TMUA Specimen Paper 1

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1 Equations · Level 3
The sum of the two values of \(x\) that satisfy the simultaneous equations \(x - 3y + 1 = 0\) and \(3x^2 - 7x y = 5\) is
A
\(-8.5\)
B
\(-7.5\)
C
\(-1.5\)
D
\(3.5\)
E
\(4.5\)
F
\(5\)
2 Trigonometric Equations · Level 3
The number of solutions in the interval \(0 \leq \theta \leq 4 \pi\) of the equation \(\sin^2 \theta + 3 \cos \theta = 3\) is
A
\(0\)
B
\(1\)
C
\(2\)
D
\(3\)
E
\(4\)
F
\(5\)
G
\(6\)
3 Coordinate Geometry · Level 3
The perpendicular bisector of the line segment joining the points \((2, -6)\) and \((5, 4)\) cuts the \(x\)-axis at the point with \(x\)-coordinate
A
\(\dfrac{1}{20}\)
B
\(\dfrac{1}{6}\)
C
\(\dfrac{1}{3}\)
D
\(\dfrac{19}{5}\)
E
\(\dfrac{41}{6}\)
4 Inequalities · Level 3
The complete set of values of \(x\) for which \((x^2 - 1)(x - 2) > 0\) is
A
\(x < -1, 1 < x < 2\)
B
\(x < -1, x > 2\)
C
\(-1 < x < 2\)
D
\(x < 1, x > 2\)
E
\(-1 < x < 1, x > 2\)
5 Exponentials and Logarithms · Level 3
Given that \(y = -\log_10 (1 - x)\) for \(x < 1\), find \(x\) in terms of \(y\).
A
\(x = -\dfrac{1}{\log_10 (1 - y)}\)
B
\(x = 1 + \log_10 y\)
C
\(x = 1 - \log_10 y\)
D
\(x = 1 - 10^{-y}\)
E
\(x = 10^{-y} - 1\)
F
\(x = 10^{1 - y}\)
6 Polynomials · Level 3
It is given that \(x + 2\) is a factor of \(x^3 + 4 c x^2 + x (c + 1)^2 - 6\). The sum of the possible values of \(c\) is
A
\(-10\)
B
\(-4\)
C
\(-1\)
D
\(4\)
E
\(10\)
7 Counting and Probabilities · Level 3
A bag contains \(n\) red balls, \(n\) yellow balls, and \(n\) blue balls. One ball is selected at random and not replaced. A second ball is then selected at random and not replaced. Each ball is equally likely to be chosen. The probability that the two balls are *not* the same colour is
A
\(\dfrac{n - 1}{3 n - 1}\)
B
\(\dfrac{2 n - 2}{3 n - 1}\)
C
\(\dfrac{2 n}{3 n - 1}\)
D
\(\dfrac{(n - 1)^3}{27 (3 n - 1)^3}\)
E
\(\dfrac{3 (n - 1)}{3 n - 1}\)
F
\(\dfrac{n^3}{27 (3 n - 1)^3}\)
8 Exponentials and Logarithms · Level 3
Given that \(a^x b^{2 x} c^{3 x} = 2\), where \(a\), \(b\), and \(c\) are positive real numbers, then \(x =\)
A
\(\log_10 \left(\dfrac{2}{a + 2 b + 3 c}\right)\)
B
\(\dfrac{\log_10 2}{\log_10 (a + 2 b + 3 c)}\)
C
\(\dfrac{2}{\log_10 (a + 2 b + 3 c)}\)
D
\(\dfrac{2}{a + 2 b + 3 c}\)
E
\(\log_10 \left(\dfrac{2}{a b^2 c^3}\right)\)
F
\(\dfrac{\log_10 2}{\log_10 (a b^2 c^3)}\)
G
\(\dfrac{2}{\log_10 (a b^2 c^3)}\)
H
\(\dfrac{2}{a b^2 c^3}\)
9 Equations · Level 3
The roots of the equation \(2x^2 - 11 x + c = 0\) differ by \(2\). The value of \(c\) is
A
\(\dfrac{105}{8}\)
B
\(\dfrac{113}{8}\)
C
\(\dfrac{117}{8}\)
D
\(\dfrac{119}{8}\)
10 Functions and Their Graphs · Level 3
The curve \(y = \cos x\) is reflected in the line \(y = 1\) and the resulting curve is then translated by \(\dfrac{\pi}{4}\) units in the positive \(x\)-direction. The equation of this new curve is
A
\(y = 2 + \cos\left(x + \dfrac{\pi}{4}\right)\)
B
\(y = 2 + \cos\left(x - \dfrac{\pi}{4}\right)\)
C
\(y = 2 - \cos\left(x + \dfrac{\pi}{4}\right)\)
D
\(y = 2 - \cos\left(x - \dfrac{\pi}{4}\right)\)
11 Exponentials and Logarithms · Level 3
The sum of the roots of the equation \(2^{2 x} - 8 \times 2^x + 15 = 0\) is
A
\(\log_10 2\)
B
\(\log_10 15\)
C
\(2 \log_10 2\)
D
\(\log_10 \left(\dfrac{15}{4}\right)\)
E
\(\dfrac{\log_10 15}{\log_10 2}\)
12 Solid Figures · Level 3
The cross-section of a triangular prism is an equilateral triangle with side \(2x\) cm. The length of the prism is \(d\) cm. Let the total surface area of the prism be \(T\) cm\(^2\). Given that the volume of the prism is \(T\) cm\(^3\), which one of the following is an expression for \(d\) in terms of \(x\)?
A
\(\dfrac{x}{2 x - 3}\)
B
\(\dfrac{3 x}{3 x - 2 \sqrt{3}}\)
C
\(\dfrac{2 x}{x - 4 \sqrt{3}}\)
D
\(\dfrac{2 x}{x - 2 \sqrt{3}}\)
E
\(\dfrac{2 x}{x - \sqrt{3}}\)
13 Polynomials · Level 3
How many real roots does the equation \(x^4 - 4 x^3 + 4 x^2 - 10 = 0\) have?
A
\(0\)
B
\(1\)
C
\(2\)
D
\(3\)
E
\(4\)
14 Exponentials and Logarithms · Level 3
\(a\), \(b\), \(x\), and \(y\) are real and positive. \(a\) and \(b\) are constants. \(x\) and \(y\) are related. A graph of \(\log y\) against \(\log x\) is drawn. For which one of the following relationships will this graph be a straight line?
A
\(y^b = a^x\)
B
\(y = a b^x\)
C
\(y^2 = a + x^b\)
D
\(y = a x^b\)
E
\(y^x = a^b\)
15 Integration · Level 3
The smallest possible value of \(\displaystyle\int_{0}^{1} (x - a)^2 d x\) as \(a\) varies is
A
\(\dfrac{1}{12}\)
B
\(\dfrac{1}{3}\)
C
\(\dfrac{1}{2}\)
D
\(\dfrac{7}{12}\)
E
\(2\)
16 Exponentials and Logarithms · Level 3
Given that \(c\) and \(d\) are non-zero integers, the expression \(\dfrac{10^{c - 2 d} \times 20^{2 c + d}}{8^c \times 125^{c + d}}\) is an integer if
A
\(c < 0\)
B
\(d < 0\)
C
\(c < 0\) and \(d < 0\)
D
\(c < 0\) and \(d > 0\)
E
\(c > 0\) and \(d < 0\)
F
\(c > 0\) and \(d > 0\)
G
\(d > 0\)
H
\(c > 0\)
17 Equations · Level 3
For what values of the non-zero real number \(a\) does the quadratic equation \(a x^2 + (a - 2) x = 2\) have real distinct roots?
A
All values of \(a\)
B
\(a = -2\)
C
\(a > -2\)
D
\(a \neq -2\)
E
No values of \(a\)
18 Trigonometric Equations · Level 3
The angle \(x\) is measured in radians and is such that \(0 \leq x \leq \pi\). The total length of any intervals for which \(-1 \leq \tan x \leq 1\) and \(\sin 2x \geq 0.5\) is
A
\(\dfrac{\pi}{12}\)
B
\(\dfrac{\pi}{6}\)
C
\(\dfrac{\pi}{4}\)
D
\(\dfrac{\pi}{3}\)
E
\(\dfrac{5 \pi}{12}\)
F
\(\dfrac{\pi}{2}\)
G
\(\dfrac{5 \pi}{6}\)
19 Sequences and Series · Level 3
A geometric series has first term \(4\) and common ratio \(r\), where \(0 < r < 1\). The first, second, and fourth terms of this geometric series form three successive terms of an arithmetic series. The sum to infinity of the geometric series is
A
\(\dfrac{1}{2} (\sqrt{5} - 1)\)
B
\(2 (3 - \sqrt{5})\)
C
\(2 (1 + \sqrt{5})\)
D
\(2 (3 + \sqrt{5})\)
20 Polynomials · Level 3
The coefficient of \(x^2\) in the expansion of \((4 - x^2)[(1 + 2x + 3x^2)^6 - (1 + 4x^3)^5]\) is
A
\(28\)
B
\(72\)
C
\(78\)
D
\(192\)
E
\(240\)
F
\(310\)
G
\(312\)

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