PENANG 2013 STPM TRIAL PAPERS for Mathematics T TERM 2

MATHEMATICS T 954/2 – SET 1

SECTION A (45 Marks) : Answer all questions in this section.

1. The function f is defined as f(x) =

>>>>−−−−

−−−−

====

<<<<−−−−

33

3

30

39

xx

x

x

xx

,

,

,

2

.

(a) Without using graphs, determine whether f is a continuous function or not. [3]

(b) Sketch the graph of f. [3]

2. Using the substitution x =y21 , show that ∫∫∫∫ −−−−

2

12 1

1

xxdx =

4ππππ . [6]

3. Given that f(x) = x + a +2

4

−−−−x

a2

, x ≠ 2, 2 < a < 3. In terms of a,

(a) find the asymptotes of y = f(x). [2]

(b) find the coordinates of the stationary points. [3]

(c) sketch the graph of y = f(x), labeling clearly the asymptotes, turning points and axial intercepts. [3]

4. The variables x and y are related by the differential equationx

y

d

d= 1 + 2x –

2x

y.

By using y = v + x2, show that the differential equation may be reduced to

xv

dd = –

2x

v . [3]

Find the solution of the differential equation given that when x = 1, y = 2. [5]

5. Given that y = sin−1 x, prove that (1 – x

2)

3

3

x

y

d

d– 3x

2

2

x

y

d

d–x

y

d

d = 0. [3]

Hence, find the Maclaurin’s series for y up to and including the term in x5. [4]

Deduce the expansion for 2x−−−−1

1. [2]

6. Sketch the curve of y = ln (x – 2). [2]

Find an approximate value for the area of the region bounded by the curve,

x-axis and the line x = 4 by using the trapezium rule with five ordinates. Give your answer correct to 3 decimal places. [4]

Hence, determine whether the estimated value is larger or smaller than the exact value. [2]

SECTION B (15 Marks) : Answer any one question in this section.

7. Sketch, on a clearly labelled diagram, the graph of the curve y = 1 +14

1

++++2x. [2]

The region R is bounded by this curve, axes and the line x =21 .

By using the substitution 2x = tan θ, find

(a) the area of the region R, [5]

(b) the volume of the solid formed when R is rotated completely about the x-axis.[8]

8. Show that the equation x3 − 6x + 1 = 0 has two positive real roots. [3]

(a) Show that the smaller positive root, αααα, lies between x = 0 and x = 1. [2]

(b) A sequence of real numbers x1, x2, x3, . . . satisfies the recurrence relation

1++++nx = 2nx(6

1 + 1) for x∈∈∈∈++++.

Use calculator to determine the behaviour of the sequence for x1 = 0. [4]

(c) Prove algebraically that, the sequence can be used to obtain the

root αααα of the equation x3 − 6x + 1 = 0. [3]

(d) Explain whether the recurrence relation in (c) can be use to estimate the larger real root. [3]

1. Not continuous 3(a) y = x + a, x = 2 (b) (2 + 2a, 2 + 5a) , (2 – 2a, 2 – 3a)

4. y = x2 +

11 −−−−xe 5. y = x +61 x

3 +

403 x

5 + . . . ; 1 +

21 x

2 +

83 x

4 + . . .

6. 1.112 ; smaller since the curve is concave downwards.

7(a) 81 (4 + ππππ) (b)

165ππππ (2 + ππππ)

8(b) sequence converges to αααα ≈≈≈≈ 0.16745

(d) Since the derivative value of the recurrence relation for the larger root is greater than 1, so it cannot be use to estimate the larger root.



1. Function f is defined by f(x) =

≥≥≥≥−−−−

<<<<−−−−

21ln

24

xx

xax

,

,

)(

2

.

Given that f’ is continuous at x = 2.

(a) Find the value of a. [3]

(b) Determine whether f is continuous at x = 2. [3]

2. A piece of wire of length d units is cut into two pieces. One piece is bent to form

a circle of radius r units, and the other piece is bent to form a regular hexagon.

Prove that, as r varies, the sum of the areas enclosed by the two shapes is a

minimum when the radius of the circle is approximately 0.076d units. [7]

3. Evaluate ∫∫∫∫ −−−−

1

02x

x

1dx. [3]

Hence, find the exact value of ∫∫∫∫1

0sin

–1 x dx. [4]

4. Find the general solution of the differential equation x

y

d

d–xy= x sec

2 x. [5]

5. Given that y = xe1sin−−−− . Show that (1 – x

2)

2

2

x

y

d

d–

x

yxd

d= y. [3]

By further differentiation of this result, find the Maclaurin’s series for y in

ascending powers of x up to and including the term in x3. [5]

Given that x is small, show that the first four terms of the series expansion for

x

e x

cos

1sin−−−−

is 1 + x + x2 +

65 x

3. [3]

6. Given the equation x2 – xe−−−− – 4 = 0.

(a) Show that the equation has only one real root. [3]

(b) Verify, by calculation that this root lies between x = 2 and x = 3. [2]

(c) Prove that, if a sequence of values given by the iterative formula

1++++nx = nxe−−−−++++4 converges, then it converges to this root. [2]

(d) Use this iterative formula to calculate the root correct to 2 decimal places. Give the result of each iteration correct to 4 decimal places. [3]


7. State the asymptotes of the graph y =4

4

−−−−

−−−−2

2

x

x )(. [2]

Find the coordinates of its stationary points and determine its nature. [8]

Sketch its graph. [3]

Hence, find the range of values of k for which the equation 4

4

−−−−

−−−−2

2

x

x )(– k = 0

has no real roots. [2]

8. (a) Solve the differential equation (1 + ye2 )x

y

d

d= ye sin x cos x,

given that y = 0 when x =6π . [6]

(b) Find the general solution of the differential equation

xx

y

d

d– y – 2x

2 + 1 = 0, expressing y in terms of x. [5]

Find the particular solution which has a stationary point on the positive x-axis.

Sketch this particular solution. [4]

1(a) a =41 (b) Not continuous 3. 1 ;

2ππππ – 1 4. y = x

tan

x + cx

5. y = 1 + x +21 x

2 +

31 x

3 + . . . 6(d) 2.0334 , 2.0325 ; 2.03

7. x = 2, x = −2, y = 1 ; (1, −3)max. , (4, 0)min. ; −3 < k < 0

8(b) y = 2x2 – 22 x + 1



1. The function f is defined by f(x) =

≥≥≥≥−−−−

<<<<−−−−

++++

22

24

6

x

x

xax

x

,

, , where a is a constant.

Find the value of a, if 2→→→→x

lim f(x) exists. [3]

With this value of a, determine whether f’ is continuous at x = 2. [3]

2. Given that y =))(( xx

x

2134

17

++++−−−−

++++. If x increases at a constant rate of 1.5 unit

per second when x = 0.25, find the rate of change of y at this instant. [6]

3. By using a suitable substitution, evaluate ∫∫∫∫ ++++

4

02)( x1

1dx. [6]

4. Using the substitution u = xy, solve the differential equation

xx

y

d

d+ y + xy

2 = 0, given that y =

21 when x = 2, expressing y in terms of x. [10]

5. Given that x is sufficiently small for x3 and higher powers of x to be neglected,

show that x

x

sin2

cos45

++++

−−−− ≈≈≈≈ 21 –

41 x +

89 x

2. [5]

6. A curve has the equation y =x

x

ln

2++++ .

(a) Show that the curve has only one stationary point, and its x-coordinate

satisfies the equation x =x

x

ln

2++++. [5]

Find the successive integers a and b such that this root lies in the

interval (a, b). [3]

(b) Use the iterative formula 1++++nx =n

nx

x

ln

2++++to determine the x-coordinate

correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [4]


7. A curve has the parametric equations

x = k + sin t and y = k cos t, where k > 0 and −ππππ ≤ t ≤ ππππ.

(a) Expressx

y

d

d in terms of t. [3]

(b) State the exact values of t at the points when the tangents are parallel

to the y-axis, and the points when the tangents are parallel to the x-axis. [4]

(c) The normal of the curve at the point where t =4ππππ has a y-intercept of −1.

Find the value of the constant k. [4]

(d) The normal intersects the curve again at point P. Using k = 1, find P. [4]

8. (a) By using the standard Maclaurin’s expansion of ex, find ∑∑∑∑

∞∞∞∞

====1!

rr1 in terms of e. [3]

(b) Given that y = tan–1 2x, show that (1 + 4x

2)

3

3

x

y

d

d+ 16x

2

2

x

y

d

d+ 8

x

y

d

d = 0. [4]

Obtain the Maclaurin’s series for tan–1 2x up to and including the term in x

3. [4]

Use the series expansion above, estimate the value of ∫∫∫∫ 51

0tan

–1 2x dx,

giving your answer as a fraction. [4]

1. a = −1 ; Not continuous 2. –169316 3. 2 ln (

35 ) –

54

4. y =

xxx

−

2ln

1 6(a) a = 4, b = 5 (b) 4.32

7(a) –k tan t (b) −2ππππ ,

2ππππ ; −ππππ, 0, ππππ (c) 1 (d) (1 –

2

1 , –2

1 )

8(a) e – 1 (b) y = 2x –38 x

3 + . . . ;

187573



1. Evaluate

−−−−

−−−−→→→→ )( xex

x

x 1

cos1lim2

0. [5]

2. Show that ∫∫∫∫21

0

sin−1(2x) dx =

41 (ππππ – 2). [5] [4,3]

3. The parametric equations of a curve are

x = ln (cos θ), y = ln (sin θ), where 0 < θ <2ππππ .

Find the equation of the tangent to the curve at the point where θ =4ππππ ,

leaving your answer in the form of y = mx + c. [6]

Show that the tangent will not meet the curve again. [4]

4. Show that the differential equation xyx

y

d

d= x

2 + 2y

2 may be reduced by

by means of the substitution y = vx to xx

y

d

d=

v

v2++++1. [3]

Hence obtain the general solution of y in the form y2 = f(x). [4]

5. Given that y = 2 tan–1

++++ 3

2

x

x. Show that (x

2 + 2x + 3)

x

y

d

d = 2. [2]

By further differentiation of the above result, find the Maclaurin’s series

expansion for y in ascending powers of x up to and including the term in x3. [5]

Hence, find the first three non-zero terms in the expansion of 32

1

++++++++ xx2 [2]

6. Show that the equation x + 4 + ln x = 0 has only one real root, and state the

successive integers a and b such that this root lies in the interval (a, b). [4]

Use the Newton-Raphson method with initial estimate xo = 0.02 to find the

real root correct to four decimal places. [4]

Give a reason why 0.5 cannot be use as the initial estimate in the

above calculation. [1] SECTION B (15 Marks) : Answer any one question in this section.

7. The diagram shows the region R bounded

by the curves y = x2 and x = (y – 2)

2 – 2

and the y-axis.

(a) Find the coordinates of the points A and B. [5]

(b) Find the area of the region R. [4]

(c) Find the volume formed when R

is rotated 2ππππ radian about the y-axis. [4]

8. Solve the differential equation (x + 1)x

y

d

d= y – y

2,

and show that the general solution can be express as y =cx

x

+

+1,

where c is a constant. [9]

Sketch the solution curve which passes through the point (–3, 2),

labelling all your intercepts and asymptotes clearly. [6]

1. –1 3. y = –x – ln 2 4. y2 = x

2(Ax

2 – 1)

5. y =32 x –

92 x

2 +

812 x

3 + . . . ;

31 –

92 x +

271 x

2 + . . . 6. a = 0, b = 1 ; 0.0180

7(a) (–1, 1), (0, 2 – 2 ) (b) 31 (7 – 24 ) (c)

301 (101 – 264 )ππππ

A

B

x

y

R

0

y = x2

x = (y – 2)2 – 2



1. Let f and g are two continuous functions in [a , b] and such that f(a) > g(a)

and f(b) < g(b). Prove that exists a value c ∈∈∈∈[a , b] such that f(c) = g(c). [5]

2. By substituting y =21 sin

2 θ, find the exact value of ∫∫∫∫ −−−−

41

0y

y

21dy. [5]

3. Diagram shows a rectangle ABCD inscribed in a semi-circle with fixed

radius r cm. Two vertices of the rectangle lie on the arc of the semi-circle.

If AB = x cm, show that the perimeter P of the

rectangle ABCD is 2x + 22 xr −−−−4 . [2]

Given that as x varies, the maximum value of P

occurs when AB : BC = 1 : k, find k. [8]

4. Find the general solution for the differential equation

x

y

d

d

21 = tan

–1 (x) –x

y, expressing y in terms of x. [8]

5. If y = cos–1 x, show that that (1 – x

2)

2

2

x

y

d

d – x

x

y

d

d= 0. [3]

Hence, find the Maclaurin’s series for y, for the first three non-zero terms. [4]

6. Given that y =x++++1

2. Show that

x

y

d

d< 0 for all x ≥ 0. [3]

By using the trapezium rule with 5 ordinates, estimate the value of I,

where I = ∫∫∫∫ ++++

3

0 x1

2dx, correct to 3 decimal places. [4]

By sketching the graph of y =x++++1

2for x ≥ 0, determine whether the

estimated value of I is larger or smaller than its actual value. [3]


7. Given that y = xe 3− sin kx, where k is a constant and thatx

y

d

d= 4 when x = 0.

(a) Find the value of k and show that2

2

x

y

d

d+ 6

x

y

d

d+ 25y = 0. [6]

(b) Find the Maclaurin’s series for y up to and including the term in x4. [5]

(c) By using the standard expansions, verify the correctness of your answer in (b). [4]

8. Given that y = [ln (1 + x)]2, show that

2

x

y

d

d=

2)( x

y

+1

4 and (1 + x)

2

2

2

x

y

d

d+ (1 + x)

x

y

d

d= 2. [5]

By further differentiation of the result above, obtain the Maclaurin’s series for

[ln (1 + x)]2 up to and including the term in x

4. [7]

Verify that the same result is obtained if the standard series expansion for

ln (1 + x) is used. [3]

2. 28

1 (ππππ – 2) 3. k = 4

4. y =2x3

1 [2x3 tan

–1 (x) + ln (1 + x) – x2 + c] 5.

2ππππ – x – x

3 + . . .

6. 3.103 ; over-estimate since curve is concave upwards.

7(a) k = 4 (b) y = 4x – 12x2 +

322 x

3 + 14x

4 + . . . 8. y = x

2 – x

3 +1211 x

4 + . . .

A D

B C

A



1. Given that f is defined as f(x) =2

4

−−−−−−−−

x

x2

, x ≠ 2.

(a) Determine whether 2→→→→x

lim f(x) exists. [3]

(b) Determine whether f is continuous at x = 2. [2]

2. The parametric equations of a curve are x = ln (2t), y = tan

–1 (2t), where t > 0.

Show that the gradient of the curve at the point where y = p is21 sin 2p. [5]

3. (a) Find the exact value of ∫∫∫∫ππππ3

02 x

x

cos

cosln )(dx. [5]

(b) Using the substitution u = 1 + cos x, show that

∫∫∫∫ ++++ x

x

cos1

2sin2dx = 4

ln

(1 + cos x) – 4 cos x + c. [5]

4 By means of the substitution z =2y

1 , show that the differential equation

2xex

y

d

d = 2xy

21−−−−2y can be reduced to the form

z−−−−1

1

x

z

d

d= –4x

2xe−−−− ,

where y > 1. [3]

Hence find the general solution of y in terms of x. [3]

Prove algebraically (not verify) that the minimum point of every member

of the family of solution curves lie on the y-axis. [3]

5. On the same axes, sketch the graphs of y = xe−−−− and y = 9 – x2. [2]

State the integer which is closest to the positive root of the equation x2 + xe−−−− = 9. [1]

Using the Newton-Raphson method, find an approximation to this root, correct to three decimal place. [4]

6. It is given that y = xcos .

(a) Show that 2

2

x

yyd

d2 +

2

x

y

d

d2 + y

2 = 0. [3]

(b) Find Maclaurin’s series for y in ascending powers of x, up to and

including the term in x2. [3]

(c) By choosing a suitable value for x, deduce the approximate relation

4 2

1 ≈≈≈≈ 1 + kππππ2, where k is a constant to be determined. [3]


7. Sketch the graphs of y =2x

x

++++1

2 and y = x for x ≥ 0, in the same diagram. [3]

The region R is bounded by the curves. Find the exact area of R. [5]

Using the substitution x = tan θ, find the exact volume of the solid formed

when R is rotated through four right angles about the x-axis. [7]

8 A curve has parametric equation x = 1 + 2 sin

θ, y = 4 + 3 cos

θ.

(a) Find the equations of the tangent and normal at the point P where θ =6ππππ . [6]

Hence, find the area of the triangle bounded by the tangent and normal at P,

as well as the y-axis. [2]

(b) Determine the rate of change of xy at θ =6ππππ , if x increases at a constant

rate of 0.1 units per second. [5]

1(a) exists (b) not continuous. 3(a) 3 (1 – ln 2) –3ππππ

4. y =22)( x

A e−−−−−−−−−−−−1

1 5. 3 ; 2.992

6(b) y = 1 –41 x

2 + . . . (b)

4ππππ ; k = –

641 7. ln 2 –

21 ;

61 (3ππππ2

– 8ππππ)

8(a) x + 2y = 13 , 4x + 2y = 3 ; 5 (b) 209




−−−−

<<<<<<<<−−−−++++

otherwise12

112

,

,

x

xx.

(a) Find −−−−−−−−→→→→ 1x

lim f(x) and ++++−−−−→→→→ 1x

lim f(x). [2]

(b) Determine whether f is continuous at x = –1. [2]

2. The volume of water in a hemispherical bowl of radius 12

cm is given by

V = 3ππππ (36x

2 – x

3), where x is the depth of the water.

(a) Using Calculus method, find the approximate amount of water necessary to raise the depth from 2

cm to 2.1

cm. [3]

(b) If water is poured in at a constant rate of 3 cm

3s

–1, find the rising

rate of the level when the depth is 3 cm. [3]

(Leave all your answers in terms of ππππ)

3. Using x = 2 cos θ, show that ∫∫∫∫ −−−− 22 xx 4

1dx = –

x

x

4

4 2−−−−+ c. [7]

4. By means of the substitution y =x1 +

z1 , show that the differential equation

x2

x

y

d

d= 1 – 2x

2y

2 can be reduced to

xz

dd =

xz4 + 2. [3]

Solve this equation and hence find the general solution of the differential equation,

x2

x

y

d

d= 1 – 2x

2y

2, expressing y in terms of x. [5]

5. By using the graphs of y = sin2x and 2y = x – 2, show that the equation

2 sin2x – x + 2 = 0 has only one real root for x > 0, and state the successive

integers a and b such that the real root lies in the interval (a, b). [4]

Use the Newton-Raphson method to find the real root correct to three decimal places [4]

6. Given that y = xe1sin−−−− , show that

(a) x

y

d

d= 1 when x = 0, [2]

(b) (1 – x2)

2

2

x

y

d

d=

x

yxd

d+ y. [3]

Hence, find the Maclaurin’s series for y, up to and including the term in x3. [4]

(c) Use the series above to estimate the value of ∫∫∫∫−−−−0.1

0

1sin xe dx,

correct to three decimal places. [3] SECTION B (15 Marks) : Answer any one question in this section.

7. A rectangular block with a square base and height 2(a – x), x < a, is inscribed

in a sphere of fixed radius a such that the vertices of the block just touch the interior of the sphere.

(a) Show that the square base has side length )( xax −22 . [3]

Hence, write down the volume of the block in terms of x and a. [2]

(b) Show that the volume of the block is largest when it is a cube. [8]

Hence, find the volume of the cube in terms of a. [2]

8. Obtain the coordinates of the turning point of the curve y =2x

x

++++4

2, for x ≥ 0. [3]

Determine the nature of the stationary point as well. [3]

Sketch the curve y =2x

x

++++4

2and the line y = x

41 on the same diagram. [2]

The region enclosed by the graphs is denoted by R. Using the substitution

x = 2 tan θ, find the volume, in terms of ππππ, of the solid generated when the

region R is rotated completely about the x-axis. [7]

1(a) 1 ; 1 (b) continuous 2(a) 4.4ππππ (b) ππππ21

1 4. y =

−+

23

311

3cxx

5. a = 3, b = 4 ; 3.869 6. y = 1 + x +21 x

2 +

31 x

3 + . . . (c) 0.105

7(a) V = 8a2x – 12ax

2 + 4x

3 (b) 3a

938

8. (2,21 )max. ; 12

1 (3ππππ2 – 8ππππ)

MATHEMATICS T 954/2 – SET MPM



−−−−

−−−−≥≥≥≥++++

otherwise1

11

,

,

x

xx

(a) Find 1

lim

−−−−→→→→x f(x). [3]

(b) Determine whether f is continuous at x = –1 . [2]

2. Find the equation of the normal to the curve with parametric equations

x = 1 – 2t and y = –2 +t2 at the point (3, –4). [6]

3. Using the substitution x = 4 sin2 u, evaluate ∫∫∫∫ −−−−

1

0 x

x

4dx. [6]

4. Show that ∫∫∫∫ −−−−

−−−− xxxx

ed

12)( =

1−−−−x

x2

. [4]

Hence, find the particular solution of the differential equation

x

y

d

d+ y

xx

x

)( 1

2

−−−−

−−−−= –

)( 1

1

−−−−xx2

which satisfies the boundary condition y =43 when x = 2. [4]

5. If y = sin–1 x, show that

2

2

x

y

d

d=

3

x

yx

d

dand

3

3

x

y

d

d=

3

x

y

d

d+

52

x

yx

d

d3 . [5]

Using Maclaurin’s theorem, express sin–1 x as a series of ascending powers

of x up to the term in x5. [6]

State the range of values of x for which the expansion is valid. [1]

6. Use the trapezium rule with subdivisions at x = 3 and x = 5 to obtain an

approximation to ∫∫∫∫ ++++

7

14

3

x

x

1dx, giving your answer correct to three places

of decimals. [4]

By evaluating the integral exactly, show that the error of the approximation is about 4.1%. [4]


7. A right circular cone of height a + x, where –a ≤ x ≤ a, is inscribed in a sphere

of constant radius a, such that the vertex and all points on the circumference

of the base lie on the surface of the sphere.

(a) Show that the volume V of the cone is given by V = ππππ31 (a – x)(a + x)

2. [3]

(b) Determine the value of x for which V is maximum and find the maximum

value of V. [6]

(c) Sketch the graph of V against x. [2]

(d) Determine the rate at which V changes when x = a21 if x is increasing at

a rate of a101 per minute. [4]

8. Two iterations suggested to estimate a root of the equation x3 – 4x

2 + 6 = 0

are 1++++nx = 4 –2nx

6 and 1++++nx =

21 (xn

3 + 6) 2

1

.

(a) Show that the equation x3 – 4x

2 + 6 = 0 has a root between 3 and 4. [3]

(b) Using sketched graphs of y = x and y = f(x) on the same axes, show that,

with initial approximation xo = 3, one of the iterations converges to the

root whereas the other does not. [6]

(c) Use the iteration which converges to the root to obtain a sequence of

iterations with xo = 3, ending the process when the difference of two

consecutive iterations is less than 0.05. [4]

(d) Determine whether the iteration used still converges to the root if the

initial approximation is xo = 4. [2]

1(a) 0 (b) continuous 2. x + y + 1 = 0 3. 31 (2ππππ – 33 )

4. y =2x

x 12 −−−− 5. y = x +

61 x

3 +

403 x

5 + . . . ; –

2ππππ < x <

2ππππ

6. 1.701

7(b) x = a31 , Vmax. =

3aππππ8132 (c)

(d) – 3aππππ401

8(c) 3.33, 3.46, 3.50 (d) Yes

•

• • •

–a x

y

0 a

3aππππ31

( a31 , 3aππππ

8132 )

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PENANG 2013 STPM TRIAL PAPERS for Mathematics T TERM 2