40% off

Early-bird ends 15 Aug, 9am

Lock in £90
← All past papers

TMUA Yotta Papers D513/11

20 questions20 marks75Updated September 2025

The TMUA Yotta Papers D513/11 paper in full: all 20 questions, each with its answer. TMUA is the Test of Mathematics for University Admission. Sit it cold under exam timing, mark it, then work back through anything you missed using the solutions below.

Download the original PDF →
Questions and answers are free. Full step-by-step worked solutions unlock with a free account. Start practising.

Question 1

1 mark
Find the sum of the x-coordinates of the six points of intersection of
y=πx(x−1)(x−2)(x−3)(x−4)(x−5)y = \pi x (x - 1) (x - 2) (x - 3) (x - 4) (x - 5)
and
y=517x+π3y = \frac{5}{17}x + \frac{\pi}{3}
  • A.-15
  • B.−44π51+1-\frac{44\pi}{51} + 1
  • C.−715-\frac{7}{15}
  • D.0
  • E.715\frac{7}{15}
  • F.44π51−1\frac{44\pi}{51} - 1
  • G.12
  • H.15

Answer: H

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 2

1 mark
A cubic function f(x)f(x) is such that f(2)=4f(2) = 4, f(3)=9f(3) = 9, f(−1)=1f(-1) = 1, and the coefficient of x3x^3 is 2. Find f(4)f(4).
  • A.-16
  • B.6
  • C.16
  • D.32
  • E.36
  • F.46

Answer: E

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 3

1 mark
Let f0(x)=xf_0(x) = x, and fn+1(x)=∣fn(x)−k∣f_{n+1}(x) = |f_n(x) - k| for non-negative integers nn, and real number kk. Let α\alpha and β\beta respectively equal the least and greatest values of xx for which fn(x)=0f_n(x) = 0. Find the value of:
∫αβfn(x)dx\int_{\alpha}^{\beta} f_n(x) dx

for n>0n > 0, in terms of nn and kk.
  • A.nk2nk^2
  • B.nk2−k2nk^2 - k^2
  • C.kn2kn^2
  • D.nknk
  • E.k(n−1)2k(n - 1)^2
  • F.nk2−1nk^2 - 1

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 4

1 mark
Non-zero integers a,ba, b and cc satisfy
abc+bc+ab+ac+a+b+c=104abc + bc + ab + ac + a + b + c = 104

What is a2+b2+c2+2(a+b+c)+1a^2 + b^2 + c^2 + 2(a + b + c) + 1?
  • A.35
  • B.54
  • C.56
  • D.81
  • E.104
  • F.105

Answer: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 5

1 mark
Let un=2un−1+7un−2u_n = 2u_{n-1} + 7u_{n-2}, where u1=4u_1 = 4 and u2=12u_2 = 12. What does the value of ukuk−1\frac{u_k}{u_{k-1}} tend towards as kk tends towards infinity?
  • A.2
  • B.3\sqrt{3}
  • C.5+1\sqrt{5} + 1
  • D.7
  • E.9
  • F.1+221 + 2\sqrt{2}
  • G.−1+3-1 + \sqrt{3}

Answer: F

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 6

1 mark
f(x)f(x) is a polynomial function defined for all real xx. Given that f(x2−12x+45)f(x^2 - 12x + 45) has two roots at x=−3x = -3 and x=15x = 15, and has a minimum value of -20, which row correctly describes f(9x2−30x+34)f(9x^2 - 30x + 34)?
Exam diagram
  • A.Roots: x=−1x = -1 and x=5x = 5. Min Value: -20
  • B.Roots: x=−43x = -\frac{4}{3} and x=143x = \frac{14}{3}. Min Value: -20
  • C.Roots: x=−23x = -\frac{2}{3} and x=163x = \frac{16}{3}. Min Value: -20
  • D.Roots: x=−1x = -1 and x=42x = 42. Min Value: -20
  • E.Roots: x=−12x = -12 and x=5x = 5. Min Value: -60
  • F.Roots: x=−1x = -1 and x=42x = 42. Min Value: -180

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 7

1 mark
How many real solutions are there to
2(27x)−32x+1−4(3x+1)+5=02(27^x) - 3^{2x+1} - 4(3^{x+1}) + 5 = 0
  • A.0
  • B.1
  • C.2
  • D.3
  • E.4
  • F.5
  • G.6

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 8

1 mark
The graph of y=tan⁡(cos⁡(sin⁡(x)))y = \tan(\cos(\sin(x))) has a period of PP and a maximum value of MM. Which row is correct?
Exam diagram
  • A.P=1,M<1P=1, M<1
  • B.P=1,M>1P=1, M>1
  • C.P=π2,M<1P=\frac{\pi}{2}, M<1
  • D.P=π2,M>1P=\frac{\pi}{2}, M>1
  • E.P=π,M<1P = \pi, M<1
  • F.P=π,M>1P = \pi, M>1
  • G.P=2π,M<1P = 2\pi, M<1
  • H.P=2π,M>1P = 2\pi, M>1

Answer: F

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 9

1 mark
How many real solutions are there to
ln⁡(sin⁡(x))=ln⁡(1−4x7π)\ln(\sin(x)) = \ln(1 - \frac{4x}{7\pi})
  • A.0
  • B.1
  • C.2
  • D.3
  • E.4
  • F.5
  • G.6
  • H.infinitely many

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 10

1 mark
A fair coin is flipped repeatedly until 4 consecutive heads are obtained. Find the expected number of coin flips.
  • A.4
  • B.14
  • C.16
  • D.30
  • E.1965\frac{196}{5}
  • F.2887\frac{288}{7}
  • G.62
  • H.64

Answer: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 11

1 mark
Worker nn, where nn is an integer, can do a task by themself in 2n2^n days. Let f(k)f(k) represent the time taken when workers 0 to kk inclusive are all working on the task simultaneously (assuming their overall speed adds up). What is the value of f−1(355512)f^{-1}(\frac{355}{512})?
  • A.8
  • B.9
  • C.16
  • D.17
  • E.64
  • F.65
  • G.Doesn't exist

Answer: G

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 12

1 mark
Which of these values is the smallest?
  • A.sin⁡(1)\sin(1)
  • B.cos⁡(12)\cos(\frac{1}{2})
  • C.0.88
  • D.13tan⁡(π3)\frac{1}{3}\tan(\frac{\pi}{3})
  • E.25\frac{2}{\sqrt{5}}

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 13

1 mark
Find the sum of the reciprocals of all of the factors of 1600.
  • A.13937\frac{1}{3937}
  • B.16003937\frac{1600}{3937}
  • C.39371600\frac{3937}{1600}
  • D.3781600\frac{378}{1600}
  • E.1600378\frac{1600}{378}
  • F.1378\frac{1}{378}

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 14

1 mark
The digital root of a number is where you find the sum of the digits of a number, then find the sum of the answer, and repeat until you get a 1-digit number. For example, to find the digital root of 9678996 you do 9+6+7+8+9+9+6=549+6+7+8+9+9+6 = 54, 5+4=95 + 4 = 9, so its digital root is 9. What's the digital root of 739357^{3935}?
  • A.1
  • B.2
  • C.3
  • D.4
  • E.5
  • F.6
  • G.7
  • H.8

Answer: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 15

1 mark
The function f(x)f(x) has the property that f(x)=f(6−x)f(x) = f(6 - x) for all real xx. Given that:
(∫23f(x)dx)2+(∫46f(x)dx)2+(∫24f(x)dx)(∫02f(x)dx)−∫363f(x)dx=−2(\int_{2}^{3} f(x) dx)^2 + (\int_{4}^{6} f(x) dx)^2 + (\int_{2}^{4} f(x) dx)(\int_{0}^{2} f(x) dx) - \int_{3}^{6} 3f(x) dx = -2

Find the sum of the possible values of ∫06f(x)dx\int_{0}^{6} f(x) dx.
  • A.-10
  • B.-5
  • C.-3
  • D.0
  • E.3
  • F.5
  • G.10
  • H.272\frac{27}{2}

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 16

1 mark
A point AA is chosen on the curve with equation:
(x−2)2+(y−3)2=4(x - 2)^2 + (y - 3)^2 = 4

and another point BB is chosen on the curve with equation:
x2+y2+8x+10y=rx^2 + y^2 + 8x + 10y = r

Find the length of the interval within the range 0<r<1250 < r < 125 for which the shortest possible distance of ABAB is less than 1.
  • A.3
  • B.5
  • C.6
  • D.49
  • E.76
  • F.117
  • G.119
  • H.120

Answer: F

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 17

1 mark
f(x)=1! x (2!) x (3!) x (4!) x (5!) x (6!) xf(x)=1!\,x^{\,(2!)\,x^{\,(3!)\,x^{\,(4!)\,x^{\,(5!)\,x^{\,(6!)\,x}}}}}


Which of these is the closest value of
f ⁣(12)f\!\left(\tfrac{1}{2}\right)
?
  • A.0
  • B.18\frac{1}{8}
  • C.14\frac{1}{4}
  • D.12\frac{1}{2}
  • E.1
  • F.2
  • G.1000000

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 18

1 mark
45 people are standing in a line. How many ways are there to choose 17 of them such that no two chosen people are next to each other, and order doesn't matter?
  • A.45!28!17!\frac{45!}{28!17!}
  • B.43!26!17!\frac{43!}{26!17!}
  • C.28!17!11!\frac{28!}{17!11!}
  • D.29!17!12!\frac{29!}{17!12!}
  • E.17!12!8!\frac{17!12!}{8!}
  • F.21745!12!\frac{2^{17}45!}{12!}

Answer: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 19

1 mark
Consider f(x)=ln⁡(x2−2x+1)f(x) = \ln(\sqrt{x^2 - 2x + 1}). Which of the following statements is true about f(x)f(x)?
  • A.It is defined for all real xx.
  • B.Defined for all real x<0x < 0 but not all real x>0x > 0.
  • C.Defined for all real x>0x > 0 but not all real x<0x < 0.
  • D.Undefined for x=ex = e.
  • E.8<f−1(2)<98 < f^{-1}(2) < 9.
  • F.f(x)=100f(x) = 100 has no solutions.

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 20

1 mark
The function F(x)F(x), where xx is a non-negative real number, is the result of subtracting the integer part of xx from xx. For example, F(3)=0F(3) = 0, F(5.43)=0.43F(5.43) = 0.43, F(π)=0.14159265...F(\pi) = 0.14159265..., Find an expression for:
∫0kF(x2)dx\int_{0}^{\sqrt{k}} F(x^2) dx

where kk is a positive integer.
  • A.13k32−F(13k32)\frac{1}{3}k^{\frac{3}{2}} - F(\frac{1}{3}k^{\frac{3}{2}})
  • B.13k32−∑n=0k−1n(n+1−n)\frac{1}{3}k^{\frac{3}{2}} - \sum_{n=0}^{k-1} n(\sqrt{n+1} - \sqrt{n})
  • C.13k32+∑n=0k−1n(n+1−n)\frac{1}{3}k^{\frac{3}{2}} + \sum_{n=0}^{k-1} n(\sqrt{n+1} - \sqrt{n})
  • D.13k32−∑n=0k−1n(n+1+n)\frac{1}{3}k^{\frac{3}{2}} - \sum_{n=0}^{k-1} n(\sqrt{n+1} + \sqrt{n})
  • E.13k32+(∑n=0k−1n)+(k−1)k\frac{1}{3}k^{\frac{3}{2}} + (\sum_{n=0}^{k-1} \sqrt{n}) + (k-1)\sqrt{k}
  • F.13k32+(∑n=0k−1n)+(1−k)k\frac{1}{3}k^{\frac{3}{2}} + (\sum_{n=0}^{k-1} \sqrt{n}) + (1-k)\sqrt{k}

Answer: F

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →