If \(\mathrm{x}=\sum\limits_{\mathrm{n}=0}^{\infty}(-1)^{\mathrm{n}} \tan ^{2 \mathrm{n}} \theta\) and \(\mathrm{y}=\sum\limits_{\mathrm{n}=0}^{\infty} \cos ^{2 \mathrm{n}} \theta,\) for \(0<\theta<\frac{\pi}{4},\) then

If \(2 \tan ^2 \theta-5 \sec \theta=1\) has exactly \(7\) solutions in the interval \(\left[0, \frac{n \pi}{2}\right]\), for the least value of \(n \in N\) then \(\sum_{\mathrm{k}=1}^{\mathrm{n}} \frac{\mathrm{k}}{2^{\mathrm{k}}}\) is equal to :

If \(\int_0^{\frac{\pi}{3}} \cos ^4 x d x=a \pi+b \sqrt{3}\), where \(a\) and \(b\) are rational numbers, then \(9 a+8 b\) is equal to :

A fair coin is tossed a fixed number of times. If the probability of getting \(7\) heads is equal to probability of getting \(9\) heads, then the probability of getting \(2\) heads is

Let \(\alpha = 3+4+8+9+13+14+\ldots\) upto 40 terms. If \((\tan\beta)^{\frac{\alpha}{1020}}\) is a root of the equation \(x^2+x-2=0\), \(\beta \in \left(0, \dfrac{\pi}{2}\right)\), then \(\sin^2\beta + 3\cos^2\beta\) is equal to:

If the distance between the plane, \(23 \mathrm{x}-10 \mathrm{y}-2 \mathrm{z}+48=0\) and the plane containing the lines \(\frac{x+1}{2}=\frac{y-3}{4}=\frac{z+1}{3}\) and \(\frac{x+3}{2}=\frac{y+2}{6}=\frac{z-1}{\lambda}(\lambda \in R)\) is equal to \(\frac{\mathrm{k}}{\sqrt{633}},\) then \(\mathrm{k}\) 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

The angle of elevation of the top of vertical tower standing on a horizontal plane is observed to be \(45^o\) from a point \(A\) on the plane. Let \(B\) be the point \(30\, m\) vertically above the point \(A\). If the angle of elevation of the top of the tower from \(B\) be \(30^o\), then the distance (in \(m\)) of the foot of the tower from the point \(A\) is:

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of \(81 \mathrm{~cm}^3 / \mathrm{min}\) and the thickness of the ice-cream layer decreases at the rate of \(\frac{1}{4 \pi} \mathrm{~cm} / \mathrm{min}\). The surface area (in \(\mathrm{cm}^2\) ) of the chocolate ball (without the ice-cream layer) is :

Let the sum of the first three terms of an \(A. P,\) be \(39\) and the sum of its last four terms be \(178.\) If the first term of this \(A.P.\) is \(10,\) then the median of the \(A.P.\) is

If the function \(f(x)=\left\{\begin{array}{cc}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}} & , x \neq 0 \\ a \log _e 2 \log _e 3 & , x=0\end{array}\right.\) is continuous at \(x=0\), then the value of \(a^2\) is equal to

The value of the integral \(\int_{-1}^{1} \log \left(x+\sqrt{x^{2}+1}\right)\, d x\) is:

Let \(a, b, c\) be the length of three sides of a triangle satisfying the condition \(\left(a^2+b^2\right) x^2-2 b(a+c)\). \(x+\left(b^2+c^2\right)=0\). If the set of all possible values of \(x\) is the interval \((\alpha, \beta)\), then \(12\left(\alpha^2+\beta^2\right)\) is equal to ...........