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Answer All Questions in This Section: 2018-1-KED-KTEK Section A (45marks)

1. This document contains 6 math problems involving functions, series, matrices, equations, and trigonometry. It asks the test taker to show work, find values, sketch graphs, and solve various mathematical expressions. 2. The first problem defines two functions f(x) and g(x) and asks to determine values of a and b and sketch the graphs. 3. Later problems involve solving equations, finding terms in sequences, reducing matrices, and proving trigonometric identities. 4. One section asks the test taker to choose one of two word problems involving linear equations and systems of equations to solve.

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Alicia Lam
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65 views3 pages

Answer All Questions in This Section: 2018-1-KED-KTEK Section A (45marks)

1. This document contains 6 math problems involving functions, series, matrices, equations, and trigonometry. It asks the test taker to show work, find values, sketch graphs, and solve various mathematical expressions. 2. The first problem defines two functions f(x) and g(x) and asks to determine values of a and b and sketch the graphs. 3. Later problems involve solving equations, finding terms in sequences, reducing matrices, and proving trigonometric identities. 4. One section asks the test taker to choose one of two word problems involving linear equations and systems of equations to solve.

Uploaded by

Alicia Lam
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
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2018-1-KED-KTEK

Section A[ 45marks]
Answer all questions in this section

1. The functions f and g are defined by

f (x)= x2−4 x +7 , x ≤ 2
and g( x )=a √ x−3+2 , x ≥ b
The function g is the inverse of f .
(a) Determine the values of a and b. [5 marks]

(b) Sketch the graphs of f and g on the same axes. [4 marks]

2. An arithmetic series has first term aand common difference d. The sum of the first 21
terms is 168.
(a) Show that a+10 d=8. [2 marks]

(b) The sum of the second term and the third term is 50. The nth term of
the series is un .

(i) Find the value of u12 . [4 marks]


21
(ii) Find the value of
∑ un. [3 marks]
n=4

−2 1 −3
3. If A= 1
[3
−1 x +6 −6 ]
x−5 and det A = 0, find the values of x. [5 marks]

4. Show that 2+i is a root of x 3−11 x +20=0. [2 marks]

Find a quadratic factor of x 3−11 x +20 with real coefficients. [2 marks]

Hence solve the equation x 3−11 x +20=0 [3 marks]

5. (a) Solve the equation giving the roots in Cartesian form.


(1+i)( z 3−i)=9+ 7 i,
[7 marks]
(b) Show the roots on an Argand diagram. [3 marks]

6. The sum of the first n terms of a sequence , , . . . is given by


u1 u2 , u3 Sn=2n 2+ kn

1
where k is a constant.

(a) Show that un =4 n+ k−2. [3 marks]


(b) Find a recurrence relation of the form un +1=f (un ). [2 marks]

Section B [15 marks]


Answer one question only in this section.

7. For the set of linear equations


3 x+ 5 y +6 z =7
x +3 y−2 z=5
2 x+ 4 y + λz=k
where λ and k are real numbers.

Show that the augmented matrix for the set of linear equations can be reduced to
1 3 −2 5
(
0 4 −12 8
0 0 λ−2 k−6 | )
[6 marks]
(a) Find the value of λ if the set of equations has no unique solution. [2 marks]

(b) Determine the values of λ and k such that the system has
(i) a unique solution,
(ii) no solution,
(iii) infinitely many solutions. [7 marks]

8. (a) Solve the equation


lo g 4 (2 x+3)+lo g 4 (2 x+ 15)=1+ lo g4 (14 x+5) [5 marks]

(b) (i) Given that 4 sin x+5 cos x =0, find the value of tan x. [2 marks]

(ii) Hence, solve the equation (1−tan x)( 4 sin x+5 cos x)=0 in the interval
0 ≤ x ≤ 36 0o , giving your values of x to the nearest 0.1o.
o
[3 marks]

(c) Prove that


16+9 si n2 θ
=5+3 cos θ
5−3 cos θ

2
Find the least value of 16+9 si n θ .
5−3 cos θ
State the exact value of θ, in radians in the interval 0 ≤ θ<2 π, at which this
least value occurs.
[5 marks]

2
“If you look the right way, you can see that the whole world is a garden.”
― Frances Hodgson Burnett, The Secret Garden

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