Modul Halus F5 (Mid Year)

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Transcript of Modul Halus F5 (Mid Year)

Name : __________________________________ Form : ________________________ R2011_1 1(a) Diagram 1 shows the relation between two sets of numbers P={ 2,1,0,1,2} Q = { 1 , 0 , 1 , 2, 3, 4 } Diagram 1 Based on the above information, the relation between set P and Q is defined by the set of ordered pairs: [ { 2, 1 } , { 1 , 1 } ,( 1, 2 ) , ( 2, 3 ) ] State ( i ) the object of 1 (ii) the image of 2 (iii) the type of relation between set P and Q[ (i) 1 (ii) 3 (iii) one to one

1(c) Diagram 3 shows the relation between two sets of numbers Set Q

3 2 1 1 2 3 Diagram43Set P

(i) Express the relation ordered pairs. (ii) the object of 1 (iii) the type of relation[ (i) (1,1],(2,2),(2,3), (3,1) (ii){1, 3} (iii) many to many )

*1(d) Diagram 4 shows the graph of f(x) = | 5 x | 1(b) Diagram 2 shows the relation between two sets of numbers P={ d,e,f} Q = { 2 , 4 , 6 , 8, 10 } Diagram 2 Based on the above information, the relation between set P and Q is defined by the set of ordered pairs [ { d, 4 } , { e , 4 } ,( f, 10 ) ] State (i) the domain (ii) the range (iii) the type of relation between set P and Q[ (i) {d,e,f } (ii) {4,10} (iii) many to one

(2,7)

y

0 State, i) Domain ii) Range iii) Type of relation

5Diagram 4

( 6,1) x

[ (i) domain 2 < x < 6 (ii) 0 < f(x) < 7, (iii) many to one ]

Halus_1

1

2(a) Given that f : x = 5x + 1 ,

2(c) Given that f : x = 6 5x ,

find ( i ) f 1 (x) (ii) f 1 (11 ) (iii) f 1 (16) = p (iv) f 1( 2w+1) = 4[(i) ( x 1 )/ 5 (ii) 2 (iii) p =3 (iv) w =10 ]

find ( i ) f 1 (x) (ii) f 1 ( 4 ) (iii) f 1 ( 9) = p (iv) f 1( 6 2 w) = 4[(i) ( x 6 )/ 5= 6 x / 5 (ii) 2 (iii) p=3 (iv) w=10 ]

2(b) Given that f : x = 4x 7

find ( i ) f 1 (x) (ii) f 1 (1 ) (iii) f 1 (5) = p *(iv) f 1( 3w+3) = w[(i) ( x +7)/4 (ii) 2 (iii) p=3 (iv) w=10 ]

2(d) Given that f : x = 4 3x

find ( i ) f 1 (x) (ii) f 1 ( 2 ) (iii) f 1 (5) = p (iv) f 1( 14 4 w) = w[(i) ( x 4) / 3 = ( 4 x )/3 (ii) 2 (iii) p=3 (iv) w=10 ]

Halus_1

2

3(a). The function f is defined by f : x

3x 2 and g:x 5x + 1. Given the composite function fg:x px + q, find the values of p and q. [ p = 15 , q = 1 ]

3(d). The function f is defined by f : x

7x + 3

and g:x mx + n . Given the composite function fg :x 10 14 x, find the values of m and n [ m = 2 , n = 1 ]

3(b). The function f is defined by f : xand g:x function of r and s.

7x + 3 6x 5. Given the composite gf :x rx + s, find the values[ r = 42 , s = 13 ]

3(e) The function f is defined by f : x

mx + 3 and g:x 3x + n . Given the composite function fg :x 12x 5, find the values of m and n. [m=4, n = 2]

3(c).The function f is defined by f : x

mx + n and g:x 5x + 1. Given the composite function fg:x 15x 2, find the values of m and n. [m=3,n=5

3(f) The function f is defined by f : x]

mx 1 and g:x 4x + 3 . Given the composite function fg :x 24x +n, find the values of m and n. [ m = 6 , n = 17 ]

Halus_1

3

For examiners use only

3*(g). The function f is defined by f: x x2 4and the function g is defined by g: x m x + n where m > 0 , n > 0 Given the composite function fg : x 4 x2 + 12 x + 5 , find the values of m and n. [ m=2 , n = 3 ]

3(i). The function f is defined by f: x mx2 +nand the function g is defined by g: x 3 x + 2 where m > 0 , n > 0 Given the composite function fg : x 18 x2 + 24 x + 11 , find the values of m and n. [ m=2 , n = 3 ]

3(h) The function f is defined by f: x 3x2 + 2and the function g is defined by g: x m x + n where m > 0 , n > 0 Given the composite function fg : x 12 x2 + 36 x + 29 , find the values of m and n. [ m=2 , n = 3 ]

3(j)The function f is defined by f: x mx2 + nand the function g is defined by g: x 5 x 3 where m > 0 , n > 0 Given the composite function fg : x 50 x2 60 x + 21 , find the values of m and n. [ m=2 , n = 3 ]

Halus_1

4

4(a). Given that 2 is a root to x2 + 2x + p = 0.Find the value of p and the other root.[ p = 8 , the other root = 4 ]

*4(d). Given that 2+p is a root to x2 + 2x 8 = 0.Find the value of p and the root of the equation [ p = 6 or 0 , roots = 4 , 2]

4(b) Given that 3 is a root to x2 + px + 12 = 0.Find the value of p and the other root.[ p = 7 , the other roots = = 4 ]

4(f). Given that 3 p is a root to x2 7x +10 = 0.Find the value of p and the root of the equation. [ p = 2 or 1 , roots = 2 , 5]

4(c).Given that p is a root to x2 + x 5p = 0.Find the value of p where p 0 and the other root. [ p = 4 , the other root = = 5 ]

4(g). Given that 2p is a root to x2 4x 12 = 0.Find the value of p and the root of the equation. [ p = 1 or 3 , roots = 2 , 6]

Halus_1

5

5(a) Diagram show the graph of f(x) = ( x + 2 )2 + 3Find the values of p and q. [ p =2,q=3]

(p,q) y

y

(p,q) 0 x

0 5(f) Diagram show the graph of f(x) = 3 ( x 2 )2 5 Find the values of p and q. [ p = y

x

5(b) Diagram show the graph of f(x) = ( x 2 )2 5Find the values of p and q. [ p =2,q=5]

2,q=5]

y

0 (p,q)

0 x (p,q)

x

5(c) Diagram show the graph of f(x) = ( x 3 ) + q2

5(g) Diagram show the graph of f(x) = 2( x 3 )2 + qFind the values of p and q. [ p =3,q=8]

Find the values of p and q. [ p =

3,q=8]

y

y

(p,8) 0 x

0 (p,8) 5(h) Diagram show thef(x) = ( x +p )2 + 7 Find the values of p and q. [ p =

xgraph of3,q=7]

5(d) Diagram show the graph of f(x) = ( x +p )2 7Find the values of p and q. [ p =3 , q = 7 ]

y

y

(3 , q ) 0 (3 , q ) x 0 x

6(a) Find the range of values of x for which ( x 3 ) ( x + 2 ) < 0 [ 2 0 , min, dy/dx = 4x 8 = 0 , x = 2 , f(2) = 2(2)2 8(2) + 5 = 3 (iii) yint = 5, min at ( 2, 3), parabolic curve]

24(g)Express function f(x)= 3 + 8x 2x2 in ( i ) a( x + b) 2 + c, hence deduce its maximum or minimum value and state the value of x when it occurs. (ii) By means of calculus, determine the maximum or minimum value of f(x) = 3 + 8x 2x2 and state the value of x then it happens. (iii) Sketch the graph of f(x)= 3 + 8x 2x2[ (i) 2( x 2 )2 + 11, max because a = 2 < 0, f(x) max = 11 when x= 2 (ii) dy/dx = 8 4x, d2y/dx2 = 4 < 0 , max, dy/dx = 4 2x = 0 , x = 2 , f(2) = 3 + 8((2) 2 (2)2 = 11 (iii) yint =3, max at ( 2,11), parabolic curve]

24(f)Express function f(x)=2 x + 14 x 1 in ( i ) a( x + b) 2 + c, hence deduce its maximum or minimum value and state the value of x when it occurs. (ii) By means of calculus, determine the maximum or minimum value of f(x) = 2x2 + 14 x 1 and state the value of x then it happens. (iii) Sketch the graph of f(x)= 2x2 + 14 x 1[ (i) 2( x +7/2 ) 25 , min because a = 2 > 0, f(x) min = 25 when x= 7/2 (ii) dy/dx = 4x +14, d2y/dx2 = 4 > 0 , min, dy/dx = 4x +14 = 0 , x = 7/2 , f( 7/2) = 2( 7/2)2 14( 7/2) 1 = 25 (iii) yint = 1 , min at ( 7/2, 25 ), parabolic curve]2

2

24(h)Express function f(x)= 9 + 14x 2x2 in ( i ) a( x + b) 2 + c, hence deduce its maximum or minimum value and state the value of x when it occurs. (ii) By means of calculus, determine the maximum or minimum value of f(x) = 9 + 14x 2x2 and state the value of x then it happens. (iii) Sketch the graph of f(x)= 9 + 14x 2x2[ (i) 2( x 7/2 )2 + 33 , max because a = 2 < 0, f(x) max = 33 when x= 7/2 (ii) dy/dx = 14 4x, d2y/dx2 = 4 < 0 , max, dy/dx = 14 4x = 0 , x = 7/2 , f(7/2) = 9 + 14((7/2) 2(7/2)2 = 33 (iii) yint = 9, max at ( 7/2,33 ), parabolic curve]

Halus_1

28

25(a) Find the equation of the curve for its gradient function is 2x + 3 and passes through (i) ( 1, 8 ) [ y = x2 + 3x + 4 ] (ii) ( 2, 8 ) [ y = x2 + 3x 2 ] (iii ) (2, 8 ) [ y = x2 + 3x + 10]

26(a) Find the equation of the tangent for curve y = 3x(2x 5) 4 and passes through (i) ( 1, 13 ) [ y = 3x 10] (ii) ( 2, 10 ) [ y = 9x 28 ] (iii) (2, 50 ) [ y = 39x 28]

25(b) Find the equation of the curve for its gradient function is 6x(x + 1) 3 and passes through (i)( 1, 9 ) [ y = 2x3 + 3x2 3x + 5 ] (ii)( 1, 9 ) [ y = 2x3 + 3x2 3x + 7 ] (iii)( 2, 3 ) [ y = 2x3 + 3x2 3x 19 ]

26(b) Find the equation of the tangent for function f(x) =6x(x + 1) 3 and passes through (i)( 1, 3 ) [y=6x9] (ii)( 1, 9 ) [ y = 18 x 9 ] (iii)( 2, 33 ) [ y = 30 x 27]

Halus_1

29

27(a) Solve0

Cos A ( 2cos A + 1 ) = 0 ; A= 90 270o ,120, 240o ]

cos 2 A + cos A + 1 = 0 for 0 A 360 0 .

27(d) Solve

cos 2 A + sin A + 2 = 0 for

( 3 2sin A ) ( 1 + sin A ) = 0 ; A= 270 ]

0 0 A 360 0 .

(b) Solve 2 cos 2 A + 3 cos A + 1 = 0 for

( 4cosA 1 ) ( cos A + 1 ) = 0 ; A= 75 31 ,28429 180o ( 3 2sin A ) ( 1 + 2sin x ) = 0 ; A= 210 ,330o ]

0 0 A 360 0 .

(e) Solve 2 cos 2 A + 4 sin A + 1 = 0 for

0 0 A 360 0 .

(c) Solve 3 cos 2 A + 5 cos A +2 = 0 for

( 6cosA 1 ) ( cos A + 1 ) = 0 ; A= 80 24 ,27936 180o

0 0 A 360 0 .

(f) Solve 3 cos 2 A + sin A 2 0 0 A 360 0 .30o

= 0 for

( 1+ 3 sinA ) ( 1 2 sin A ) = 0 ; A= 199 28 ,34032

, 150

o

Halus_1

30

(c) Given that tan C = n and C is an acute 28(a) Given that sin A = p and A is an acute angle, express the following in terms of p. (i) tan A ( ii ) cos A (iii) sin 2A ( iv ) cos 2A[ (i) p / ( 1 p ) ( ii) (iv) 1 2p2 )2

angle, express the following in terms of n. ( i ) cos C ( ii ) sin C (iii) sin 2C (iv) cos 2C (v) tan 2C[ (i) 1/ ( 1 + n2 ) ( ii) n / ( 1 + n2 ) (iii) 2n/( 1 + n2 ) (iv) 2/(1+n2 ) 1 =( 1 n2]/( 1+n2) (v) 2n/(1-n2)

( 1