The length of longer diagonal of the parallelogram constructed on 5a + 2b and a - 3b, if it is given that $\left|\mathrm{a}\right| = 2\sqrt{2}, \left|\mathrm{b}\right| = 3$and the angle between a and b is $\frac{\mathrm{\pi }}{4}$, is

• 15

• $\sqrt{113}$

• $\sqrt{593}$

• $\sqrt{369}$

If the gradient of the tangent at any point (x, y) of acurve which passes through the point $\left(1, \frac{\mathrm{\pi }}{4}\right)$ is $\left\{\frac{\mathrm{y}}{\mathrm{x}} - {\sin }^2\left(\frac{{\displaystyle \mathrm{y}}}{{\displaystyle \mathrm{x}}}\right)\right\}$, then the equation of the curve is

• $\mathrm{y} = {\cot }^{-1}\left({\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)$

• $\mathrm{y} = {\cot }^{-1}\left({\log }_{\mathrm{e}}\left(\frac{\mathrm{x}}{\mathrm{e}}\right)\right)$

• $\mathrm{y} = {\mathrm{xcot}}^{-1}\left({\log }_{\mathrm{e}}\left(\mathrm{ex}\right)\right)$

• $\mathrm{y} = {\cot }^{-1}\left({\log }_{\mathrm{e}}\left(\frac{\mathrm{e}}{\mathrm{x}}\right)\right)$

If a plane meets the coordinate axes at A, B and C in such a way that the centroid of $∆$ABC is at the point (1, 2, 3), then equation of the plane is

• $\frac{\mathrm{x}}{1} + \frac{\mathrm{y}}{2} + \frac{\mathrm{z}}{3} = 1$

• $\frac{\mathrm{x}}{3} + \frac{\mathrm{y}}{6} + \frac{\mathrm{z}}{9} = 1$

• $\frac{\mathrm{x}}{1} + \frac{\mathrm{y}}{2} + \frac{\mathrm{z}}{3} = \frac{1}{3}$

• None of these

If 3p and 4p are resultant of force 5p,then angle between 3p and 5p is

• ${\sin }^{-1}\left(\frac{3}{5}\right)$

• ${\sin }^{-1}\left(\frac{4}{5}\right)$

• $90°$

• None of these

If A(- 2, 1), B(2, 3) and C( - 2, - 4) are three points. Then the angle between BA and BC is

• ${\tan }^{-1}\left(\frac{2}{3}\right)$

• ${\tan }^{-1}\left(\frac{3}{2}\right)$

• ${\tan }^{-1}\left(\frac{1}{3}\right)$

• ${\tan }^{-1}\left(\frac{1}{2}\right)$

In a $∆\mathrm{ABC}$, $\frac{\left(\mathrm{a} + \mathrm{b} + \mathrm{c}\right)\left(\mathrm{b} + \mathrm{c} - \mathrm{a}\right)\left(\mathrm{c} + \mathrm{a} - \mathrm{b}\right)\left(\mathrm{a} + \mathrm{b} - \mathrm{c}\right)}{4{\mathrm{b}}^2{\mathrm{c}}^2}$ equals

• ${\cos }^2\left(\mathrm{A}\right)$

• ${\cos }^2\left(\mathrm{B}\right)$

• ${\sin }^2\left(\mathrm{A}\right)$

• ${\sin }^2\left(\mathrm{B}\right)$

417.

The angle between the lines whose direction cosines satisfy the equations l + m + n = 0, l² + m² - n² = 0 is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

The plane through the point (- 1, - 1, - 1) and containing the line of intersection of the planes $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$ = 0 and $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$ = 0 is

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + 5\hat{\mathrm{j}} - 5\hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$

If $\mathrm{\theta }$ is the angle between the lines AB and AC where A, B and C are the three points with coordinates (1, 2, - 1), (2, 0, 3), (3, - 1, 2) respectively, then $\sqrt{462} \cos \left(\mathrm{\theta }\right)$ is equal to

• 20

• 10

• 30

• 40

Let the pairs $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{d}}$ each determine a plane. Then the planes are parallel, if :

• $\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}\right) \mathrm{and} \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{d}}\right) = \overrightarrow{0}$

• $\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}\right) . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{d}}\right) = 0$

• $\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) \times \left(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{d}}\right) = \overrightarrow{0}$

• $\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) . \left(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{d}}\right) = 0$