GCE ADVANCED LEVEL - MATH WITH STATISTICS - PAPER 2

Type
gce a/l
Option
Country
Cameroon
Year
2019
Duration
Min
Coefficient
Marks
100
Subject
PURE MATHS WITH STATISTICS
Paper 1 :
Exercise 1 : (100 Marks)
Problem : (100 Marks)
Part 1 :
1.
a)

Find the complementary function of the differential equation

2d2ydx2-dydx-3y=5e-x,

in the form y=f(x)

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b)

Hence find the particular integral and the general solution in the form y=f(x)

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2.
a)

Express f(x) in partial fractions where

f(x)=2x3+x+2(x2+1)(x+1)(x-2), 

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b)

Hence, or otherwise, show that

01f(x)dx=-112[13ln2+π]

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3.
a)

Solve the equation

tanh-1x-2x+1=ln2.

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b)

Show that the set {1,2,4,8} under x15, multiplication mod15, forms a group.

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4.
a)

Given that the matrix M is defined by

M=3n3n-2n02n, for all n1.

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b)

A curve is given by the parametric equations

x=t2, y=t(1-13t2), 0t3.

Show that length of the curve is 23.

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5.
a)

Show that the curve with polar coordinates (r,θ) where

r=4-3+3sinθ, θnπ+(-1)nπ2, n

is a parabola, P in the (x,y) plane

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b)

Show that the point (2,56) lies on P and find the equation of the tangent to  Pat this point.

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6.
a)

By the use of the Chinese Remainder Theorem, or otherwise, solve the system of congruences

x3(mod4)x4(mod7)

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b)

A complex number z is defined by z=12(cosθ+isinθ), such that 

zn=12n(cosnθ+isinnθ).

Using De Moivre's theorem, or otherwise, show that

i.

r=014rsin2rθ is a convergent geometric progression.

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ii.

r=014rsin2rθ=4sin2θ17-16cos2θ.

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7.

A transformation, f, on a complex plane is defined by

z'=2z+3-4i

a)

Find the image of the point z=2-i

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b)

Determine the invariant point of f in the form a+ib,  a,b

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c)

Show that f is a similarity transformation (similitude), stating its radius.

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d)

Give the geometrical interpretation of f.

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8.

Given two vectors

a=αi-j-4k and b=3i+2j+(1+2β)k, α, β axb=3i-21j+6k

a)

Calculate the values of the real constants α and β.

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b)

By using the values of α and β, state the vectors a and b.

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c)

Show that a and b are linearly independent.

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d)

Find the Cartesian equation of the plane containing a and b.

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9.

A function, f, is defined by

f(x) = 1(1 + ex )2

a)

`Find the domain of f.

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b)

Find the intercept(s) of the curve y=f(x).

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c)

Find

limx-f(x) and limx+f(x)  

and state the asymptotes of the curves y = f(x).

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d)

Determine f'(x) and f''(x).

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e)

Prove that there are no turning points.

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f)

Prove, also, that -ln2,49 is the only point of inflexion.

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g)

Obtain the intervals on which f is concave up and intervals on which f is concave down.

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h)

Obtain a variation table for f.

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i)

Sketch the curve, y=f(x).

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10.

Two sequences, (un) and (vn), for n  are defined as follows:

u0=3un+1=12(un+vn)and v0=4vn+1=12(un+1+vn)

a)

Calculate u1, v1, u2 and v2

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b)

Another sequence (wn) is defined by

wn=vn-un, n.

Show that (wn) is a convergent geometric sequence.

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c)

Express wn as a function of n and obtain its limit.

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d)

Study the sense of variation (monotony) of (un) and (vn).

What can you deduce.

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e)

Consider another sequence , tn, defined by

tn=un+2vn3, n

Show that (tn) is a constant sequence.

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f)

Hence, obtain the limits of the sequences (un) and (vn)

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