Quiz about Road to A

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

Accepted Answer

I and II

Answer

I, III and IV

Answer

IV only

Answer

All of the above

Question 2

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 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 $$\left|\frac{2x-1}{x-1}\right|\ge3$$

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 4

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

Accepted Answer

$$x=-1$$

Answer

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

Answer

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

Answer

$$x=1$$

Question 5

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 6

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 7

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

Accepted Answer

$$\frac{\ln2}{2}$$

Answer

$$\frac{\ln4}{2}$$

Answer

$$-\ln2$$

Answer

$$\ln2$$

Question 8

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

Accepted Answer

-2

Answer

0

Answer

-3

Answer

Undefined

Question 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

Accepted Answer

I and II

Answer

II and III

Answer

I, II and III

Answer

IV only

Question 10

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 11

Given that $$y=\frac{P}{x^2}+\frac{3Q}{x}$$ where P and Q are constants and $$x e0$$ . Find $$\frac{d^2y}{dx^2}$$ .

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 12

Find $$f$$ if $$f$$ .

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 13

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

Accepted Answer

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

Answer

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

Answer

$$6\sec^23x$$

Answer

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

Question 14

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 15

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 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 $$\left(-\frac{1}{3},\frac{5}{2}\right)$$ .

Accepted Answer

27/16

Answer

-27/16

Answer

1/3

Answer

-1/3

Question 17

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 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).

Accepted Answer

$$-\frac{108}{125}$$

Answer

$$-\frac{324}{25}$$

Answer

$$\frac{108}{125}$$

Answer

$$\frac{324}{25}$$

Question 19

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

Accepted Answer

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

Answer

x=-1, x=1

Answer

x=0, x=2

Answer

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

Question 20

The derivative of function f is given by $$f$$ . 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

Accepted Answer

II and III

Answer

I and III

Answer

II and IV

Answer

I and IV

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