Free Printable Road to A

1.

Find the values of a and b that satisfy the equation $2+ai=\frac{6-2i}{b+i}$ .

I. $a=-2,\ b=2$

II. $a=-4,\ b=1$

III. $a=0,\ b=1$

IV. $a=-1,\ b=3$

a)

I and II

b)

I, III and IV

c)

IV only

d)

All of the above

2.

Solve the inequality $5x>2x^2+3$ .

a)

$\left(-\infty,\infty\right)$

b)

$\left[1,\frac{3}{2}\right]$

c)

$\left(1,\frac{3}{2}\right)$

d)

$\left(-\infty,1\right)\cup\left(\frac{3}{2},\infty\right)$

3.

Given $\frac{2x-1}{x-1}-3\ge0$ and the solution is $\left(1,2\right]$ . Given $\frac{2x-1}{x-1}+3>\le0$ and the solution is $\left[\frac{4}{5},1\right)$ . Find $\left|\frac{2x-1}{x-1}\right|\ge3$

a)

$\left[\frac{4}{5},1\right)\cup\left(1,2\right]$

b)

$\left(\frac{4}{5},1\right)\cup\left(1,2\right)$

c)

$\left(-\infty,\infty\right)$

d)

$\left(-\infty,\frac{4}{5}\right)\cup\left(1,2\right)$

4.

Solve the equation $2\left(3^{2x+1}\right)+7\left(3^x\right)-3=0$ .

a)

$x=-1$

b)

$x=-1,\ x=-\frac{3}{2}$

c)

$x=1,\ x=-\frac{3}{2}$

d)

$x=1$

5.

Find the domain of $f\left(x\right)=\frac{4x-1}{\sqrt[]{4-x}}$ .

a)

$\left(-\infty,-4\right)\cup\left(4,\infty\right)$

b)

$\left(-\infty,4\right)$

c)

$\left(-2,2\right)$

d)

$\left[-2,2\right]$

6.

Find $g\left(x\right)$ if $\left(g\circ f\right)\left(x\right)=\frac{1+2x}{2+x},\ x\ne-2$ and $f\left(x\right)=3+2x$ .

a)

$g\left(x\right)=\frac{2\left(x+2\right)}{x+1}$

b)

$g\left(x\right)=\frac{2\left(x-2\right)}{x+1}$

c)

$g\left(x\right)=\frac{2\left(x+2\right)}{x-1}$

d)

$g\left(x\right)=\frac{2\left(x-2\right)}{x-1}$

7.

Given $h\left(x\right)=e^{2x}+3,\ x\in R$ . Find $h^{-1}\left(5\right)$ .

a)

$\frac{\ln4}{2}$

b)

$-\ln2$

c)

$\ln2$

d)

$\frac{\ln2}{2}$

8.

Find $\lim_{x\rightarrow3}\frac{x^2-8x+15}{x-3}$ .

a)

-2

b)

0

c)

-3

d)

Undefined

9.

Find the horizontal asymptote(s) (if any) of $f\left(x\right)=\frac{\sqrt[]{4x^2+1}}{2x+3}$ .

I. $y=1$

II. $y=-1$

III. $y=0$

IV. No horizontal asymptote

a)

I and II

b)

II and III

c)

I, II and III

d)

IV only

10.

Determine the correct method to determine the derivative of $f\left(x\right)=\sqrt[]{x-1}$ by using the first principles.

a)

$\lim_{x\rightarrow0}\frac{\sqrt[]{x-1+h}-\sqrt[]{x-1}}{h}$

b)

$\lim_{h\rightarrow0}\frac{\sqrt[]{x-1+h}-\sqrt[]{x-1}}{h}$

c)

$\lim_{x\rightarrow0}\frac{\sqrt[]{x-1-h}-\sqrt[]{x-1}}{h}$

d)

$\lim_{h\rightarrow0}\frac{\sqrt[]{x-1-h}-\sqrt[]{x-1}}{h}$

11.

Given that $y=\frac{P}{x^2}+\frac{3Q}{x}$ where P and Q are constants and $x\ne0$ . Find $\frac{d^2y}{dx^2}$ .

a)

$\frac{d^2y}{dx^2}=\frac{6}{x^4}\left(P+Qx\right)$

b)

$\frac{d^2y}{dx^2}=\frac{2}{x^2}\left(P+\frac{3Q}{x}\right)$

c)

$\frac{d^2y}{dx^2}=\frac{6}{x^4}\left(3Q-\frac{4P}{x}\right)$

d)

$\frac{d^2y}{dx^2}=\frac{6}{x^3}\left(Q-\frac{P}{x}\right)$

12.

Find $f'''\left(x\right)$ if $f''\left(x\right)=4e^{1-2x}$ .

a)

$8e^{1-2x}$

b)

$-4e^{1-2x}$

c)

$4e^{1-2x}$

d)

$-8e^{1-2x}$

13.

Find the gradient of the curve $y=6\tan\frac{1}{3}x$

a)

$6\sec^2\frac{1}{3}x$

b)

$2\sec^2\frac{1}{3}x$

c)

$6\sec^23x$

d)

$3\sec^2\frac{1}{3}x$

14.

Which is the correct formula to find the value of $f'\left(3\right)$ for $f\left(x\right)=\frac{1}{\sqrt[]{2x+3}}$ .

a)

$\lim_{x\rightarrow3}\frac{1}{3}\left(\frac{\sqrt[]{2x+3}-3}{x-3}\right)$

b)

$\lim_{x\rightarrow3}\left(\frac{3-\sqrt[]{2x+3}}{x-3}\right)$

c)

$\lim_{x\rightarrow3}\frac{1}{x-3}\left(\frac{3-\sqrt[]{2x+3}}{3\sqrt[]{2x+3}}\right)$

d)

$\lim_{x\rightarrow3}\frac{1}{x-3}\left(\frac{3-\sqrt[]{2x+3}}{\sqrt[]{2x+3}}\right)$

15.

Find the derivative of $y=\ln\sqrt[]{2x+3}$ with respect to x in terms of y.

a)

$\frac{dy}{dx}=2e^y$

b)

$\frac{dy}{dx}=2e^{-y}$

c)

$\frac{dy}{dx}=e^{-2y}$

d)

$\frac{dy}{dx}=e^{2y}$

16.

Given that $x=\frac{1}{1-t^2}$ and $y=\frac{1+t^2}{t}$ , where t is a non-zero parameter. Which of the following are the value(s) of $\frac{dy}{dx}$ at the point $\left(-\frac{1}{3},\frac{5}{2}\right)$ .

a)

-27/16

b)

1/3

c)

27/16

d)

-1/3

17.

Given $y=3^{2x^2+x}$ , find $\frac{dy}{dx}$ .

a)

$\ln3\left(3^{2x^2+x}\right)$

b)

$\ln3\left(3^{2x^2+2}\right)\left(4x+1\right)$

c)

$\left(3^{2x^2+x}\right)\left(4x+1\right)$

d)

$\ln3\left(4x+1\right)$

18.

Given $6x^2+\left(6y^2-9x\right)\frac{dy}{dx}-9y=0$ is a first derivative of a function, find $\frac{d^2y}{dx^2}$ if $\frac{dy}{dx}=\frac{4}{5}$ is at point (1,2).

a)

$-\frac{324}{25}$

b)

$-\frac{108}{125}$

c)

$\frac{108}{125}$

d)

$\frac{324}{25}$

19.

Find the critical numbers of the curve $f\left(x\right)=x^4-2x^2$ .

a)

x=-1, x=1

b)

x=-1, x=0, x=1

c)

x=0, x=2

d)

x=-1, x=1, x=2

20.

The derivative of function f is given by $f'\left(x\right)=3\left(x+1\right)\left(x-5\right)$ . Determine the x-coordinate if the stationary point exists for the function f and state its nature.

I. x=5, local maximum

II. x=-1, local maximum

III. x=5, local minimum

IV. x=-1, local maximum

a)

I and III

b)

II and IV

c)

I and IV

d)

II and III

Road to A

Total questions: 20

Worksheet time: 1hrs 10mins

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