Year 1 - Pure

Question 1

4.5 What does the transformation of y = f(x + a) do?

Answer 1

Translates a by the column vector (-a,0)

Question 2

4.5 What does the transformation of y = f(x) + a do?

Answer 2

Translates a by the column vector (0, a)

Question 3

4.5 What happens to any asymptotes when a function is translated?

Answer 3

The asymptote is also translated

Question 4

4.6 What does multiplying a constant outside a function do?
e.g. y = af(x)

Answer 4

Stretches y = f(x) by a scale factor a in the vertical direction (multiply the y coordinate by a)

Question 5

4.6 What does multiplying a constant inside a function do?
e.g. y = f(ax)

Answer 5

Stretches the graph y = f(x) by a scale factor of 1/a in the horizontal direction (all x coordinates are multiplied by 1/a)

Question 6

4.6 What does y = f(-x) do to the graph of y = f(x)?

Answer 6

Reflects the graph in the y-axis

Question 7

4.6 What does y = -f(x) do to the graph of y = f(x)

Answer 7

Reflects the graph in the x-axis

Question 8

5.1-5.4 How do you find the distance between two given points?

Answer 8

sqrt((Δx)^2 + (Δy)^2)

Question 9

5.1-5.4 What do the equations of 2 parallel lines have in common?

Answer 9

They have the same gradient

Question 10

5.1-5.4 How are the equations of 2 perpendicular lines different from each other?

Answer 10

Their gradients are negative reciprocals of each other (their product is -1)

Question 11

5.1-5.4 What is the point slope formula?

Answer 11

y-y1=m(x-x1)

Question 12

6.1 What is the midpoint of a line segment?

Answer 12

M((x1+x2)/2 , (y1+y2)/2)

Question 13

6.2 What is the general formula for a circle which has a centre not at the origin?

Answer 13

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

Centre (a,b), radius r

Question 14

6.3 Describe the different situations where a line intersects a circle (reference the discriminant)

Answer 14

2 intersections - b^2 -4ac > 0

Tangent - b^2 -4ac = 0

No intersections - b^2 -4ac < 0

Question 15

6.4 What is the property of a tangent of a circle relating to the radius?

Answer 15

A tangent is always perpendicular to the radius at the point of intersection

Question 16

6.4 What is the property of a chord relating to a circle?

Answer 16

The perpendicular bisector of a chord of a circle will always go through the centre of the circle

Question 17

6.5 What is a circumcircle?

Answer 17

A circle where the vertices of a triangle are on the circumference

Question 18

6.5 What is the centre of a circumcircle called and what property does it have relating to the sides of the triangle?

Answer 18

Circumcentre - it is the point where the perpendicular bisectors of the sides of the triangle intersect

Question 19

6.5 What is the circle theorem relating to a semi circle?

Answer 19

The angle subtended by the diameter of the circle makes a right angle at the circumference

Question 20

7.3 What is the factor theorem?

Answer 20

If f(p)=0 is a root of a polynomial, then (x-p) is a factor of the polynomial
OR
If (x-p) is a factor of the polynomial, then f(p)=0

Question 21

7.4 What are the methods of proof?

Answer 21

By deduction, by exhaustion, by counter example, by contradiction

Question 22

9.6 Describe the features of the graph of y = sin(x)

Answer 22

• Repeats every 360 degrees
• Crosses x-axis at -360, -180, 0, 180, & 360 degrees
• Max. = 1, Min. = -1
• sin(-x) = -sin(x)

Question 23

9.6 Describe the features of the graph of y = cos(x)

Answer 23

• Repeats every 360 degrees
• Crosses x-axis at -270, -90, 90, & 270 degrees
  -Max. =1, Min. = -1
• cos(-x) = cos(x)

Question 24

9.6 Describe the features of the graph of y = tan(x)

Answer 24

• Repeats every 180 degrees
  -Crosses x-axis at -360, -180, 0, 180, & 360 degrees
• Max. = ∞, Min. = -∞
• Asymptotes at: -270, -90, 90, & 270 degrees

Question 25

10.1 What do the 4 quadrants of the CAST diagram mean?

Answer 25

1st quadrant - All are +ves
2nd quadrant - Only sin is +ve
3rd quadrant - Only tan is +ve
4th quadrant - Only cos is +ve

Question 26

10.1 How do you find the other values for sin(x) if given the principal value?

Answer 26

sin(x) = sin(180-x)
OR
sin(x) = cos(90-x)

Question 27

10.1 How do you find the other values for cos(x) if given the principal value?

Answer 27

cos(x) = cos(360-x)

Question 28

10.1 How do you find the other values for tan(x) if given the principal value?

Answer 28

tan(x) = tan(180+x)
- Or just add/subtract multiples of 180 to x to find infinite solutions

Question 29

10.2 Give all the exact trig values for sin, cos, and tan for 0, 30, 45, 60, & 90 degrees

Answer 29

sin0 = 0, sin30 = 1/2, sin45 = sqrt(2)/2, sin 60 = sqrt(3)/2, sin90 = 1

cos0 = 1, cos30 = sqrt(3)/2, cos45 = sqrt(2)/2, cos60 = 1/2, cos90 = 0

tan0 = 0, tan30 = 1/sqrt(3), tan45 = 1, tan60 = sqrt(3), tan90 = undefined

Question 30

10.3 Give the trig identity that involves sin^2(x) and cos^2(x)

Answer 30

sin^2(x) + cos^2(x) ≡ 1

Question 31

10.3 Give the trig identity that involves sin(x) and cos(x)

Answer 31

(sin(x))/(cos(x)) = tan(x)

Question 32

10.4 What happens when you put the inverse of a trig function into a calculator?

Answer 32

It gives the principal value

Question 33

10.5 Describe how you would solve: sin4x = 0

Answer 33

1. Adjust the range to match the argument
2. Draw the graph y=sinx and find the number of solutions within this range
3. Manipulate to find the solutions equal to x (in this case divide by 4)

Question 34

10.6 Describe how you would solve this: sin^2 (x) = 3/4

Answer 34

1. Adjust the range to match the argument (not applicable in this case)
2. Take the square root on both sides, so you have either sinx= +sqrt(3)/2 or sinx= -sqrt(3)/2
3. Sketch the graph y=sinx and find the number of solutions in the range
4. Manipulate the solutions to find x

Question 35

11.1 How do you represent parallel vectors?

Answer 35

Any vector parallel to the vector a can be represented by λa where λ is a non-zero scalar

Question 36

11.2 What happens when you multiply a column vector by a value?

What happens when you add two column vectors together?

Answer 36

λ(q, r) = (λq, λr)

(p, q) + (r, s) = (p+r, q+s)

Question 37

11.3 How do you find the magnitude of a vector?

Answer 37

The magnitude of vector a = mod(a) = sqrt(x^2 + y^2)

Question 38

11.3 How do you find the unit vector in the direction a?

Answer 38

a/(mod(a)), e.g. If mod(a) = 5 then a unit vector in the direction of a is a/5

Question 39

11.4 What is a position vector?

Answer 39

A point P with coordinates (p,q) has a position vector O->P = p𝐢 + q𝐣 = (p q)