Product Law

Q.

If $f\left(x\right)=\frac{ax+b}{cx+d}$, then $fof\left(x\right)=x$ provided that:

1. $d=-a$

2. $d=a$

3. $a=b=c=d=1$

4. $a=b=1$

Q.

$(x-3)(x-4)=\frac{34}{{33}^2}$.Solve

Q.

The value of $\left(2.{}^1{P}_0-3.{}^2{P}_1+4.{}^3{P}_2-.......upto{51}^{st}term\right)+\left(1!-2!+3!-.......upto{51}^{st}term\right)$ is equal to

1. $1-51\left(51!\right)$

2. $1+52!$

3. $1$

4. $1+51!$

Q.

If the function $f\left(x\right)$ defined by $f\left(x\right)=\frac{x^{100}}{100}+\frac{x^{99}}{99}+...+\frac{x^2}{2}+x+1$, then $f\left(0\right)$ is equal to:

1. $100f\left(0\right)$

2. $100$

3. $1$

4. $-1$

Q.

$\frac{2^{288}}{2^n}=512$ . What Is the Value of $n$ ?

Q.

Question 3 (iii)
Simplify:

$\frac{{11}^{\frac{1}{2}}}{{11}^{\frac{1}{4}}}$

Q.

When $a=3$, $b=0$, and $c=-2$ , find the value of :

$a^3+b^3+c^3-3abc$

Q.

If ${}^{n}C_{12}={}^{n}C_6$, then ${}^{n}C_2$ is equal to

1. $172$

2. $153$

3. $306$

4. $2556$

Q.

$1m^3=\_\_\_\_\_\_\_c{m}^3$

Q. If $\frac{9^n\times 3^2\times (3^{-n/2}{)}^{-2}-(27{)}^n}{3^{3m}\times 2^3}$=$\frac{1}{27}$, prove that m - n = 1.

Q.

For which value of $x$, the function$f\left(x\right)=x^2–2x$ is decreasing

1. $x>1$

2. $x>2$

3. $x<1$

4. $x<2$

Q.

$x^{m{x}^{m{x}^{m{x}^{......}}}}=y^{n{y}^{n{y}^{n{y}^{......}}}}$, then $\frac{dy}{dx}$is equal to:

1. $\raisebox{1ex}{$y$}\!\left/ \!\raisebox{-1ex}{$x$}\right.$

2. $\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$y$}\right.$

3. $\raisebox{1ex}{$my$}\!\left/ \!\raisebox{-1ex}{$nx$}\right.$

4. $\raisebox{1ex}{$ny$}\!\left/ \!\raisebox{-1ex}{$mx$}\right.$

Q.

If $1{\mathrm{m}}^2=\mathrm{x}{\mathrm{mm}}^2$, then the value of $\mathrm{x}$ is

1. $1000$

2. $10000$

3. $100000$

4. $1000000$

Q.

An integer $m$is said to be related to another integer $n$if m is a multiple of $n$then the relation is

1. $Reflexiveandsymmetric$

2. $Reflexiveandtransitive$

3. $Symmetricandtransitive$

4. $Equivalencerelation$

Q. If $\frac{9^n\times 3^2\times (3^{-n/2}{)}^{-2}-(27{)}^n}{3^{3m}\times 2^3}$=$\frac{1}{27}$, prove that m - n = 1.

Q.

If $C_r={}^{2n+1}C_r$ then, $C_0-{C_1}^2+{C_2}^2+....+(-1){{}^{2n+1}C^2}_{2n+1}$ is equal to?

1. ${\left({}^{2n+1}C_n-{}^{2n+1}C_{2n+1}\right)}^2$

2. ${}^{2n}C_n$

3. $\frac{1}{n}\left({}^{2n}C_n\right)$

4. $0$

Q.

The set of possible values of $x$ is such that $5^x+{(2\sqrt{3})}^{2x}-169$is always positive is

1. $[3,\infty )$

2. $[2,\infty )$

3. $(2,\infty )$

4. None of these

Q.

The product $2^{\frac{1}{4}}\times 4^{\frac{1}{16}}\times 8^{\frac{1}{48}}\times {16}^{\frac{1}{128}}......\infty$ is equal to

1. $2^{\frac{1}{4}}$

2. $2$

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

4. $1$

Q.

If $y=x^2+\frac{1}{x^2+{\displaystyle \frac{1}{x^2+{\displaystyle \frac{1}{x^2.....\infty }}}}}$ then $\frac{dy}{dx}$ is equal to:

1. $\frac{2xy}{2y-x^2}$

2. $\frac{xy}{y+x^2}$

3. $\frac{xy}{y-x^2}$

4. $\frac{2x}{2+{\displaystyle \frac{x^2}{y}}}$

Q.

Find the value of m, if $(-10{)}^{m+1}$ × $(-10{)}^5$ = $(-10{)}^7$.

1. -1

2. 3

3. 2

4. 1

Q. Which of the following equations are correct?

$\left(i\right)(a+b+c{)}^2=a^2+b^2+c^2+2(ab+bc+ca)$
$\left(ii\right)(a+b{)}^3=a^3+b^3+3ab(a-b)$
$\left(iii\right)(a-b{)}^3=a^3-b^3-3ab(a+b)$
$\left(iv\right)a^2-b^2=(a+b)(a-b)$

1. (iv) and (iii)
2. (i) and (iv)
3. (ii), (iii) and (iv)
4. (iv) only

Q. If $4^x+2^{2x-1}=3^{x+\frac{1}{2}}+3^{x-\frac{1}{2}}$, then x is equal to

1. $\frac{1}{2}$
2. $\frac{3}{2}$
3. $\frac{5}{2}$
4. $1$

Q. Solve for $x$ in the equation $3^{(2+x)}+3^{(1+x)}=2916.$

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

Q. If $2^x-2^{x-2}=192$, then the value of $x$ is

1. $5$
2. $7$
3. $6$
4. $8$

Q.

A rectangular block has 20, 25 and 30 as length, width and height respectively. If the sides of the block are extended by the same length x, what is the equation for the new volume?

1. V(x)=$x^3+65x^2+1850x+15000$

2. V(x)=$x^3+75x^2+185x+15000$

3. V(x)=$x^3+75x^2+1850x+15000$

4. V(x)=$x^3+75x^2+1850x+1500$

Q. If ${\left(\frac{a}{b}\right)}^{x-1}={\left(\frac{b}{a}\right)}^{x-3}$, then find $x$.

1. $2$
2. $1$
3. $3$
4. $4$

Q.

Solve

$3^{x^2}:3^x=9:1$.

1. (2, -1)

2. (2, -2)

3. (1, -1)

4. (1, -2)

Q. Simplify
(i) $\frac{{11}^{\frac{1}{2}}}{{11}^{\frac{1}{4}}}$ .
(ii) $7^{\frac{1}{2}}{.8}^{\frac{1}{2}}$

Q. If $4^x-4^{x-1}=24$, then the value of $(2x{)}^x$ is:

1. $(5{)}^{\frac{3}{2}}$
2. 4
3. $(5{)}^{\frac{5}{2}}$
4. $\frac{1}{5}$

Q.

Question 62

In the following question, fill in the blanks to make the statements true.

By solving $(6^0-7^0)\times (6^0+7^0)$, we get ___ .