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TMUA 2017 D513/01

20 questions20 marks75Updated July 2025

The TMUA 2017 D513/01 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.

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Question 1

1 mark
THIS IS WRONG, PLEASE REVIEW AGAIN!
Given that dydx=3x2−2−3xx3\frac{dy}{dx} = 3x^2 - \frac{2-3x}{x^3}, x≠0x ≠ 0
and y=5y = 5 when x=1x = 1, find yy in terms of xx.
  • A.
    y=13x3+x−2−3x−1+623y = \frac{1}{3}x^3 + x^{-2} - 3x^{-1} + 6\frac{2}{3}
  • B.
    y=x3+12x−2−3x−1+612y = x^3 + \frac{1}{2}x^{-2} - 3x^{-1} + 6\frac{1}{2}
  • C.
    y=x3+x−2−3x−1+6y = x^3 + x^{-2} - 3x^{-1} + 6
  • D.
    y=x3+x−2−x−1+4y = x^3 + x^{-2} - x^{-1} + 4
  • E.
    y=x3+2x−2−x−1+3y = x^3 + 2x^{-2} - x^{-1} + 3
  • F.
    y=3x3+x−2−x−1+2y = 3x^3 + x^{-2} - x^{-1} + 2

Answer: C

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Question 2

1 mark
The function ff is given by
f(x)=(2x−12x2)2f(x) = \left(\frac{2}{x} - \frac{1}{2x^2}\right)^2
(x≠0x ≠ 0)
What is the value of f′′(1)f''(1)?

  • A.-3
  • B.-1
  • C.5
  • D.17
  • E.29
  • F.80

Answer: C

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Question 3

1 mark
A line ll has equation y=6−2xy = 6-2x
A second line is perpendicular to ll and passes through the point (−6,0)(-6, 0).
Find the area of the region enclosed by the two lines and the xx-axis.
  • A.161516\frac{1}{5}
  • B.18
  • C.213521\frac{3}{5}
  • D.27
  • E.401240\frac{1}{2}

Answer: A

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Question 4

1 mark
When (3x2+8x−3)(3x^2 + 8x - 3) is multiplied by (px−1)(px - 1) and the resulting product is divided by
(x+1)(x + 1), the remainder is 2424.
What is the value of pp?
  • A.-4
  • B.2
  • C.4
  • D.87\frac{8}{7}
  • E.114\frac{11}{4}

Answer: B

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Question 5

1 mark
SS is the complete set of values of xx which satisfy both the inequalities
x2−8x+12<0x^2 - 8x + 12 < 0 and 2x+1>92x + 1 > 9
The set SS can also be represented as a single inequality.
Which one of the following single inequalities represents the set SS?
  • A.(x2−8x+12)(2x+1)<0(x^2 - 8x + 12)(2x + 1) < 0
  • B.(x2−8x+12)(2x+1)>0(x^2 - 8x + 12)(2x + 1) > 0
  • C.x2−10x+24<0x^2 - 10x + 24 < 0
  • D.x2−10x+24>0x^2 - 10x + 24 > 0
  • E.x2−6x+8<0x^2 - 6x + 8 < 0
  • F.x2−6x+8>0x^2 - 6x + 8 > 0
  • G.x<2x < 2
  • H.x>6x > 6

Answer: C

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Question 6

1 mark
A tangent to the circle x2+y2=144x^2 + y^2 = 144 passes through the point (20,0)(20, 0) and crosses the positive yy-axis.
What is the value of yy at the point where the tangent meets the yy-axis?
  • A.12
  • B.15
  • C.493\frac{49}{3}
  • D.20
  • E.643\frac{64}{3}
  • F.803\frac{80}{3}

Answer: B

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Question 7

1 mark
The first three terms of an arithmetic progression are pp, qq and p2p^2 respectively, where
p<0p < 0
The first three terms of a geometric progression are pp, p2p^2 and qq respectively.
Find the sum of the first 1010 terms of the arithmetic progression.
  • A.
    238\frac{23}{8}
  • B.
    958\frac{95}{8}
  • C.
    1158\frac{115}{8}
  • D.
    1858\frac{185}{8}

Answer: B

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Question 8

1 mark
Find the complete set of values of xx, with 0≤x≤π0 \leq x \leq \pi, for which
(1−2sin⁡x)cos⁡x≥0(1-2 \sin x) \cos x \geq 0
  • A.0≤x≤π6,π2≤x≤5π60 \leq x \leq \frac{\pi}{6},\quad \frac{\pi}{2} \leq x \leq \frac{5\pi}{6}
  • B.0≤x≤π6,5π6≤x≤π0 \leq x \leq \frac{\pi}{6},\quad \frac{5\pi}{6} \leq x \leq \pi
  • C.π6≤x≤π2,5π6≤x≤π\frac{\pi}{6} \leq x \leq \frac{\pi}{2},\quad \frac{5\pi}{6} \leq x \leq \pi
  • D.π6≤x≤5π6\frac{\pi}{6} \leq x \leq \frac{5\pi}{6}

Answer: A

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Question 9

1 mark
A circle has equation x2+y2−18x−22y+178=0x^2 + y^2 - 18x - 22y + 178 = 0
A regular hexagon is drawn inside this circle so that the vertices of the hexagon touch the circle.
What is the area of the hexagon?
  • A.6
  • B.636\sqrt{3}
  • C.18
  • D.18318\sqrt{3}
  • E.36
  • F.36336\sqrt{3}
  • G.48
  • H.48348\sqrt{3}

Answer: F

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Question 10

1 mark
A curve CC has equation y=f(x)y = f(x) where
f(x)=p3−6p2x+3px2−x3f(x) = p^3 - 6p^2x + 3px^2 - x^3
and pp is real.
The gradient of the normal to the curve CC at the point where x=−1x = -1 is MM.
What is the greatest possible value of MM as pp varies?
  • A.
    −32-\frac{3}{2}
  • B.
    −23-\frac{2}{3}
  • C.
    −12-\frac{1}{2}
  • D.
    14\frac{1}{4}
  • E.
    23\frac{2}{3}
  • F.
    32\frac{3}{2}

Answer: E

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Question 11

1 mark
The sequence xnx_n is defined by the rules
x1=7x_1 = 7
xn+1=23xn−535xn+1x_{n+1} = \frac{23x_n - 53}{5x_n + 1}

The first three terms in the sequence are 7,3,17, 3, 1
What is the value of x100x_{100}?
  • A.-5
  • B.0
  • C.1
  • D.3
  • E.7

Answer: A

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Question 12

1 mark
The polynomial function f(x)f(x) is such that f(x)>0f(x) > 0 for all values of xx.
Given ∫24f(x) dx=A\int_2^4 f(x) \,dx = A, which one of the following statements must be correct?
  • A.
    ∫02[f(x+2)+1] dx=A+1\int_0^2 [f(x + 2) + 1] \,dx = A + 1
  • B.
    ∫02[f(x+2)+1] dx=A+2\int_0^2 [f(x + 2) + 1] \,dx = A + 2
  • C.
    ∫24[f(x+2)+1] dx=A+1\int_2^4 [f(x + 2) + 1] \,dx = A + 1
  • D.
    ∫24[f(x+2)+1] dx=A+2\int_2^4 [f(x + 2) + 1] \,dx = A + 2
  • E.
    ∫46[f(x+2)+1] dx=A+1\int_4^6 [f(x + 2) + 1] \,dx = A + 1
  • F.
    ∫46[f(x+2)+1] dx=A+2\int_4^6 [f(x + 2) + 1] \,dx = A + 2

Answer: B

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Question 13

1 mark
In the expansion of (a+bx)5(a + bx)^5 the coefficient of x4x^4 is 88 times the coefficient of x2x^2.
Given that aa and bb are non-zero positive integers, what is the smallest possible
value of a+ba + b?
  • A.3
  • B.4
  • C.5
  • D.9
  • E.13
  • F.17

Answer: C

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Question 14

1 mark
The solution of the simultaneous equations
2x+3×22y=32^x + 3 \times 2^{2y} = 3
22x−9×22y=62^{2x} - 9 \times 2^{2y} = 6
is x=px = p, y=qy = q.
Find the value of p−qp - q
  • A.512\frac{5}{12}
  • B.73\frac{7}{3}
  • C.log⁡2512\log_2 \frac{5}{12}
  • D.log⁡273\log_2 \frac{7}{3}
  • E.log⁡29\log_2 9
  • F.log⁡215\log_2 15

Answer: F

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Question 15

1 mark
It is given that f(x)=−2x2+10f(x) = -2x^2 + 10
Consider the following three curves:
(1)(1) y=f(x)y = f(x)
(2)(2) y=f(x+1)y = f(x + 1)
(3)(3) the curve y=f(x+1)y = f(x + 1) reflected in the line y=6y = 6
The trapezium rule is used to estimate the area under each of these three curves between
x=0x = 0 and x=1x = 1.
State whether the trapezium rule gives an overestimate or underestimate for each of these areas.
Exam diagram
  • A.underestimate, underestimate, underestimate
  • B.underestimate, underestimate, overestimate
  • C.underestimate, overestimate, underestimate
  • D.underestimate, overestimate, overestimate
  • E.overestimate, underestimate, underestimate
  • F.overestimate, underestimate, overestimate
  • G.overestimate, overestimate, underestimate
  • H.overestimate, overestimate, overestimate

Answer: B

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Question 16

1 mark
The functions ff and gg are given by f(x)=3x2+12x+4f(x) = 3x^2 + 12x + 4 and g(x)=x3+6x2+9x−8g(x) = x^3 + 6x^2 + 9x - 8.
What is the complete set of values of xx for which one of the functions is increasing and the
other decreasing?
  • A.x≥−1x \geq -1
  • B.x≤−1x \leq -1
  • C.−3≤x≤−2-3 \leq x \leq -2, x≥−1x \geq -1
  • D.x≤−2x \leq -2, x≥−1x \geq -1
  • E.x≤−3x \leq -3, −2≤x≤−1-2 \leq x \leq -1
  • F.x≤−3x \leq -3, x≥−2x \geq -2
  • G.−2≤x≤−1-2 \leq x \leq -1

Answer: E

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Question 17

1 mark
The two functions F(n)F(n) and G(n)G(n) are defined as follows for positive integers nn:
F(x)=1n∫0n(n−x) dxF(x) = \frac{1}{n}\int_0^n (n - x) \,dx

G(n)=∑r=1nF(r)G(n) = \sum_{r=1}^n F(r)

What is the smallest positive integer nn such that G(n)>150G(n) > 150?
  • A.22
  • B.23
  • C.24
  • D.25
  • E.26

Answer: D

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Question 18

1 mark
The graph of y=log⁡10xy = \log_{10}x is translated in the positive yy-direction by 22 units.
This translation is equivalent to a stretch of factor kk parallel to the xx-axis.
What is the value of kk?
  • A.0.01
  • B.log⁡102\log_{10} 2
  • C.0.5
  • D.2
  • E.log⁡210\log_2 10
  • F.100

Answer: A

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Question 19

1 mark
The set of solutions to the inequality x2+bx+c<0x^2 + bx + c < 0 is the interval p<x<qp < x < q
where b,c,pb, c, p and qq are real constants with c<0c < 0.
In terms of p,qp, q and cc, what is the set of solutions to the inequality x2+bcx+c3<0x^2 + bcx + c^3 < 0?
  • A.pc<x<qc\frac{p}{c} < x < \frac{q}{c}
  • B.qc<x<pc\frac{q}{c} < x < \frac{p}{c}
  • C.pc<x<qcpc < x < qc
  • D.qc<x<pcqc < x < pc
  • E.pc2<x<qc2pc^2 < x < qc^2
  • F.qc2<x<pc2qc^2 < x < pc^2

Answer: D

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Question 20

1 mark
The lengths of the sides QRQR, RPRP and PQPQ in triangle PQRPQR are aa, a+da + d and a+2da + 2d respectively, where aa and dd are positive and such that 3d>2a3d > 2a.
What is the full range, in degrees, of possible values for angle PRQPRQ?
  • A.0<angle PRQ<600 < \text{angle } PRQ < 60
  • B.0<angle PRQ<1200 < \text{angle } PRQ < 120
  • C.60<angle PRQ<12060 < \text{angle } PRQ < 120
  • D.60<angle PRQ<18060 < \text{angle } PRQ < 180
  • E.120<angle PRQ<180120 < \text{angle } PRQ < 180

Answer: E

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