha } \\ 
  {\cos {\mkern 1mu} \alpha }&{ - \sin {\mkern 1mu} \alpha }&{ - \cos {\mkern 1mu} \alpha } 
\end{array}} \right| = 0\) has only one solution lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)

Let \(\alpha \in R\) be such that the function \(f(x)=\left\{\begin{array}{ll} \frac{\cos ^{-1}\left(1-\{x\}^{2}\right) \sin ^{-1}(1-\{x\})}{\{x\}-\{x\}^{3}}, & x \neq 0 \\ \alpha, & x=0 \end{array}\right.\) is continuous at \(x=0,\) where \(\{x\}=x-[x],[x]\) is the greatest integer less than or equal to \(X\). Then :

If three positive numbers \(a, b\) and \(c\) are in \(A.P.\) such that \(abc\, = 8\), then the minimum possible value of \(b\) is

If the system of equations \(x + y+  z = 5\) ; \(x + 2y + 3z = 9\) ; \(x + 3y + \alpha z = \beta \) has infinitely many solutions, then \(\beta  - \alpha \) equals