If the line of intersection of the planes 2x + 3y + z = 1 and x + 3y + 2 = 2 makes angle $\mathrm{\alpha }$ with positive direction of x  axis, then $\cos \left(\mathrm{\alpha }\right)$ will be equal to

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{\sqrt{5}}$

• $\frac{1}{\sqrt{7}}$

• $\frac{1}{\sqrt{3}}$

The value of $\int \frac{\cos \left(\sqrt{\mathrm{x}}\right)}{\sqrt{\mathrm{x}}}\mathrm{dx}$ will be

• $2\sin \left(\sqrt{\mathrm{x}}\right) + \mathrm{c}$

• $2\cos \left(\sqrt{\mathrm{x}}\right) + \mathrm{c}$

• $2\sin \left(\mathrm{x}\right) + \mathrm{c}$

• $\sqrt{2}\sin \left(\mathrm{x}\right) + \mathrm{c}$

The value of ${\int }_2^3\frac{\sqrt{\mathrm{x}}}{\sqrt{5 - \mathrm{x}} + \sqrt{\mathrm{x}}}\mathrm{dx}$ will be

• $\frac{\sqrt{3}}{2}$

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{2}$

• $\frac{1}{\sqrt{3}}$

The area common to the curves y² = x and x² = y will be

• 1 sq unit

• $\frac{2}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{1}{4} \mathrm{sq} \mathrm{unit}$

• $\frac{1}{3} \mathrm{sq} \mathrm{unit}$

If x + y $\le$ 2; x $\ge$ 20, y $\ge$ 20, then the point, at which the maximumvalue of 3x + 2y is attained, will be

• (0, 0)

• $\left(\frac{1}{2}, \frac{{\displaystyle 1}}{{\displaystyle 2}}\right)$

• (2, 0)

• (0, 2)

The maximum value of P = 6x + 8y, if 2x + y $\le$ 30, x + 2y $\le$ 24; x $\ge$ 20, y $\ge$ 20, will be

• 90

• 120

• 96

• 240

Regression of saving (s) of a family on income y may be expressed as s = a + $\frac{\mathrm{y}}{\mathrm{m}}$, where a and m are constants. In a random sample of 100 families the variance of savings is one quarter of the variance of incomes and the correlation coefficient is found to be 0.8, the value of m is

• 0.8

• 1.25

• 0.25

• None of these

The integral ${\int }_1^{10}{\mathrm{x}}^3\mathrm{dx}$ is approximately evaluated by Trapezoidal rule ${\int }_1^{10}{\mathrm{x}}^3\mathrm{dx}$ = $3\left[\frac{1 + {10}^3}{2} + \mathrm{\alpha } + 7^3\right]$ for  n = 3, then the value of $\mathrm{\alpha }$ is

• 4³

• 4²

• 5³

• None of these

A fair coin is tossed n times. If the probability of getting 7 heads is equal to the probability of getting 9 heads, then the value of n will be

• 8

• 13

• 15

• None of these

50.

The probabilities of solving a question by three students
are $\frac{1}{2}, \frac{1}{4}, \frac{1}{6}$ respectively. What is the probability that the question is solved ?

• $\frac{35}{48}$

• $\frac{1}{48}$

• $\frac{11}{16}$

• $\frac{2}{11}$