1. The number of values of $x$ in the internal $[0,3 \pi]$ satisfying the equation $2 \sin ^{2} x+5 \sin x-3=0$ is

(a). $6$

(b). $1$

(c). $2$

(d). $4$

Show Answer

Solution :

We have $2 \sin ^{2} x+5 \sin x-3=0$

$2 \sin ^{2} x+6 \sin x-\sin x-3=0$

$(2 \sin x-1)(\sin x+3)=0$

$\sin \mathrm{x}=\frac{1}{2} \quad(\because \sin \mathrm{x} \neq 3) \quad(\because-1 \leq \sin \mathrm{x} \leq 1)$

$y=\sin x$ and $y=\frac{1}{2}$ intersect in 4 points

$\therefore$ number of values of $\mathrm{x}$ in the internal $[0,3 \pi]$ is $4.$

2. The number of values of $\theta$ in the internal $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ such that $\theta \neq \frac{\mathrm{n} \pi}{5}$ for $\mathrm{n}=0, \pm 1, \pm 2$ and $\tan \theta$ $=\cot 5 \theta$ as well as $\sin 2 \theta=\cos 4 \theta$ is

3. The number of all possible values of $\theta$, where $0<\theta<\pi$, for which the system of equaitons $(\mathrm{y}+\mathrm{z}) \cos 3 \theta=(\mathrm{xyz}) \sin 3 \theta$

$x \sin 3 \theta=\frac{2 \cos 3 \theta}{y}+\frac{2 \sin 3 \theta}{z}$

and $(x y z) \sin 3 \theta=(y+2 z) \cos 3 \theta+y \sin 3 \theta$

have a solution $\left(\mathrm{x} _{0}, \mathrm{y} _{0}, \mathrm{z} _{0}\right)$ with $\mathrm{y} _{0} \mathrm{z} _{0} \neq 0$ is

4. The solution of the equation $4 \sin ^{4} x+\cos ^{4} x=1$ is

(a). $x=2 n \pi$

(b). $\mathrm{x}=\mathrm{n} \pi+1$

(c). $x=(n+2) \pi$

(d). none of these

5. The solution of the equation $[\sin x+\cos x]^{1+\sin 2 x}=2,-\pi \leq x \leq \pi$ is

(a). $\frac{\pi}{2}$

(b). $\pi$

(c). $\frac{\pi}{4}$

(d). $\frac{3 \pi}{4}$

6. The most general solutions $2^{\sin x}+2^{\cos x}=2^{1-\frac{1}{\sqrt{2}}}$ are

(a). $\mathrm{n} \pi-\frac{\pi}{4}$

(b). $\mathrm{n} \pi+\frac{\pi}{4}$

(c). $ \mathrm{n} \pi+(-1)^{\mathrm{n}} \frac{\pi}{4}$

(d). $2 \mathrm{n} \pi \pm \frac{\pi}{4}$

7. The set of values of $\theta$ satisfying the inequation $2 \sin ^{2} \theta-5 \sin \theta+2>0$, where $0<\theta<2 \pi$ is

(a). $\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, 2 \pi\right)$

(b). $\left[0, \frac{\pi}{6}\right] \cap\left[\frac{5 \pi}{6}, 2 \pi\right]$

(c). $\left[0, \frac{\pi}{3}\right] \cup\left[\frac{2 \pi}{3}, 2 \pi\right]$

(d). none of these

Show Answer

Solution : Given inequation $2 \sin ^{2} \theta-5 \sin \theta+2>0$

$2 \sin ^{2} \theta-4 \sin \theta-\sin \theta+2>0$

$2 \sin \theta(\sin \theta-2)-1(\sin \theta-2)>0$

$(2 \sin \theta-1)(\sin \theta-2)>0$

$\Rightarrow \sin \theta<\frac{1}{2}$ or $\sin \theta>2$

but $\sin \theta>2$ not possible

$\therefore \sin \theta<\frac{1}{2}$

If $\mathrm{x} \varepsilon\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, \pi\right)$ for $\mathrm{x} \in(0,2 \pi)$

$\therefore \sin \theta<\frac{1}{2} \Rightarrow \theta \in\left(0 \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, \pi\right)$ for $\theta \in(0,2 \pi)$.

Correct option is ’d’

8. Passage

Suppose equation is $f(x)-g(x)=0$ or $f(x)=g(x)=y$ say. Then draw the graphs of $y=f(x)$ and $y=g(x)$. If graphs of $\mathrm{y}=\mathrm{f}(\mathrm{x})$ and $\mathrm{y}=\mathrm{g}(\mathrm{x})$ cuts at one, two, three …., no points, then number of solutions are one, two, three, … zero respectively.

On the basis of above information, answer the following questions.

1. The number of solution of $\sin x=\frac{|x|}{10}$ is

(a). 4

(b). 6

(c). 8

(d). none of these

2. Total number of solutions of the equation $3 x+2 \tan x=\frac{5 \pi}{2}$ in $x \varepsilon[0,2 \pi]$ is equal to

(a). 1

(b). 2

(c). 3

(d). 4

3. Total number of solutions of $\sin \{x\}=\cos \{x\}$, where $\{$.$\}$ denotes the fractional part, in $[0, 2 \pi]$ is

(a). 3

(b). 5

(c). 7

(d). none of these

4. If $1-\sin \mathrm{x}=\frac{\sqrt{3}}{2}\left|\mathrm{x}-\frac{\pi}{2}\right|+\mathrm{a}$ has no solution when $\mathrm{a} \varepsilon \mathrm{R}^{+}$then

(a). $ \mathrm{a} \in \mathrm{R}^{+}$

(b). $ \mathrm{a}>\frac{3}{2}+\frac{\pi}{\sqrt{3}}$

(c). $\mathrm{a} \in\left(0, \frac{3}{2}+\frac{\pi}{\sqrt{3}}\right)$

(d). $ \mathrm{a} \in\left(\frac{3}{2}, \frac{3}{2}+\frac{\pi}{\sqrt{3}}\right)$

Show Answer

Solution :

1. Graphs of $y=\sin x$ and $y=\frac{|x|}{10}$ meet exactly six times. Hence no of solutions $=6$

2. We have $3 x+2 \tan x=\frac{5 \pi}{2}$ in $x \varepsilon[0,2 \pi]$

$\tan \mathrm{x}=\frac{5 \pi}{4}-\frac{3 \mathrm{x}}{2}$

Graphs of $y=\frac{5 \pi}{4}-\frac{3 \pi}{2}$ and $y=\tan x$ meet exactly three times in $[0,2 \pi]$

Thus number of solution $=3$

3. $\sin \{x\}=\cos \{x\}$

Graphs of $y=\sin \{x\}$ and $y=\cos \{x\}$ meet excatly 7 times in $[0,2 \pi]$

4. slope of $y=1-\sin x$ is $-\cos x$

slope of $\mathrm{y}=\frac{\sqrt{3}}{2}\left|\mathrm{x}-\frac{\pi}{2}\right|+$ a for $\mathrm{x}>\frac{\pi}{2}$ is $\frac{\sqrt{3}}{2}$

$\cos x=-\frac{\sqrt{3}}{2} \Rightarrow x=\frac{7 \pi}{6} \Rightarrow P=\left(\frac{7 \pi}{6}, \frac{3}{2}\right)$

If $\mathrm{y}=\frac{\sqrt{3}}{2}\left|\mathrm{x}-\frac{\pi}{2}\right|+\mathrm{a}$ passes throgh ’ $\mathrm{P}$ ’ then

$\mathrm{a}=\frac{3}{2}-\frac{\pi}{\sqrt{3}}$

$\Rightarrow \mathrm{a}>\frac{3}{2}-\frac{\pi}{\sqrt{3}}$

Practice questions

1. $\sin \theta+\sqrt{3} \cos \theta=6 x-x^{2}-11,0 \leq \theta \leq 4 \pi, x \in R$, holds for

(a). no values of $x$ and $\theta$

(b). one value of $x$ and two values of $\theta$

(c). two values of $x$ and two values of $\theta$

(d). two points of values of $(x, \theta)$

2. For $0 \leq x \leq 2 \pi$, then $2^{\operatorname{cosec}^{2} x} \sqrt{\frac{1}{2} y^{2}-y}+1 \leq \sqrt{2}$

(a). is satisfied by exactly one value of $y$

(b). is satisfied by exactly two value of $x$

(c). is satisfied by $\mathrm{x}$ for which $\cos \mathrm{x}=0$

(d). is satisfied by $x$ for which $\sin x=0$

3. Let $\tan x-\tan ^{2} x>0$ and $|2 \sin x|<1$. Then the intersection of which of the following two sets satisfies both the inequalities?

(a). $ \mathrm{x}>\mathrm{n} \pi, \mathrm{n} \in \mathrm{Z}$

(b). $ \mathrm{x}>\mathrm{n} \pi-\frac{\pi}{6}, \mathrm{n} \in \mathrm{Z}$

(c). $ \mathrm{x}<\mathrm{n} \pi-\frac{\pi}{4}, \mathrm{n} \in \mathrm{Z}$

(d). $ \mathrm{x}<\mathrm{n} \pi+\frac{\pi}{6}, \mathrm{n} \in \mathrm{Z}$

4. The equation $(\operatorname{cosp}-1) \mathrm{x}^{2}+(\cos p) \mathrm{x}+\sin \mathrm{p}=0$ in the varibale $\mathrm{x}$ has real roots. Then $\mathrm{p}$ can take any value in the interval

(a). $(0,2 \pi)$

(b). $(-\pi, 0)$

(c). $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

(d). $(0, \pi)$

5. Let $2 \sin ^{2} \mathrm{x}+3 \sin \mathrm{x}-2>0$ and $\mathrm{x}^{2}-\mathrm{x}-2<0$ ( $\mathrm{x}$ is measured in radians). Then $\mathrm{x}$ lies in the interval

(a). $\left(\frac{\pi}{6}, \frac{5 \pi}{6}\right)$

(b). $\left(-1, \frac{5 \pi}{6}\right)$

(c). $(-1,2)$

(d). $\left(\frac{\pi}{6}, 2\right)$

6. If $\left(\operatorname{cosec}^{2} \theta-4\right) x^{2}+(\cot \theta+\sqrt{3}) x+\cos ^{2} \frac{3 \pi}{2}=0$ holds true for all real $x$, then the most general values of $\theta$ can be given by ( $\varepsilon$ Z )

(a). $2 \mathrm{n} \pi+\frac{11 \pi}{6}$

(b). $2 n \pi+\frac{5 \pi}{6}$

(c). $ 2 n \pi \pm \frac{7 \pi}{6}$

(d). $\mathrm{n} \pi \pm \frac{11 \pi}{6}$

7. If $\sin ^{4} x+\cos ^{4} y+2=4 \sin x \cos y$ and $0 \leq x, y \leq \frac{\pi}{2}$, then $\sin x+\cos y$ is equal to

(a). $-2$

(b). $0$

(c). $2$

(d). $\frac{3}{2}$

8. The solution of the equation $\sin ^{10} \mathrm{x}+\cos ^{10} \mathrm{x}=\frac{29}{16} \cos ^{4} 2 \mathrm{x}$ is

(a). $\mathrm{x}=\frac{\mathrm{n} \pi}{4}+\frac{\pi}{8}, \mathrm{n} \varepsilon \mathrm{I}$

(b). $\mathrm{x}=\mathrm{n} \pi+\frac{\pi}{4}, \mathrm{n} \varepsilon \mathrm{I}$

(c). $ \mathrm{x}=2 \mathrm{n} \pi+\frac{\pi}{2}, \mathrm{n} \varepsilon \mathrm{I}$

(d). none of these

9. The most general values of $x$ for which $\sin x+\cos x=\min \left\{1, a^{2}-4 a+6\right\}$, a $R$ are given by

(a). $2 \mathrm{n} \pi, \mathrm{n} \varepsilon \mathrm{N}$

(b). $ 2 \mathrm{n} \pi+\frac{\pi}{2}, \mathrm{n} \varepsilon \mathrm{N}$

(c). $ \mathrm{n} \pi+(-1)^{\mathrm{n}} \frac{\pi}{4}-\frac{\pi}{4}, \mathrm{n} \varepsilon \mathrm{N}$

(d). none of these

10. Number of solutions of the equations $y=\frac{1}{3}[\sin x+[\sin x]]$ and $[y+[y]]=2 \cos x$ where $[$.$]$ denotes the greatest integer function is

(a). 0

(b). 1

(c). 2

(d). infinite

11. If $|\cos x|^{\sin ^{2} x-\frac{3}{2} \sin x+\frac{1}{2}}=1$, then possible values of $x$ are

(a). $\mathrm{n} \pi$ or $\mathrm{n} \pi+(-1)^{\mathrm{n}} \frac{\pi}{6}, \mathrm{n} \varepsilon \mathrm{I}$

(b). $n \pi$ or $2 n \pi+\frac{\pi}{2}$ or $n \pi+(-1)^{n} \frac{\pi}{6}$, $n \varepsilon Z$

(c). $\mathrm{n} \pi+(-1)^{\mathrm{n}} \frac{\pi}{6}, \mathrm{n} \varepsilon$ I

(d). $\mathrm{n} \pi, \mathrm{n} \varepsilon \mathrm{I}$

12. If $\left(\cos ^{2} x+\frac{1}{\cos ^{2} x}\right)\left(1+\tan ^{2} 2 y\right)(3+\sin 3 z)=4$, then

(a). $x$ may be multiple of $\pi$

(b). $x$ cannot be an even multiple of $\pi$

(c). $ \mathrm{z}$ can be a multiple of $\pi$

(d). $y$ can be a multiple of $\frac{\pi}{2}$

13. Matrix match type

14. If $3^{\sin 2 x+2 \cos ^{2} x}+3^{1-\sin 2 x+2 \cos ^{2} x}=28$, then the value of $x$ are given by

(a). $\tan x=1$

(b). $\tan x=-1$

(c). $\cos x=0$

(d). none of these

15. If $e^{\left\{\left(\sin ^{2} x+\sin ^{4} x+\sin ^{6} x+\ldots \infty\right) \log _{c}{ }^{2}\right\}}$ satisfies the equation $x^{2}-9 x+8=0$, then value of $\frac{\cos x}{\cos x+\sin x}$, $0<\mathrm{x}<\frac{\pi}{2}$ is

(a). $\frac{1}{2}(\sqrt{3}+1)$

(b). $\frac{1}{2}(\sqrt{3}-1)$

(c). $\frac{1}{2}$

(d). $0$

Integer type questions

16. If $\alpha, \beta$ satisfy the equaiton $12 \sin \alpha+5 \cos \alpha=2 \beta^{2}-8 \beta+21$, then $\beta^{2}-2 \beta$, is_______

17. The value of $x$ and $y$ satisfy the equaiton $\tan ^{2}(x+y)+\cot ^{2}(x+y)=1-2 x-x^{2}$, then the value of $x^{2}-$ $3 x+2$ is __________

18. The integral value of $p$ for which $\sqrt{\mathrm{p}} \cos x-2 \sin x=\sqrt{2}+\sqrt{2-p}$ has a solution, is __________