The equation of the tangent of the ellipse \(4 x^2+9 y^2=36\) at the end of the latusrectum lying in the second quadrant, is

If \(\vec{a}, \vec{b}, \vec{c}\) are unit vectors such that \(\vec{a} \cdot \vec{b}=\vec{a} \cdot \vec{c}=0\) and the angle between \(\vec{b}\) and \(\vec{c}\) is \(\pi / 3\), then \(\vec{a}\) is equal to

If the line joining \((2,3,-1)\) and \((3,5,-3)\) is perpendicular to the line joining \(A(1,2,3)\) and \(B(\alpha, \beta, \gamma)\) then a possible point for \(B\) is

If $x$-coordinate of a point $P$ on the line joining the points $Q\left(2,2,1\right)$ and $R\left(5,2,-2\right)$ is $4$, then the $y$-coordinate of $P=$

Let \(A=\left[\begin{array}{lll}n & 0 & 0 \\ 0 & n & 0 \\ 0 & 0 & n\end{array}\right]\) and \(B=\left[\begin{array}{lll}0 & 0 & n \\ 0 & n & 0 \\ n & 0 & 0\end{array}\right]\). Then, \(A^2+B^2+A B=\)

Let ' \(\mathrm{O}\) ' be the origin, \(\mathrm{A}\) and \(\mathrm{B}\) be two points with position vectors \(-3 \hat{i}-3 \hat{j}+4 \hat{k}\) and \(4 \hat{i}-4 \hat{j}-3 \hat{k}\) respectively. Let \(P\) be a point such that the line drawn through \(P\) parallel to \(\overrightarrow{\mathrm{OB}}\) meets \(\mathrm{OA}\) in \(\mathrm{L}\) and another line through \(\mathrm{P}\) parallel to \(\overrightarrow{\mathrm{OA}}\) meets \(\mathrm{OB}\) in \(\mathrm{M}\). If \(\mathrm{L}\) divides \(\mathrm{OA}\) in the ratio \(2: 3\) and \(\mathrm{M}\) divides \(\mathrm{OB}\) in the ratio \(3: 2\), then the distance from 0 to \(\mathrm{P}\) is

\(T_m\) denotes the number of triangles that can be formed with the vertices of a regular polygon of \(m\) sides. If \(T_{m+1}-T_m=15\), then \(m\) is equal to

How many distinct alkenes obtained from the 3-Bromo3-methylhexane upon treatment with alc. \(\mathrm{KOH}\) ?

If \(f(t)=\frac{1+\operatorname{cosec} t}{1-\operatorname{cosec} t}\) for \(0 < t < \frac{\pi}{2}\) and \(f^{\prime}(t)=f(t) g(t)\), then \(g(t)=\)

The differential equation representing the family of circles having their centres on Y-axis is \(\left(y_1=\frac{d y}{d x}\right.\) and \(\left.y_2=\frac{d^2 y}{d x^2}\right)\)

If the line \(5 x-2 y-6=0\) is a tangent to the hyperbola \(5 x^2-k y^2=12\). then the equation of the normal to this hyperbola at the point \((\sqrt{6}, p)(p \lt 0)\) is

77. If \(5 f(x)+3 f\left(\frac{1}{x}\right)=2-\frac{1}{x}, x \neq 0\), then \(\int_1^2 f\left(\frac{1}{x}\right) d x=\)

The focal length of a thin converging lens in air is 20 cm . When the lens is immersed in a liquid, it behaves like a concave lens of power 1 D . If the refractive index of the material of the lens is 1.5 , the refractive index of the liquid is