Logarithmic Inequalities

Q.

If $f\left(1\right)=3,f\left(1\right)=2$, then $\frac{d}{dx}(\log f(e^x+2x\left)\right)atx=0$ is equal to

Q.

If $\frac{2x}{\left(2x^2+5x+2\right)}>\frac{1}{(x+1)}$, then

1. $–2>x>–1$

2. $–2\ge x\ge –1$

3. $–2<x<–1$

4. $–2<x\le –1$

Q.

Let $\overrightarrow{a}$, $\overrightarrow{b}$ and $\overrightarrow{c}$ be three unit vectors such that ${\left|\overrightarrow{a}-\overrightarrow{b}\right|}^2+{\left|\overrightarrow{a}-\overrightarrow{c}\right|}^2=8$ , then ${\left|\overrightarrow{a}+2\overrightarrow{b}\right|}^2+{\left|\overrightarrow{a}+2\overrightarrow{c}\right|}^2$is equal to

Q.

There are $6$ batsmen, $7$ bowlers, and $2$ wickets keepers. Find the number of ways of forming a team of $11$ players having at least $4$ batsmen, at least $5$ bowlers, and at least $1$ wicket keeper.

1. $567$

2. $525$

3. $462$

4. $777$

Q.

The domain of the function $f\left(x\right)={\log }_{10}\left(\sqrt{x-4}+\sqrt{6-x}\right)$ is

1. $\left[4,6\right]$

2. $\left(-\infty ,6\right)$

3. $[2,3)$

4. None of these

Q.

Find the value of $\log 1000$.

Q.

$\frac{d\left\{{\left(\sin \left(x\right)\right)}^{\log \left(x\right)}\right\}}{dx}=$

1. ${\left(\sin \left(x\right)\right)}^{\log \left(x\right)}\left\{\left[\frac{\log \left(\sin \left(x\right)\right)}{x}\right]+\cot \left(x\right)\right\}$

2. ${\left(\sin \left(x\right)\right)}^{\log \left(x\right)}\left\{\left[\frac{\log \left(\sin \left(x\right)\right)}{x}\right]+\cot \left(x\right)\log \left(x\right)\right\}$

3. ${\left(\sin \left(x\right)\right)}^{\log \left(x\right)}\left\{\left[\frac{\log \left(\sin \left(x\right)\right)}{x}\right]+\log \left(x\right)\right\}$

4. none of these

Q. The solution set of ${\log }_{|\sin x|}(x^2-8x+23)>\frac{3}{{\log }_2|\sin x|}$ contains

1. $(3,\pi )$
2. $(\pi ,\frac{3\pi }{2})$
3. $(\frac{3\pi }{2},\frac{7\pi }{4}]$
4. $(\pi ,5)-\left\{\frac{3\pi }{2}\right\}$

Q.

The number of ways of selecting $10$ balls out of an unlimited number of white, red, blue and green balls is

1. $256$

2. $286$

3. $84$

4. none of these

Q.

In the mean value theorem $f\left(b\right)-f\left(a\right)=(b-a)f\left(c\right)$, if $a=4,b=9$and $f\left(x\right)=\sqrt{x}$, then the value of $c$is

1. $8.00$

2. 

3. 

4. 

Q. If ${\log }_{0.04}(x-1)\ge {\log }_{0.2}(x-1),$ then $x$ belongs to the interval

1. $(1,2]$
2. $(-\infty ,2]$
3. $[2,\infty )$
4. none of these

Q.

If $f\left(x\right)=\frac{x^2-1}{x^2+1}$, for every real number $x$, then the minimum value of $f$

1. Does not exist bacause $f$ is unbounded.

2. Is not attained even though $f$ is bounded.

3. Is equal to $1$.

4. Is equal to $-1$.

Q. The set of real values of $x$ satisfying ${\log }_{0.2}\left(\frac{x+2}{x}\right)\le 1$ is

1. $(-\infty ,-\frac{5}{2}]\cup (0,\infty )$
2. $[\frac{5}{2},\infty )$
3. $(-\infty ,-2)\cup (0,\infty )$
4. None of these

Q.

Let $f\left(x\right)=-1+\left|x-1\right|,-1\le x\le 3$ and $g\left(x\right)=2-\left|x+1\right|,-2\le x\le 2$. Then $f\circ g\left(x\right)=$

1. $x+1$

2. $-1-x$

3. $x-1$

4. $$\begin{gathered} x+1if-2\le x\le -1 \\ -1-xif-1\le x\le 0 \\ x-1if0\le x\le 2 \end{gathered}$$

Q. Let $a,b\in R^{+}$ such that ${\log }_{27}a+{\log }_{9}b=\frac{7}{2}$ and ${\log }_{27}b+{\log }_{9}a=\frac{2}{3}.$ Then the value of $ab$ is

1. $2^7$
2. $7^2$
3. $3^5$
4. $5^3$

Q. If ${\log }_2\left(\frac{x-1}{x-2}\right)>0$, then

1. $x<2$
2. $x>2$
3. $x<1$
4. $x>1$

Q.

There are $600$ students in the school. If $400$ of them can speak Telugu, and $300$ can speak Hindi, then the number of students who can speak both Telugu and Hindi is:

1. $100$

2. $200$

3. $300$

4. $400$

Q. The solution set of inequality $x^{({\log }_{10}x{)}^2-3({\log }_{10}x)+1}>1000$ is

1. $(1000,\infty )$
2. $(100,\infty )$
3. $(0,1000)$
4. $(-1000,\infty )$

Q. The number $N=6{\log }_{10}2+{\log }_{10}31$ lies between two successive integers whose sum is equal to

1. $5$
2. $6$
3. $7$
4. $9$

Q. Consider the logarithmic inequality $1+{\log }_5(x^2+1)\ge {\log }_5(a{x}^2+4x+a)$ for all real values of $x.$ The number of integers which $a$ cannot take, is

Q. The solution set of ${\log }_{(1-x)}(x-2)\ge -1$ is

1. $x\in (1,2)$
2. $x=0$
3. $x=\varphi$
4. $x\in (-\infty ,1)\cup (2,\infty )$

Q.

The value of ${\int }_2^3\frac{x+1}{x^2(x-1)}dx$ is

1. $\log \left(\frac{16}{9}\right)+\frac{1}{6}$

2. $\log \left(\frac{16}{9}\right)-\frac{1}{6}$

3. $2\log \left(2\right)-\frac{1}{6}$

4. $\log \left(\frac{4}{3}\right)-\frac{1}{6}$

Q. The values of $x$ for which the logarithmic function $f\left(x\right)={\log }_{|-x^2+2x-4|}\left(x\right(3-x\left)\right)$ is defined, is

1. $(0,3)$
2. $(0,3)-\left\{1\right\}$
3. $R^{+}-\left\{1\right\}$
4. $(-\infty ,0)\cup (3,\infty )$

Q. The least integer greater than ${\log }_{2}15\cdot {\log }_{1/6}2\cdot {\log }_3\frac{1}{6}$ is

Q. If $n$ is a natural number such that $n=p_1^{a_1}.p_2^{a_2}\cdots p_k^{a_k}$ and $p_1,p_2,p_3,\cdots ,p_k$ are distrinct prime numbers, then ${\log }_{e}n\ge$

1. $k\log 2$
2. $k\log 3$
3. $k\log 4$
4. none of these

Q.

A person goes in for an examination in which there are four papers with a maximum of $m$ marks from each paper.

The number of ways in which one can get $2m$marks is?

1. ${}^{2m+3}C_3$

2. $\frac{1}{3}(m+1)(2m^2+4m+1)$

3. $\frac{1}{3}(m+1)(2m^2+4m+3)$

4. None of these

Q.

Solution set of the inequality $lo{g}_{0.8}\left(lo{g}_6\frac{x^2+x}{x+4}\right)<0$

1. (-4, -3)

2. $(-3,4)\cup (8,\infty )$
   

3. $(-3,\infty )$
   

4. $(-4,-3)\cup (8,\infty )$

Q.

The complex number $\left(\frac{1+2i}{1-2i}\right)$ lies in which quadrant of the complex plane.

1. First

2. Second

3. Third

4. Fourth

Q. Sum of integers satisfying $\sqrt{{\log }_{2}x-1}-\frac{1}{2}{\log }_2\left(x^3\right)+2>0$ is

Q. The solution set of ${\log }_{1/3}(2^{x+2}-4^x)\ge -2$, is

1. $(-\infty ,2-\sqrt{13})$
2. $(-\infty ,2+\sqrt{13})$
3. $(-\infty ,2)$
4. $R$