Condition for Monotonically Increasing Function

Q.

Let $a,b,c\in R$ be all non-zero and satisfy $a^3+b^3+c^3=2$. If the matrix $A=\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]$, satisfies $A^{T}A=I$, then value of $abc$can be:

1. $\frac{2}{3}$

2. $3$

3. $-\frac{1}{3}$

4. $\frac{1}{3}$

Q.

If $A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]then{A}^{100}=$

1. $2^{100}A$

2. $2^{99}A$

3. $100A$

4. $299A$

Q. The function $f\left(x\right)=x^3-6x^2+ax+b$ is such that $f\left(2\right)=f\left(4\right)=0$. Consider two statements.

$\left(S1\right)$ There exists $x_1,x_2\in (2,4),x_1<x_2$, such that $f^{\prime }\left(x_1\right)=-1$ and $f^{\prime }\left(x_2\right)=0$

$\left(S2\right)$ There exists $x_3,x_4\in (2,4),x_1<x_4$, such that $f$ is decreasing in $(2,x_4)$, increasing in $(x_4,4)$ and $2f^{\prime }\left(x_3\right)=\sqrt{3}f\left(x_4\right)$

1. Both $\left(S1\right)$ and $\left(S2\right)$ are true
2. Both $\left(S1\right)$ and $\left(S2\right)$ are false
3. $\left(S1\right)$ is true and $\left(S2\right)$ is false
4. $\left(S1\right)$ is false and $\left(S2\right)$ is true

Q. Let $f\left(x\right)=\frac{x}{\sqrt{a^2+x^2}}-\frac{d-x}{\sqrt{b^2+(d-x{)}^2}},x\in R$,

where $a,b$ and $d$ are non-zero real constants, Then :

1. $f$ is an increasing function of $x$
2. $f$ is a decreasing function of $x$
3. $f$ is neither increasing nor decreasing function of $x$
4. $f^{\prime }$ is not a continuous function of $x$

Q. Let $f,g:R\to R$ be defined by $f\left(x\right)=x^2-\frac{\cos x}{2}$ and $g\left(x\right)=\frac{x\sin x}{2}$. Then

1. $f\left(x\right)=g\left(x\right)$ has exactly one solution.
2. $f\left(x\right)=g\left(x\right)$ has exactly two solutions.
3. $f\left(x\right)-g\left(x\right)\to \infty$ as $x\to -\infty$
4. The minimum value of $f\left(g\right(x\left)\right)$ is $-1.$

Q. $P$ is the extremity of the latus rectum of ellipse $3x^2+4y^2=48$ in the first quadrant. The eccentric angle of $P$ is

1. $\frac{\pi }{8}$
2. $\frac{3\pi }{4}$
3. $\frac{\pi }{3}$
4. $\frac{2\pi }{3}$

Q.

If the ${\mathrm{p}}^{\mathrm{th}}$, ${\mathrm{q}}^{\mathrm{th}}$ and ${\mathrm{r}}^{\mathrm{th}}$ terms of a GP are $a,b$and $c$ respectively. Prove that ${\mathrm{a}}^{\mathrm{q}-\mathrm{r}}{\mathrm{b}}^{\mathrm{r}-\mathrm{p}}{\mathrm{c}}^{\mathrm{p}-\mathrm{q}}=1$

Q.

In the interval $[0,1]$ the function $x^2–x+1$ is:

1. Increasing

2. Decreasing

3. Neither increasing nor decreasing

4. None of the above

Q.

On the interval $(1,3)$ the function $f\left(x\right)=3x+\left(\frac{2}{x}\right)$is

1. Strictly decreasing

2. Strictly increasing

3. Decreasing in (2, 3) only

4. Neither increasing nor decreasing

Q.

If $f$ and $g$ are differentiable functions such that $f\left(2\right)=3,f\left(2\right)=1,f\left(3\right)=7,g\left(2\right)=-5$ and $g\left(2\right)=2$, then find the numbers indicated in problem $\left(fg\right)\left(2\right)$.

Q. A function $f\left(x\right)$ is monotonic increasing or decreasing according as $f^{\prime }\left(x\right)>0\text{or}{f}^{\prime }\left(x\right)<0.$ Let us consider a function $f\left(x\right)={\int }_0^x\{2\sqrt{2}{\sin }^{2}t+(2-\sqrt{2})\sin t-1\}dt$

$0<x<2\pi .$ Then

1. b
2. c
3. a
4. d

Q. For the function y = f(x) the formal definition of derivative is $f^{\prime }\left(x\right)={lim}_{\Delta x\to 0}f(X+\Delta X)-f\left(X\right).$

1. True
2. False

Q. The value of ${\left(\frac{1+\sqrt{3}i}{1-\sqrt{3}i}\right)}^{64}+{\left(\frac{1-\sqrt{3}i}{1+\sqrt{3}i}\right)}^{64}$ is

1. $0$ (zero)
2. $-1$
3. $1$
4. $i$

Q. Assertion :If $\left[x\right]$ denotes the integral part of $x$, then domain of the function $f\left(x\right)=g\left(x\right)+h\left(x\right)$, where $g\left(x\right)=\frac{\sqrt{3-x}}{(x-1)(x-2)(x-3)}$ and $h\left(x\right)={\sin }^{-1}\left[\frac{3x-2}{2}\right]$ is $[0,2)-\left\{1\right\}$ Reason: Domain of $h\left(x\right)$ is $[0,2)$

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion is incorrect but Reason is correct

Q. $f\left(x\right)=cos\left(\frac{\pi }{x}\right),x\ne 0$ increases in the interval

1. 
2. 
3. None of these
4. 

Q. Show that $f\left(x\right)=\cos x$ is an decreasing function on $(0,\pi )$ increasing in $(-\pi ,0)$ and neither increasing nor decreasing in $(-\pi ,\pi )$.

Q. Let $f\left(x\right)$ be defined by $f\left(x\right)=\{\begin{array}{c}\sin 2xif0<x\le \pi /6\\ ax+bif\pi /6<x\le 1\end{array}$.

1. $a=1,b=1/\sqrt{2}+\pi /6$
2. $a=1/\sqrt{2},b=1/\sqrt{2}$
3. $a=1,b=\sqrt{3}/2-\pi /6$
4. $a=\sqrt{3}/2,b=\sqrt{3}/2+\pi /6$

Q.

Which of the following function is a monotonically increasing function?

1. $f\left(x\right)=x^3+2x$

2. $f\left(x\right)=x^3+2x$

3. $f\left(x\right)=x^2+2x$

4. $f\left(x\right)=cosx$

Q.

Which of the following condition is true for a monotonically increasing function f(x), which is differentiable in its domain?

1. f’(x) > 0

2. f’(x) ≥ 0

3. f’(x) < 0

4. f’(x) ≤ 0

Q. Let $f\left(x\right)=\frac{x}{\sqrt{a^2+x^2}}-\frac{d-x}{\sqrt{b^2+(d-x{)}^2}},x\in R$,

where $a,b$ and $d$ are non-zero real constants, then :

1. $f$ is an increasing function of $x$
2. $f$ is neither increasing nor decreasing function of $x$
3. $f$ is a decreasing function of $x$
4. $f^{\prime }$ is not a continuous function of $x$

Q.

Which of the following condition is true for a monotonically increasing function f(x), which is differentiable in its domain?

1. $f^{\prime }\left(x\right)>0$

2. $f^{\prime }\left(x\right)\ge 0$

3. $f^{\prime }\left(x\right)<0$

4. $f^{\prime }\left(x\right)\le 0$

Q. Show that $f\left(x\right)=\sin x$ is increasing on $(0,\frac{\pi }{2})$ and decreasing on $(\frac{\pi }{2},\pi )$ and neither inreasing nor decreasing in $(0,\pi )$.

Q. $f\left(x\right)=cos\left(\frac{\pi }{x}\right),x\ne 0$ increases in the interval

1. 
2. 
3. 
4. None of these

Q. $y=e^{asin-1}x\Rightarrow (1-x^2)y_n+2-(2n+1)x{y}_n+1$ is equal to

1. $-(n^2-a^2)y^2$
2. $(n^2+a^2)y_n$
3. $(n^2-a^2)y_n$
4. $-(n^2+a^2)y_n$

Q. Show that the function $x^2-x+1$ is neither increasing nor decreasing on $(0,1)$.

Q.

Which of the following function is a monotonically increasing function?

1. f(x) = x² +2x

2. f(x) = x³ + 2x

3. f(x) = x³ + 2x

4. f(x) = cos x

Q. Is the function defined by $f\left(x\right)=x^2-\sin x+5$ continuous at $x=\pi$?

Q. Assertion (A): Let $f:R\to R$ be a function such that $f\left(X\right)=X^3+X^2+3X+\sin X$, then $f$ is one to one .
Reason (R): $f\left(x\right)$ is neither increasing nor decreasing function.

1. Both (A) and (R) are true and (R) is the correct explanation of (A).
2. (A) is true but (R) is false.
3. (A) is false but (R) is true.
4. Both (A) and (R) are true and (R) is not the correct explanation of (A).

Q. If $f:[1,10]\to [1,10]$ is a non-decreasing function and $g:[1,10]\to [1,10]$ is a non-increasing function. Let $h\left(x\right)=f\left(g\right(x\left)\right)$ with $h\left(1\right)=1$, then $h\left(2\right)$

1. is equal to $1$
2. is not defined
3. lies in $(1,2)$
4. is more than $2$

Q. Let $f\left(x\right)=\int \frac{x}{2}{h}^{\prime }\left(x\right)$, where $h\left(x\right)=\frac{x^2-1}{x^2}$. If $f\left(2\right)=\frac{1}{2}$ and $g_r\left(x\right)=\underset{rtimes}{\underset{}{f\left(f\right(f(....f(x)....)\left)\right)}}$ i.e., $g_1\left(x\right)=f\left(x\right),g_2\left(x\right)=f\left(f\right(x\left)\right)$ and so on, then identify the CORRECT statement(s).

1. $\frac{d}{dx}\left(g_{(3n-2)}\right(x\left)\right)=1$ (whenever exists) for $n\in N$
2. $\frac{d}{dx}\left(g_{\left(3n\right)}\right(x\left)\right)=1$ (whenever exists) for $n\in N$
3. $\underset{x\to 1}{lim}\frac{\sum _{r=1}^{100}\left(g_{3r}\right(x){)}^r+x-101}{x-1}=5050$
4. Equation of the tangent to the curve $y=g_{80}\left(x\right)$ when $x=2$ is $x-y-3=0$