MAT 2023 MCQ

10 questions40 marks60Updated July 2026

The MAT 2023 MCQ paper in full: all 10 questions, each with its answer. Sit it cold under exam timing, mark it, then work back through anything you missed using the solutions below.

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Question 1

4 marks

The function

p(x) = x^3

is an example of a polynomial with the property that there is a point on the graph

y = p(x)

with zero derivative which is neither a local maximum nor a local minimum. Which one of the following polynomials has the same property?

A. $y = x^3 - 3x^2 + x$
B. $y = x^3 - 3x^2 + 2x$
C. $y = x^3 - 3x^2 + 3x$
D. $y = x^3 - 3x^2 + 4x$
E. $y = x^3 - 3x^2 + 5x$

Answer: C

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Question 2

4 marks

Which of the following numbers is the smallest?

A. $\log_{10} \left( \sqrt[10]{10^{11}} \right)$
B. $\frac{\pi}{2}$
C. $\frac{11}{9}$
D. $\sqrt{\frac{3}{2}}$
E. $\sqrt{3} \cos(44^\circ)$

Answer: A

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Question 3

4 marks

The sum

\sum_{k=1}^n k^3 = \frac{n^2(n+1)^2}{4}

. It follows that

1^3 + 3^3 + 5^3 + 7^3 + \cdots + 19^3

is equal to

A.19,800
B.19,900
C.20,000
D.20,100
E.20,200

Answer: C

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Question 4

4 marks

All even square numbers are multiples of 4. All odd square numbers are one more than a multiple of 4. It follows that the number of positive integer solutions

(x, y)

to the equation

x^2 + 3y^2 = 4442

A.0
B.1
C.2
D.3
E.4442

Answer: C

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Question 5

4 marks

Let

d(n)

be the number of digits in a positive integer

n

(with

n

written in the usual decimal notation). For example

d(2) = 1, d(103) = 3

, and

d(10^6) = 7

. Define the sequence

s_n = (20)^{-d(n)}

. What is the sum

\sum_{n=1}^\infty s_n

equal to?

A. $\frac{1}{2}$
B. $\frac{4}{5}$
C. $\frac{9}{10}$
D.1
E. $\frac{9}{5}$

Answer: A

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Question 6

4 marks

For two vectors

\begin{pmatrix} a \ b \end{pmatrix}

and

\begin{pmatrix} c \ d \end{pmatrix}

with integer components (positive or negative or zero), we define the function
f

\left(\begin{pmatrix} a \ b \end{pmatrix}, \begin{pmatrix} c \ d \end{pmatrix}\right) = \begin{pmatrix} ac + bd \ ad + bc + 2bd \end{pmatrix}

.
How many vectors

\begin{pmatrix} a \ b \end{pmatrix}

with integer components are there such that
f

\left(\begin{pmatrix} a \ b \end{pmatrix}, \begin{pmatrix} a \ b \end{pmatrix}\right) = \begin{pmatrix} 2 \ 0 \end{pmatrix}

A.0
B.1
C.2
D.3
E.Infinitely many

Answer: B

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Question 7

4 marks

For a pair of integers

x

and

y

with

x \ge 0

and

y > 0

, we define

f(x, y) = \frac{1}{2}(x+y)(x+y+1) + y

. What is the set of possible values that

f(x, y)

can take?

A.All positive integers.
B.All positive even integers.
C.All positive integers except for odd prime numbers.
D.All positive integers that are triangular numbers (those which are the sum of the first $k$ positive integers for some $k \ge 1$ ).
E.All positive integers except for the triangular numbers.

Answer: B

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Question 8

4 marks

Let

p(x) = 2x^4 - 3x^3 - 5x^2 + 2x + 2

. Given that the

y = mx

, with

m

a real number, crosses the curve

y = p(x)

at four distinct points, let the

x

-coordinates of those points be

x_1, x_2, x_3

, and

x_4

. The product

x_1x_2x_3x_4

is equal to

A.0
B.1
C.2
D.3
E.Not enough information

Answer: E

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Question 9

4 marks

Consider the nine lines

y = 2x + 1, y = 2x + 2, \dots, y = 2x + 9

and the seven lines

y = -x + 1, y = -x + 2, \dots, y = -x + 7

. How many distinct points are there at which the line

y = 1 - 10x

crosses one or more of the other lines?

A.12
B.13
C.14
D.15
E.16

Answer: B

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Question 10

4 marks

Which of the following is the graph of

y(y^3 + 4y^2x + 4x^3) = x^2(1 - x^2 - 6y^2)

A.Graph (a)
B.Graph (b)
C.Graph (c)
D.Graph (d)
E.Graph (e)

Answer: A

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