Which of the following is not always true?

• ${\left|\mathrm{a} + \mathrm{b}\right|}^2 = {\left|\mathrm{a}\right|}^2 + {\left|\mathrm{b}\right|}^2$, if a and b are perpendicular to each other

• $\left|\mathrm{a} + \mathrm{\lambda b}\right| \ge \left|\mathrm{a}\right| \mathrm{for} \mathrm{all} \mathrm{\lambda } \in \mathrm{R}$,  if a and b are perpendicular to each other

• ${\left|\mathrm{a} + \mathrm{b}\right|}^2 +\hspace{0.17em}{\left|\mathrm{a} - \mathrm{b}\right|}^2 = 2\left({\left|\mathrm{a}\right|}^2 + {\left|\mathrm{b}\right|}^2\right)$

• $\left|\mathrm{a} + \mathrm{\lambda b}\right| \ge \left|\mathrm{a}\right| \mathrm{for} \mathrm{all} \mathrm{\lambda } \in \mathrm{R}$, if a is parallel to b

If the four points with position vectors $- 2\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \mathrm{\lambda }\hat{\mathrm{j}} + \hat{\mathrm{k}}$ are coplanar, then $\mathrm{\lambda }$ is equal to

• 1

• 2

• - 1

• 0

The area of the region bounded by the curve $\mathrm{y} = {\mathrm{x}}^3$, its tangent at (1, 1) and X-axis, is

• $\frac{1}{12} \mathrm{sq} \mathrm{units}$

• $\frac{1}{6} \mathrm{sq} \mathrm{units}$

• $\frac{2}{17}$ sq units

• $\frac{2}{15}$ sq units

The value of $\mathrm{\lambda }$ for which the straight line $\frac{\mathrm{x} - \mathrm{\lambda }}{3} = \frac{\mathrm{y} - 1}{2 + \mathrm{\lambda }} = \frac{\mathrm{z} - 3}{- 1}$ may lie on the plane x - 2y = 0, is

• 2

• 0

• $- \frac{1}{2}$

• $\mathrm{there} \mathrm{is} \mathrm{no} \mathrm{such} \mathrm{\lambda }$

Area of the region bounded by $\mathrm{y} = \left|\mathrm{x}\right| \mathrm{and} \mathrm{y} = - \left|\mathrm{x}\right|$ + 2 is

• 4 sq units

• 3 sq units

• 2 sq units

• 1 sq units

Let f(x) denotes the fractional part of a real number x. Then, the value of ${\int }_0^{\sqrt{3}}\mathrm{f}\left({\mathrm{x}}^2\right)d\mathrm{x}$

• $2\sqrt{3} - \sqrt{2} - 1$

• 0

• $\sqrt{2} - \sqrt{3} + 1$

• $\sqrt{3} - \sqrt{2} + 1$

The value of $\int \frac{\left(\mathrm{x} - 2\right)}{{\left\{{\left(\mathrm{x} - 2\right)}^2 {\left(\mathrm{x} + 3\right)}^7\right\}}^{1/3}}\mathrm{dx}$

• $\frac{3}{20}{\left(\frac{\mathrm{x} - 2}{\mathrm{x} +\hspace{0.17em}3}\right)}^{4/3} + \mathrm{C}$

• $\frac{3}{20}{\left(\frac{\mathrm{x} - 2}{\mathrm{x} +\hspace{0.17em}3}\right)}^{3/4} + \mathrm{C}$

• $\frac{5}{12}{\left(\frac{\mathrm{x} - 2}{\mathrm{x} +\hspace{0.17em}3}\right)}^{4/3} + \mathrm{C}$

• $\frac{3}{20}{\left(\frac{\mathrm{x} - 2}{\mathrm{x} +\hspace{0.17em}3}\right)}^{5/3} + \mathrm{C}$

If 5 distinct balls are placed at random into 5 cells, then the probability that exactly one cell remains empty, is

• 48/125

• 12/125

• 8/125

• 1/125

A survey of people in a given region showed that 20% were smokers. The probability of death due to lung cancer, given that a person smoked, "was 10 times the probability of death due to lung cancer, given that a person did not smoke. If the probability of death due to lung cancer in the region is 0.006. What is the probability of death due to lung cancer given that a person is a smoker?

• 1/140

• 1/70

• 3/140

• 1/10

80.

Suppose a machine produces metal parts that contain some defective parts with probability 0.05. How many parts should be produced in order that the probability of atleast one part being defective is 1/2 or more? (Given that, log₁₀ 95 = 1.977 and log₁₀2 = 0.3)

• 11

• 12

• 15

• 14