ESAT Mock Maths 2 Esat-maths2-bank-1

15 questions15 marks40Updated August 2026

The ESAT Mock Maths 2 Esat-maths2-bank-1 paper in full: all 15 questions, each with its answer. ESAT is the Engineering and Science Admissions Test. 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

1 mark

Consider the simultaneous equations:

x + 2y = 5

x^2 + y^2 = 10

What is the sum of the possible values of

y

A. $-4$
B. $2$
C. $4$
D. $5$
E. $10$

Answer: C

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Question 2

1 mark

The function

f

is defined for all real numbers

x

f(x) = \sqrt{x^2 - 4|x| + 4}

For how many distinct real values of

a

does the equation

f(x) = |x - a|

have infinitely many solutions?

A. $0$
B. $1$
C. $2$
D. $3$
E. $4$

Answer: C

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Question 3

1 mark

The sequence

x_n

is defined by the following rules:

x_1 = 3

x_{n+1} = \frac{1}{1 - x_n}

for

n \ge 1

What is the value of the sum

\sum_{n=1}^{91} x_n

A. $91$
B. $95$
C. $97.5$
D. $98$
E. $273$

Answer: D

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Question 4

1 mark

Let

S_n

denote the sum of the first

n

positive integers. Let

T_n

denote the sum of the next

n

positive integers, such that

T_n = (n+1) + (n+2) + \dots + 2n

. If

T_n - S_n = 625

, what is the value of

n

A. $20$
B. $25$
C. $35$
D. $50$
E. $625$

Answer: B

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Question 5

1 mark

The coefficient of

x^2

in the expansion of

(1 + kx)^5

in ascending powers of

x

40

, where

k

is a constant. What is the coefficient of

x^4

in this expansion?

A. $5$
B. $20$
C. $40$
D. $80$
E. $160$

Answer: D

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Question 6

1 mark

The line

L

has the equation

(2k + 1)x + (k - 1)y = 3k + 3

, where

k

is a real constant. It can be shown that all such lines pass through a fixed point

P

. Find the equation of the perpendicular bisector of the line segment joining

P

to the point

Q(4, 5)

A. $x + 3y - 9 = 0$
B. $3x - y - 7 = 0$
C. $x + 3y - 19 = 0$
D. $x - 3y + 3 = 0$
E. $3x + y - 11 = 0$

Answer: A

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Question 7

1 mark

A circle

C

has radius 4 and is tangent to the line

y = \sqrt{3}x

at the origin

(0,0)

. The centre of the circle lies in the second quadrant where

x < 0

and

y > 0

. A horizontal chord

OA

is drawn from the origin to a point

A

on the circle. A point

B

lies on the circumference of the circle such that the area of the triangle

OAB

is maximized. What is the maximum possible area of triangle

OAB

A. $4\sqrt{3}$
B. $8\sqrt{3}$
C. $12\sqrt{3}$
D. $16\sqrt{3}$
E. $24\sqrt{3}$

Answer: C

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Question 8

1 mark

A sector of a circle with radius

r

and angle

\theta

radians has a perimeter of

20\,\text{cm}

and an area of

24\,\text{cm}^2

. Which one of the following is a possible value for

r

A. $2\,\text{cm}$
B. $4\,\text{cm}$
C. $5\,\text{cm}$
D. $8\,\text{cm}$
E. $10\,\text{cm}$

Answer: B

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Question 9

1 mark

Find the complete set of values of

x

in the interval

0 \leq x \leq \pi

for which the inequality

\sin(2x) > \sin(x)

is satisfied.

A. $0 < x < \frac{\pi}{6}$
B. $0 < x < \frac{\pi}{3}$
C. $\frac{\pi}{6} < x < \frac{\pi}{3}$
D. $\frac{\pi}{3} < x < \frac{\pi}{2}$
E. $\frac{\pi}{3} < x < \pi$

Answer: B

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Question 10

1 mark

Find the sum of all possible values of

x

in the range

0 \le x \le \frac{\pi}{2}

which satisfy the equation

8 \sin^4 x - 10 \sin^2 x + 3 = 0

A. $\frac{5\pi}{12}$
B. $\frac{\pi}{2}$
C. $\frac{7\pi}{12}$
D. $\frac{2\pi}{3}$
E. $\frac{3\pi}{4}$

Answer: C

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Question 11

1 mark

The curve

C_1

has the equation

y = 4^x + 3(2^x) + 5

.

The curve

C_2

is obtained by reflecting the graph of

y = 2^{x+1}

in the

x

-axis and then translating it by

k

units in the positive

y

-direction, where

k

is a real constant.

For which set of values of

k

do the curves

C_1

and

C_2

have at least one point of intersection?

A. $k > 0$
B. $k > 5$
C. $k \ge 5$
D. $k > -1.25$
E. $k > 11$

Answer: B

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Question 12

1 mark

The real numbers

x

and

y

satisfy the simultaneous equations:

\log_2 x + \log_2 y = 3

\log_2 (x + y) = 2 \log_2 3

What is the value of

x^2 + y^2

A. $20$
B. $48$
C. $65$
D. $73$
E. $81$

Answer: C

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Question 13

1 mark

The function

f

is defined for

x > 0

f(x) = \frac{k}{x} + x^2

where

k

is a non-zero constant. The tangents to the graph

y = f(x)

x = 1

and

x = 2

intersect at a point on the

y

-axis. What is the value of

f''(1)

A. $-8$
B. $-4$
C. $0$
D. $4$
E. $8$

Answer: B

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Question 14

1 mark

The curve with equation

y = x^3 - 3x^2 - 9x + k

, where

k

is a real constant, has two stationary points. The line segment connecting these two stationary points passes through the origin

(0,0)

.

What is the value of

k

A. $-21$
B. $0$
C. $3$
D. $11$
E. $27$

Answer: C

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Question 15

1 mark

For a particular value of the constant

k

, the definite integral

\int_0^3 (x^2 - kx) \, dx

is equal to zero.

What is the total area enclosed between the curve

y = x^2 - kx

, the

x

-axis, and the lines

x = 0

and

x = 3

A. $0$
B. $\frac{4}{3}$
C. $\frac{8}{3}$
D. $4$
E. $9$

Answer: C

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