Advanced level 2023 south west regional mock pure mathematics with statistics 2

Advanced level 2023 south west regional mock pure mathematics with statistics 2

Advanced level 2023 south west regional mock pure mathematics with statistics 2

(i) The coefficient of x3 in the expansion of P(x), where P(x)- (5 + 2x )(
Also, the remainder when x3 + kx + m is divided by x — 3 is 59.
Find the values of the constants k and m.
(li) The function f is defined by f(x) = xz + 2kx + 3fe + 4 = 0, where k Is a constant .
(a) Express f(x) in the form (x + p)2 + <7, where p and q are constants.
*
Hence or otherwise,
(b) find the range of values of k for which the equation f(x) = 0 has no real solutions.

2(i) (a) Prove that tan-1x- tan”1y = tan”1(““)
Hence,
(b) Find the value of 6 given that tan”14- tan”1 (^) = 0-^
(ii) Find the general solution of the equation cos 20 = sin 0

B( D
t 0 5 10 15 20 25 30
v 0 1.22 2.28 6.11
The table above shows the speed of a train v at intervals of 5 seconds. Given that the speed
is related to the time by the equation v = Vl.2f -1, 0 < L < 30,
(a ) Copy and complete the table giving your values to 2 decimal places.
The distance 5 meters travelled by the train in 30 seconds Is given by S = Jp°(Vl.2f — 1)dt.
(b) Using the trapezium rule with all the values from your table, estimate the value of S. (3 marks)
(ii) A curve Is defined by the equation sin 2x cos x + cos y = V2 .
(a ) Verify that the point P lies on the curve.
i d>’
( b) Find — in terms of x and y. Hence, find the gradient of the curve at the point P.