Kulai 2023 Q
Kulai 2023 Q
Kulai 2023 Q
2 2𝑛+1
(a) The sum of the first n terms of a series is given as 10 − 3𝑛−1 . Show that the
2 𝑛
𝑛𝑡ℎ term of the series is 3 (3) and show it is a geometric progression. [4 marks]
2 [2 marks]
(b) Express 𝑟 2 +4𝑟+3 in partial fraction.
𝑛
2
Hence , find 𝑠𝑛 = ∑ 2 , simplify your answer in 2 single fractions. [3 marks]
𝑟=1 𝑟 +4𝑟+3
[2 marks]
Find the sum to infinity of 𝑆𝑛 .
3. 1 0 1
Given Matrix M =(7 3 𝑥 + 2) and determinant M = 4, find the value of x.
2 𝑥−1 2
14 6 10
Hence, deduce the determinant of matrix N = ( 1 0 1 ). [6 marks]
4 4 4
4. 3
The complex number w has modulus 2 and argument and the complex
4
1
number z has modulus 2 and argument .
3
(a) Find the modulus and argument wz. [3 marks]
(b) By expressing w and z in the form x + iy, hence, find the exact real and
imaginary parts of wz. [4 marks]
6. The points A and B have position vectors 2i -2j + 4k and -2i +4j -2k respectively,
relative to the origin O. The point P and Q are such that ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ and
𝑂𝑃 = 2𝑂𝐴
⃗⃗⃗⃗⃗⃗ = −3𝑂𝐵
𝑂𝑄 ⃗⃗⃗⃗⃗ .
(b) Find the angle OPQ, giving your answer to the nearest 0.1 [3 marks]
Section B [15 marks]
You may answer all the questions, but only the first answer will be marked.
1 0 1
(a) Find the inverse of the matrix 2 0 1 in term of k using the elementary
7
k 1 0
row operations. [5 marks]
1 2 𝑝 −3 2 −1
A = (𝑞 3 2) , B =(−1 0 1)
1 0 1 3 −2 −1
Find the value of k, p and q such that AB = kI. Where I is the 3 x 3 identity matrix.
Hence, solve the system of linear equation below.
x + 2y +pz = 6
qx + 3y +2z = -3 [10marks]
-x – z = -2
8 The points P, Q, R and S have position vectors , relative to origin O are i +2j –k,
-i +bj +3k, 2i +j +4k and i +j +k, where b is a constant. Given that ⃗⃗⃗⃗⃗
𝑂𝑃 is
⃗⃗⃗⃗⃗⃗
perpendicular to 𝑂𝑄.