If the sum and product of the first three term in an \(A.P\). are \(33\) and \(1155\), respectively, then a value of its \(11^{th}\) tern is

Let \(k\) be a non-zero real number  If \(f(x) = {\rm{ }}\left\{ {\begin{array}{*{20}{c}}
{\frac{{\left( {{e^x} - 1} \right)^2}}{{\sin {\mkern 1mu} \left( {\frac{x}{k}} \right){\mkern 1mu} \log {\mkern 1mu} \left( {1 + \frac{x}{4}} \right)}}{\mkern 1mu} ,{\mkern 1mu} x \ne 0}\\
{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} 12{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} ,x{\mkern 1mu} {\mkern 1mu}  = 0{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} }
\end{array}} \right.\)  is a continuous function then the value of \(k\) is

If for \(x, y \in {R}, x>0,\) \(y=\log _{10} x+\log _{10} x^{1 / 3}+\log _{10} x^{1 / 9}+\ldots . .\) upto \(\infty\) terms and \(\frac{2+4+6+\ldots+2 \mathrm{y}}{3+6+9+\ldots+3 \mathrm{y}}=\frac{4}{\log _{10} \mathrm{x}}\), then the ordered pair \((x, y)\) is equal to :

Let m and \(\mathrm{n},(\mathrm{m} \lt \mathrm{n})\) be two 2-digit numbers. Then the total numbers of pairs \((m, n)\), such that \(\operatorname{gcd}(m, n)=6\), is ________

Let \(f(x)\) and \(g(x)\) be two functions satisfying \(f\left(x^{2}\right)\) \(+g(4-x)=4 x^{3}\) and \(g(4-x)+g(x)=0\), then the value of \(\int_{-4}^{4} f(x)^{2} d x\) is

If \(\left|\begin{array}{ccc}x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2\end{array}\right|=\frac{9}{8}(103 x+81)\), then \(\lambda\), \(\frac{\lambda}{3}\) are the roots of the equation

The diameter of the circle, whose centre lies on the line \(x+y=2\) in the first quadrant and which touches both the lines \(x=3\) and \(y=2,\) is

Let \(P\) be the foot of the perpendicular from the point \(Q(10,-3,-1)\) on the line \(\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z+1}{-2}\). Then the area of the right angled triangle \(P Q R\), where \(R\) is the point \((3,-2,1)\), is

The sum of the abosolute maximum and minimum values of the function \(f(x)=\left|x^2-5 x+6\right|-3 x+2\) in the interval \([-1,3]\) is equal to :

The number of distinct solutions of the equation \(\log _{\frac{1}{2}}|\sin x|=2-\log _{\frac{1}{2}}|\cos x|\) in the interval \([0,2 \pi],\) is

The number of distinct real roots of the equation \(3 x^{4}+4 x^{3}-12 x^{2}+4=0\) is ..... .

Let \(f : R \to R\) be a function such that \(f(2 - x)\, = f(2 + x)\) and \(f(4 -x)\, = f(4 + x)\), for all \(x \in  R\) and \(\int\limits_0^2 {f\left( x \right)\,dx = 5} \) . Then the value of \(\int\limits_{10}^{50} {f\left( x \right)\,\,dx} \) is

Two poles, \(\mathrm{AB}\) of length \(a\) metres and \(\mathrm{CD}\) of length \(\mathrm{a}+\mathrm{b}(\mathrm{b} \neq \mathrm{a})\) metres are erected at the same horizontal level with bases at \(\mathrm{B}\) and \(\mathrm{D} .\) If \(\mathrm{BD}=\mathrm{x}\) and \(\tan \angle\,ACB=\frac{1}{2}\), then: