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

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

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

The polynomial p(*) = 2*3 4- px~ — qx + 6 where p and q are constants leaves a remainder of —4 when divided
by ( x — l). Given that (2x — l) is a factor of p(x ),
(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 log3 x — log27 X3.
(ii) Find real values of x for which 3X — 7 — 18(3~x). 4 . t
(iii) Determine the range of values of x for which the expansion of is valid.
(2, 4, 4)marks
d y k
3. (i) Given that = ,x & 1show that — = where k is a constant.
(tf-D1 Ax (*~1)3
(ii) The function f is defined on the set R, of real numbers by f(x )= x 3 + 3x 2 + 4x —12.
(a) Show that f(x) = 0 has a root between 0 and 2.
(b) Show also that if f(x) = 0,then x = 4(33+-xx), x* -3.
(4, 5) marks
4. (i) The functions gand h are defined as shown below
g: x -4—x+3, x G H,x* —3
x+1
x £ & 2 .
(a) Find g o h stating its domain.
Show that h( x) 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) ~PAQ
(b) ~ p -» Q
(c) Draw a truth table for the statement rv p y h:x -4
x-2′
(lmark)
(1 mark)
(7, 4) marks
5. (i) Given that f(0) ~ V3 cos Q -sin & ,
(a) Write f(0) in the form R cos(_d + a), where R, is a positive real number and a an acute angle
(b) Hence find the general solution of the equation if(0) = \/3
(ii) Given that tan 15 = x and tan-17 = y , show that y – x — tan 118 —
(6, 3) marks
7x+1
~ 2
in partial fractions.
(((ii b-))1find Hence 4- xthe )~equation show =-ythat +of 1.JaHence 2curve f(x)sketch which dx =this passes ^In curve through ^ the point (2,-2) and satisfies the differential equation