ROAD TO A+DD MATH

SATURDAY 21 AUGUST 2021

10:30 AM - 4:00 PM

SBPI GOMBAK & SEKOLAH SULTAN ALAM SHAH

SCHOOL OF GLOBAL EXCELLENCE

Xtrovian & Marshals 1822

NAMA :_____________ 1. 2. FUNCTIONS

KELAS :____________ x h(x)

TARGET e 5• 01 3.

PENTAKSIRAN (Gr) ;__ 4 • • 25 12 The following information is about the

TOPIC : -4 • • 16 m4 function h and the compositefunction h2

-5 • 56

✓ Functions h:x→ ax +b, where a and b are

✓ Quadratic Functions Set A Set B Diagram above shows the linear function h constants and a > 0

✓ Indices , Surds & (a) State the value of m

In diagram above, set B shows the images (b) Using the function notation, express h h2 :x→ 36x – 35

Logarithms of certain elements of set A.

✓ SOT in terms of x. Find the value of a and b.

(a) State the type of relation between A and [2 marks] [3 marks]

B.

Ans: (a) m = 3, (b) h(x) = x + 1 Ans: a = 6, b = –5

(b) Using the function notation, write a

relation between set A and set B.

[2 marks]

Ans: (b) f : x → x2

I 23

2 A d d i ti ona l M a th ematics S A S – I N TE R GOM 2021

4. 5. 6. 7.

Diagram below shows therelation between

s • set M and set N. hg Given f : x → x − 3 , find the value of x

r • xyz

Set Y such that f(x)= 5

q •• 8

• • –3 5 [2 marks]

p 5 •s • –1 Ans: x = 8 or –2

6• • 1 2

24 6 7• • 3

Set X In diagram above, the function h maps x to y

•4 and the function g maps y to z.

Diagram above shows the relation between Determine

(a ) h-1(5)

set X and set Y in the graph form. Set M Set N (b) gh(2)

Sta te

(a)The object of q, Sta te [2 marks]

(a) the object of –1, Ans: (a) 2, (b) 8

(b)The codomain of the relation. (b) the range of therelation.

[2 marks]

[2 marks]

Ans: (a) 2, 6 (b) {p, q, r, s} Ans: (a) –3, (b) {–3, –1, 1, 3}

45 67

8. 9. Given the function g:x→3x – 1, find 10. Given the funtions g : x→x – 8 and 11. The inverse function h-1 is defined by

(a) g(2),

y (b) the value of p when g-1(p) = 11. h: x → x , x 2 , find the valueof h−1 : x → 2 , x 3. Find

3x − 2 −

y = f(x) [3 marks] hg(10) 3 3 x

Ans: (a) 5 (b) 32

[3 marks] (a) h(x),

(b) the value of x such that h(x) = –5.

1 Ans: 1 [4 marks]

0t 5x

2

Diagram above shows the graph of the Ans: (a) 3x − 2 0 , (b) 1

h(x) = x , x 4

function f (x) = 2x −1 , for the domain

0 x 5.

Sta te

(a) the value of t,

(b) the range of f(x) corresponding to the

given domain. [3 marks]

Ans: (a) t = 1 , (b) 0 f(x) 9

2

8 9 10 u

4 A d d i ti ona l M a th ematics S A S – I N TE R GOM 2021

12. Given that f : x → x + 5 , find 13. 14 . Given the functions g:x→2x– 3 and 15 .

Given that f : x → 3x – 2 and h:x→4x, find

(a) f(3)

(b) the value of k such that g: x → x +1, find (a) hg(x),

5

2 f −1(k) = f (3) . (b) the va lue of x if hg(x) = 1 g(x) .

2

[3 marks] (a) f −1(x) , [4 marks]

Ans: (a) 8, (b) k = 9 [1 mark] Ans: (a) 8x – 12, (b) 3

(b) f −1g(x) , 2

[2 marks]

(c) h(x) such that hg(x) = 2x + 6

[3 marks]

Ans: a) x + 2 , (b) x +15 ,

3 15

(c) h(x )= 10x - 4

12 13 M 15

16 . 17 .

I

2

6 A d d i ti ona l M a th ematics S A S – I N TE R GOM 2021

18. 19.

34

INDICES, SURD & LOG

1. Solve the equation 2x + 4 – 2x + 3 = 1 2 . Given the value of 9(3n−1) = 27n , 3. Solve the equation 23x = 8 + 23x−1 . 4 . Solve the equation [3 marks]

[3 marks] [4 marks] 2 + log3 (x – 1) = log3 x

find the value of n.

Ans: x = –3 [3 marks] Ans: 4 Ans: x = 1 1

3 8

Ans: n = 0.5

56 78

5 . Given that log4 x = log2 3 , find the valueof x. 6 . Given that logm 2 = p 7 . Given that logm 2 = p 8.

[3 marks] and and

logm 3 = r, Express logm 3 = r, Express

Ans: x = 9

27m 27m

log m 4 in terms of p log m 4 in terms of p

and r. [4 marks] and r. [4 marks]

Ans: 3r – 2p + 1 Ans: 3r – 2p + 1

9 w 11 12

9. 10. 11.

13 14 18

12.

l

13.

2

14.

3

15.

4

16.

5

QUADRATIC FUNCTIONS

1 . Solve the quadratic equation 2 .A quadratic equation 3.The quadratic equation hx2 + kx + 3 = 4. It is given that –1 is one of the roots

x(2x – 5) = 2x – 1. Give your answer 0, where h and k are constants, has two of the quadratic equation x2 – 4x – p =

correct to three decimal places. x2 + px + 9 = 2x has two equal roots. equal roots. Express h in terms of k. 0.

Find the possible values of p. Find the value of p.

[2 marks] [4 marks]

Ans: 3.351 or 0.149 [3 marks] [2 marks]

Ans: p = 8, –4 Ans: h = k2 Ans: p = 5

12

67 8 9

5. The straight line of y = 5x – 1 does 9 . Find the range of the values of x for 10. 11. The quadratic function f(x) = –x2 +

not intersect the curve y = 2x2 + x + p. (2x −1)(x + 4) 4 + x . The quadratic equation x2 − 5x + 6 = 0 4x + a2, where a is a constant, has

Find the range of values of p. has roots h and k, where h > k. maximum value of 8.

[2 marks] (a) Find Find the values of a.

[3 marks] Ans: x < –4, x > 1

Ans: p > 1 (i) the value of h and of k, [3 marks]

11 (ii) the range of x if Ans: – 2, 2

10

x2 − 5x + 6 0 13

[5 marks]

(b) Using the values of h and k from

(a), form the quadratic equation

which has roots h + 2 and 3k – 2.

[2 marks]

Ans: (a)(i) h=3, k=2 (ii) x<2, x>3,

(b) x2 – 9x + 20 = 0

(2

12. The quadratic function f(x) = –x2 + 13 . A quadratic equation 14 . 15.

4x – 3 can be express in the form of x2 + 4(3x + k) = 0 , where k is a

f(x) = –(x – 2)2 + k, where k is a y y

constant. constant, has roots p and 2p, p 0 .

(a) Find the value of k, y = f(x)

(b) Sketch the graph of the function f(x) (a) Find the value of p and of k. [5

m a rks] y = f(x)

on the given axes.

[4 marks] (b) Hence, form the quadratic equation ●• x x

which has the roots p – 1 and p + 6. (1, q) 01 5

Ans: (a) k = 1 [3 marks] 0

y = –4

(b) f(x) Ans: (a) p = –4, k = 8, (b) Diagram above shows the graph of a

x2 + 3x −10 = 0 quadratic functions Diagram above shows the graph of a

f(x)= 3(x + p)2 + 2, where p is a quadratic function y = f(x). The straight

constant. Thecurve y = f(x) has the line y = –4 is a tangent to the curve y =

minimum point (1, q), where q is a f(x).

(a) Write the equation of the axis of

constant. State

(a) the value of p, symmetry of the curve.

(b) the value of q, (b) Express f(x) in the form

(c) the equation of the axis of (x + b)2 + c, where b and c are

symmetry. constant

[3 marks] [3 marks]

Ans: (a) p = –1, (b) q = 2, (c) x = 1 Ans: (a) x = 3, (b) f(x)=(x – 3)2 - 4

0x

14 15 Iz

16 . The quadratic function 17. 18 . Diagram below shows thegraph of 19 . Diagram below shows thecurve of

a quadratic function f(x) = (x + 3)2 + 2k a quadratic function f(x) = –x2 + kx – 5.

f (x) = x2 + 2x − 4 y – 6, where k is a constant. The curve has a maximum point at B(2,

can be expressed in the form

y = f(x) y p) and intersects the f(x)-axis at point A.

f (x) = (x + m)2 − n , where m and n are f(x) = (x + 3)2 + 2k – 6

constants. Find thevalueof m and n. x f(x)

01 5 –

[3 marks] 0 B(2▪,p) x

Ans: m = 1, n = 5 y = –4

A

Diagram above shows the graph of a

quadratic function y = f(x). The straight –4 x (a) Find the coordinateof A. [1 mark]

line y = –4 is a tangent to the curve y = 0

f(x). (b) By using the method of completing

(c) Write the equation of the axis of (a) State the equation of the axis of

the square, find the value of k and

symmetry of the curve. symmetry of the curve. of p.

(d) Express f(x) in the form (b) Given that the minimum value of

the function is 4, find the value of k. [4 marks]

(x + b)2 + c, where b and c are

constant [3 marks] (c) Determine therange of values of x,

if f(x) ≥ –5. [2

[3 marks] Ans: x = –3, (b) k = 5

Ans: (a) x = 3, (b) f(x)=(x – 3)2 - 4 m a rks]

Ans: (a) A(0, –5), (b) k = 4, p = –1,

3y 5 6

20.

7

SOLUTION OF TRIANGLES

1.

8

2.

9

3.

10

4.

It

5.

12

13

8.

( Ll