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Section A [ 45 marks ] Answer all questions in this section

1. (a) Determine the value of a if

3 +ai is a real number and find this number. 1− 3 i

(b) If z is a complex number such that

z = 1,

find the real part of

[5]

[4]

1 . 1 −z

1 −x in ascending powers of x up to and including the term in x 2 . 1 +2 x In an attempt to estimate the value of 2 , a student substituted x = −1 in the

2. Expand

above expansion. Explain why this is wrong. 1 By putting x = in the expansion, show that 9 integer to be determined.

2≈

A where A is a positive 1296 [8]

3. Verify the identity 2r −1 2r +1 2 − ≡ . Hence, using the method of differences, r ( r −1) r (r +1) ( r −1)(r +1)

prove n

that

2

3

2n + 1

∑ (r −1)(r +1) = 2 − n(n +1) . Deduce the sum of the infinite series r =2

1 1 1 1 + + +...... + +...... 1×3 2 ×4 3 ×5 (n −1)(n +1)

[6]

4. Express

3 cos θ +sin θ in

the form r cos(θ −α) where r > 0 and 0 < α < 90 o .

Hence, find the least value of of θ

1 and the corresponding value 5 + 3 cos θ + sin θ

[6]

5. The line y =2 x − a intersects parabola y 2 = 4ax at P ( ap 2 ,2ap ) and Q ( aq 2 ,2aq ) .

Find (a) the values of p + q and pq.

[6]

(b) the coordinates of the mid-point of PQ.

[2]

6. A straight line l has equation r = (1, 2, -5) + t(2, -3, 1), the planes π1 and π2 have equations 2x + 5y - 3z = 6 and 4x + 3y = 8 respectively. (a) find the position vector of the point of intersection of l and π1

[4]

(b) find the angle between the planes π1 and π2 . [4]

Section B [ 15 marks ] Answer any one question in this section 1. A system if linear equation is given by

λx + y + z =1 x + λy + z =λ

x + y + λz =λ2 , where

λ is a constant.

(a) Determine the value of λ for which the system has a unique solution,

[9]

infinitely many solution and no solution. (b) Find the unique solution in terms of λ .

[6]

2. The position vectors a, b and c of three points A, B and C respectively are given by a = 2i + j - k b = 3i + j + 2k c = 2i - 2j + 2k (a) Find a unit vector parallel to 2a - b + c

[3]

(b) Calculate the acute angle between b and 2a - b + c

[3]

(c) Find the vector of the form λi + µj − k perpendicular to both a and b

[2]

(d) Determine the position vector of the point D which is such that ABCD is a

[3]

parallelogram having BD as a diagonal (e) Calculate the area of the parallelogram ABCD Answers 1. (a) -3, real number = 5 39 2 x +...... 2. 1 − x + 8 8

(b)

3

−

1 2

1 1

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1. (a) Determine the value of a if

3 +ai is a real number and find this number. 1− 3 i

(b) If z is a complex number such that

z = 1,

find the real part of

[5]

[4]

1 . 1 −z

1 −x in ascending powers of x up to and including the term in x 2 . 1 +2 x In an attempt to estimate the value of 2 , a student substituted x = −1 in the

2. Expand

above expansion. Explain why this is wrong. 1 By putting x = in the expansion, show that 9 integer to be determined.

2≈

A where A is a positive 1296 [8]

3. Verify the identity 2r −1 2r +1 2 − ≡ . Hence, using the method of differences, r ( r −1) r (r +1) ( r −1)(r +1)

prove n

that

2

3

2n + 1

∑ (r −1)(r +1) = 2 − n(n +1) . Deduce the sum of the infinite series r =2

1 1 1 1 + + +...... + +...... 1×3 2 ×4 3 ×5 (n −1)(n +1)

[6]

4. Express

3 cos θ +sin θ in

the form r cos(θ −α) where r > 0 and 0 < α < 90 o .

Hence, find the least value of of θ

1 and the corresponding value 5 + 3 cos θ + sin θ

[6]

5. The line y =2 x − a intersects parabola y 2 = 4ax at P ( ap 2 ,2ap ) and Q ( aq 2 ,2aq ) .

Find (a) the values of p + q and pq.

[6]

(b) the coordinates of the mid-point of PQ.

[2]

6. A straight line l has equation r = (1, 2, -5) + t(2, -3, 1), the planes π1 and π2 have equations 2x + 5y - 3z = 6 and 4x + 3y = 8 respectively. (a) find the position vector of the point of intersection of l and π1

[4]

(b) find the angle between the planes π1 and π2 . [4]

Section B [ 15 marks ] Answer any one question in this section 1. A system if linear equation is given by

λx + y + z =1 x + λy + z =λ

x + y + λz =λ2 , where

λ is a constant.

(a) Determine the value of λ for which the system has a unique solution,

[9]

infinitely many solution and no solution. (b) Find the unique solution in terms of λ .

[6]

2. The position vectors a, b and c of three points A, B and C respectively are given by a = 2i + j - k b = 3i + j + 2k c = 2i - 2j + 2k (a) Find a unit vector parallel to 2a - b + c

[3]

(b) Calculate the acute angle between b and 2a - b + c

[3]

(c) Find the vector of the form λi + µj − k perpendicular to both a and b

[2]

(d) Determine the position vector of the point D which is such that ABCD is a

[3]

parallelogram having BD as a diagonal (e) Calculate the area of the parallelogram ABCD Answers 1. (a) -3, real number = 5 39 2 x +...... 2. 1 − x + 8 8

(b)

3

−

1 2

1 1

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