MAT1510 EXAM PACK 2026

MAT1510
EXAM PACK




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ASSIGNMENT 02
MAT1510 EXAM 2025
Due date: Friday, 31 May 2024
Total Marks: 100
UNIQUE ASSIGNMENT NUMBER: 186115

ONLY FOR YEAR MODULE

This assignment covers chapter 2 of the prescribed book as well as the study guide

DO NOT USE A CALCULATOR.

Question 1: 13 Marks

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give
an example that disproves the statement.

(1.1) If f is a function, then f (s + t) = f (s) + f (t). (3)

(1.2) If f (s) = f (t), then s = t. (2)

(1.3) If f is a function, then f (3x) = 3f (x). (3)

(1.4) A vertical line intersects the graph of a function at most once. (2)
1
(1.5) If f is one-to-one, then f −1 (x) = . (3)
f (x)

Question 2: 9 Marks

The perimeter of a rectangle is 16 meters.

(2.1) If the length of one of the sides of the rectangle is (1 + x) meters, express the area A of the (4)
rectangle in terms of x.

(2.2) Calculate the maximum area of the rectangle. (3)

(2.3) What are the dimensions of the rectangle when its area is a maximum? (2)

Question 3: 6 Marks

Suppose a stone is thrown vertically upwards with a velocity of u meters per second. Then its height is h (in
meters) after t seconds is given by the formula
h = ut − 4.8t 2 .


(3.1) Suppose the stone is thrown upwards with a velocity of 24 meters per second. Sketch the (5)
graph of the function defined by
h = ut − 4.8t 2 .
Label the axes properly, and show the coordinates of the critical points on the graph clearly.


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(3.2) What is the maximum height that the stone reaches? (1)


Question 4: 6 Marks

Sketch the graph of the function g(x) which is piecewise-defined by

(4.1) (4)

2
−x + 2x + 3
 if x < 1
g(x) = 4 if x = 1

 2
x − 2x + 5 if x > 1


(4.2) Explain why g is called a function. (1)

(4.3) Is g a one-to-one function? Give a reason for your answer. (1)


Question 5: 20 Marks




The sketch shows the graph of the functions f and g. Function f is defined by

y = f (x) = m|x − p| + q


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and g is defined by

y = g(x) = ax 2 + bx + c
S is the salient point of the graph of f , and T is the turning point of the graph of g. The two graphs intersect
each other at T and R.

(5.1) Determine the value of m, p and q and then write down the equation of f . (4)

(5.2) Describe the steps of the transformation process that you would apply to the graph of f to (3)
obtain the graph of y = 5|x|.

(5.3) Find the values of a, b and c and then write down the equation of g. (4)

(5.4) Determine the coordinates of T . (2)

(5.5) Use the graph of f and g to solve the inequality (3)
f (x)
<1 for x ∈ (−2, 4).
g(x)
(Do not solve the inequality algebraically.)

(5.6) Suppose the function d describes the vertical distance between the graph of f and g on the
interval [−2, xT ] ( where xT is the x-coordinate of T ).
(a) Complete and simplify the equation (2)

d(x) = ................ for x ∈ [−2, xT ].
(b) What is the maximum vertical distance between the graphs of f and g on the interval (2)
[−2, xT ].

Question 6: 11 Marks

Suppose a function g is defined by

y = g(x) = 7 + 6x − x 2

(6.1) Restrict the domain of g such that the function gr defined by (2)
gr (x) = g (x) for all x ∈ Dgr ,
is one-to-one function, and such that the domain Dgr contains only positive numbers.

(6.2) Determine the equation of the inverse function gr−1 and the set Dg −1 . (3)
r


(6.3) Show that (6)
gr ◦ gr−1


(x) = x for x ∈ Dg −1
r

and
gr−1 ◦ gr (x) = x


for x ∈ Dgr .



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