Let \(y=y(x)\) be the solution of the differential equation \(\left(x-x^{3}\right) d y=\left(y+y x^{2}-3 x^{4}\right) d x, x>2\). If \(y(3)=3\), then \(y(4)\) is equal to :

If the mean and variance of the following data: \(6,10,7,13, a, 12, b, 12\) are 9 and \(\frac{37}{4}\) respectively, then \((a-b)^{2}\) is equal to:

Let \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) be three points on the parabola \(y^2=6 x\) and let the line segment \(A B\) meet the line \(L\) through \(\mathrm{C}\) parallel to the \(\mathrm{x}\)-axis at the point \(\mathrm{D}\). Let \(\mathrm{M}\) and \(\mathrm{N}\) respectively be the feet of the perpendiculars from \(\mathrm{A}\) and \(\mathrm{B}\) on \(\mathrm{L}\). Then \(\left(\frac{\mathrm{AM} \cdot \mathrm{BN}}{\mathrm{CD}}\right)^2\) is equal to ...........

Let \(f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in R\) Then \(f^{\prime}(10)\) is equal to ..............

If \(0 < x < \frac{1}{\sqrt{2}}\) and \(\frac{\sin ^{-1} x}{\alpha}=\frac{\cos ^{-1} x}{\beta}\), then a value of \(\sin \left(\frac{2 \pi \alpha}{\alpha+\beta}\right)\) is\(......\)

If the system of linear equations  \(2 x + y - z =7\) ; \(x-3 y+2 z=1\)  ; \(x +4 y +\delta z = k\), where \(\delta, k \in R\)  has infinitely many solutions, then \(\delta+ k\) is equal to

For a triangle ABC, let \( \vec{p}=\vec{BC}, \vec{q}=\vec{CA} \) and \( \vec{r}=\vec{BA} \). If \( |\vec{p}|=2\sqrt{3}, |\vec{q}|=2 \) and \( cos\hat{\theta}=\frac{1}{\sqrt{3}} \) where θ is the angle between \( \vec{P} \) and \( \vec{q} \) then \( |\vec{p}\times(\vec{q}-3\vec{r})|^{2}+3|\vec{r}|^{2} \) is equal to :

Let \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b\) be an ellipse, whose eccentricity is \(\frac{1}{\sqrt{2}}\) and the length of the latus rectum is \(\sqrt{14}\). Then the square of the eccentricity of \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) is :

Suppose that the number of terms in an A.P. is \(2 k, k \in N\). If the sum of all odd terms of the A.P. is 40 , the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then k is equal to :

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:

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to

Let \(T\) be the tangent to the ellipse \(E: x^{2}+4 y^{2}=5\) at the point \(P(1,1)\). If the area of the region bounded by the tangent \(T\), ellipse \(E\), lines \(x=1\) and \(x=\sqrt{5}\) is \(\alpha \sqrt{5}+\beta+\gamma \cos ^{-1}\left(\frac{1}{\sqrt{5}}\right)\), then \(|\alpha+\beta+\gamma|\) is equal to \(....\)

Let \(S=\{4,6,9\}\) and \(T=\{9,10,11, \ldots, 1000\}\). If \(A=\left\{a_{1}+a_{2}+\ldots+a_{k}: k \in N, a_{1}, a_{2}, a_{3}, \ldots, a_{k} \in S\right\}\) then the sum of all the elements in the set \(T - A\) is equal to \(......\)