If the curve \(x^{2}+2 y^{2}=2\) intersects the line \(x + y =1\) at two points \(P\) and \(Q ,\) then the angle subtended by the line segment \(PQ\) at the origin is ...... .

Let \(A=\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]\). If \(M\) and \(N\) are two matrices given by \(M =\sum \limits_{ k =1}^{10} A ^{2 k }\) and \(N =\sum \limits_{ k =1}^{10} A ^{2 k -1}\) then \(MN ^{2}\) is

Let PQ and MN be two straight lines touching the circle \( x^{2}+y^{2}-4x-6y-3=0 \) at the points A and B respectively. Let O be the centre of the circle and \( \angle AOB=\pi/3. \) Then the locus of the point of intersection of the lines PQ and MN is:

A circle with centre \((2,3)\) and radius \(4\) intersects the line \(x + y =3\) at the points \(P\) and \(Q\). If the tangents at \(P\) and \(Q\) intersect at the point \(S(\alpha, \beta)\), then \(4 \alpha-7 \beta\) is equal to \(........\).

Let \(A\) be a \(2 \times 2\) matrix with \(\operatorname{det}(A)=-1\) and det \((( A + I )(\operatorname{Adj}( A )+ I ))=4\). Then the sum of the diagonal elements of \(A\) can be.

Let the position vectors of two points \(P\) and \(Q\) be \(3 \hat{ i }-\hat{ j }+2 \hat{ k }\) and \(\hat{ i }+2 \hat{ j }-4 \hat{ k },\) respectively. Let \(R\) and \(S\) be two points such that the direction ratios of lines \(PR\) and \(QS\) are \((4,-1,2)\) and \((-2,1,-2),\) respectively. Let lines \(PR\) and \(QS\) intersect at \(T\). If the vector \(\overline{ TA }\) is perpendicular to both \(\overline{ PR }\) and \(\overline{ QS }\) and the length of vector \(\overline{ TA }\) is \(\sqrt{5}\) units, then the modulus of a position vector of \(A\) is

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

Consider the circle \(C: x^2+y^2-6x-8y-11=0\). Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle \(x^2+y^2-\alpha x - \beta y - \gamma = 0\), then \(\alpha + \beta + 2\gamma\) is equal to ________.

In an examination of Mathematics paper, there are \(20\) questions of equal marks and the question paper is divided into three sections : \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\). A student is required to attempt total \(15\) questions taking at least \(4\) questions from each section. If section \(A\) has \(8\) questions, section \(\mathrm{B}\) has \(6\) questions and section \(\mathrm{C}\) has \(6\) questions, then the total number of ways a student can select \(15\) questions is

Let \(\alpha \in(0,1)\) and \(\beta=\log _{ c }(1-\alpha)\). Let \(P_n(x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\ldots . .+\frac{x^n}{n}, x \in(0,1)\) Then the integral \(\int \limits_0^\alpha \frac{ t ^{50}}{1- t } dt\) is equal to

The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is:

Let \(\vec{a}, \vec{b}, \vec{c}\) be three vectors mutually perpendicular to each other and have same magnitude. If a vector \(\overrightarrow{\mathrm{r}}\) satisfies. \(\overrightarrow{\mathrm{a}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{a}}\}+\overrightarrow{\mathrm{b}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{c}}) \times \overrightarrow{\mathrm{b}}\}+\overrightarrow{\mathrm{c}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{a}}) \times \overrightarrow{\mathrm{c}}\}=\overrightarrow{0}\) then \(\overrightarrow{\mathrm{r}}\) is equal to:

The function \(f(x)=2 x+3(x)^{\frac{2}{3}}, x \in \mathbb{R}\), has