There is an error of $\pm 0.04$cm in the measurement of the diameter of a sphere. When radius is 10cm, the percentage error in the volume of the sphere is

• $\pm 1.2$

• $\pm 1.0$

• $\pm 0.8$

• $\pm 0.6$

$\mathrm{The} \mathrm{function} \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^3 +\hspace{0.17em}{\mathrm{ax}}^2 + \mathrm{bx} +\hspace{0.17em}\mathrm{c}, {\mathrm{a}}^2 \le 3\mathrm{b} \mathrm{has}$

• one maximum value

• one minimum value

• no extreme value

• one maximum and one minimum value

$\mathrm{The} \mathrm{maximum} \mathrm{value} \mathrm{of} \frac{\log \left(\mathrm{x}\right)}{\mathrm{x}}, 0 < \mathrm{x} < \infty \mathrm{is}$

• $\infty$

• e

• 1

• e ^{- 1}

$\mathrm{z} = \tan \left(\mathrm{y} +\hspace{0.17em}\mathrm{ax}\right) +\hspace{0.17em}\sqrt{\mathrm{y} - \mathrm{ax}} \Rightarrow {\mathrm{z}}_{\mathrm{xx}} - {\mathrm{a}}^2{\mathrm{z}}_{\mathrm{yy}} =?$

• 0

• 2

• ${\mathrm{z}}_{\mathrm{x}} + {\mathrm{z}}_{\mathrm{y}} = 0$

• ${\mathrm{z}}_{\mathrm{x}}{\mathrm{z}}_{\mathrm{y}}$

In a quadrilateral ABCD, the point P divides DC in the ratio 1 : 2 and Q is the mid point of AC. If$\overrightarrow{\mathrm{AB}} + 2\overrightarrow{\mathrm{AD}} + \overrightarrow{\mathrm{BC}} - 2\overrightarrow{\mathrm{DC}} = \mathrm{k}\overrightarrow{\mathrm{PQ}},$ then k is equal

• - 6

• - 4

• 6

• 4

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

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

$$\begin{gathered} \mathrm{If} \overrightarrow{\mathrm{a}} = - \hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}}, \mathrm{and} \overrightarrow{\mathrm{c}} = - 2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 3\hat{\mathrm{k}}, \\ \mathrm{then} \mathrm{the} \mathrm{angle} \mathrm{between} 2\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{c}} \mathrm{and} \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} \mathrm{is} \end{gathered}$$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{2}$

$$\begin{gathered} \mathrm{If} {\mathrm{m}}_1, {\mathrm{m}}_2, {\mathrm{m}}_3 \mathrm{and} {\mathrm{m}}_4 \mathrm{are} \mathrm{respectively} \mathrm{the} \mathrm{magnitudes} \mathrm{of} \mathrm{the} \mathrm{vectors} \\ {\overrightarrow{\mathrm{a}}}_1 = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}, {\overrightarrow{\mathrm{a}}}_2 = 3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} - 4\hat{\mathrm{k}}, \\ {\overrightarrow{\mathrm{a}}}_3 = \hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} {\overrightarrow{\mathrm{a}}}_4 = - \hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}, \mathrm{then} \mathrm{the} \mathrm{correct} \mathrm{order} \mathrm{of} {\mathrm{m}}_1, {\mathrm{m}}_2, {\mathrm{m}}_3 \mathrm{and} {\mathrm{m}}_4 \mathrm{is} \end{gathered}$$

• ${\mathrm{m}}_3 < {\mathrm{m}}_1 < {\mathrm{m}}_4 < {\mathrm{m}}_2$

• ${\mathrm{m}}_3 < {\mathrm{m}}_1 < {\mathrm{m}}_2 < {\mathrm{m}}_4$

• ${\mathrm{m}}_3 < {\mathrm{m}}_4 < {\mathrm{m}}_1 < {\mathrm{m}}_2$

• ${\mathrm{m}}_3 < {\mathrm{m}}_4 < {\mathrm{m}}_2 < {\mathrm{m}}_1$

$$\begin{gathered} \mathrm{Suppose} \overrightarrow{\mathrm{a}} = \mathrm{\lambda }\hat{\mathrm{i}} - 7\hat{\mathrm{j}} + 3\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = \mathrm{\lambda }\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\mathrm{\lambda }\hat{\mathrm{k}}. \mathrm{If} \\ \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{a} \mathrm{and} \mathrm{b} \mathrm{is} \mathrm{greater} \mathrm{than} 90°, \mathrm{then} \mathrm{A} \mathrm{satisfies} \mathrm{the} \mathrm{inequality} \end{gathered}$$

• $- 7 < \mathrm{\lambda } < 1$

• $\mathrm{\lambda } > 1$

• $1 < \mathrm{\lambda } < 7$

• $- 5 < \mathrm{\lambda } < 1$

70.

$$\begin{gathered} \mathrm{The} \mathrm{volume} \mathrm{of} \mathrm{the} \mathrm{tetrahedron} \mathrm{having} \mathrm{the} \mathrm{edge} \\ \hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}, \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \hat{\mathrm{i}} - \hat{\mathrm{j}} + \mathrm{\lambda }\hat{\mathrm{k}} \mathrm{as} \mathrm{coterminous}, \mathrm{is} \frac{2}{3}\mathrm{cubic} \mathrm{unit}. \mathrm{Then} \mathrm{\lambda } = ? \end{gathered}$$

• 1

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