Algebra for Advanced Diploma in Engineering: Master Equations and Inequalities

Question 1

Solve the equation: 3x + 5 = 16

Accepted Answer

x = 4

Answer

x = 3

Answer

x = 5

Answer

x = 6

Question 2

Factorise the expression: x^2 - 4

Accepted Answer

(x - 2)(x + 2)

Answer

(x - 4)(x + 4)

Answer

(x - 1)(x + 1)

Answer

(x - 3)(x + 3)

Question 3

Find the x-intercept of the line y = 2x - 4

Accepted Answer

2

Answer

-2

Answer

-4

Answer

4

Question 4

Solve the linear equation: 2x + 5 = 13.

Accepted Answer

x = 4

Answer

x = 6

Answer

x = 8

Answer

x = 10

Question 5

Factorise the expression: x^2 - 9.

Accepted Answer

(x - 3)(x + 3)

Answer

(x - 9)(x + 1)

Answer

(x - 3)^2

Answer

Cannot be factorised

Question 6

Solve for x in the exponential equation: 2^x = 16.

Accepted Answer

x = 4

Answer

x = 2

Answer

x = 8

Answer

x = 16

Question 7

Which of the following graphs represents the equation y = x^2?

Accepted Answer

A parabola opening upwards

Answer

A parabola opening downwards

Answer

A straight line with a positive slope

Answer

A straight line with a negative slope

Question 8

Solve the linear equation: 3x + 5 = 14

Accepted Answer

x = 9

Answer

x = 3

Answer

x = 5

Answer

x = 11

Question 9

Identify the quadratic equation:

Accepted Answer

x^2 + 3x - 4 = 0

Answer

x + 2 = 0

Answer

2x + 1 = 0

Answer

x^3 - 2x^2 + 3x - 4 = 0

Question 10

Calculate the volume of a cube with a side length of 5 cm:

Accepted Answer

125 cm^3

Answer

25 cm^3

Answer

225 cm^3

Answer

375 cm^3

Question 11

Find the derivative of the function f(x) = x^2 + 2x:

Accepted Answer

f'(x) = 2x + 2

Answer

f'(x) = 2x

Answer

f'(x) = x^2

Answer

f'(x) = 2

Question 12

Which of the following is not a characteristic of an algebraic equation?

Accepted Answer

It is guaranteed to have a solution.

Answer

It may represent a real-world situation.

Answer

It must have an equal sign.

Answer

It expresses a relationship between unknown values.

Question 13

Determine the solution set for the inequality 'x - 3 > 7':

Accepted Answer

{x | x > 4}

Answer

{x | x < 4}

Answer

{x | x > 10}

Answer

{x | x < 10}

Question 14

Find the equation of the line with a slope of -1 and a y-intercept of 2

Accepted Answer

y = -x + 2

Answer

y = x + 2

Answer

y = -x - 2

Answer

y = x - 2

Question 15

Factor the expression: x^2 - 4x + 4

Accepted Answer

(x - 2)^2

Answer

(x + 2)^2

Answer

x(x - 4)

Answer

x(x + 4)

Question 16

Find the domain of the function: f(x) = sqrt(x - 2)

Accepted Answer

[2, ∞)

Answer

(-∞, 2]

Answer

[0, ∞)

Answer

(2, ∞)

Question 17

Using the quotient rule of logarithms, simplify the expression: log(x² / y)

Accepted Answer

2log(x) - log(y)

Answer

log(2x) - log(y)

Answer

log(x) - log(y)

Answer

log(x²y)

Question 18

Which of the following is NOT an algebraic expression?

Accepted Answer

The speed of a moving vehicle (km/h)

Answer

2x + 3

Answer

sin(θ)

Answer

(a + b)^2

Question 19

Factorise the expression: x^2 - 9

Accepted Answer

(x + 3)(x - 3)

Answer

(x - 9)(x + 9)

Answer

(x + 3)^2

Answer

(x - 3)(x + 9)

Question 20

Solve the inequality: 2x + 5 < 13

Accepted Answer

x < 4

Answer

x > 4

Answer

x < -4

Answer

x > -4

Question 21

Which of the following is the equation of a straight line with a gradient of -2 and y-intercept of 3?

Accepted Answer

y = -2x + 3

Answer

y = -2x - 3

Answer

y = 2x + 3

Answer

y = 2x - 3

Question 22

Solve the system of equations: 2x + 3y = 11, x - y = 1

Accepted Answer

x = 3, y = 2

Answer

x = 2, y = 3

Answer

x = -3, y = 2

Answer

x = 3, y = -2

Question 23

Simplify the expression: √(16x^6y^4)

Accepted Answer

4x^3y^2

Answer

8x^6y^4

Answer

2x^3y^2

Answer

4x^6y^2

Question 24

Which of the following is a polynomial?

Accepted Answer

5x^3 - 2x^2 + 3x - 7

Answer

e^x

Answer

1 / x

Answer

cos(x)

Question 25

A cylindrical tank has a radius of 5 meters and a height of 10 meters.  Water is being pumped into the tank at a rate of 2 cubic meters per minute.  How long will it take to fill the tank completely? (Use π ≈ 3.14)

Accepted Answer

785 minutes

Answer

314 minutes

Answer

157 minutes

Question 26

Solve the following inequality for *x*: 2*x* - 5 > 11

Accepted Answer

x > 8

Answer

x > 3

Answer

x < 8

Question 27

The resistance (R) in an electrical circuit is inversely proportional to the cross-sectional area (A) of the wire. If the resistance is 10 ohms when the area is 2 square millimeters, what is the resistance when the area is 5 square millimeters?

Accepted Answer

4 ohms

Answer

50 ohms

Answer

20 ohms

Answer

2.5 ohms

Question 28

Solve the following equation for *x*: 3*x* + 5 = 14

Accepted Answer

x = 3

Answer

x = -6

Answer

x = 6

Answer

x = -3

Question 29

A beam is supported at its ends, and a load is applied at the center. The deflection (*y*) of the beam at a distance (*x*) from the left support is given by the equation *y* = *k* *x* (*L* - *x*), where *k* and *L* are constants. If the maximum deflection occurs at *x* = *L*/2, what is the value of *k*?

Accepted Answer

4*y*/L²

Answer

8*y*/L²

Answer

*y*/L²

Answer

2*y*/L²

Question 30

Solve the following inequality for *x*: 2*x* - 3 > 7

Accepted Answer

x > 5

Answer

x > -5

Answer

x < -5

Answer

x < 5

Question 31

Simplify the following expression: 3(2*x* - 5) + 4*x*

Accepted Answer

10*x* - 15

Answer

10*x* - 5

Answer

7*x* - 15

Answer

6*x* - 15

Question 32

Solve the equation for x: 5x - 2 = 13

Accepted Answer

x = 3

Answer

x = -3

Answer

x = 2

Answer

x = 4

Question 33

A company manufactures widgets. The cost of producing x widgets is given by the equation C(x) = 5x + 100.  The company sells each widget for £8. Calculate the profit from selling 100 widgets.

Accepted Answer

£200

Answer

£700

Answer

£100

Answer

£300

Question 34

A machine part requires a length, x,  to be within 0.5 cm of 10 cm. Which inequality represents the acceptable lengths for this part?

Accepted Answer

9.5 ≤ x ≤ 10.5

Answer

x ≤ 9.5

Answer

x ≥ 10.5

Answer

9.5 < x < 10.5

Question 35

The formula for the volume of a cylinder is V = πr²h, where r is the radius and h is the height. If a cylinder has a radius of 5 cm and a volume of 250π cm³, what is the height?

Accepted Answer

10 cm

Answer

12.5 cm

Answer

20 cm

Answer

5 cm

Question 36

A beam is subjected to a load that increases linearly along its length. The load is 10 kN at the start of the beam (x = 0 metres) and 20 kN at the end (x= 1 metre). Which equation represents the load (L) in terms of the length (x) in kN?

Accepted Answer

L = 10 + 10x

Answer

L = 20 - 10x

Answer

L = 20x

Answer

L = 10x

Question 37

An electrical circuit consists of two resistors connected in series.  The resistances are R₁ = 10 Ω and R₂ = 20 Ω.  Calculate the total resistance (R) of the circuit.

Accepted Answer

30 Ω

Answer

6.67 Ω

Answer

15 Ω

Answer

20 Ω

Question 38

Two machines are working simultaneously to produce units. Machine A produces x units per hour, while Machine B produces y units per hour. What is the algebraic expression for the total number of units (T) produced by both machines in 4 hours?

Accepted Answer

T = 4(x + y)

Answer

T = 4xy

Answer

T = x + 4y

Answer

T = 4x + y

Question 39

Find the values of x and y that satisfy the following system of equations: 2x + 3y = 10 and x - y = 1

Accepted Answer

x = 2, y = 1

Answer

x = 2.5, y = 1.5

Answer

x = 1, y = 0

Answer

x = 3, y = 2

Question 40

Determine the solution to the system of equations: x + y = 5 and x - y = 1

Accepted Answer

x = 3, y = 2

Answer

x = 1, y = 4

Answer

x = 4, y = 1

Answer

x = 2, y = 3

Question 41

Factor the quadratic expression completely: x² - 9

Accepted Answer

(x + 3)(x - 3)

Answer

(x + 3)²

Answer

(x + 9)(x - 1)

Answer

(x - 3)²

Question 42

Simplify the expression: (x + 2)^2 - (x - 2)^2

Accepted Answer

8x

Answer

32x

Answer

4x

Answer

16x