cameroon gce advanced level June 2022 pure maths with mechanics 2
cameroon gce advanced level June 2022 pure maths with mechanics 2
(i) Given that the roots of the equation 3×2 – 4x + 2 = 0 are a and /?,
(a) show that (“+ [fi +~) ~ 3 •
(h) find an equation,with integral coefficients.whose roots arc^ and —.
(ii) The polynomial /’(x) is such that /’(x) = ax3 + bx2 — 2x- 6. Given that /'(x) leaves a remainder
of 10 when divided by x — 2 and that 2x — 3 is a factor of P(x),
find the values of the constants a and b.
1.
2. (i) A function f:K — 1} —> M — {1}, is defined by f(x) =
(a) Show that f is surjective.
(b) find the composite function (f°f)00< stating its domain. (c) Determine the parity of (f°f)(x). t . l-x (ii) Expand as a series in ascending power of X as far as term in x2. (l +2x)3 (4 marks) ‘Hieposition vectorsofthe points A,D andC area,b andc respectively, where=i+ 5j- 3k, b = —3i — j + 7k and c =i+ j.Given that A/is the midpoint of AB, find (a) an equation of the line AB, in the form r = a + tb, (b) .the position vector of M,. (c) the Cartesian equation of the plane ABC, • ‘ : J • (d) the value of the constant a for which the line r = j + 2k + A(2i+ aj + k) is parallel to the plane ABC. J. cos 0 sin 0 4. Given that f(0) = j-tan 0 * l-cot 0 ’ (a) prove that f(0) = cos 0 + sin 6. •s (b) express f(0) in the form Rsin(0 + A), where R > 0 and A an acute angle.
(c) find the general solutions of the equation f(0) =1.
(i) Hie sum of the sixth and the eighth term of an AP is 40.The seventh term is 4 times the second
term. Find,
(a) the first term and the common difference,
(b) the least number of terms required for the sum of the progression to exceed 1000.
5.
(fed)) yx == (1sinx + t)3(lnx and),yfind = t^+.t 4, find
^ in terms of t.
6. (i) Given that 2 and w are two complex numbers, solve the simultaneous equations
3z + w = 9 +lli
iw — z = —8 — 2i, leaving the answers in the form a + bi.
(ii) Lei 14 be the set ofnatural numbers. A relation R is defined on M x 14 by (a,b)R(c,d) o – =-
Show that R is an equivalence relation.
