The curve satisfying the differential equation, \((x^2 -y^2) \,dx + 2xydy\, = 0\) and passing through the point \((1 , 1 )\) is

The value of the integral \(\displaystyle\int_{\pi/6}^{\pi/3} \left(\dfrac{4 - \csc^2 x}{\cos^4 x}\right) dx\) is:

Let \(\vec{a}=2 \hat{i}-\hat{j}+5 \hat{k}\) and \(\vec{b}=\alpha \hat{i}+\beta \hat{j}+2 \hat{k}\). If \(((\vec{a} \times \vec{b}) \times \hat{i}) \cdot \hat{k}=\frac{23}{2}\), then \(|\vec{b} \times 2 \hat{j}|\) is equal to.

Let \(\mathrm{Q}\) and \(\mathrm{R}\) be the feet of perpendiculars from the point \(\mathrm{P}(\mathrm{a}, \mathrm{a}, \mathrm{a})\) on the lines \(\mathrm{x}=\mathrm{y}, \mathrm{z}=1\) and \(\mathrm{x}=-\mathrm{y}\), \(\mathrm{z}=-1\) respectively. If \(\angle \mathrm{QPR}\) is a right angle, then \(12 \mathrm{a}^2\) is equal to

If a tangent to the circle \(x^2 + y^2 = 1\) intersects the coordinate axes at distinct points \(P\) and \(Q,\) then the locus of the mid-point of \(PQ\) is

The number of terms common to the two A.P.'s \(3,7,11, \ldots ., 407\) and \(2,9,16, \ldots . .709\) is

Let the circle \(x^{2}+y^{2}=4\) intersect x-axis at the points \(A(a,0), a>0\) and \(B(b,0)\). Let \(P(2 \cos\alpha, 2 \sin\alpha), 0<\alpha<\frac{\pi}{2}\) and \(Q(2 \cos\beta, 2 \sin\beta)\) be two points such that \((\alpha-\beta)=\frac{\pi}{2}.\) Then the point of intersection of AQ and BP lies on:

Let \(\mathrm{ABC}\) be a triangle of area \(15 \sqrt{2}\) and the vectors \(\overrightarrow{\mathrm{AB}}=\hat{i}+2 \hat{j}-7 \hat{k}, \quad \overrightarrow{B C}=a \hat{i}+b \hat{j}+c \hat{k}\) and \(\overrightarrow{\mathrm{AC}}=6 \hat{\mathrm{i}}+\mathrm{d} \hat{\mathrm{j}}-2 \hat{\mathrm{k}}, \mathrm{d}>0\). Then the square of the length of the largest side of the triangle \(\mathrm{ABC}\) is ...........

Let the observations \(\mathrm{x}_{\mathrm{i}}(1 \leq \mathrm{i} \leq 10)\) satisfy the equations, \(\sum\limits_{i=1}^{10}\left(x_{i}-5\right)=10\) and \(\sum\limits_{i=1}^{10}\left(x_{i}-5\right)^{2}=40\) If \(\mu\) and \(\lambda\) are the mean and the variance of the observations, \(\mathrm{x}_{1}-3, \mathrm{x}_{2}-3, \ldots ., \mathrm{x}_{10}-3,\) then the ordered pair \((\mu, \lambda)\) is equal to :

Consider the function \(f:(0, \infty) \rightarrow R\) defined by \(f(x)=e^{-\left|\log _e x\right|}\). If \(m\) and \(n\) be respectively the number of points at which \(f\) is not continuous and \(f\) is not differentiable, then \(\mathrm{m}+\mathrm{n}\) is

For two non-zero complex number \(z_1\) and \(z_2\), if \(\operatorname{Re}\left(z_1 z_2\right)=0\) and \(\operatorname{Re}\left(z_1+z_2\right)=0\), then which of the following are possible ? \((A)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((B)\) \(\operatorname{Im}\left(z_1\right) < 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((C)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) < 0\) \((D)\) \(\operatorname{Im}\left( z _1\right) < 0\) and \(\operatorname{Im}\left( z _2\right) < 0\) Choose the correct answer from the options given below :

Let \( f(x)=x^{3}+x^{2}f^{\prime}(1)+2x~f^{\prime\prime}(2)+f^{\prime\prime\prime}(3), x\in R. \) Then the value of \(f^{\prime}(5)\) is :

The probability that two randomly selected subsets of the set \(\{1,2,3,4,5\}\) have exactly two elements in their intersection, is :