Five numbers are in \(A.P.\), whose sum is \(25\) and product is \(2520 .\) If one of these five numbers is \(-\frac{1}{2},\) then the greatest number amongst them is

The integral \(\int\limits_{7\pi /4}^{7\pi /3} {\sqrt {{{\tan }^2}\,x}\,dx } \) is equal to

The locus of the mid-points of the perpendiculars drawn from points on the line, \(\mathrm{x}=2 \mathrm{y}\) to the line \(\mathrm{x}=\mathrm{y}\) is 

If \(\int(\frac{1-5~cos^{2}x}{sin^{5}x~cos^{2}x})dx=f(x)+C\) where C is the constant of integration, then \(f(\frac{\pi}{6})-f(\frac{\pi}{4})\) is equal to

Let the curve \(z(1+i)+\bar{z}(1-i)=4, z \in \mathrm{C}\), divide the region \(|z-3| \leq 1\) into two parts of areas \(\alpha\) and \(\beta\). Then \(|\alpha-\beta|\) equals :

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

If the co-efficient of \(x^9\) in \(\left(\alpha x^3+\frac{1}{\beta x}\right)^{11}\) and the co-efficient of \(x^{-9}\) in \(\left(\alpha x-\frac{1}{\beta x^3}\right)^{11}\) are equal, then \((\alpha \beta)^2\) is equal to \(.............\).

Let the coefficients of three consecutive terms \(T_r, T_{r+1}\) and \(T_{r+2}\) in the binomial expansion of \((a+b)^{12}\) be in a G.P. and let \(p\) be the number of all possible values of \(r\). Let \(q\) be the sum of all rational terms in the binomial expansion of \((\sqrt[4]{3}+\sqrt[3]{4})^{12}\). Then \(\mathrm{p}+\mathrm{q}\) is equal to :

The angle of elevation of a jet plane from a point \(A\) on the ground is \(60^{\circ}\). After a flight of \(20\, seconds\) at the speed of \(432\, km / hour\), the angle of elevation changes to \(30^{\circ}\). If the jet plane is flying at a constant height, then its height is ..... \(m.\)

Let the ellipse \(\mathrm{E}_1: \frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1, \mathrm{a}\gt\mathrm{b}\) and \(\mathrm{E}_2: \frac{x^2}{\mathrm{~A}^2}+\frac{y^2}{\mathrm{~B}^2}=1, \mathrm{~A} \lt \mathrm{B}\) have same eccentricity \(\frac{1}{\sqrt{3}}\). Let the product of their lengths of latus rectums be \(\frac{32}{\sqrt{3}}\), and the distance between the foci of \(E_1\) be 4. If \(E_1\) and \(E_2\) meet at \(A, B, C\) and \(D\), then the area of the quadrilateral \(A B C D\) equals :

Let a set \(A=A_{1} \cup A_{2} \cup \ldots \cup A_{k,} \quad\) where \(A_{ i } \cap A _{ j }=\phi\) for \(i \neq j 1 \leq i , j \leq k\). Define the relation \(R\) from \(A\) to \(A\) by \(R=\left\{(x, y): y \in A_{i}\right.\) if and only if \(\left.x \in A_{i}, 1 \leq i \leq k\right\}\). Then, \(R\) is

Let a curve \(y=y(x)\) be given by the solution of the differential equation \(\cos \left(\frac{1}{2} \cos ^{-1}\left(e^{-x}\right)\right) d x=\sqrt{e^{2 x}-1} \,d y\) If it intersects \(y\)-axis at \(y=-1\), and the intersection point of the curve with \(x\)-axis is \((\alpha, 0)\) the \(\mathrm{e}^{\alpha}\) is equal to \(.....\)

Let three vectors \(\overrightarrow{\mathrm{a}}=\alpha \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}\), \(\vec{b}=5 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{c}=x \hat{i}+y \hat{j}+z \hat{k}\) from a triangle such that \(\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}\) and the area of the triangle is \(5 \sqrt{6}\). if \(\alpha\) is a positive real number, then \(|\overrightarrow{\mathrm{c}}|^2\) is :