# A level north west regional mock gce 2022 mathematics with statistics 2

### A level north west regional mock gce 2022 mathematics with statistics 2

A level north west regional mock gce 2022 mathematics with statistics 2

The polynomial p(x) = 2*3 + V* 1 – qx + 6 where p and q are constants leaves a remainder of -4 when divided
by ( x — 1). Given that (2x — l) is a factor of p(*),
(a) find the values of the constants p and q .
(b) factorize p(x) completely.
(c) solve the equation p(x) = 0.
1.
(5, 3, 3) marks
2. (i) Show that log3x = iogZ7 x3′-
(ii) Find real values of x for which 3* – 7 = 18(3“*). •.* •
(iii) Determine the range of values of x for which the expansion of J (x2-—Jr 12 is valid.
(2, 4, 4)marks
5*2-1QX+S dy k
*3. (i) Given thaty= (.*-!)= ,x =£ 1 show that — = (*-l)3 where k is a constant.
(ii) The function f is defined on the set R, of real numbers by f(x ) = x 3 -r 3x 2 + 4*- 12.
(a) Show that f( x ) = 0 has a root between 0 and 2.
dx
. f t
(b) Show also that if f(x) = G/then x = J4(34 3-*x), x =£ — 3.
(4, 5) marks
4. (i) The functions gand h are defined as shown below
U s *-3
h: x -4*41 x Elk, x 2 .
(a) Find g oh stating its domain.
Show that h(^r) is not surjective.
(ii) let P: John is sick and Q: John will play the game
write in ordinary English the following logical statements
(a) ~P/\Q
(b) P -4 Q
(c) Draw a truth table for the statement ~ P Q (2 marks)
x—2*

5. (i) Given that f(0) = V3 cos9 — sin 9 ,
(a) Write f(0) in the form R cos(0 4- a), where R5 is a positive real number and a an acute angle
(b) Hence find the general solution of the equation f{6) =\[3
(ii) Given that tan 1S = x and tan 1 7 = y , show that y — x = tan T18 -i-
(6, 3) marks
6. (i) (a) Express f( x ) = —— in partial fractions.

(b) Hence show that J2 f( x )d x = -In J
(ii) Find the equation of a curve which passes through the point (2, —2) and satisfies the differential equation