Monotonicity in an Interval

Q. If $f\text{'}\left(x\right)=|x|-\left\{x\right\}$ where $x$ denotes the fractional part of $x$, then $f\left(x\right)$ is decreasing in

1. $(-\frac{1}{2},0)$
2. $(-\frac{1}{2},2)$
3. $(1,2)$
4. $(\frac{1}{2},\infty )$

Q. The domain of the function $f\left(x\right)=\sqrt{\frac{\left(x+1\right)\left(x-3\right)}{x-2}}$is
(a) [−1, 2) ∪ [3, ∞)
(b) (−1, 2) ∪ [3, ∞)
(c) [−1, 2] ∪ [3, ∞)
(d) None of these

Q. $\underset{x\to 0}{\lim }\frac{1-\cos 4x}{x^2}$

Q. $\underset{x\to 0}{\lim }\frac{\left(1-\cos 2x\right)\sin 5x}{x^2\sin 3x}=$

(a) 10/3
(b) 3/10
(c) 6/5
(d) 5/6

Q. Find the real values of θ for which the complex number $\frac{1+i\mathrm{cos\theta }}{1-2i\mathrm{cos\theta }}$ is purely real.

Q. $\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{\sqrt{2}-\sqrt{1+\sin x}}{{\cos }^{2}x}$

Q. Prove that:
${\sin }^2\left(\frac{\pi }{8}+\frac{x}{2}\right)-{\sin }^2\left(\frac{\pi }{8}-\frac{x}{2}\right)=\frac{1}{\sqrt{2}}\sin x$

Q. The domain of definition of the function $f\left(x\right)=\sqrt{\frac{x-2}{x+2}}+\sqrt{\frac{1-x}{1+x}}$is
(a) (−∞, −2] ∪ [2, ∞)
(b) [−1, 1]
(c) ϕ
(d) None of these

Q. If function $f\left(x\right)=\{\begin{array}{cc}1,& x=1\\ a^2-3a+x^2,& x>1\end{array}$ has a local maximum at $x=1,$ then the set of values of $a$ is

1. $(3,\infty )$
2. $(-\infty ,0)$
3. $(-\infty ,3)$
4. $(0,3)$

Q. Sketch the graphs of the following functions:

(i) f(x) = 2 sin x, 0 ≤ x ≤ π
(ii) $g\left(x\right)=3\sin \left(x-\frac{\pi }{4}\right),0\le x\le \frac{5\pi }{4}$
(iii) $h\left(x\right)=2\sin 3x,0\le x\le \frac{2\pi }{3}$
(iv) $\mathrm{\varphi }\left(x\right)=2\sin \left(2x-\frac{\pi }{3}\right),0\le x\le \frac{7\pi }{5}$
(v) $\mathrm{\psi }\left(x\right)=4\sin 3\left(x-\frac{\pi }{4}\right),0\le x\le 2\pi$
(vi) $\mathrm{\theta }\left(x\right)=\sin \left(\frac{x}{2}-\frac{\pi }{4}\right),0\le x\le 4\pi$
(vii) $u\left(x\right)={\sin }^{2}x,0\le x\le 2\pi v\left(x\right)=\left|\sin x\right|,0\le x\le 2\pi$
(viii) f(x) = 2 sin πx, 0 ≤ x ≤ 2

Q. If $f\left(x\right)=2x+{\cot }^{-1}x+\log \left(\sqrt{1+x^2}-x\right)$, then f(x)

1. increases in $\left(0,\infty \right)$
2. decreases in $\left(0,\infty \right)$
3. neither increases nor decreases in $\left(0,\infty \right)$
4. increases in $\left(-\infty ,\infty \right)$

Q. Find the radian measure corresponding to the following degree measures:
(i) 300°
(ii) 35°
(iii) −56°
(iv) 135°
(v) −300°
(vi) 7° 30'
(vii) 125° 30'
(viii) −47° 30'

Q. The general value of x satisfying the equation $\sqrt{3}\sin x+\cos x=\sqrt{3}$ is given by
(a) $x=n\pi +{\left(-1\right)}^n\frac{\pi }{4}+\frac{\pi }{3},n\in Z$

(b) $x=n\pi +{\left(-1\right)}^n\frac{\pi }{3}+\frac{\pi }{6},n\in Z$

(c) $x=n\pi \pm \frac{\pi }{6},n\in Z$

(d) $x=n\pi \pm \frac{\pi }{3},n\in Z$

Q. (x sin x + cos x ) (e^x + x² log x)

Q. ${\sin }^{3}x+{\sin }^3\left(\frac{2\pi }{3}+x\right)+{\sin }^3\left(\frac{4\pi }{3}+x\right)=-\frac{3}{4}\sin 3x$

Q. $\underset{x\to 0}{\lim }\frac{1-\cos 2x}{x}\mathrm{is}$

(a) 0
(b) 1
(c) 2
(d) 4

Q. How to solve the indefinite integral of (x^4 +1)/(x^6 +1)

Q. What is the differentiation of f(x)=y-2÷(3-x).

Q. Convert the following products into sum or difference. If angles are given in degrees, evaluate from tables 2 cos 5 theta cos 3 theta?

Q. for what value of thita is the expression 5+2icos thita/4-3icos thita purely real?

Q. Let f (x) = sinx + ax + b. Then f(x) = 0 has

1. only one real root which is positive if $a>1,b<0$
2. only one real root which is negative if $a>1,b<0$
3. only one real root which is negative if $a<-1,b>0$
4. CAN'T SAY ANYTHING

Q. $\underset{x\to \infty }{\lim }{a}^x\sin \left(\frac{b}{a^x}\right),a,b>1$ is equal to

(a) b
(b) a
(c) a log_e b
(d) b log_e a

Q. Prove the following identities (1-16)
${\sin }^{6}x+{\cos }^{6}x=1-3{\sin }^{2}x{\cos }^{2}x$

Q. x⁴ (5 sin x − 3 cos x)

Q. Prove that 4 cos 12 degrees × cos 48 degrees × cos 72 degrees = cos 36 degrees.