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U6z - Maths Test 3

This document is a mathematics test for U6 - ZIMSEC students, consisting of various problems including partial fractions, parametric equations, derivatives, and geometric progressions. It covers topics such as integration, function definitions, and graphical analysis. The test is designed to assess students' understanding and application of mathematical concepts over a duration of 1 hour and 30 minutes.

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Tinashe Ruzvidzo
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56 views2 pages

U6z - Maths Test 3

This document is a mathematics test for U6 - ZIMSEC students, consisting of various problems including partial fractions, parametric equations, derivatives, and geometric progressions. It covers topics such as integration, function definitions, and graphical analysis. The test is designed to assess students' understanding and application of mathematical concepts over a duration of 1 hour and 30 minutes.

Uploaded by

Tinashe Ruzvidzo
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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QUINTON SENIOR SCHOOL

U6 - ZIMSEC

MATHEMATICS - TEST 3 1hr 30mins

𝑥 3 +4𝑥 2 +3
1. a) Express in partial fractions: . [6]
𝑥 2 +2𝑥−3
b) Hence, or otherwise, evaluate the exact value of:
3 𝑥 3 +4𝑥 2 +3
∫2 𝑥 2 +2𝑥−3 d(𝑥) [4]

2. A curve is defined by parametric equations:


X = (√3)sin2t , y = 4cos2t , 0≤t≤π
𝑑𝑦
a) Show that 𝑑𝑥 = k(√3)tan2t, where k is a constant to be determined. [5]
𝜋
b) Find the equation, in the form of the tangent to C at the point where t = 3 . [3]

c) Find the Cartesian equation of C. [3]

3. Given that f(x) = tanx, x


𝑠𝑖𝑛𝑥
a) By expressing f(x) as , show that f'(x) = 1 + tan2x. [3]
𝑐𝑜𝑠𝑥
b) Obtain the expressions for f''(x) and f'''(x). [3]
𝑥³
c) Show that the series expansion for tanx is x + , if x is small enough for terms
3
in x4 and higher powers to be neglected. [3]

4. The function f, g and h are defined as:


f : x → x2 + 2x x ∈ R, x ≥ - 1
g : x → 4x + 1 x ∈ R, -2 ≤ x ≤ 3
1
h:x→ x ∈ R, x≠3
𝑥−3
a) Find: (i) gh(x), stating clearly its domain. [3]
(ii) f -1(x). [4]

b) (i) Sketch the graph of y = |𝑔(𝑥)|, showing clearly the intercept and end points[2]
(ii) Hence, state the range of |𝑔(𝑥)|. [1]

5. The first, second and forth terms of a convergent Geometric progression are
consecutive terms of an arithmetic progression. Prove that the common ratio of the
− 1 ± √5
geometric progression is . [5]
2

6. The variables x and y are connected by the equation y = a𝑒 𝑏𝑥 , where a and b are
constants. When the values of lny are plotted against values of x, a straight line is
obtained passing through the points (0 ; 𝑙𝑛2) and (3 ; 𝑙𝑛5). Calculate the values of
a and b in exact form. [5]

7. a. Sketch on the same axes, the graphs of: y = 𝑒 3𝑥 and y = 2x + 10. [2]
b. Show that the equations has a root between -5 and -4 [2]
c. Taking 𝑥1 = - 4, use the Newton-Raphson method once to find the second
approximation 𝑥2 , giving your answer correct to 5 decimal places. [3]

2 1
8. Show that ∫0 𝑥𝑒 2𝑥 𝑑𝑥 . = 4 (3𝑒 4 + 1). [3]

TOTAL - 60

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