HPC SEM1 practice 1

1. Question

Which of the following numbers is undefined?

[math]\log_21[/math]

[math]\log_22[/math]

[math]\log_12[/math]

[math]\ln1[/math]

2. Question

What is the transformation from [math]y=\log_3x[/math] to [math]y=\log_3(2x-2)[/math]

Horizontal compress by a factor of 2, and then translate 2 units left.

Horizontal stretch by a factor of 2, and then translate 2 units left.

Horizontal compress by a factor of 2, and then translate 1 units left.

Horizontal stretch by a factor of 2, and then translate 1 units left.

3. Question

What is the solution to the equation [math]6^{x-2}=2^x[/math]

[math]2\log_36[/math]

[math]\log_62[/math]

[math]\log_36+2[/math]

No solution

4. Question

Which of the following functions does not have inverse function?

[math]f(x)=\ln(-x)[/math]

[math]f(x)=(x-1)^3+1[/math]

[math]f(x)=\sqrt[3]{2x-5}[/math]

[math]f(x)=\sqrt{1+x^2}[/math]

5. Question

Which of the following function is not always decreasing?

[math]y=-2^x[/math]

[math]y=2^{-x}[/math]

[math]y=\log_2(-x)[/math]

[math]y=-\log_2\left(\dfrac{1}{x}\right)[/math]

6. Question

What is the domain of [math]f(x)=\dfrac{5}{\sqrt{5-x}}[/math]?

[math](-\infty,5][/math]

[math](-\infty,5)[/math]

[math][5,\infty)[/math]

[math](5,\infty)[/math]

7. Question

What is the range of the function [math]f(x)=\sqrt{x^2+2x+3}[/math]?

[math][0,\infty)[/math]

[math][\sqrt{2},\infty)[/math]

[math][\sqrt{3},\infty)[/math]

[math](-\infty,\infty)[/math]

8. Question

What is the value of [math]\tan(315^\circ)[/math]?

9. Question

If [math]\cos\theta=-\dfrac{1}{2}[/math], which of the following numbers cannot be [math]\theta[/math]?

[math]\dfrac{2\pi}{3}[/math]

[math]\dfrac{-2\pi}{3}[/math]

[math]\dfrac{5\pi}{3}[/math]

[math]\dfrac{-4\pi}{3}[/math]

10. Question

What is the reference angle of [math]-2[/math]?

[math]-2[/math]

[math]2[/math]

[math]2-\pi[/math]

[math]\pi-2[/math]

11. Question

If [math]\tan x=6[/math] where [math]x\in[\pi, 2\pi][/math], what is the value of [math]\sin x[/math]?

[math]\dfrac{1}{\sqrt{37}}[/math]

[math]-\dfrac{1}{\sqrt{37}}[/math]

[math]\dfrac{6}{\sqrt{37}}[/math]

[math]-\dfrac{6}{\sqrt{37}}[/math]

12. Question

The angular velocity of a Ferris wheel with radius 50 meters is 2.4 rad/hr, what is its speed, in meters per hour?

13. Question

What is the perimeter of a sector with radius 6 and central angle 2?

14. Question

What is the period of the function [math]y=3\cot(\pi x-2)+4[/math]?

15. Question

Which of the following equations has no solutions?

[math]\sin x=\dfrac{1}{3}[/math]

[math]\tan x=\dfrac{1}{3}[/math]

[math]\sec x=\dfrac{1}{3}[/math]

[math]\cos^2 x=\dfrac{1}{3}[/math]

16. Question

Which of the following functions represents the graph above?

[math]y=\sec\left(2x-\dfrac{2\pi}{3}\right)-1[/math]

[math]y=\sec\left(2\left(x-\dfrac{2\pi}{3}\right)\right)-1[/math]

[math]y=\csc\left(2\left(x-\dfrac{\pi}{3}\right)\right)-1[/math]

[math]y=\csc\left(2x-\dfrac{\pi}{3}\right)-1[/math]

17. Question

Which of the following expressions is not equal to 1?

[math]\cos^2x+\sin^2x[/math]

[math]\tan^2x-\sec^2x[/math]

[math]\csc^2x+\cot^2x[/math]

[math]1+\sin40^\circ-\cos50^\circ[/math]

18. Question

If [math]\sin x=m[/math] where [math]x[/math] is obtuse, which of the following expressions is equivalent to [math]\sin2x[/math]

[math]2m[/math]

[math]-m\sqrt{1-m^2}[/math]

[math]2m\sqrt{1-m^2}[/math]

[math]-2m\sqrt{1-m^2}[/math]

19. Question

Find the largest number.

[math]\sin^{-1}1[/math]

[math]\cos^{-1}1[/math]

[math]\tan^{-1}1[/math]

[math]1[/math]

20. Question

What is the domain of the function [math]f(x)=\sin^-1(\cos(x+3))[/math]

[math]\mathbb{R}[/math]

[math][-1,1][/math]

[math]\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right][/math]

[math][-4,-2][/math]

21. Question

How many solutions does the equation [math]\pi^{\sin x}=e[/math] have?

Zero

One

Two

More than two

22. Question

What is the value of [math]\tan^{-1}(\tan(-2))[/math]?

[math]-2[/math]

[math]-2-\pi[/math]

[math]-2+\pi[/math]

[math]2[/math]

23. Question

Solve the equation [math]\tan(3x)+1=0[/math]

[math]\pi+[/math] [math]k\pi[/math] where [math]k[/math] is an integer.

24. Question

Evaluate [math]-4\tan^{-1}\left(-2\sin\left(\dfrac{1}{2}\cos^{-1}\left(\dfrac{1}{2}\right)\right)\right)[/math]

[math]0[/math]

[math]\pi[/math]