A survey shows that 63% of the peoplein a city read newspaper A whereas 76% read newspaper B. If x% of the people read both the newspapers, then a possible value of x can be :

• 65

• 55

• 37

• 29

Two vertical poles AB = 15m and CD = 10m are standing apart on a horizontal ground with points A and C on the ground. If P is the point of intersection of BC and AD, then the height of P (in m) above the line AC is:

• 10/3

• 6

• 5

• 20/3

Let [t] denote the greatest integer $\le$ t. Then the equation in x, [x]² + 2[x + 2] – 7 = 0 has :

• no integral solution

• infinitely many solutions

• exactly two solutions

• exactly four integral solutions

$\mathrm{If} \left(\mathrm{a} + \sqrt{2}\mathrm{bcos}\left(\mathrm{x}\right)\right)\left(\mathrm{a} - \sqrt{2}\mathrm{bcos}\left(\mathrm{y}\right)\right) = {\mathrm{a}}^2 - {\mathrm{b}}^2, \mathrm{where} \mathrm{a} > \mathrm{b}, \mathrm{then} \frac{\mathrm{dx}}{\mathrm{dy}} \mathrm{at} \left(\frac{\mathrm{\pi }}{4}, \frac{\mathrm{\pi }}{4}\right) \mathrm{is} :$

• $\frac{2\mathrm{a} + \mathrm{b}}{2\mathrm{a} - \mathrm{b}}$

• $\frac{\mathrm{a} + \mathrm{b}}{\mathrm{a} - \mathrm{b}}$

• $\frac{\mathrm{a} - \mathrm{b}}{\mathrm{a} + \mathrm{b}}$

• $\frac{\mathrm{a} - 2\mathrm{b}}{\mathrm{a} + 2\mathrm{b}}$

Let x₀ be the point of local maxima of $$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right), \mathrm{where} \overrightarrow{\mathrm{a}} = \mathrm{x}\hat{\mathrm{i}} - 2\hat{\mathrm{j}} +\hspace{0.17em}3\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = - 2\hat{\mathrm{i}} + \mathrm{x}\hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \\ \overrightarrow{\mathrm{c}} = 7\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + \mathrm{x}\hat{\mathrm{k}}. \mathrm{Then} \mathrm{the} \mathrm{value} \mathrm{of} \overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{a}} \mathrm{at} \mathrm{x} = {\mathrm{x}}_0 \end{gathered}$$

is :

• - 22

• 14

•  - 4

•  - 30

$\mathrm{Let} \mathrm{f}\left(\mathrm{x}\right) = \int \frac{\sqrt{\mathrm{x}}}{{\left(1 + \mathrm{x}\right)}^2}\mathrm{dx} \left(\mathrm{x} \ge 0\right). \mathrm{Then} \mathrm{f}\left(3\right) - \mathrm{f}\left(1\right) = ?$

• $- \frac{\mathrm{\pi }}{6} + \frac{1}{2} + \frac{\sqrt{3}}{4}$

• $- \frac{\mathrm{\pi }}{12} + \frac{1}{2} + \frac{\sqrt{3}}{4}$

• $\frac{\mathrm{\pi }}{12} + \frac{1}{2} - \frac{\sqrt{3}}{4}$

• $\frac{\mathrm{\pi }}{6} + \frac{1}{2} - \frac{\sqrt{3}}{4}$

$$\begin{gathered} \mathrm{Let} \mathrm{f} \mathrm{be} \mathrm{the} \mathrm{twice} \mathrm{differential} \mathrm{function} \mathrm{on} \left(1, 6\right). \mathrm{If} \mathrm{f}\left(2\right) = 8, \mathrm{f}\text{'}\left(2\right) = 5, \\ \mathrm{f}\text{'}\left(\mathrm{x}\right) \ge 1 \mathrm{and} \mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right) \ge 4, \mathrm{for} \mathrm{all} \mathrm{x} \in \left(1, 6\right), \mathrm{then} : \end{gathered}$$

• $\mathrm{f}\left(5\right) \hspace{0.17em}+ \mathrm{f}\text{'}\left(5\right) \ge 28$

• $\mathrm{f}\text{'}\left(5\right) + \mathrm{f}\text{'}\text{'}\left(5\right) \le 20$

• $\mathrm{f}\left(5\right) \le 10$

• $\mathrm{f}\left(5\right) + \mathrm{f}\text{'}\left(5\right) \le 26$

18.

$\mathrm{Let} {\left(2{\mathrm{x}}^2 + 3\mathrm{x} + 4\right)}^{10} = \sum _{\mathrm{r} = 0}^{20} {\mathrm{a}}_{\mathrm{r}}{\mathrm{x}}^{\mathrm{r}}. \mathrm{Then} \frac{{\mathrm{a}}_7}{{\mathrm{a}}_{13}} = ?$

Suppose a differentiable function f(x) satisfies the identity f(x + y) = f(x) + f(y) + xy² + x²y, for all real x and y. If

$\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{x}} = 1$, then f'(3) is equal to :

If the system of equations

x – 2y + 3z = 9

2x + y + z = b

x – 7y + az = 24,

has infinitely many solutions, then a – b is equal to :