Core 3 basics revision quiz

Apr2

Core 3 basic facts quiz

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C3's where it suddenly gets serious. If you don't know the basics, you're not going to do well - so here's a chance to test yourself on the basics.

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Question 1

What is $\frac{dy}{dx}$ if $y = \tan(x)$?

A	-$\cot(x)$
B	$\tan^2(x)$
C	$\cot(x)$
D	$\sec(x)\tan(x)$
E	$\sec^2(x)$
F	$\sec(x)$

Question 2

What's another way to write $\cosec^2(x)$?

A	$\tan^2(x) - 1$ Hint: No.
B	$1-\cot^2(x)$ Hint: No.
C	$1+\tan^2(x)$ Hint: No - that's $sec^2(x)$
D	$1-\tan^2(x)$ Hint: Nope.
E	$\cot^2(x) - 1$ Hint: No.
F	$1+\cot^2(x)$ Hint: Correct

Question 2 Explanation:

$\cosec^2(x) = \frac{1}{\sin^2(x)} = \frac{\sin^2(x) + \cos^2(x)}{\sin^2(x)} = 1 + \cot^2(x)$.

Question 3

What's the derivative of $y = 4x \sin(x)$?

A	$\frac{dy}{dx} = 4\cos(x)$
B	$\frac{dy}{dx} = 4\sin(x) - 4x\cos(x)$
C	$\frac{dy}{dx} = -4\cos(x)$
D	$\frac{dy}{dx} = -4\sin(x) + 4x\cos(x)$ Hint: What's the derivative of $y = 4x \sin(x)$?
E	$\frac{dy}{dx} = 4\sin(x) + 4x\cos(x)$

Question 4

What do you get if you differentiate $\frac{\ln(x)}{x}$?

A	$\frac{1}{x^2}$
B	$\frac{1 - x\ln(x)}{x}$
C	$\frac{1 - x\ln(x)}{x^2}$
D	$\ln(x)$
E	$\frac{1 - \ln(x)}{x^2}$

Question 5

Given $T = 20 + 80e^{-kt}$, and that $T=50$ when $t=60$, find the exact value of $k$.

A	$k = \frac{8}{3}\ln(60)$
B	$k = 60\left(\ln(8) - \ln(3)\right)$
C	$k = {60}\left(\ln(3) - \ln(8)\right)$
D	$k = \frac{1}{60}\left(\ln(3) - \ln(8)\right)$
E	$k = \frac{1}{60}\left(\ln(8) - \ln(3)\right)$

Question 6

How else could you write the expression $3\sin(x) + 4\cos(x)$?

A	$5\sin(x + 37º)$
B	$3.5\cos(x - 53º)$
C	$3.5\sin(x + 37º)$
D	$5\sin(x + 53º)$
E	$5\cos(x + 37º)$

Question 7

Let $f$ be the function $f(x) = \sqrt{x^2-4} + 2$, $2 \le x \le 6$. What is the range of $f$?

A	$0 \le f(x) \le 4 \sqrt{2}$
B	$2 \le f(x) \le 4\sqrt{2} + 2$
C	$2 \le x \le 6$
D	$f(x) \ge 2$
E	$2 \le f(x) \le 6$

Question 8

What is the gradient of $y = 6e^{3x^2 - 3}$ at the point where $x=2$?

A	$y = 72e^{9}$
B	$\frac{dy}{dx} = 6e^{3x^2 - 3}$
C	$\frac{dy}{dx} = 36xe^{3x^2 - 3}$
D	$\frac{dy}{dx} = 6e^{6x}$
E	$y=6e^9$
F	$\frac{dy}{dx} = 6e^{12}$

Question 9

Where do the lines $y=x$ and $y = | 2x-1 |$ cross?

A	$(1,1)$ and $(\frac{1}{3}, \frac{1}{3})$
B	$(1,1)$, $(1,-1)$, $(\frac{1}{3}, \frac{1}{3})$ and $(\frac{1}{3}, -\frac{1}{3})$
C	$(0,0)$, $\frac{1}{2},0)$ and $(0,1)$
D	$(1,1)$ Where do the lines $y=x$ and $y = \| 2x-1 \|$ cross?
E	$(\frac{1}{3}, \frac{1}{3})$

Question 10

Solve $\sin(2x) = \cos(x)$ for $0 \le x \lt 2\pi$. (Note: this question is Core 4 on the AQA specification. Thanks to Prajwal for pointing this out.)

A	30º, 90º, 150º, 270º
B	60º, 90º, 120º, 270º
C	$\frac{\pi}{6}$, $\frac{\pi}{2}$, $\frac{5\pi}{6}, $\frac{3\pi}{2}$
D	$\frac{\pi}{3}$, $\frac{\pi}{2}$, $\frac{2\pi}{3}, $\frac{3\pi}{2}$
E	$\frac{\pi}{6}$, $\frac{5\pi}{6}.

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Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

2 comments on “Core 3 basics revision quiz”

Jasmine Jones
January 6, 2014 at 7:05 pm

I hope this is good! Absolutely bricking my mocks!

Reply →
- Colin
  January 6, 2014 at 9:35 pm
  
  Good luck, Jasmine – let me know if there are any topics you’d like me to cover!
  
  Reply →

Core 3 basic facts quiz

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