Key knowledge for Core

Question 1

What happens in differentiation?

Answer 1

You times the coefficient by the power and then subtract one from the power.

Eg.
f(x) = 3x^4

f’(x) = 12x^3

Question 2

What happens in integration?

Answer 2

You add one to the power and then divide the coefficient by the new power.

f’(x) = 12x^3

f(x) = (12x^4) / 4 = 3x^4

Question 3

When an equation has no real roots, how would you show this using the discriminant?

Answer 3

b^2 - 4ac < 0

Question 4

When an equation has real roots, how would you show this using the discriminant?

Answer 4

b^2 - 4ac > 0

Question 5

When an equation has repeated roots, how would you show this using the discriminant?

Answer 5

b^2 - 4ac = 0

Question 6

If you had to work out the value of r is a geometric series, but you only had the third and fifth term?

third term = x
fifth term = y

Answer 6

(ar^4)/(ar^2) = y/x

simplifies to r^2 = y/x

r = √(y/x)

Question 7

How would you cancel out e^x?

Answer 7

Times everything by ln.

Question 8

How would you cancel out ln(x)?

Answer 8

Times everything by e.

Question 9

What differentiation rule would you use to work out the differential of the following equation, y = x^(2)sin(2x)

Answer 9

Product rule.

UV’ + U’V

Question 10

What differentiation rule would you use to work out the differential of the following equation, y = e^(2x+1)

Answer 10

Chain rule.

dy/du) x (du/dx

Question 11

What differentiation rule would you use to work out the differential of the following equation, y = (2x+1)/(3x+7)

Answer 11

Quotient rule.
f(x) = U/V
f’(x) = (VU’ - UV’)/(V^2)

Question 12

What are the ways in which you can solve a quadratic equation?

Answer 12

• Factorising
• Quadratic formula
• Completing the square

Question 13

What is the shape of a negative quadratic equation

(-x^2)?

Answer 13

∩

Question 14

What is the shape of a positive quadratic equation

(x^2)?

Answer 14

∪

Question 15

In simultaneous equations if you have a linear and a quadratic equation, what would you do to solve it?

Answer 15

Substitute the linear equation into the quadratic equation.

Question 16

When you multiply or divide an inequality by a negative number, what do you need to do?

Answer 16

Change the inequality sign to its opposite.
Eg.
(-y > 3) x (-1) = y < -3

Question 17

If you have an equation of a line and need to find the equation of the normal to that line, what do you need to do to the gradient?

Answer 17

-(1/m)

Change the sign of the gradient and flip it.

Question 18

If you had a question such as divide (x^3 - 1) by (x - 1), what would you need to include to make it possible?

Answer 18

0x^2 + 0x
………….______________
(x - 1) √x^3 + 0x^2 + 0x - 1

Question 19

When using the factor theorem, if you are told f(a) is a factor, what should the equation equal when you substitute it in?

Answer 19

0
eg.
You are told that (x-2) is a factor of x^3 + x^2 - 4x - 4, so substitute,(x - 2 = 0 => x = 2), f(2) = (2)^3 + (2)^2 - 4(2) - 4 => 0, hence it is a factor.

Question 20

If you are told to solve the following equation, what would you do?

3^(2x) + 3^(x+1) - 10 = 0

Answer 20

1) State y = 3^x
2) 3^(2x) = (3^x)(3^x) and 3^(x+1) = (3^x)(3^1)
3) Substitute y into the equation
4) y^2 + 3y - 10 = 0
5) Then solve

Question 21

When using binomial expansion on an equation such as (3 + x)^4, what do you need to do?

Answer 21

Factor out 3 so you form (1 + x/3)^4, but you need to remember to put whatever you factor out to the power of the expansion, so it should be (3^4)(1 + x/3)^4

Question 22

How do you show if a point is a maximum point?

Answer 22

d^(2)y/dx^(2) < 0

Question 23

How do you show if a point is a minimum point?

Answer 23

d^(2)y/dx^(2) > 0

Question 24

What is 1/sec(x) equivalent to?

Answer 24

cos(x)

Question 25

Work out the range of k, when f(x) = k has no solutions and f(x) = (2√5)cos(2θ - 26.7).

Answer 25

1) Refer to cos(x) graph, which has asymptotes x = ±1
2) cos(2θ - 26.7) has no effect on the asymptotes, but the 2√5 does
3) the new asymptotes would be x = ±2√5
4) So the range of values to which have no solutions are k > 2√5 and k < -2√5

Question 26

When you are told to find the equation of the circle, and are presented with an equation such as
x^2 + y^2 + 4x - 2y - 11 = 0
What would you do?

Answer 26

1) Group the x’s together (x^2 + 4x) and the y’s together (y^2 - 2y)
2) Complete the square for both x and y
3) Write in the form (x±a)^2 + (y±b)^2 = r^2

Question 27

What is the inverse function of e^x?

Answer 27

ln(x)

Question 28

To work out the value of α in
Rcos(2ө - α) = 4cos(2ө) + 2 sin(2ө)
What would you do?

Answer 28

1) cos(2ө - α) = cos(2ө)cos(α) + sin(2ө)sin(α)
2) Equate coefficients, so cos(α) = 4 and sin(α) = 2
3) Form tan(α), by dividing sin(α) by cos(α).
Hence tan(α) = 1/2
4) Solve tan(α) = 1/2, α = (tan^(-1)(1/2) = 26.6 degrees

Question 29

What would you do to find the cartesian equation of the following parametric equations?
x = 2t
y = t^2

Answer 29

1) Make t the subject of one of the equations, (the best one to use in this case would be x =2t)
2) t = x/2, Substitute t into the y equation and simplify.
y = (x/2)^2 => y = (x^2) / 4

Question 30

To find the area under a curve given parametrically, what formula do you need to use?

Eg.
x = 5t^2
y = t^3

Answer 30

The formula you need to remember is ∫y (dx/dt) dt
Eg.
y = t^3
dx/dt = 10t

∫(t^3)(10t) = 10t^4

Question 31

How do you find the gradient on the curve that is given parametrically?

Eg.
Find the gradient at the point P where t = 2, on the curve given by x = t^3 + t and y = t^2 + 1.

Answer 31

1) Chain rule, so dy/dx = (dy/dt) X (dt/dx)
2) dy/dt = 2t
3) dx/dt = 3t^2 + 1 => dt/dx = 1/((3t^2) + 1)
4) dy/dx = (2t) X (1/((3t^2) + 1)) = 2t/((3t^2) + 1)
5) Substitute t = 2, and solve
6) dy/dx = 4/13

dt/dx = 1/(dx/dt)

Question 32

What would be the answer to the differential of y^3?

Answer 32

3y^2(dy/dx)

Question 33

What would be the answer to the differential of

y^2 + y

Answer 33

2y(dy/dx) + 1(dy/dx)

Question 34

Differentiate the implicit equation

x^3 + x + y^3 + 3y = 6

Answer 34

3x^2 + 1 + 3y^2(dy/dx) + 3(dy/dx) = 0

Question 35

In vectors how do you work out IaI?

eg.

a = i + 2j + 4k

Answer 35

IaI = √(1)^2 + (2)^2 + (4)^2

IaI = √1+4+16 = √21

Question 36

If you the non-zero vectors a and b are perpendicular, what is a.b equal to?

Answer 36

a.b = 0

Question 37

If you the non-zero vectors a and b are parallel, what is a.b equal to?

Answer 37

a.b = IaI IbI