A Bit of A2 Maths Each Day (C3, C4 and M1)

A Bit of A2 Maths Each Day (C3, C4 and M1)

A BIT OF A2 MATHS EACH DAY (C3, C4 AND M1)

MONDAY / TUESDAY / WEDNESDAY / THURSDAY / FRIDAY / SATURDAY / SUNDAY
11th April / 24th March / 25th March / 26th March / 27th March
BACK TO SCHOOL!!
Cars A and B are approaching the end of a race. A is 1.5km from the finish, is travelling at a speed of 30ms-1 and is accelerating uniformly at 0.7ms-2.
B is 210m behind A, is travelling at a speed of 40ms-1 and is accelerating at 0.5ms-2.
(a) Show that B overtakes A 285m before the finish.
(b) Calculate the difference in time between the arrivals of the two cars at the finish. / HAPPY EASTER!
Get revising!
A little bit, often, makes a big difference
/ MAUNDY THURSDAY
The functions f and g are defined by


(a) Find the exact value of fg(4)
(b) Find the inverse function, f-1(x), stating its domain.
(c) Sketch the graph of y=|g(x)|. Indicate clearly the equation of the vertical asymptote and the coordinates of the point at which the graph crosses the y-axis.
(d) Find the exact values of x for which
/ GOOD FRIDAY
(a) Prove that, when , the value of is exactly equal to cos30°.
(b) i. Expand , in ascending powers of x up to and including the term in x3, writing your answer in as simple a form as possible.
ii. Use your expansion to find an
approximation for cos30°.
(c) Find the percentage error in your approximation for the value of cos30°. / EASTER SUNDAY
Two ships, P and Q, are travelling at night with constant velocities. At midnight, P is at the point with position vector (20i + 10j)km relative to a fixed origin O. At the same time, Q is at the point with position vector (14i – 6j)km. Three hours later, P is at the point with position vector (29i + 34j)km. The ship Q travels with velocity 12j kmh-1. At time t hours after midnight, the position vectors of P and Q are p km and q km respectively. Find:
(a) the velocity of P, in terms of i and j.
(b) expressions for p and q, in terms of t, i, and j.
At time t hours after midnight, the distance between P and Q is d km.
(c) By finding an expression for , show that d2 = 25t2 – 92t + 292.
Weather conditions are such that an observer on P can only see the lights on Q when the distance between P and Q is 15km or less. Given that when t = 1, the lights on Q move into sight of the observer,
(d) find the time, to the nearest minute, at which the lights on Q move out of sight of the observer.
28th March / 29th March / 30th March / 31st March / 1st April / 2nd April / 3rd April
(a) Given that 2sin(θ+30)° = cos(θ+60)°, find the exact value of tanθ.
(b) i. Using the identity
cos(A+B)≡cosAcosB – sinAsinB
prove that cos2A≡1 – 2sin2A
ii. Hence solve, for 0 ≤ x ≤2π,
cos2x = sinx,
giving your answers in terms of π.
iii. Show that
sin2y tany + cos2y ≡ 1 for 0 ≤ y < ½π / The curve y(x + y)2 + 15 = 3x3 has a tangent at the point (2, 1).
Find the point where the tangent intersects the x-axis.
Also find the angle that the tangent makes with the x-axis. / The functions f and g are defined by
f(x) = 2x + ln2,
g(x) = e2x,
(a) Prove that the composite function gf is
gf(x) = 4e4x,
(b) Sketch the curve with equation
y = gf(x) and show the coordinates of
the point where the curve cuts the y-
axis.
(c) Write down the range of gf
(d) Find the value of x for which

Giving your answer to 3 s.f. / (a) Show, by using the substitution
x = sinθ, that, for |x| < 1,

where c is an arbitrary constant.
(b) Show that the exact value of

can be written as

where p and q are integers to be found. / (a) Differentiate with respect to x
i. e3x(sinx + 2cosx) ii. x3ln(5x + 2)
Given that
(b) show that
(c) Hence find and the values of x for which / A truck, A, of mass 3 tonnes moves on straight horizontal rails. It collides with a truck B, of mass 1 tonne. Immediately before the collision the speed of A is 3ms-1, the speed of B is 4ms-1, and the trucks are moving towards each other. The trucks become coupled to form a single body, C, which continues to move on the rails.
(a) Find the speed and direction of C after the collision.
(b) Find the magnitude of the impulse exerted by B on A in the collision.
(c) State a modelling assumption which you have made about the trucks in your solution.
Immediately after the collision a constant braking force of magnitude 250N is applied to C. It comes to rest in a distance d metres.
(d) Find the value of d.
4th April / 5th April / 6th April / 7th April / 8th April / 9th April / 10th April

This diagram shows part of the curve with equation y = x2 + 2. The finite region R is bounded by the curve, the x-axis and the lines x = 0 and x = 2.
(a) Use the trapezium rule with four equal strips to estimate the area of R.
(b) State, with a reason, whether your answer to (a) is an under or over-estimate of the area of R.
(c) Using integration, find the volume of the solid generated when R is rotated through 360° about the x-axis, giving your answer in terms of π. /
(a) Show that the equation f(x) = 0 has a root between 1 and 2.
(b) Show that the iterative formula

Can be used to solve the equation f(x) = 0.
(c) Using x0 = 1.5 calculate the values of x1, x2, x3 and x4, giving your final answer to 3 d.p.
(d) Verify that your final answer to (c) is a root to the equation f(x) = 0 correct to 3 d.p. / A table top in the shape of a parallelogram is made from two types of wood. The design is shown in the diagram. The area inside the ellipse is made from one type of wood, and the surrounding area is made from a 2nd type of wood. The ellipse has parametric equations
x = 5cosθ, y = 4sinθ, 0 ≤ x < 2π
The parallelogram consists of four line segments which are tangents to the ellipse at the points where θ=α, θ=-α, θ=π-α, θ=-π+α.
(a) Find an equation of the tangent to the ellipse at (5cosα,4sinα), and show that it can be written in the form 5ysinα + 4xcosα = 20.
(b) Find by integration the area enclosed by the ellipse.
(c) Hence show that the area enclosed between the ellipse and the parallelogram is
(d) Given that 0<απ/4, find the value of α for which the areas of the two types of wood are equal. / f(x) = 5cosx + 12sinx
Given that f(x) = Rcos(x – α), where

(a) Find the value of R and the value of αto 3 d.p.
(b) Hence, solve the equation
5cosx + 12sinx = 6 for 0 ≤ x ≤ 2π
(c) i. Write down the maximum value of
5cosx + 12sinx.
ii. Find the smallest positive value of x
for which this maximum value
occurs. / Referred to an origin O, the points A, B and C have position vectors (9i – 2j + k), (6i + 2j + 6k) and (3i + pj + qk) respectively, where p and q are constants.
(a) Find, in vector form, an equation of the line l which passes through A and B.
Given that C lies on l,
(b) Find the value of p and the value of q.
(c) Calculate, in degrees, the acute angle between OC and AB.
The point D lies on AB and is such that OD is perpendicular to AB.
(d) Find the position vector of D. / A uniform plank, AB, has mass 40kg and length 4m.
It is supported in a horizontal position by two smooth pivots, one at the end A, the other at the point C of the plank where AC = 3m.
A man of mass 80kg stands on the plank which remains in equilibrium.
The magnitudes of the reactions at the two pivots are each equal to Rnewtons.
By modelling the plank as a rod and the man as a particle, find
(a) the value of R
(b) the distance of the man from A.
The man moves towards B.
(c) Determine how far he can get from A before the plank begins to tilt.

SOLUTION TO EACH QUESTION WILL BE POSTED ON MY TWITTER FEED (@mrchadburn) AND SCHOOLS (@ASCHSMaths) THE DAY AFTER IT APPEARS ON THE CALENDAR