If $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} + \mathrm{\alpha }\hat{\mathrm{j}}$ are coplanar vectors, then te value of $\mathrm{\alpha }$ is :

• $- \frac{4}{3}$

• $\frac{3}{4}$

• $\frac{4}{3}$

• 2

If the position vectors of the vertices A, B, C of a triangle ABC are $7\hat{\mathrm{j}} + 10\hat{\mathrm{k}}$, $- \hat{\mathrm{i}} + 6\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ and $- 4\hat{\mathrm{i}} + 9\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ respectively, the triangle is :

• equilateral

• isosceless

• scalene

• right angled and isosceless also

The differential equation of all straight lines passing through origin is :

• $\mathrm{y} = \sqrt{\mathrm{x}}\frac{\mathrm{dy}}{\mathrm{dx}}$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{y} + \mathrm{x}$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{y} - \mathrm{x}$

• None of these

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are three unit vectors such that  $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}$, where $\overrightarrow{0}$ is null vector, then $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{a}} = \overrightarrow{0}$ is :

• - 3

• - 2

• $- \frac{3}{2}$

• 0

25.

The equation of the bisector of the acute angles between the lines 3x - 4y + 7=0 and 12x + 5y - 2 = 0 is :

• 99x - 27y - 81 = 0

• 11x - 3y + 9 = 0

• 21x + 77y - 101 = 0

• 21x + 77y + 101 = 0

To reduce the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{Py} = \mathrm{Q}\left(\mathrm{x}\right) . {\mathrm{y}}^{\mathrm{n}}$ to the linear form, the substitution is :

• $\mathrm{v} = \frac{1}{{\mathrm{y}}^{\mathrm{n}}}$

• $\mathrm{v} = \frac{1}{{\mathrm{y}}^{\mathrm{n} - 1}}$

• v = yⁿ

• v = y^{n - 1}

The area of the parallelogram whose adjacent sides are $\hat{\mathrm{i}} - \hat{\mathrm{k}}$ and $2\hat{\mathrm{j}} +\hspace{0.17em}3\hat{\mathrm{k}}$ is :

• 2

• 4

• $\sqrt{17}$

• $2\sqrt{13}$

Integrating factor of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} +\hspace{0.17em}\mathrm{P}\left(\mathrm{x}\right)\mathrm{y} = \mathrm{Q}\left(\mathrm{x}\right)$ is :

• $\int \mathrm{P} \mathrm{dx}$

• $\int \mathrm{Q} \mathrm{dx}$

• ${\mathrm{e}}^{\int \mathrm{P} \mathrm{dx}}$

• ${\mathrm{e}}^{\int \mathrm{Q} \mathrm{dx}}$