Rational Function

Q.

$(3\mathrm{x}+4)(3\mathrm{x}-5)$ how to do it by using identities?

Q. The range of $f\left(x\right)=\frac{x^2-81}{x-9}$ is

1. $R-\left\{18\right\}$
2. $R-\left\{9\right\}$
3. $R-\{\pm 9\}$
4. $R-\{\pm 18\}$

Q. Let $f\left(x\right)$ be a cubic polynomial with $f\left(1\right)=-10,$ $f(-1)=6,$ and has a local minima at $x=1$, and $f^{\prime }\left(x\right)$ has a local minima at $x=-1.$ Then $f\left(3\right)$ is equal to

Q. The range of $f\left(x\right)=\frac{x^2+14x+9}{x^2+2x+3},x\in R$ is

1. $R-\{1,3\}$
2. $[-5,4]$
3. $R$
4. $R-\{-5,4\}$

Q. Suppose $f\left(x\right)=ax+b$ and $g\left(x\right)=bx+a,$ where $a,b$ are positive integers $a>b$. If $f\left(g\right(50\left)\right)-g\left(f\right(50\left)\right)=28,$ then the possible values of $ab$ is/are

1. $210$
2. $12$
3. $240$
4. $280$

Q. The range of $f\left(x\right)=\frac{x^2-81}{x-9}$ is

1. $R-\left\{18\right\}$
2. $R-\left\{9\right\}$
3. $R-\{\pm 9\}$
4. $R-\{\pm 18\}$

Q. Find the adjoint of the matrix $A=\left[\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right]$ and hence show that $A\left(\mathrm{adj}A\right)=\left|A\right|I_3$.

Q. If $f\left(x\right)=(a{x}^2+b{)}^3,b\in R,a\in R-\{0\}$ and $g\left(x\right)$ is a function such that $f\left(g\right(x\left)\right)=g\left(f\right(x\left)\right)=x,$ then $g\left(x\right)=$
(Given that $f$ and $g$ are bijective functions)

1. $\sqrt{\frac{x^{1/3}+b}{a}}$
2. $\sqrt{\frac{x^{1/3}-b}{a}}$
3. $\sqrt{\frac{x^3-b}{a}}$
4. $\sqrt{\frac{x^3+b}{a}}$

Q. Three pipes $X,$ $Y$ and $Z$ can fill a tank from empty to full in $10$ minutes, $20$ minutes, and $40$ minutes respectively. When the tank is empty, all the three pipes are opened. $X,$ $Y$ and $Z$ discharge chemical solutions $P,$ $Q$ and $R$ respectively. What is the proportion of the solution $Q$ in the liquid in the tank after $5$ minutes?

1. $\frac{7}{8}$
2. $\frac{2}{7}$
3. $\frac{5}{7}$
4. $\frac{1}{4}$

Q.

$\sum _{k=1}^{2n+1}{(-1)}^{k-1}{\left(k\right)}^2$equals

1. $(n-1)(2n-1)$

2. $(n+1)(2n+1)$

3. $(n+1)(2n-1)$

4. $(n-1)(2n+1)$

Q. The range of $f\left(x\right)=\frac{x^2+14x+9}{x^2+2x+3},x\in R$ is

1. $R-\{1,3\}$
2. $[-5,4]$
3. $R$
4. $R-\{-5,4\}$

Q. If LMVT holds for the function $f\left(x\right)=\ln x$ on the interval $[1,3]$ at $x=c,$ then which of the following is not true

1. $c=2{\log }_{3}e$
2. $3^c=e^2$
3. $c=\frac{1}{2}{\log }_{3}e$
4. $1<c<2$

Q. Let for all $x>0,f\left(x\right)=\underset{n\to \infty }{lim}n(x^{1/n}-1)$, then

1. $f\left(x\right)+f\left(\frac{1}{x}\right)=1$
2. $f\left(xy\right)=f\left(x\right)+f\left(y\right)$
3. $f\left(xy\right)=xf\left(y\right)+yf\left(x\right)$
4. $f\left(xy\right)=xf\left(x\right)+yf\left(y\right)$

Q. The range of $\frac{3x^2+9x+17}{3x^2+9x+7}$ is

1. $[1,41]$
2. $(0,10]$
3. $[11,\infty )$
4. $(1,41]$

Q. The interval where the function $\log (1+x)$ is continuous, is

1. $(0,\infty )$
2. $(-1,\infty )$
3. $(-\infty ,-1)$
4. None of the above

Q. The domain of definition of $f\left(x\right)=\sqrt{\frac{1-|x|}{2-|x|}}$ is

1. $(-\infty ,-1)\cup (2,\infty )$
2. $[-1,1]\cup (2,\infty )\cup (-\infty ,-2)$
3. $(-\infty ,1)\cup (2,\infty )$
4. $[-1,1]\cup (2,\infty )$

Q. The domain of the function $f\left(x\right)=2^{{\log }_{2}x}+\frac{2x+3}{x^2-4x+3}$ is

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

Q. Which of the following options is/are true for the function $f\left(x\right)=-\frac{1}{x}$

1. Domain is $R-\left\{0\right\}$
2. Range is $R-\left\{0\right\}$
3. Range is $R$
4. Graph is
   
   

Q. If $(5x+\frac{1}{5x})=7$; find the value of $(125x^3+\frac{1}{125x^3})$.

Q. At $60$ seconds, how far had this object traveled?

1. $10$ m
2. $20$ m
3. $30$ m
4. $40$ m

Q. The domain of the function $\frac{1}{\sqrt{x-|x|}}$ is

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

Q. The number of values of $x$ in the interval $[0,1]$ for which the function $f\left(x\right)=\frac{x+3}{|{\log }_{2}x|-6}$ is not defined are

1. $2$
2. $3$
3. $4$
4. Function exists everywhere $\forall x\in [0,1]$

Q. Let $f\left(x\right)=\frac{3}{x-2}-\frac{1}{x+3}$ and $g\left(x\right)=\frac{x^2-4x+19}{x^2+x-6}$. If $f\left(x\right)=g\left(x\right)$, then $x=$

1. $2$
2. $3$
3. $4$
4. $6$

Q. Range of $\frac{x^4-81}{x-3}$ is

1. $R-\left\{3\right\}$
2. $R-\left\{81\right\}$
3. $R$
4. $R-\left\{108\right\}$

Q. If $\left(f\right(x)-1){(x^2+x+1)}^2-\left(f\right(x)+1)(x^4+x^2+1)=0$ is true for all $x\in R-\left\{0\right\},$ then which of the following statements is (are) CORRECT ?

1. $\left|f\right(x\left)\right|\ge 2,\forall x\in R-\left\{0\right\}$
2. $f\left(x\right)$ has local maximum at $x=-1$
3. $f\left(x\right)$ has local minimum at $x=1$
4. $\underset{-\pi }{\overset{\pi }{\int }}\left(\cos x\right)f\left(x\right)dx=0$

Q. $f$ is a function defined as $\sum _{k=1}^{n}f(a+k)=16(2^n-1)$ and $f(x+y)=f\left(x\right).f\left(y\right)$ and $f\left(1\right)=2$ then integral value of $a$

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

Q. If $h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$, where $f\left(x\right)=x^2-2x$ and $g\left(x\right)=x^3-3x^2+2x,$ then the domain of $h\left(x\right)$ is

1. $R-\left\{2\right\}$
2. $R-\left\{1\right\}$
3. $R-\{0,2\}$
4. $R-\{0,1,2\}$

Q. If $\frac{3x+2}{(x+1)(2x^2+3)}=\frac{A}{(x+1)}+\frac{Bx+C}{(2x^2+3)}$ then $A+C-B=$

1. 0
2. 2
3. 5
4. 3

Q. Which of the following options is/are true for the function $f\left(x\right)=-\frac{1}{x}$

1. Domain is $R-\left\{0\right\}$
2. Range is $R-\left\{0\right\}$
3. Range is $R$
4. Graph is
   
   

Q. If f' (x) = 8x³ − 2x, f(2) = 8, find f(x)