ADVANCED ALGEBRA – 2023/2024 PAST QUESTIONS AND ANSWERS

This Article shows Past Questions and Answer workings in Advanced Algebra for Higher National Diploma (HND) Students of Mechanical Engineering Technology Auchi Polytechnic, Auchi.

This Past Questions and Answers is helpful to all students in School of Engineering offering this general course; Advance Algebra (MTH 311). For those currently writing their on going exams in the Department of Mechanical Engineering Technology, this is the past questions and answers for you to study.

N.B – These questions and answers are uploaded only as a guide and ease for your studies.

Below are the Questions:

AUCHI POLYTECHNIC, AUCHI

SCHOOL OF GENERAL STUDIES

DEPARTMENT OF BASIC SCIENCES

FIRST SEMESTER EXAMINATION 2023/2024 ACADEMIC SESSION

DEPARTMENT: MECHANICAL ENGINEERING TECHNOLOGY                   CLASS: HNDI

COURSE: ADVANCED ALGEBRA (MTH311)                                DATE: 11 March. 2024

EXAMINER: MRS. ALIU, KA                                              DURATION: 2 HOURS 30 MINS

INSTRUCTION: ATTEMPTANY FIVE (5) QUESTIONS

QUESTION ONE

A. Find the scalar (or dot) product of vectors A and B, if A=5i + 4j + 2k and B = 3i – j + 6

B. Find the angle between the vectors A = 4i + 2j – 5k and B = 3i + j – 2k

C. State the laws of Vector Algebra

QUESTION TWO

A. Evaluate (4 + 3i)(2 – 7i)(5 + 2i)

B. Simplify: 4 – 5i / 3 + 2i

C. Find | 5 – 4i | | 4 + 3i |

QUESTION THREE

Solve the equation using Determinants and Gaussian Elimination method;

2x + y – 5z = 11; x – y + z =6; 4x + 2y – 3z = –8

QUESTION FOUR

A. Using logarithmic form of the inverse hyperbolic function, evaluate:

(i) sin h -¹ 1.475  (ii) cosh-¹ 2.364

B. Evaluate (i) cosh 0.684 (i) sin 1.478

QUESTION FIVE

A. Investigate the convergence or otherwise of the series:  1/1 + 3/2 + 5/2² + 7/2³ + ….

B. Expand (1+ x) using the Binomial series.

QUESTION SIX

Determine the eigenvalues and vectors for the equation A•x= πx where A= (2 0 1; -1 4 -1; -1 2 0)

QUESTION SEVEN

A. Find the first four(4) termns of the series e× sinh x

B. Express ^Sin (x + h)as a series of powers of h and evaluate sin 44° correct to 5d.p

C. Find a series for In 1+ x using the Maclaurin’s series

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Study the Answers Available!

ANSWER TO MTH PAST QUESTIONS (2023/2024)

QUESTION ONE

A. Solving…

A · B = (5i + 4j + 2k) · (3i – j + 6k)

Recall,

i x i = 1 , j x j = 1, k x k = 1 so therefore,

= (5 x 3) + (4 x (-1)) + (2 x 6)

= 15 – 4 + 12

= 23

So, the scalar or dot product of A and B is 23.

B. Solving…

let’s find the dot product of A and B:

A · B = (4i + 2j – 5k) · (3i + j – 2k)

= (4×3) + (2×1) + (-5x-2)

= 12 + 2 + 10

= 24

Now, let’s find the magnitudes of A and B:

|A| = √((4)^2 + (2)^2 + (-5)^2) = √(16 + 4 + 25) = √45

|B| = √((3)^2 + (1)^2 + (-2)^2) = √(9 + 1 + 4) = √14

The angle between them is: A · B = |A| · |B| cos(θ)

cos(θ) = (A · B) / (|A| |B|)

cos(θ) = 24 / (√45 x √14)

θ = cos -¹ (24 / (√45 x √14))

θ = cos -¹ (24 / (6.7082 x 3.7417))

θ = cos -¹ (24 / 25.1)

θ = cos -¹ (0.9562)

θ = 17.02°

θ ≈ 17°

So, the angle between A and B is approximately 17°.

C. The laws of vector algebra are:

I. Commutative Law:

a + b = b + a (vector addition is commutative)

a · b = b · a (vector dot product is commutative)

II. Associative Law:

(a + b) + c = a + (b + c) (vector addition is associative)

(a · b) · c = a · (b · c) (vector dot product is associative)

III. Distributive Law:

a · (b + c) = a · b + a · c (vector dot product distributes over vector addition)

a + (b + c) = (a + b) + c (vector addition distributes over itself) etc

QUESTION TWO

A. Solving….

1. Multiply the first two brackets:

(4 + 3i)(2 – 7i) = 8 – 28i + 6i – 21i^2

= 8 – 22i – 21(-1) (Recall; i^2 = -1)

= 8 – 22i + 21

= 29 – 22i

Now, Multiply the result by the third bracket:

(29 – 22i)(5 + 2i) = 145 + 58i – 110i – 44i^2

= 145 + 58i – 110i – 44(-1)

= 145 – 52i + 44

= 189 – 52i

So, (4 + 3i)(2 – 7i)(5 + 2i) = 189 – 52i

B. Solving…

Divide the numerator and denominator of the fraction by the common denominator (3 + 2i) just like surd.

(4 – 5i) / (3 + 2i) = ((4 – 5i) (3 – 2i)) / ((3 + 2i) (3 – 2i))

= (12 – 8i – 15i + 10i^2) / (9 + 6i – 6i – 4i^2)

= (12 – 23i – 10) / (9 + 4)

= (2 – 23i) / 13

 Now, simplify the entire expression:

4 – (2 – 23i) / 13

= 4 – 2/13 + 23i/13

= (52 – 2) / 13 + 23i/13

= 50/13 + 23i/13

So, 4 – 5i / 3 + 2i = 50/13 + 23i/13

C. Solving…

Firstly, Find the absolute value of 5 – 4i:

|5 – 4i| = √(5² + (-4)²) = √(25 + 16) = √41

Now, Find the absolute value of 4 + 3i:

|4 + 3i| = √(4² + 3²) = √(16 + 9) = √25 = 5

Multiply the absolute values:

|5 – 4i| |4 + 3i| = √41 × 5 = 5√41

So, |5 – 4i| |4 + 3i| = 5√41

QUESTION THREE

Solve the equation using Determinants and Gaussian Elimination method;

2x + y – 5z = 11; x – y + z =6; 4x + 2y – 3z = –8

QUESTION FOUR

A. Using logarithmic form of the inverse hyperbolic function, evaluate:

(i) sin h -¹ 1.475

= sinh^{-1}(1.475)

= ln(1.475 + √(1.475^2 + 1))

= ln(1.475 + √(2.187 + 1))

= ln(1.475 + √3.187)

= ln(1.475 + 1.785)

= ln(3.26)

(ii) cosh-¹ 2.364

(ii) cosh^{-1}(2.364)

= ln(2.364 + √(2.364^2 – 1))

= ln(2.364 + √(5.565 – 1))

= ln(2.364 + √4.565)

= ln(2.364 + 2.134)

= ln(4.498)

B. Solving…

Using the definitions of hyperbolic cosine (cosh) and sine (sin):

(i) cosh(0.684)

= (e^0.684 + e^(-0.684)) / 2

≈ (2.001 + 0.499) / 2

≈ 1.250

(ii) sin(1.478)

= sin(1.478 radians)

≈ -0.981 (since 1.478 radians is in the 4th quadrant, where sine is negative)

≈ -0.981