1. a) find m when both lines are perpendicular, m=2
b)find x coordinate of intercepts (x=-5/2)
2. two equations to solve First was a=1/4 and 0?
second was definitely ±√2
Forgot the order so here are some of my answers:
Question on circles (centre of 2,-4) and radius of √28
the line x=k intersected at a tangent, find exact values of k, k=±√2
part c) was 2±2√7 or something like that
Queston on factorisation:
a) Prove (x-4) is a factor
b) prove the curve has two distinct roots (got x-4 as a repeated root) and -3/2 as the other root
c) Suggest the number of roots the graph f(x) - 2 had, should be three root as graph is translated by vector
(0, -2)
d)Find the value of k when f(x+k) crosses the orgin got k=4 which translated 4 to the left, or k=-3/2 which moves it 1.5 to the right
Question on binomial expansion:
a) First three terms of (2+3/4x)6 i think
b) Explain how you would use your answer to approximate (1.925)6
Question on trig:
a) Show that x = 2√3 when the area of the triangle is 18
b) Second question, find the length of AC, which was √84, simplifies to 2√21
Other trig question:
a) Had to show that the trig identity (10sin2(x) - 7cos(x) + 2) / (3 + 2cos(x)) = 4 - 5cos(x)
b) solve to get tan=-5/3. Solutions are 301 and 121 degrees in the range 0≤x<360
Question on quadratics with tin as a model of a quadratic
a) Find the value when n=1
b) State the max. amount of tin they could mine
c) Careful here. Asked for the amount mined in 2023, not amount up to 2023. So you find the amount up to 2024 and subtract from the amount up to 2023. Total was like around 100 tonnes.
d) Limitation of the model
Question on definite integrals:
Had to find the x- coordinate for when x is a minimum. X=4
You then had to use this value and x=0 as your limits to differentiate the curve to get 256/3
Question on linear modelling about trees and their height and time
a) Find the relationship between H and t
H=0.31t+1.42
b) The original height was 140 cm = 1.4m, comment on the suitability of the model
Proof:
prove n³+2 is not divisble by 8.
I proved it by counter example and when n=2n (an even number) and when n=(2n+1) an odd number and got an expression that was not wholly divisible by 8.
The proof asked for all n ∈ Z+, someone on here said you had to use (2n)² and (2n+1)² for your values of n to ensure they meet the criterion.
Question on graphs:
sketch a graph of k²/x + 1, stating asymptote equation.
Asymptote at y=1
Question on exponentials
a) initial value was like 5700+2300 = 8000
part b differential the exponent to get what they asked
part c, find the time when it is valued at £500, got 8 years 3 months
d) limitation of the model
Last question on vectors:
a) state what is meant by non-zero or something to that extent
b)
|a + b| = |a| + |b|
m = |6| and |m-n|=3 and the angle between them is 30, had to find the angle between |m| and |n|
I got 15 degrees using the sin rule, however some people got an angle of 135, which is 180-(15+30)