MathematicsT STPM Baharu PERAK 2012

2012 TRIAL STPM BAHARU MATHEMATICS T SMK ACS IPOH,PERAK

SectionSectionSectionSection AAAA[45 marks]Answer allallallall questions in this section.

1. The functions f and g are defined as

f : xx1

→ , x +ℜ∈ andandandand g : x xln→ , x +ℜ∈

(a) State the ranges of f and g . [2[2[2[2 marksmarksmarksmarks](b) If h is the composite function gof, find the function h. [2[2[2[2 marks]marks]marks]marks](c) Show that h(x) + g(x) = 0. [1[1[1[1 mark]mark]mark]mark]

2. (a) Given ∑∞

=⎟⎠⎞

⎜⎝⎛

0 31

i

i

p = ( )∑=

+10

3

31j

j where p is a constant.

Find the value of p. [3[3[3[3 marks]marks]marks]marks]

(b) Use the binomial theorem to expandxx

−+

11 as a series of ascending powers

of x up to and including the term in x2 , where x <1.

Hence, by substituting x =101 , show that

20066311 ≈ . [6[6[6[6 marks]marks]marks]marks]

3.3.3.3. Matrices PPPP, QQQQ and RRRR are⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−322211210

,⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−−−15228387

333414and

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−

−

262333666

respectively. Find121 PQRPQRPQRPQR and deduce RRRR-1-1-1-1 . [4[4[4[4 marks]marks]marks]marks]

Hence, solve the system of linear equations x + y – z = 8-4x +4 y +4 z = 0x + 3y +z =10 [5[5[5[5 marks]marks]marks]marks]

4. Given that the real numbers r and θ , where r > 0 , - πθπ << ,r cosθ + 2r2 cos2θ + 3r3 cos3θ =0 and r sinθ + 2r2sin 2θ + 3r3 sin3θ =0.By writing z = r (cosθ + i sinθ ) and using De Moivre’s Theorem, show that

z = )21(31 i±− . [5[5[5[5 marks]marks]marks]marks]

Determine the value of r and the two possible values of tanθ . [4[4[4[4 marks]marks]marks]marks]

5. A curve has the parametric equations x = 8 cosθ + 3 and y = 4 3 sinθwhere πθπ ≤<− . Show that the curve is an ellipse and find its vertices, centreand foci. Sketch the ellipse. [9[9[9[9 marks]marks]marks]marks]

6. Find the value of α for which the vectors aaaa ==== 3iiii ---- 4jjjj ++++ kkkk and bbbb = iiii ++++ 2jjjj + α kkkkare perpendicular. Hence, find aaaa ---- bbbb . [4[4[4[4 marks]marks]marks]marks]


SectionSectionSectionSection BBBB [15 marks]Answer any oneoneoneone question in this section.

7. (a) Solve for x the equation sin 2x = 3 cos2x, giving all solutions between 0o and360o correct to the nearest 0.1o. [5[5[5[5 marks]marks]marks]marks](b) Express 2 cos x + 5 sin x in the form r cos ( x - )α , where r>0 and

0<α < π21 . Hence, find the maximum value of 2 cos x + 5 sin x and the

corresponding value of x in the interval π20 ≤≤ x . [6[6[6[6 marks]marks]marks]marks]

Sketch the curve y = 2 cos x + 5 sin x for π20 ≤≤ x . [2[2[2[2 marks]marks]marks]marks]By drawing an appropriate line on the graph, determine the number of rootsof the equation 2 cos x + 5 sin x = 1, in the interval π20 ≤≤ x . [2[2[2[2 marks]marks]marks]marks]

8. The line l has equation rrrr =⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

012

505

λ , ℜ∈λ .

(a) Show that l lies in the plane whose equation is

rrrr .... 5021

−=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−. [3[3[3[3 marks]marks]marks]marks]

(b) Find the position vector of A, the foot of the perpendicular from the origin O to l.[4[4[4[4 marks]marks]marks]marks]

(c) Find an equation of the plane containing O and l. [4[4[4[4 marks]marks]marks]marks](d) Find the position vector of the point P where l meets the plane π whose

equation is rrrr .... .11221

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛[4[4[4[4 marks]marks]marks]marks]

MATHEMATICALMATHEMATICALMATHEMATICALMATHEMATICAL FORMULAEFORMULAEFORMULAEFORMULAEBinomial expansions

,......21

)( 221 nrrnnnnn bbarn

ban

ban

aba ++⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+=+ −−− where n∈ZZZZ++++

...,!

)1)...(1(...!2)1(1)1( 2 +

+−−++

−++=+ rn x

rrnnnxnnnxx where ∈n QQQQ ,,,, x <<<<1

ConicsParabola with vertex ( h, k ), focus ( a+h, k ) and directrix x = - a + h

( y – k )2 = 4 a ( x – h )Ellipse with centre ( h, k ) and foci (- c + h, k ) , ( c + h, k ), where c2= a2 – b2

1)()(2

2

2

2

=−

+−

bky

ahx

Hyperbola with centre ( h, k ) and foci (- c + h, k ), ( c + h, k ), where c2= a2+ b2

1)()(2

2

2

2

=−

−−

bky

ahx


1. The functions f and g are defined as f : xx1

→ , x ∈ ℝ AND g : x xln→ , x ∈ ℝ+

(a) State the ranges of f and g . [2 marks](b) If h is the composite function gof, find the function h. [2 marks](c) Show that h(x) + g(x) = 0. [1 mark]

1(a) Rf = {y : y > 0}

Rg = {y : y ∈ }

B1

B1

(b) gof(x) = g

= ln or – ln xM1

h : x� – ln x , x > 0 A1(c) h (x) + g(x) = – ln x + ln x

= 0A1

[5]

2. (a) Given ∑∞

=⎟⎠⎞

⎜⎝⎛

0 31

i

i

p = ( )∑=

+10

3

31j

j where p is a constant.

Find the value of p. [3 marks]

(b) Use the binomial theorem to expandxx

−+

11 as a series of ascending powers of x up to

and including the term in x2 , where x <1.

Hence, by substituting x =101 , show that

20066311 ≈ . [6 marks]

2(a)= OR

= 8 + 3{3 + 4 + 5 + . . . + 9 + 10}

B1 Either one

= 164 M1

p = A1

(b) ×= ×

= ×M1M1 Correct binomial

expansion

≈ 1 + x + ½x2 A1

= B1


≈ 1 + +½

≈ 3 ×

≈

M1

A1

Subst. x =

[9]

3 Matrices PPPP, QQQQ and RRRR are⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−322211210

,⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−−−15228387

333414and

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−

−

262333666

respectively. Find121PQRPQRPQRPQR and deduce RRRR-1-1-1-1 . [4 marks]

Hence, solve the system of linear equations x + y – z = 8-4x +4 y +4 z = 0x + 3y +z =10 [5 marks]

3PQRPQRPQRPQR =

= M1 Correct QR

=

orororor PQRPQRPQRPQR =

= M1 Correct PQ

=

∴ PQRPQRPQRPQR =A1

PQRPQRPQRPQR = - 9 IIII

RRRR-1 = - PQPQPQPQ

= -

=

M1

A1By adjusting the system of linear equations : B1 Able to rearrange the

system of equations


B1

M1 Reject all othermethods.

= A1

∴ x = 1, y = 4, z = -3

A1 [9]

4 Given that the real numbers r and θ , where r > 0 , - πθπ << ,r cosθ + 2r2 cos2θ + 3r3 cos3θ = 0 and r sinθ + 2r2sin 2θ + 3r3 sin3θ = 0.

By writing z = r (cosθ + i sinθ ) and using De Moivre’s Theorem, show that z = )21(31 i±− .

[5 marks]Determine the value of r and the two possible values of tanθ . [4 marks]

4 r cosθ + 2r2 cos2θ + 3r3 cos3θ = 0 (1)r sinθ + 2r2sin 2θ + 3r3 sin3θ = 0. (2)By operating (1) + i(2),r (cosθ + isinθ ) + 2r2(cos2θ + isin 2θ ) + 3r3(cos3θ + isin3θ ) =0

M1

r (cosθ + isinθ ) + 2r2(cosθ + isinθ )2 + 3r3(cosθ + isinθ ) 3 = 0 M1 Using De Moivre’stheorem

∴ z + 2z2 + 3z3 = 0

A1

OR z = r (cos θ + i sin θ)z2 = r2(cos 2θ + i sin 2θ) and z3 = r 3(cos 3θ + i sin 3θ) M1 Using De Moivre’s

theorem2z2 = 2r2(cos 2θ + i sin 2θ) and 3z3 = 3r 3(cos 3θ + i sin 3θ) M1

∴ z + 2z2 + 3z3 = 0

A1


z (3z2 + 2z + 1) = 0

z ≠ 0, z =

=

M1

A1

| z | = B1

tan θ = or tan θ = M1

tan θ = - tan θ = A1A1 [9]

5 A curve has the parametric equations x = 8 cosθ + 3 and y = 4 3 sinθ whereπθπ ≤<− .

Show that the curve is an ellipse and find its vertices, centre and foci. Sketch the ellipse.[9 marks]

5 By using cos θ = and sin θ = with sin2 θ + cos2 θ = 1, B1 Eliminatingparameter θ

+ = 1M1

Using trig. identity

+ = 1

∴ the curve is an ellipse.

A1

Centre is (3, 0) A1The vertices are (-5, 0) and (11, 0) A1A1The foci are (7, 0) and (-1, 0) A1

x

y

(3, 0)● ●● ●●

●

●

(7, 0) (11, 0)(-1, 0)(-5, 0)

(3, 4 )

(3, - 4 )

D1

D1

Ellipse shape(dependent on thecorrect equationabove)

All ‘his’ points shownand labeled.

[9]


6 Find the value of α for which the vectors aaaa ==== 3iiii ---- 4jjjj ++++ kkkk and bbbb = iiii ++++ 2jjjj + α kkkk areperpendicular. Hence, find | aaaa ---- bbbb |. [4 marks]

6 By using aaaa • bbbb = 0,

• = 0

α = 5

M1

A1

aaaa - bbbb = - M1

=

|||| aaaa – bbbb | == A1 Accept 7.48

[4]7 (a) Solve for x the equation sin 2x = 3 cos2x, giving all solutions between 0o and 360o

correct to the nearest 0.1o. [5 marks]

(e) Express 2 cos x + 5 sin x in the form r cos ( x - )α , where r >0 and 0<α < π21 .

Hence, find the maximum value of 2 cos x + 5 sin x and the corresponding value ofx in the interval π20 ≤≤ x . [6 marks]Sketch the curve y = 2 cos x + 5 sin x for π20 ≤≤ x . [2 marks]By drawing an appropriate line on the graph, determine the number of roots of theequation 2 cos x + 5 sin x = 1, in the interval π20 ≤≤ x . [2 marks]

7(a) 2sin x cos x - 3cos2x = 0 B1cos x (2sin x - 3cos x) = 0 M1 Reject division by cos

x

cos x = 0 or tan x =M1

x = 90 °, 270 ° or x = 56.3° , 236.3 ° A1

∴ x = 56.3°, 90 °, 236.3 °, 270 °

A1 All correct

(b) 2cos x + sin x ≡ r cos x cos α + r sin x sin αr sin α =r cos α = 2

B1 Both

r = 3, tan α = M1

r = 3, α = 0.841 radians. A1 Both


∴ 2cos x + sin x = 3 cos (x – 0.841)

A1

max value is 3 B1when cos (x – 0.841) = 1 � x = 0.841radians. B1

y

x

3

2

-3

y =1

● ●

●(0.841, 3)

(2π, 2)

D1

D1

M1

A cosine curve with amax. and a min.turning point.

Initial and final pointsand ‘his’ max. point.line y = 1

There are 2 roots. A1 Dependent oncorrect graph.[15]

8 The line l has equation rrrr =⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

012

505

λ , λ ∈ ℝ.

(a) Show that l lies in the plane whose equation is rrrr .... 5021

−=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−. [3 marks]

(b) Find the position vector of A, the foot of the perpendicular from the origin O to l. [4 marks](c) Find an equation of the plane containing O and l. [4 marks]

(d) Find the position vector of the point P where l meets the plane π whose equation is rrrr .... .11221

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

[4 marks]8(a)

rrrr = + λ and rrrr • = -5

By using rrrr • nnnn = d,

• = -5 B1


-5 – 2λ + 2λ = - 5 for all values of λ ∈ ℝ.

M1

∴ line l lies on the plane.

A1

(b)

= + λ1 , λ1∈ ℝ

• = 0

• = 0 B1

10 + 4 λ1 + λ1 = 0� λ1 = -2

M1

= - 2

=

M1

A1

(c)nnnn = ×

=M1

= A1

∴ equation of plane is rrrr •••• = ••••

rrrr •••• = 0 or – 5x + 10y + 5z = 0

M1

A1 Accept rrrr •••• = 0

(d) Since P is on line l,


= + λ2 , λ2∈ ℝ

Given that rrrr .... .11221

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

.... .11221

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

B1

5 + 2 λ2 + 2 λ2 + 10 = 11 M1λ2 = - 1 A1

= - 1

=A1 [15]

MathematicsT STPM Baharu PERAK 2012

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