Three points $\mathrm{O}(0,0), \mathrm{P}\left(\mathrm{a}, \mathrm{a}^2\right), \mathrm{Q}\left(-\mathrm{b}, \mathrm{b}^2\right), \mathrm{a}>0, \mathrm{~b}>0$, are on the parabola $y=x^2$. Let $S_1$ be the area of...

The lines $\mathrm{L}_1, \mathrm{~L}_2, \ldots, \mathrm{I}_{20}$ are distinct. For $\mathrm{n}=1,2,3, \ldots, 10$ all the lines $\mathrm{L}_{2 \mathrm{n}-1}$ are parallel to...

Let $\vec{a}=\hat{\imath}+\hat{\jmath}+\hat{k}, \vec{b}=-\hat{\imath}-8 \hat{\jmath}+2 \hat{k}$ and $\overrightarrow{\mathrm{c}}=4 \hat{\imath}+\mathrm{c}_2 \hat{\jmath}+\mathrm{c}_3 \hat{\mathrm{k}}$ be three vectors such that $\vec{b} \times \vec{a}=\vec{c} \times \vec{a}$....

Let C be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x+y=2$ intersects the...

If $\lambda>0$, let $\theta$ be the angle between the vectors $\vec{a}=\hat{\imath}+\lambda \hat{\jmath}-3 \hat{k}$ and $\vec{b}=3 \hat{\imath}-\hat{\jmath}+2 \hat{k}$. If the vectors...

If the function $f(x)=\left\{\begin{array}{ll}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}} & , x \neq 0 \\ \log _e 2 \log _e 3 & , x=0\end{array}\right.$...

If $y=\frac{(\sqrt{x}+1)\left(x^2-\sqrt{x}\right)}{x \sqrt{x}+x+\sqrt{x}}+\frac{1}{15}\left(3 \cos ^2 x-5\right) \cos ^3 x$, then $96 y^{\prime}\left(\frac{\pi}{6}\right)$ is equal to :

Vanillin compound obtained from vanilla beans, has total sum of oxygen atoms and  electrons is__

Let $f:(0, \infty) \rightarrow R$ and $F(x)=\int_0^x t f(t) d t$. If $F\left(x^2\right)=\mathrm{x}^4+\mathrm{x}^5$, then $\sum_{\mathrm{r}=1}^{12} \mathrm{f}\left(\mathrm{r}^2\right)$ is equal to :

2.7 Kg of each of water and acetic acid are mixed, The freezing point of the solution will be $...

Let $A=I_2-2 M^{\mathrm{T}}$, where M is real matrix of order $2 \times 1$ such that the relation $\mathrm{M}^{\mathrm{T}} \mathrm{M}=\mathrm{I}_1$ holds....

A first row transition metal with highest enthalpy of atomisation, upon reaction with oxygen at high temperature forms oxides of...

The total number of 'sigma' and 'Pi' bonds in 2-oxohex-4-ynoic acid is____.

Consider the following reaction, the rate expression of which is given below
$...

From 6.55 g of aniline, the maximum amount of acetanilide that can be prepared will be $\_\_\_\_$ $\times 10^{-1} \mathrm{~g}$.

Three moles of an ideal gas are compressed isothermally from 60 L to 20 L using constant pressure of 5...

Number of compounds/species from the following with non-zero dipole moment is $\_\_\_\_$ $...

The maximum number of orbitals which can identified with $\mathrm{n}=4$ and $\mathrm{m}_l=0$ is $\_\_\_\_$