61.

$\mathrm{If} \mathrm{a} = 2\hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{k}}, \mathrm{b} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \mathrm{c} = 4\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 7\hat{\mathrm{k}}$, then the vector r satisfying r x b = c x b and r · a = 0 is

• $\hat{\mathrm{i}} + 8\hat{\mathrm{j}} +\hspace{0.17em}2\hat{\mathrm{k}}$

• $\hat{\mathrm{i}} - 8\hat{\mathrm{j}} +\hspace{0.17em}2\hat{\mathrm{k}}$

• $\hat{\mathrm{i}} - 8\hat{\mathrm{j}} -\hspace{0.17em}2\hat{\mathrm{k}}$

• $- \hat{\mathrm{i}} - 8\hat{\mathrm{j}} +\hspace{0.17em}2\hat{\mathrm{k}}$

If a, b and c are three vectors such that $\left|\mathrm{a}\right| = 1, \left|\mathrm{b}\right| = 2, \left|\mathrm{c}\right| = 3 \mathrm{and} \mathrm{a} . \mathrm{b} = \mathrm{b} . \mathrm{c} = \mathrm{c} . \mathrm{a} = 0, \mathrm{then} \left|\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]\right| = ?$ is equal to

• 2

• 3

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• 5

$\mathrm{If} \left[\mathrm{a} \times \mathrm{b} \mathrm{b} \times \mathrm{c} \mathrm{c} \times \mathrm{a}\right] = \mathrm{\lambda }{\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]}^2, \mathrm{then} \mathrm{\lambda } \mathrm{is} \mathrm{equal} \mathrm{to}$

• 0

• 1

• 2

• 3

The cartesion equation of the plane passing through the point (3, - 2, - 1) and parallel to the vectors $\mathrm{b} = \hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 4\hat{\mathrm{k}} \mathrm{and} \mathrm{c} = 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 5\hat{\mathrm{k}} \mathrm{is}$

• 2x - 17y - 8z + 63 = 0 

• 3x + 17y + 8z + 36 = 0

• 2x + 17y + 8z + 36 = 0

• 3x - 16y + 8z - 63 = 0

The probability distribution of a random variable is given below

Then P(0 ) < X < 5) =?

• $\frac{1}{10}$

• $\frac{3}{10}$

• $\frac{8}{10}$

If (2, - 1, 2) and (K, 3, 5) are the triads of direction ratios of two lines and the angle between them is 45°, then the value of K is

• 2

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• 6

The length of perpendicular from the origin to the plane which makes intercepts $\frac{1}{3}, \frac{1}{4} \mathrm{and} \frac{1}{5}$ respectively on the coordinate axes is

• $\frac{1}{5\sqrt{2}}$

• $\frac{1}{10}$

• $5\sqrt{2}$

• 5

The volume of sphere is increasing at the rate of 1200 cu cm/s. The rate of increase in its surface area when the radius is 10 cm is

• 120 sqcm/s

• 240 sqcm/s

• 200 sqcm/s

• 100 sqcm/s

The slope of the tangent to the curve

$\mathrm{y} = {\int }_0^{\mathrm{x}}\frac{1}{1 + {\mathrm{t}}^3}d\mathrm{t} \mathrm{at} \mathrm{the} \mathrm{point}, \mathrm{where} \mathrm{x} = 1 \mathrm{is}$

• $\frac{1}{4}$

• $\frac{1}{3}$

• $\frac{1}{2}$

• 1

$\int \frac{\mathrm{dx}}{\left(\mathrm{x} - 1\right)\sqrt{{\mathrm{x}}^2 - 1}}$ is equal to

• $- \sqrt{\frac{\mathrm{x} - 1}{\mathrm{x} + 1}} + \mathrm{C}$

• $\sqrt{\frac{\mathrm{x} - 1}{{\mathrm{x}}^2 + 1}} + \mathrm{C}$

• $- \sqrt{\frac{\mathrm{x} + 1}{\mathrm{x} - 1}} + \mathrm{C}$

• $\sqrt{\frac{{\mathrm{x}}^2 + 1}{\mathrm{x} - 1}} + \mathrm{C}$