mathematics t for new stpm syallabus

Mathematics (T) Worked Examples

Loo Soo Yong

1/27/2013

This document contains various questions with detailed workings and explanations from the new STPM syllabus. A PDF version of this document may be obtained from Dropbox.

Mathematics (T)

(c) 2012-2013 All rights reserved

Binomial Expansions

1. Expand ( ) up to and include the term . Hence, by subsisting , find an estimate

for

, and give your answer to 6 decimal places.

Solution:

( ) ( )( ) ( )( )

( )

( )( )( )

( )

( )

Substituting ... ( ) ( ) ( ) ( )

2. ( ) is given by the expression ( ) √

(a) Show that ( )can be written as ( ) .

/

(b) When is so small that and higher powers of x is ignored, show that

.

(c) Hence, state the range of values of x for which the expansion is valid.

(d) By substituting into the equation, find an estimate for √ , giving your answer in the

form

, where p and q are integers to be determined.

Solution:

(a) ( ) ( )

( ) 0 .

/1

( )

.

/

( ) .

/

(shown)

(b) ( ) 6

.

/

.

/.

/

.

/

.

/.

/.

/

.

/ 7

( ) 0

1

Mathematics (T)


( )

(shown)

(c) The expansion is valid for |

| i.e. .

(d) √

( )

( )

3. For , the expansion of ( ) in ascending powers of x is given by

where k and n are integers. Show that and

Solution:

(a) ( ) ( )( )

( )

By comparing coefficient of and

( )( )( )

( )

.

/

( )

(shown)

Subst ...

(shown)

Mathematics (T)


4. By taking the natural logarithm to both sides or otherwise, show that

( ) ( ) can be

written in the form of ( ) ( ) ( ) ( ) , where A, B, p, q, r and s are

constants to be determined. [No credit will be given if the derivative is obtained using product and

chain rule]

Solution:

Let ( ) ( )

Taking natural log to both sides, ,( ) ( ) -

Using properties of logarithms, ( ) ( )

Differentiating y with x,

0

( )1 0

( )1

0

1

( ) ( ) 0

1

( ) ( ) ( ) ( )

Therefore, A=8, B=6, p=3, q=6, r=4, s=5.

5. Express ( )

( )( )( ) in partial fractions. If x is so small that and higher powers of x are

ignored, show that ( )

. Hence, use the quadratic approximation to estimate

the value of ∫ ( )

, giving your answer to 4 decimal places.

Solution:

( )( )( )

( )( ) ( )( ) ( )( )

Letting

Letting

Letting

Mathematics (T)


If x is so small that and greater powers of x are ignored,

( ) ( ) ( )

( ) [ .

/

] [ .

/

]

0 ( )( ) ( )( )

( ) 1

[ ( ) .

/

( )( )

.

/ ] ,

( ( ) .

/

( )( )

.

/ )]

, -

0

1 0

.

/1

( )

.

/

( )

(shown)

∫ ( )

∫

∫

∫

∫

0

1

∫

∫

0

1

0

1

0

1

0

( )

( )1

0

( )

( ) 1

0

( )

( ) 1

( )

Separate the constants first for

easier integration.

Mathematics (T)


6. Show that ( )

Hence by using the information given,

(a) Expand

( )√ in ascending powers of x, up to and including the term .

(b) State the set values of x for which the expansion is valid.

(c) By taking

, find an estimate for √ , giving your answer to 4 decimal places.

Solution:

( ) .

/

( )

( ( ) .

/

( )( )

.

/

( )( )( )

.

/ )

( )

.

/

( )

(shown)

(a)

( )√ ( )(

)(

√ )

( ) ( )

.

/4 .

/ ( )

.

/.

/

( ) 5

(

)(

)

(b) The expansion is valid for |

| | | .

| | | |

From the graph below,

-2

2

Therefore the set values of x are * | |

+

(c) Let

,

( .

/)√ .

/

.

/

.

/

√

√

You only need to expand

until 𝑥 as the question

specifies you to do so.

Mathematics (T)


7. Consider the curve

(a) Find the equation of this tangent at the point where – .

(b) Find the coordinates of the point where this tangent meets the curve again.

Solution:

(a)

At point ,

( ) ( )

When ( ) ( ) ( )

Therefore the coordinates is ( )

The equation of the tangent, ( ) ( )( )

(b) ( )

( )

Subst. (2) to (1),

Factoring, ( ) ( )

(rejected) or

When ( ) ( ) ( )

Therefore the point is ( ).

8. The polynomial ( ) ( ) leaves a remainder of when divided by ( ) and a

remainder of when divided by ( ). Find the real numbers a and b.

Solution:

( )

( )

( )

( )

Mathematics (T)


( )

( )

Subst ( ) into ( ),

Subst

into (1)

.

9. Two complex numbers are defined as

and

. Find the real numbers a and b such

that .

Solution:

.

/

.

/

Given ,

.

/ .

/

If there is a complex number as the

denominator, always multiply by its

conjugate.

Mathematics (T)


Comparing real and imaginary parts,

( )

( )

Subst. (2) to (1),

Subst into (2)

10. A curve is defined parametrically as

(a) Find the gradient of the curve where

(b) Hence, find the equation of the tangent to the curve at the point where

Solution:

(a)

.

/ .

/

( ) .

( ) /

( )

( )

When

( )

( )

(b) When ,

Mathematics (T)


( ) ( )

When

The point is ( )

The equation of the tangent is ( )

11. Find the set values of x for which

√

Solution:

√

√

(√ ) √

Let √

( )( )

Therefore,

√

However, we need to enforce the condition that .

Therefore, the solution set is *

+

If 𝑎 𝑏 then

𝑎

𝑏

Since

𝑥 √𝑥 is undefined for

𝑥 , hence this condition

must be enforced.

x is always positive, because of

squaring a number always produces a

positive number. The minimum value

of 𝑥 is zero.

Mathematics (T)


12. Let f to be a cubic polynomial. Given that ( ) , ( ) , ( ) ( ) and ( )

find the polynomial ( ).

Solution:

Let ( )

( ) ( ) ( ) ( )

( )

( ) ( ) ( )

( )

( )

( ) ( )

( )

( )

( )

( )

( ) ( )

Subst.

into (1),

( )

13. Show that is a root to the equation . Hence find the other two

solutions in exact form.

An arithmetic sequence has p as its common difference. A geometric sequence has p as its common

ratio. Both sequences has 1 as the first term.

Mathematics (T)


(a) Write down the first four terms of each sequence.

(b) If the sum of the third and fourth terms of the arithmetic sequence is equal to the sum of the

third and fourth terms of the geometric sequence, find the possible values of p.

(c) Hence, state the value of p for which the geometric sequence has a sum to infinity, and find the

value, expressing your answer in the form of √ , where a, b are constants.

(d) For the same value stated in (c), find the sum of the first 20 terms of the arithmetic sequence,

giving your answer in the form √ where p and q are constants.

Solution:

is a root, therefore,

( ) (shown)

( ) is a factor.

( )( )

Comparing ,

.

( )( )

√ ( )( )

√

√

(a) Arithmetic sequence,

Geometric sequence,

(b) ( ) ( )

( )( )

√

√

(c) For the geometric sequence to have a sum to infinity, | | . Therefore, √

√

( √ )

Mathematics (T)


√

√ .

√

√ /

( √ )

√

(d) Sum of first 20 terms of the arithmetic sequence

0 ( ) ( )

√

1

0 . √

/1

0

( √ )1

( √ )

√

14. A curve, C is defined implicitly as

(a) Show that the tangent at point (

) has gradient

(b) The line cuts the curve at point A(

) and at point B. Determine the coordinates of point

B.

Find, in the form of . / .

/

(i) The equation of the tangent at A.

(ii) The equation of the normal at B.

(c) Hence, find the acute angle between tangent at A and the normal at B.

Solution:

(a)

Differentiating y with x, .

/ ( ) .

/

At .

/, .

/

.

/

(shown)

(b) (i) The equation of the tangent is 4

5 .

/

(ii) Subst. into

( )

( ) .

/ ( )

( )

At B, ( ) .

/ ( ) ( ) ( ) .

/

This point is rejected due to

.

/ is point A.

Mathematics (T)


Gradient of the normal

Gradient of the normal at point B

The equation of the normal at B . / .

/

(c) Acute angle between two lines . / .

/

√ ( ) √

The acute angle between two lines is

15. By using de Moivre’s theorem, prove that Hence,

(a) Show that one of the roots of the equation is

and express the other

roots in trigonometric form.

(b) Deduce that

√ √ and find an exact expression for

Solution:

( )

( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

Comparing real parts,

( ) ( )

( )

(shown)

( 𝜃 𝑖 𝜃) may be written as

(𝑐 𝑖𝑠) for simplicity.

Use the binomial theorem.

Mathematics (T)


(a) Let and consider

Solving ,

The solutions are

(b) Let

, .

/ .

/ .

/

.

/ .

/

.

/

√( ) ( )( )

( )

√

√

√

.

/ √ √

√ ( √ )

√ √ (shown)

.

/ (

)

Since .

/ .

/

.

/ .

/

.

/ .

√ √ /

.

/

( √ )

.

/

√

.

/

√

.

/

√

.

/

√ √

.

/

√ √

.

/

√ √

Value of 𝜃 must be within

𝜋 𝜃 𝜋

The positive root is rejected because of the

requirement of the question.

(𝑎 𝑏) 𝑎 𝑏 𝑎 𝑏

Use the identity

.𝜋

/ is on the first quadrant, therefore .

𝜋

/ is

positive.

Use the identity

𝜃 𝜃 ≡

Mathematics (T)


16. A curve is given as . Find the gradient of tangent of the curve at the point where

and

Solution:

When

The point is ( )

.

/

At (1,1), .

/

17. Let ( ) √

. Find the set values of x for which f is real and finite.

Solution:

Let ( )

√

𝑓(𝑥) is only defined for 𝑥

Mathematics (T)


4 √

54 √

5

- + - +

√

√

The set values of x is * √

√

+

18. Find, in the form of the equation of the tangent to the curve at the point

with x-coordinate .

Solution:

When ,

The coordinates is ( )

.

/ ( )( )

At the x-coordinate

Equation of tangent, ( )

Use the product rule

𝑑

𝑑𝑥 𝑥

𝑥

Use the formula 𝑦 𝑦 𝑚(𝑥 𝑥 )

Mathematics (T)


19. The equation of a curve is .

(a) Show that

(b) Show also, if

then

(c) Hence, find the coordinates of the point on the curve where the tangent is parallel to the x-axis.

Solution:

(a)

.

/ ( ) .

/ ( )

.

/ .

/

.

/ ( )

(shown)

(b) When

( )

Since (shown)

(c) Subst. into

( ) ( )

Subst. into ,

The coordinates is ( )

20. The expansion ( ) in ascending powers of x, as far as the term in is .

Given , find the value of p and the value of A.

Solution:

( ) .

/

( .

/

( )( )

.

/

.

/

Using a combination of implicit

differentiation and product rule

Since 𝑑𝑦

𝑑𝑥 is undefined for 𝑦

therefore the solution 𝑦 is

rejected.

Do not write it as . 𝑝

𝑥/

( 𝑥)𝑛 𝑛𝑥

𝑛(𝑛 )𝑥

Mathematics (T)


Comparing the coefficient of and ,

.

/

21. Write down an identity for and use this result to show that

.

(a) It is given that

and

√ . Without using a calculator, show that

.

(b) Hence, show that the solutions to the equation ( )

for are

√

or

√ ( √ )

.

Solution:

( ) ( )

.

/( )

( )

(

)

(shown)

Mathematics (T)


(a)

√

√

√

From

.

/ .

/

.

/

(shown)

(b) Let

√

Subst. into

( )

It is given that

Let

From part (b), it is known that

( ) is a factor of the cubic equation.

( )( )

Comparing ,

( )( )

√( ) ( )( )

( )

√

√ or

√

You may draw a triangle and use

Pythagoras’ Theorem to find the

value of 𝜃

Mathematics (T)


22. Expand .

/ .

/ Hence or otherwise, expand .

/ .

/

.

(a) By using de Moivre’s theorem, if show that

and find a

similar expression for

.

(b) Hence, express in the form of , where

are constants to be determined.

(c) By using the result in (b), find ∫

Solution:

.

/ .

/

.

/ .

/ .

/ .

/ .

/

0.

/ .

/1

.

/

Mathematics (T)


.

/ .

/

.

/ .

/

.

/

.

/

(a)

(1)

( ) ( )

(2)

(1)+(2):

(shown)

(1)-(2):

(shown)

(b)

( ) ( ) .

/ .

/

( )( ) .

/

0.

/ .

/ .

/ 1

,( ) ( ) ( ) -

( )

(c) ∫ ∫

∫

∫

∫ ∫

(

)

(

)

(

)

Using de Moivre’s Theorem

( 𝜃) 𝜃

𝑎𝜃

𝑎 𝑎𝜃 𝑐

Mathematics (T)


23. ( ) is defined as ( )

(a) Find ( )

(b) Hence, find the possible values of for which ( )

Solution:

(a) ( )

( )

( )

(b) ( )

( )( )

or

24. A curve is defined parametrically as and ,

(a) Find

and

in terms of t. Show that

( )

and hence deduce that the curve has no

turning points.

(b) Find, in exact form, the equation of the normal of the curve at the point where .

Solution:

(a)

.

/

( ) .

/

( )

(shown)

To find the turning point

( )

( )

or

The curve has no turning points.

(b) When ,

( )

You do not need to solve for x as the

question requires the values of 𝑥

only.

𝑑

𝑑𝑥 𝑥 𝑥

𝑑

𝑑𝑥 𝑥 𝑥

𝑐𝑜𝑠(𝑥)

Mathematics (T)


Gradient of normal

When

Equation of the normal ( )

( )

25. A curve C is defined as .

(a) Find

in terms of

(b) Hence, find the equation of the tangent to the curve C at the point where

Solution:

(a)

.

/ ( )

, .

/ ( )-

, .

/ )-

(b) When

, .

/

.

/

When

.

/ ,(

) .

/ .

/

-

.

/ .

/

Equation of tangent, .

/ .

/

Mathematics (T)


Newton-Raphson method

The Newton-Raphson method is used when an equation ( ) cannot be solved using simple

algebraic methods. The formula for Newton-Raphson method is given by

( )

( )

Consider the graph of the function ( ) for the interval , -

( )

a b

From the graph, it is known that ( ) for and ( ) for . There is a change in sign

of ( ) for the equation. Hence, a root exists in the interval , - for the equation ( ) . Refer to

question 26 for an example.

26. Show that the equation has a root between the interval , -. Hence, by using

Newton-Raphson method with as the first approximation, find the root of the equation,

giving your answer to 5 decimal places.

Solution:

Let ( )

( ) ( )( )

( )

( )

( )

Since there is a change in sign, therefore a root exists between 1 and 2.

Formula for Newton-Raphson method: ( )

( )

( ) .

/

𝑓(𝑥)

𝑓(𝑥)

Mathematics (T)


( )

Therefore, the root of the equation is

27. A function is defined parametrically as

.

(a) Find

in terms of

(b) It is given that

. Show that

(c) Show that the equation has a root between 0 and 1.

(d) Hence, by using Newton-Raphson method with as the first approximation, find the root

of the equation , giving your answer correct to 5 decimal places.

Solution:

(a)

.

/ .

/

.

/

( ) .

/

( )

(b)

(shown)

(c) Let ( )

( )

( )

Since there is a change in sign, therefore there is a root between 0 and 1.

The full working is not required

when finding the root

Stop when the value starts to

converge

Mathematics (T)


(d) ( )

( )

( )

The root of the equation is

28. The parametric equations of curve C are

(a) Show that the normal to C at the point with parameter p has equation

.

(b) The normal to C at the point P intersects the x-axis at A and the y-axis at B. Given that O is the

origin and find the value of p.

Solution:

(a)

.

/

.

/

Gradient of the normal

At the point

Equation of normal is

( )

(shown)

(b) The x-intercept is given by point A.

x-intercept,

( )

The y-intercept is given by point B.

y-intercept,

Mathematics (T)


( ( ))

Distance of OA √,( ) ( )- , -

( )

Distance of OB √, - ,( ) -

Since

, -

Using a calculator, we know that is a root of the equation. Therefore is a factor.

( )( )

Comparing

( )( )

29. The function defined on the domain .

/ is given by

( )

(a) Find ( ) in terms of x.

The x-coordinate of the maximum point is denoted by .

(b) Show that

(c) Verify the root lies between 1.27 and 1.28.

Solution:

(a) ( )

, ( )-

( ), ( )- .

/ ( )

( ) , ( )- 0

1

( ) 0

1

(b) To find the maximum point, ( ) when .

When

0

1

(undefined)

(shown)

Mathematics (T)


(c) Let ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

Since there is a change of sign of ( ) therefore a root exists between 1.27 and 1.28.

30. Show that

simplifies to a constant, and find the constant.

Solution:

( ) ( )

( )( )

( )

(shown)

31. Show that ( ) ( ). Hence or otherwise, find, in terms of a and b, the

three values of x for which ( ) ( ) ( )

Solution:

LHS: , -

( ) ( ) (shown)

Given that ( ) ( )

Replacing p with ( ) and q with ( )...,

( ) ( ) ( ) ( )( )( )

( ) ( ) ( ) ( )( )( )

( ) ( ) ( )( )( ) ( )

𝛼 must be in radians.

Mathematics (T)


Hence, ( )( )( ) ( ) ( )

( )( )( )

32. Solve the following system of linear equations without using a calculator.

Solution:

(

| )

(

|

)

(

|

)

(

|

)

Express the system of linear

equations in the form of an

augmented matrix

Using elementary row

operations

Mathematics (T)


33. ( ) . Show that a real number, exists such that for all values of x,

( ( )) ( )

Solution:

( )

( )

( ( )) ( )

( )

( )

( )

34. Solve the equation for , giving your answer in terms of .

Solution:

.

/ .

/ .

/ .

/

( )

Mathematics (T)


35. In the binomial expansion of ( ) | |

the coefficient of is equal to the coefficient of

, and the coefficient of is positive. Find the value of p.

Solution:

( ) , ( )( ) ( )( )

( )

( )( )

( )

( )( )( )

( ) -

Since the coefficient of is equal to the coefficient of

( )

( )

( )( )( )

( )

( )

( )( )( )

( )

( )( )( )

( ),

( )( )-

( ) 0

( )1

( ) 0

1

( )

(rejected)

( )( )

Since the coefficient of is given by ( )( )

( )

When

Coefficient of is .

/.

/.

/

( ) (rejected)

Mathematics (T)


When

Coefficient of .

/.

/.

/

( )

36. The curve C has parametric equations

Find the values of t at the points where the normal at C at cuts C again.

Solution:

.

/

.

/

Normal

When

Gradient of normal

Mathematics (T)


When

( ) ( )

( )

The point is ( ).

The equation of normal ( )

Since the equation of the normal cuts the curve again,

( )

Since it is known that lies on the curve, is a factor.

Factoring gives ( )( )( )

37. Find the coordinates of the turning points of the curve and determine their

nature.

Solution:


.

/ .

/ ( )

( ) .

/

Substitute the parametric

equations into 𝑦 𝑥

Mathematics (T)


To find the turning point,

When

( ) ( )

( )( )

√

When

When √

.

/ .

/ ( )


.

/ .

/ .

/ .

/ .

/ ( )

Since

.

/ .

/

.

/ .

/

( )

( )

Mathematics (T)


When

( )

(maximum)

When √

( √

)

√

(min)

( ) is a maximum point, . √

/ is a minimum point.

38. Find ∫

√ by using the substitution √ .

Solution:

Let √

√

√

∫

( )

∫

∫

∫

| |

√ |√ |

Mathematics (T)


Maclaurin’s Theorem

Certain functions, for example and

can be expressed in the form of a polynomial. The Maclaurin’s Theorem (or sometimes referred as

Taylor series) is given by

( ) ( ) ( )

( )

( )

( )

The approximation becomes more accurate when more terms are included in the expansion.

The Maclaurin’s series is used for:

Finding approximate values for an integral

Evaluating limits

Approximating the value of a function

Refer to question 39 and 40 for an example.

39. If is so small that and higher powers of may be ignored, expand as a polynomial in .

Solution:

Using Maclaurin’s Theorem,

( ) ( ) ( )

( )

( )

Let ( )

( )

( )

( )

( )

( )

( )

( )

( )

Mathematics (T)


40. Given that ( ) ( ) show that ( ) ( )( ) ( ( )) Expand

( ) up to and include the term . Hence, find an approximation for (

) giving your

answer to 4 decimal places.

Solution:

Let ( ) ( )

( ) ( )

( )

( )

( )

( )

( )

( )

( ) ( )( )

( )

( ) ( )( )

( )

Differentiating ( ) with respect to x,

( ) ( )( ) ( ( ))

( ) ( )( ) ( ( ))

( )( )

Using Maclaurin’s Theorem,

( ) ( ) ( )

( )

( )

( )

( ) ( )

( )

( )

Mathematics (T)


When

,

.

/

.

/

.

/

41. Show that ∫

where is an integer to be found.

Solution:

Let

( )≡

≡ ( ) ( )( )

Let

Compare

Compare

∫

≡ ∫ .

/

∫

∫ .

/

, | |- , | |-

, - , -

Mathematics (T)


42. By using a suitable substitution or otherwise, find ∫

( ) .

Solution:

Let

( )

∫

( ) 0

( )1

∫

∫( )

0

1

43. Show that ( )( ) ≡ .

Hence, evaluate

( )( )

Solution:

LHS, ( )( )

,( ) ( )-,( ) ( )-

( ) ( )

( ) ( )

( ) ( )

(RHS) (shown)

( )( )

( )

Mathematics (T)


0

1

0

1 0

1

0

.

/

.

/1 0

1

44. By using the substitution , for

, show that

√

where are constants to be determined. Hence evaluate

√

Solution:

When

√

When

√

( )( )( )

Mathematics (T)


√

( )

( )( )

( )

( )

, -

20

1 0

13

6√

√

7

√

Mathematics (T)


mathematics t for new stpm syallabus

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