The minimum distance of a point on the curve \(y = x^2 - 4\) from the origin is

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the \(i^{\text {th }}\) roll than the number obtained in the \((i-1)^{\text {th }}\) roll, \(i=2,3\), is equal to :

The angle of elevation of the top of a tower from a point \(A\) due north of it is \(\alpha\) and from a point \(B\) at a distance of \(9\) units due west of \(A\) is \(\cos ^{-1}\left(\frac{3}{\sqrt{13}}\right)\). If the distance of the point \(B\) from the tower is \(15\) units, then \(\cot \alpha\) 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

\(\smallint \frac{{dx}}{{{x^2}{{\left( {{x^4} + 1} \right)}^{\frac{3}{4}}}}} = \)

Let \(A\) and \(B\) be two finite sets with \(m\) and \(n\) elements respectively. The total number of subsets of the set \(A\) is 56 more than the total number of subsets of \(B\). Then the distance of the point \(P ( m , n )\) from the point \(Q (-2,-3)\) is

Consider the three planes  \(P_{1}: 3 x+15 y+21 z=9\)  ;  \(P _{2}: x -3 y - z =5,\) and ;  \(P_{3}: 2 x+10 y+14 z=5\). Then, which one of the following is true ?

Let \(y = y ( x )\) be the solution of the differential equation \(x d y-y d x=\sqrt{\left(x^{2}-y^{2}\right)} d x, x \geq 1\), with \(y (1)=0 .\) If the area bounded by the line \(x =1, x = e ^{\pi}, y =0\) and \(y = y ( x )\) is \(\alpha e ^{2 \pi}+\beta\) then the value of \(10(\alpha+\beta)\) is equal to ....... .

If the data \(x_1, x_2, ...., x_{10}\) is such that the mean of first four of these is \(11\), the mean of the remaining six is \(16\) and the sum of squares of all of these is \(2,000\); then the standard deviation of this data is

Let \(\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{c}}=\beta \hat{\mathrm{j}}-\hat{\mathrm{k}}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\) for which the area of the parallelogram of diagonals \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) is \(\frac{\sqrt{21}}{2}\), be \(\left(\alpha_1, \beta_1\right)\) and \(\left(\alpha_2, \beta_2\right)\). Then \(\alpha_1^2+\beta_1^2-\alpha_2 \beta_2\) is equal to

The distance of the point \((1,-2,3)\) from the plane \(x-y+z=5\) measured parallel to the line \(\frac{x}{2}=\frac{y}{3}=\frac{z}{-6}\) is

The area of the triangle with vertices \(A ( z ), B ( iz )\) and \(C(z+i z)\) is

Let \(\alpha, \beta ; \alpha>\beta\), be the roots of the equation \(x^2-\sqrt{2} x-\sqrt{3}=0\). Let \(P_n=\alpha^n-\beta^n, n \in N\). Then \((11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}\) is equal to :