If \(a, b\) and \(c\) be three distinct numbers in \(G.P.\) and \(a + b + c = xb\) then \(x\) can not be

Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be a function defined \(f(x)=\frac{x}{\left(1+x^4\right)^{1 / 4}}\) and \(g(x)=f(f(f(f(x))))\) then \(18 \int_0^{\sqrt{2 \sqrt{5}}} x^2 g(x) d x\)

If the foot of the perpendicular from the point \(A (-1,4,3)\) on the plane \(P : 2 x + my + nz =4\), is \(\left(-2, \frac{7}{2}, \frac{3}{2}\right)\), then the distance of the point \(A\) from the plane \(P\), measured parallel to a line with direction ratios \(3,-1,-4\), is equal to.

If the following system of linear equations \(2 x+y+z=5\) \(x-y+z=3\) \(x+y+a z=b\) has no solution, then :

If \(\lim _{x \rightarrow 0} \frac{\sin ^{-1} x-\tan ^{-1} x}{3 x^{3}}\) is equal to \(L,\) then the value of \((6 L +1)\) is

Let \(f: R \rightarrow R\) be a function defined as \(f(x)=a \sin \left(\frac{\pi[x]}{2}\right)+[2-x], a \in R\), where [t] is the greatest integer less than or equal to \(t\). If \(\lim _{x \rightarrow-1} f(x)\) exists, then the value of \(\int_{0}^{4} f(x) d x\) is equal to.

For \(\alpha, \beta, \gamma, \in \mathbf{R}\), if \(\lim _{x \rightarrow 0} \frac{x^2 \sin \alpha x+(\gamma-1) e^{x^2}}{\sin 2 x-\beta x}=3\), then \(\beta+\gamma-\alpha\) is equal to:

Let the solution \(y=y(x)\) of the differential equation \(\frac{d y}{d x}-y=1+4 \sin x\) satisfy \(y(\pi)=1\). Then \(y\left(\frac{\pi}{2}\right)+10\) is equal to ...........

Let \({ }^{n} C_{r}\) denote the binomial coefficient of \(x^{r}\) in the expansion of \((1+ x )^{ n }.\) If \(\sum_{ k =0}^{10}\left(2^{2}+3 k \right){ }^{ n } C _{ k }=\alpha .3^{10}+\beta \cdot 2^{10}, \alpha, \beta \in R\) then \(\alpha+\beta\) is equal to ....... .

Let \(L_1: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}), \lambda \in \mathrm{R}\) \(L_2: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(3 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{p} \hat{\mathrm{k}}), \mu \in \mathrm{R}\) and  \(L_3: \overrightarrow{\mathrm{r}}=\delta(\ell \hat{\mathrm{i}}+\mathrm{m} \hat{\mathrm{j}}+\mathrm{n} \hat{\mathrm{k}}) \delta \in \mathrm{R}\). Be three lines such that \(\mathrm{L}_1\) is perpendicular to \(\mathrm{L}_2\) and \(L_3\) is perpendicular to both \(L_1\) and \(L_2\). Then the point which lies on \(\mathrm{L}_3\) is

Let the area of the triangle formed by a straight Line L: \(\mathrm{x}+\mathrm{by}+\mathrm{c}=0\) with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line L makes an angle of \(45^{\circ}\) with the positive x -axis, then the value of \(\mathrm{b}^2+\mathrm{c}^2\) is:

Let the equation of two diameters of a circle \(x ^{2}+ y ^{2}\) \(-2 x +2 fy +1=0\) be \(2 px - y =1\) and \(2 x + py =4 p\). Then the slope \(m \in(0, \infty)\) of the tangent to the hyperbola \(3 x^{2}-y^{2}=3\) passing through the centre of the circle is equal to \(......\)

The value of \(\lim _{x \rightarrow 0} 2\left(\frac{1-\cos x \sqrt{\cos 2 x} \sqrt[3]{\cos 3 x} \ldots . . \sqrt[10]{\cos 10 x}}{x^2}\right)\) is ............