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A-Level Maths > Year 1 Pure Facts/Rules > Flashcards

Year 1 Pure Facts/Rules Flashcards

1
Q

What is the quadratic formula?

A

x=(-b±√(b^2-4ac))/2a

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2
Q

What is the discriminant?

A

b^2-4ac

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3
Q

How do you determine if a quadratic equation has no real roots?

A

The discriminant is negative
ie b^2-4ac<0
or b^2<4ac

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4
Q

How do you determine if a quadratic equation has equal (repeated) real roots?

A

The discriminant equals zero
ie b^2-4ac=0
or b^2=4ac

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5
Q

How do you determine if a quadratic equation has two real roots?

A

The discriminant is positive
ie b^2-4ac>0
or b^2>4ac

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6
Q

If you are sketching a cubic graph that has a positive x3 term, where does your sketch start - bottom left or top left?

A

Bottom left

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7
Q

How do we find the points at which a quadratic function crosses the x-axis?

A

Put y = 0 and solve the quadratic equation by factorising, completing the square or using the formula. The x-axis crossing points are the solutions (the ‘roots’) of the equation.

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8
Q

If we have completed the square to get
〖y= (x+3)〗^2+19
what are the coordinates of the minimum point of this quadratic function?

A

(-3,19)

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9
Q

Factorise x^2-y^2

A

(x-y)(x+y)

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10
Q

Complete the square: x^2-8x

A

(x-4)^2-16

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11
Q

What is the value of 〖64〗^(1/3)?

A

4

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12
Q

What is the value of 〖11〗^(-2)?

A

1/121

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13
Q

Simplify √50

A

5√2

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14
Q

How would you rationalise the denominator of 1/(a+√b)?

A

Multiply by (a-√b)/(a-√b)

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15
Q

How do you find the gradient of the straight line joining two points?

A

m= rise/run=(change in y)/(change in x)= (y_2-y_1)/(x_2-x_1 )

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16
Q

What method should you use to solve simultaneous equations where one is linear and one is quadratic?

A

Substitution

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17
Q

When solving an inequality, what must you do if you multiply or divide both sides by a negative number?

A

Turn the inequality sign around

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18
Q

What must you do when solving a quadratic inequality?

A

Draw a sketch

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19
Q

If the straight line L has gradient m, what is the gradient of the straight line perpendicular to L?

A

-1/m (ie the negative reciprocal)

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20
Q

If a tangent to a curve at the point P has gradient-2/3, what is the gradient of the normal to the curve at the point P?

A

3/2

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21
Q

The point A(3,5) lies on the curve y = f(x).

What are the new coordinates of the point A under the transformation y = f(-x)?

A

( -3 , 5 )

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22
Q

The point A(3,5) lies on the curve y = f(x).

What are the new coordinates of the point A under the transformation y = f(x+3)?

A

( 0 , 5 )

23
Q

The point A(3,5) lies on the curve y = f(x).

What are the new coordinates of the point A under the transformation y = f(3x)?

A

( 1 , 5 )

24
Q

The point A(3,5) lies on the curve y = f(x).

What are the new coordinates of the point A under the transformation y = 3f(x)?

A

( 3 , 15 )

25
Q

The point A(3,5) lies on the curve y = f(x).

What are the new coordinates of the point A under the transformation y = f(x)+3?

A

( 3 , 8 )

26
Q

The point A(3,5) lies on the curve y = f(x).

What are the new coordinates of the point A under the transformation y = -f(x)?

A

( 3 , -5 )

27
Q

Write as a single log: logax - logay

A

loga(x/y)

28
Q

How do we write tanx in terms of sinx and cosx?

A

tanx= sinx/cosx

29
Q

What is the factor theorem?

A

If (x-a) is a factor of f(x) then f(a) = 0

30
Q

What is the Sine Rule?

A

a/sinA=b/sinB=c/sinC

31
Q

What is the Cosine Rule?

A

a^2=b^2+c^2-2bcCosA

32
Q

What is the name of a line that touches a circle at one point only?

A

Tangent

33
Q

What is the equation of a circle with centre (a,b) and radius r?

A

(x-a)^2+(y-b)^2=r^2

34
Q

What is the magnitude of vector 3i+4j?

A

√(3^2+4^2 )=5

35
Q

Given the position vectors of A and B, how would you find the vector (AB) ⃗?

A

(OB) ⃗-(OA) ⃗ or b - a

36
Q

What is the coefficient of x^2 in the expression 〖5x〗^3+〖3x〗^2/4+x/2?

A

3/4

37
Q

What notation do we use for ‘3 factorial’? How is it calculated and what is its value?

A

3! = 3 x 2 x 1 = 6

38
Q

What notation do we use for ‘5 choose 2’? How is it calculated?

A

(■(5@2)) or 5C2 = 5!/2!3!

39
Q

Write 2^3 = 8 in log form

A

log28 = 3

40
Q

What does 〖sin〗^2 x+〖cos〗^2 x equal?

A

1

41
Q

Write as a single log: logax + logay

A

logaxy

42
Q

When is a function increasing?

A

When the gradient is positive

43
Q

What is the value of loga1?

A

0

44
Q

What is the value of logaa?

A

1

45
Q

Write logaxn without an index

A

nlogax

46
Q

What is the period of the Sine and Cosine functions?

A

360o

47
Q

When is a function decreasing?

A

When the gradient is negative

48
Q

Write log416 = 2 in index form

A

4^2 = 16

49
Q

How do we find stationary points?

A

Find the point(s) where the gradient equals zero.

50
Q

What does the second derivative tell us about the nature of a stationary point?

A

If the second derivative is positive, it is a minimum. If the second derivative is negative, it is a maximum. (If the second derivative is zero, it could be a minimum, maximum or point of inflexion).

51
Q

How do we find the area under a curve between x = a and x = b?

A

Integrate the function and evaluate it between the limits a and b.

52
Q

What is the midpoint of the points (a, b) and (c, d)?

A

((a+c)/2 ,(b+d)/2)

53
Q

What is the inverse of y=e^x?

A

lnx

54
Q

How do you find the area of a triangle?

A

Area= 1/2 absinC where C is the included angle

or Area= 1/2 b × h where h is the perpendicular height

A-Level Maths (3 decks)

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