The differential equation of the family of parabolas with vertex at (0, - 1) and having axis along the Y-axis is

• yy' + 2xy + 1 = 0

• xy' + y + 1 = 0

• xy' - 2y - 2

• xy' - y - 1 = 0

$\mathrm{The} \mathrm{solution} \mathrm{of} \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{y} + {\mathrm{e}}^{\raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.} \mathrm{with} \mathrm{y}\left(1\right) = 0 \mathrm{is}$

• $1 = \log \left(\mathrm{x}\right) + {\mathrm{e}}^{ \raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.}$

• $\log \left(\mathrm{x}\right) = {\mathrm{e}}^{ - \raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.}$

• $1 = 2\log \left(\mathrm{x}\right) + {\mathrm{e}}^{ - \raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.}$

• $\log \left(\mathrm{x}\right) + {\mathrm{e}}^{ - \raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.} = 1$

The solution of cos(y) + (xsin(y) - 1)dy/dx = 0 is, 

• $\mathrm{xsec}\left(\mathrm{y}\right) = \tan \left(\mathrm{y}\right) + \mathrm{C}$

• $\tan \left(\mathrm{y}\right) - \sec \left(\mathrm{y}\right) = \mathrm{Cx}$

• $\tan \left(\mathrm{y}\right) + \sec \left(\mathrm{y}\right) = \mathrm{Cx}$

• $\mathrm{xsec}\left(\mathrm{y}\right) + \tan \left(\mathrm{y}\right) = \mathrm{C}$

74.

Three non-zero non-collinear vectors $\hat{\mathrm{a}}, \hat{\mathrm{b}} \mathrm{and} \hat{\mathrm{c}}$ are such that $\hat{\mathrm{a}} + 3\hat{\mathrm{b}}$ is collinear with $\hat{\mathrm{c}}$, while $\hat{\mathrm{c}}$ is $3\hat{\mathrm{b}} + 2\hat{\mathrm{c}}$ collinear with a. Then $\hat{\mathrm{a}} + 3\hat{\mathrm{b}} + 2\hat{\mathrm{c}}$ equals

• 0

• $2\hat{\mathrm{a}}$

• $3\hat{\mathrm{b}}$

• $4\hat{\mathrm{c}}$

If $\hat{\mathrm{a}}, \hat{\mathrm{b}} \mathrm{and} \hat{\mathrm{c}}$ are non-coplanar vectors and if $\hat{\mathrm{d}}$ is such that $\hat{\mathrm{d}} = \frac{1}{\mathrm{x}}\left(\hat{\mathrm{a}} + \hat{\mathrm{b}} + \hat{\mathrm{c}}\right)$ and $\hat{\mathrm{d}} = \frac{1}{\mathrm{y}}\left(\hat{\mathrm{b}} + \hat{\mathrm{c}} + \hat{\mathrm{d}}\right)$ where x and y are non-zero real numbers, then $\frac{1}{\mathrm{xy}}\left(\hat{\mathrm{a}} + \hat{\mathrm{b}} + \hat{\mathrm{c}} + \hat{\mathrm{d}}\right)$ equals to

• 3c

• - a

• 0

• 2a

The angle between the lines $\hat{\mathrm{r}} = \left(2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) + \mathrm{\lambda }\left(\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$ and $\hat{\mathrm{r}} = \left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right) + \mathrm{\mu }\left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}\right)$ is

• $\frac{\mathrm{\pi }}{2}$

• ${\cos }^{-1}\left(\frac{9}{\sqrt{91}}\right)$

• ${\cos }^{-1}\left(\frac{7}{\sqrt{84}}\right)$

• $\frac{\mathrm{\pi }}{3}$

If a, b and c are vectors with magnitudes 2, 3 and 4 respectively, then the best upper bound of ${\left|\hat{\mathrm{a}} - \hat{\mathrm{b}}\right|}^2 + {\left|\hat{\mathrm{b}} - \hat{\mathrm{c}}\right|}^2 + {\left|\hat{\mathrm{c}} - \hat{\mathrm{a}}\right|}^2$ among the given values is

• 93

• 97

• 87

• 90

If x, y and z are non-zero real numbers and $\hat{\mathrm{a}} = \mathrm{x}\hat{\mathrm{i}} + 2\hat{\mathrm{j}}$, $\hat{\mathrm{b}} = \mathrm{y}\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and $\hat{\mathrm{c}} = \mathrm{x}\hat{\mathrm{i}} + \mathrm{y}\hat{\mathrm{j}} + \mathrm{z}\hat{\mathrm{k}}$ are such that $\hat{\mathrm{a}} \times \hat{\mathrm{b}} = \mathrm{z}\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}}$, then $\left[\hat{\mathrm{a}} \hat{\mathrm{b}} \hat{\mathrm{c}}\right]$ equals to

• 3

• 10

• 9

• 6

If x is real, then the minimum value of y = $\frac{{\mathrm{x}}^2 - \mathrm{x} + 1}{{\mathrm{x}}^2 + \mathrm{x} + 1} \mathrm{is}$

• 3

• $\frac{1}{3}$

• $\frac{1}{2}$

• 2

The locus of the centroid of the triangle with vertices at (acos(θ), asin(θ)), (bsin(θ), - bcos(θ)) and (1, 0) is (here, θ is a parameter)

• ${\left(3\mathrm{x} + 1\right)}^2 + 9{\mathrm{y}}^2 = {\mathrm{a}}^2 + {\mathrm{b}}^2$

• ${\left(3\mathrm{x} - 1\right)}^2 + 9{\mathrm{y}}^2 = {\mathrm{a}}^2 - {\mathrm{b}}^2$

• ${\left(3\mathrm{x} - 1\right)}^2 + 9{\mathrm{y}}^2 = {\mathrm{a}}^2 + {\mathrm{b}}^2$

• ${\left(3\mathrm{x} + 1\right)}^2 + 9{\mathrm{y}}^2 = {\mathrm{a}}^2 - {\mathrm{b}}^2$