Logarithms

Q. The value of $(0.16{)}^{{\log }_{2.5}(\frac{1}{3}+\frac{1}{3^2}+\cdots \infty )}$ is

Q.

The solution set of the equation $pq{x}^2-{(p+q)}^{2}x+{(p+q)}^2=0$ is

1. $\left\{\frac{p}{q},\frac{q}{p}\right\}$

2. $\left\{pq,\frac{p}{q}\right\}$

3. $\left\{\frac{q}{p},pq\right\}$

4. $\left\{\frac{p+q}{p},\frac{p+q}{q}\right\}$

5. $\left\{\frac{p-q}{p},\frac{p-q}{q}\right\}$

Q.

The cube root of $0.008$ is

Q.

The number of solutions of the equation $\sqrt{x+1}-\sqrt{x-1}=\sqrt{4x-1}$ is / are

1. $2$

2. $0$

3. $3$

4. $1$

Q.

If the functions $f,g$ and $h$ are defined from the set of real numbers $R\to R$ such that $f\left(x\right)=x^2-1$, $g\left(x\right)=\sqrt{x^2+1}$ and $h\left(x\right)=\left\{\begin{array}{ll}0,& \mathrm{if}x\le 0\\ x,& \mathrm{if}x\ge 0\end{array}\right.$ then $h\circ \left(f\circ g\right)\left(x\right)$ is defined by

1. $1$

2. $0$

3. $-1$

4. $x^2$

Q. The value of ${(3^{\sqrt{{\log }_{3}4}}-4^{\sqrt{{\log }_{4}3}})}^2$ is

Q. Integrate the function $\frac{1}{\sqrt{8+3x-x^2}}$

Q. If $x_1$ and $x_2$ are two real solutions of the equation $(x{)}^{\ln x^2}=e^{18},$ then the product $(x_1.x_2)$ equals

1. $\frac{\left({\cot }^2{5}^{\circ }\right)\left({\cos }^2{5}^{\circ }\right)}{{\cot }^2{5}^{\circ }-{\cos }^2{5}^{\circ }}$
2. $\frac{\left({\tan }^2{10}^{\circ }\right)\left({\sin }^2{10}^{\circ }\right)}{{\tan }^2{10}^{\circ }-{\sin }^2{10}^{\circ }}$
3. $\sec 0+\sec \frac{\pi }{7}+\sec \frac{2\pi }{7}+\sec \frac{3\pi }{7}+\sec \frac{4\pi }{7}+\sec \frac{5\pi }{7}+\sec \frac{6\pi }{7}=1$
4. $\sqrt{1-{\sin }^2{110}^{\circ }}\cdot \sec {110}^{\circ }$

Q. The natural number $x,$ satisfying ${\log }_{10}(x^2-12x+36)=2$ is

Q. Consider the equation $(1+a+b{)}^2=3(1+a^2+b^2)$, where $a,b$ are real numbers. Then

1. there are infinitely many solution pairs $(a,b)$
2. there is no solution pair $(a,b)$
3. there are exactly two solution pairs $(a,b)$
4. there is exactly one solution pair $(a,b)$

Q. If $x_1$ and $x_2$ are two real solutions of the equation $(x{)}^{\ln x^2}=e^{18},$ then the product $(x_1.x_2)$ equals

1. $\frac{\left({\cot }^2{5}^{\circ }\right)\left({\cos }^2{5}^{\circ }\right)}{{\cot }^2{5}^{\circ }-{\cos }^2{5}^{\circ }}$
2. $\frac{\left({\tan }^2{10}^{\circ }\right)\left({\sin }^2{10}^{\circ }\right)}{{\tan }^2{10}^{\circ }-{\sin }^2{10}^{\circ }}$
3. $\sec 0+\sec \frac{\pi }{7}+\sec \frac{2\pi }{7}+\sec \frac{3\pi }{7}+\sec \frac{4\pi }{7}+\sec \frac{5\pi }{7}+\sec \frac{6\pi }{7}=1$
4. $\sqrt{1-{\sin }^2{110}^{\circ }}\cdot \sec {110}^{\circ }$

Q. If $\int \frac{2x+3}{x^2-5x+6}dx=9ln(x-3)-7ln(x-2)+A$, then A=

1. 5 ln (x-2) +constant
2. -4 ln (x-3)+constant
3. Constant
4. None of these
   

Q. The rationalising factor of $2\sqrt[3]{35}$ is _________.

1. $\sqrt[3]{35^2}$
2. $5^3$
3. $5^2$
4. $\sqrt[3]{35}$

Q. The value of ${\left(0.16\right)}^{{\log }_{2.5}(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^2}+\cdots \text{to}\infty )}$ is equal to

Q.

Integrate the function: $\frac{1}{\sqrt{(x-1)(x-2)}}$

Q. Integrate the rational function: $\frac{x}{(x+1)(x+2)}$

Q.

The set of real values of x satisfying the equation
${|x-1|}^{lo{g}_3\left(x^2\right)-2lo{g}_x\left(9\right)}=(x-1{)}^7$
is

1. {2}
   

2. {27, 81}
   

3. {81}
   

4. {2, 81}

Q. The range of $g\left(x\right)=\left|f\left|\frac{x}{2}\right|\right|$ is

1. $[1,\infty )$
2. $[2,\infty )$
3. $[0,\infty )$
4. None of these

Q. Let $S_n\left(x\right)={\log }_{a^{1/2}}x+{\log }_{a^{1/3}}x+{\log }_{a^{1/6}}x+{\log }_{a^{1/11}}x+{\log }_{a^{1/18}}x+{\log }_{a^{1/27}}x+\cdots \text{up to}n\text{-terms},$ where $a>1.$ If $S_{24}\left(x\right)=1093$ and $S_{12}\left(2x\right)=265,$ then value of $a$ is equal to

Q. If $A$ is the solution set of the equation ${\log }_{x}2\cdot {\log }_{2x}2={\log }_{4x}2$ and $B$ is the solution set of the equation $x^{{\log }_x(3-x{)}^2}=25,$ then $n(A\cup B)$ is equal to

1. $n(A\times B)$
2. $n(A-B)+n(B-A)$
3. $n((A-B)\times (B-A))$
4. $n(A\cup B)-n(A\cap B)$

Q. If $\frac{{\log }_{10}a}{12}=\frac{{\log }_{10}b}{21}=\frac{{\log }_{10}c}{15},$ then $bc$ is equal to

1. $a$
2. $a^2$
3. $a^3$
4. $a^4$

Q.

$C_1+2C_2+3C_3+4C_4+......+n{C}_n$ =

1. 

2. 

3. 

4. 

Q. $If{\int }_2^e[\frac{1}{logx}-\frac{1}{(logx{)}^2}]dx=\alpha +\frac{\beta }{log2},then$

1. $\alpha =-e,\beta =2$
2. $\alpha =e,\beta =-2$
3. $\alpha =e,\beta =2$
4. $\alpha =-e,\beta =-2$

Q. If $y^{\prime }-3y^{\prime }+2y=0$ where $y\left(0\right)=1,y^{\prime }\left(0\right)=0$, then the value of y at $x=log2$ is

1. $1$
2. $-1$
3. $0$
4. $2$

Q. If ${\log }_{4}A={\log }_{6}B={\log }_9(A+B)$, then $\left[\frac{4B}{A}\right]$ (where $[.]$ represents the greatest integer function) equals

Q. $\frac{d}{dx}{\log }_{|x|}e=$

1. $e^x$
2. $\frac{1}{(\log x{)}^2}$
3. $\frac{-1}{x(\log |x|{)}^2}$
4. $\frac{1}{|x|}$

Q. Solve the following inequality:
$\frac{x^2-7|x|+10}{x^2-6x+9}<0$

Q. Which of the following, when simplified reduces to unity?

1. ${\log }_{10}5.{\log }_{10}20+{\left({\log }_{10}2\right)}^2$
2. $\frac{2\log 2+\log 3}{\log 48-\log 4}$
3. $-{\log }_5{\log }_3\sqrt{\sqrt[5]{59}}$
4. $\frac{1}{6}{\log }_{\sqrt{3}/2}\left(\frac{64}{27}\right)$

Q. If the sum of two numbers $p$ and $q$ is $\sqrt{10}$ and their difference is $\sqrt{6}$, then the value of ${\log }_{q}p$ is

1. $-1$
2. $0$
3. $1$
4. $2$

Q. If $A$ is the solution set of the equation ${\log }_{x}2\cdot {\log }_{2x}2={\log }_{4x}2$ and $B$ is the solution set of the equation $x^{{\log }_x(3-x{)}^2}=25,$ then $n(A\cup B)$ is equal to

1. $n(A\times B)$
2. $n(A-B)+n(B-A)$
3. $n((A-B)\times (B-A))$
4. $n(A\cup B)-n(A\cap B)$