I'm going to run a little experiment for a while. Every Tuesday until the exams, I'm going to put out a ten-question quiz on one of the A-level modules. I'd love to have feedback on whether you find them useful, how I can make them better, and what else I ought to cover! Here's the first one - it's on Core 2.

Core 2 basic facts quiz

This quiz takes you through some of the most important rules and identities for C2. Mastering this stuff won't guarantee you a good grade in your exam, but it'll give you a great foundation to work from.
(It's designed for OCR, but works just as well for the other boards).

Start

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Question 1

What's the simplest way you could write $\log_{4}(8) + \log_{4}(16)$?

A

$\log_{4}(128)$

Hint:

It's good, but it's not right!

B

3.5

Hint:

Yep! $4^{\frac{3}{2}} = 8$, so $\log_{4}(8) = \frac{3}{2}$.

C

$\log_{4}(32)$

Hint:

No! What are you playing at?!

D

$\log_{16}(128)$

Hint:

No, don't mess with the bases.

E

$\log_{4}(8) + 2$

Hint:

Close, but no cigar.

Question 2

Why does $\int_{-5}^{5} x^2 - 4x dx$ not give you the area under the curve $y = $x^2 - 4x$ between the lines $x=-5$ and $x=5$?

A

You don't integrate to find an area.

Hint:

Yes you do.

B

You can't have a negative limit.

Hint:

Yes you can.

C

The integral goes to infinity

Hint:

No it doesn't.

D

The limits are wrong.

Hint:

No they're not.

E

The curve goes below the $x$-axis and you need to account for that.

Hint:

That's right!

Question 3

You know $\frac{dy}{dx} = 5x^4$. What is $y$?

A

$20x^5 + c$

Hint:

No - you need to divide by the new power when you integrate.

B

$20x^3 + c$

Hint:

No - you've tried to differentiate when you needed to integrate.

C

$20x^3$

Hint:

No, that would be $\frac{d^2 y}{dx^2}$ - you've differentiated.

D

$x^5$

Hint:

Something missing!

E

$x^5 + c$

Hint:

Yep - you know the derivative, so you have to integrate to get rid of it.

Question 4

How else could you write $sin^2(x)$?

A

$1 + \cos(x)$

B

$\cos(x)$

C

$1 - \cos^2(x)$

Hint:

Correct! Because $\cos^2(x) + \sin^2(x) \equiv 1$

D

$1 + \cos^2(x)$

E

$1 - \cos(x)$

Question 4 Explanation:

$\cos^2(x) + \sin^2(x) = 1$ is the most important identity in A-level maths. It should be tattooed on your eyelids.

Question 5

How do you find the remainder given by a polynomial when it's divided by $(x - 4)$?

A

Work out $f(4)$

Hint:

Correct

B

Guess

Hint:

Come on, now, you're not trying.

C

Work out $f(-4)$

Hint:

No - that would be for dividing by $(x + 4)$

D

Do long division

Hint:

You could. But you shouldn't.

E

Put it in the calculator.

Hint:

How!?

Question 6

What's the area of a triangle, if you know two sides $a$ and $b$ and the angle $C$ between them?

A

$\frac{1}{2}ab\sin(C)$

Hint:

Boom!

B

$\frac{1}{2}abC$

Hint:

Not that simple, either.

C

$abC$

Hint:

It's not that simple.

D

$\frac{1}{2} bh$

Hint:

No, you don't know $h$.

E

$\frac{1}{2}ab\cos(C)$

Hint:

Nope. Wrong function!

Question 7

If you know two sides of a triangle and the angle between them, which rule do you use to find the remaining side?

A

A slide rule

Hint:

Possibly, if it's the 1960s. *Checks calendar*. Nope, it's not.

B

Rule 34

Hint:

*Looks sternly over glasses.* No.

C

Sine rule

Hint:

No - that won't help you here!

D

Heron's rule

Hint:

Good guess. Wrong.

E

Cosine rule

Hint:

Yep!

Question 8

For $0 \le x \lt 360º$, which values of $x$ satisfy $\cos^2(x) = \frac{1}{9}$?

A

$70.5º$ and $109.5º$ only

Hint:

No.

B

$70.5º$ and $289.5º$ only

Hint:

No.

C

$70.5º$ and $250.5º$ only

Hint:

No.

D

$70.5º$ only

Hint:

No.

E

$70.5º$, $109.5º$, $250.5º$, and $289.5º$

Hint:

Yes!

Question 8 Explanation:

You get $\cos(x) = \pm \frac{1}{3}$, which gives you results in all four quadrants.

Question 9

What's the formula for arclength, if you know the radius $r$ and the angle $\theta$ in radians?

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.

## Omole Musah-Eroje

Hi. Thank’s for the quiz, it’s really helpful.

However on question 10,

“If you know two sides of a triangle and the angle between them, which rule do you use?”,

you don’t say whether we are finding an angle or a side. [Spoilers redacted — CB]

## Colin

You’re quite right — I’ve now corrected the question. Thanks!

## AH

Hahahahahhahahaha Rule 34. Nice.

## Colin

Ha! I’d forgotten about that 🙂