Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Trigonometric Functions

Multiple Choice Questions

111.

If a = $\hat{i} - \hat{j} + 2 \hat{k}$ and b = $2 \hat{i} - \hat{j} + \hat{k}$ , then the angle $θ$ between a and b is given by

$\tan^{- 1} (1)$
$\sin^{- 1} (\frac{1}{2})$
$\sec^{- 1} (1)$
$\tan^{- 1} (\frac{1}{\sqrt{3}})$

112.
At t = 0, the function f(t) = $\frac{\sin (t)}{t}$ has

a minimum

a discontinuity

a point of inflexion

a maximum

a maximum

$Given, f (t) = \frac{\sin (t)}{t}$

At t = 0, first we will check continuity of the function

Now, LHL = f(0 - h)

$= \lim_{h \to 0} \frac{\sin (0 - h)}{(0 - h)} = \lim_{h \to 0} \frac{- \sin (h)}{- h} = 1$

$RHL = f (0 + h) = \lim_{h \to 0} \frac{\sin (0 + h)}{(0 + h)} = \lim_{h \to 0} \frac{\sin (h)}{h} = 1$

and f(0) = 1

Since, LHL = RHL = f(0)

So, the function is continuous att = 0.

Now, we check the function is maximum or minimum

$f' (t) = \frac{1}{t} \cos (t) - \frac{1}{t^{2}} \sin (t) and f'' (t) = \frac{- 1}{t} \sin (t) - \frac{1}{t^{2}} \cos (t) - \frac{1}{t^{2}} \cos (t) + \frac{2}{t^{3}} \sin (t) = \frac{- \sin (t)}{t} - \frac{2 \cos (t)}{t^{2}} + \frac{2 \sin (t)}{t^{3}}$

For maximum or minimum value of f(x), put

$f' (x) = 0 \Rightarrow \frac{\cos (t)}{t} - \frac{\sin (t)}{t^{2}} = 0 \Rightarrow \frac{\tan (t)}{t} = 1$

$Now, \lim_{t \to 0} f'' (t) = - \lim_{t \to 0} (\frac{\sin (t)}{t}) - 2 \lim_{t \to 0} (\frac{tcos (t) - \sin (t)}{t^{3}}) [\frac{0}{0} form] = - 1 - 2 \lim_{t \to 0} (\frac{\cos (t) - tsin (t) - \cos (t)}{3 t^{2}}) [∵ using L' Hospital rule] = - 1 + \frac{2}{3} \lim_{t \to 0} \frac{\sin (t)}{t} = - 1 + \frac{2}{3} \times 1 = \frac{- 1}{3} < 0$

So, function f(t) is maximum at t = 0.

113.

If $∆_{r} = |\begin{matrix} 2 r - 1 & {}^{m}C_{r} & 1 \\ m^{2} - 1 & 2^{m} & m + 1 \\ \sin^{2} (m^{2}) & \sin^{2} (m) & \sin^{2} (m + 1) \end{matrix}|$ , then the value of $\sum_{r = 0}^{m} ∆_{r}$

1
0
2
None of these

114.

If $\cos (α) + isin (α), b = \cos (β) + isin (β), c = \cos (γ) + isin (γ)$ and $\frac{b}{c} + \frac{c}{a} + \frac{a}{b} = 1$ , then $\cos (β - γ) + \cos (γ - α) + \cos (α - β)$ is equal to

$\frac{3}{2}$
- $\frac{3}{2}$
0
1

115.

The maximum value of 4 sin²(x) - 12sin(x) + 7 is

25
4
does not exist
None of the above

116.

A line making angles 45° and 60° with the positive directions of the axes of x and y makes with the positive direction of z-axis, an angle of

60°
120°
60° or 120°
None of these

117.

If I = $[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ , J = $[\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}]$ and $B = [\begin{matrix} \cos (θ) & \sin (θ) \\ - \sin (θ) & \cos (θ) \end{matrix}]$ , then B is equal to