TMUA Mock Exam B – Paper 1
1 / 20
Given that \( \sin x = \frac{1}{3} \), find the product of all possible exact values of \( \cos x + \tan x \).
\(\pm \frac{5\sqrt{2}}{12}\)
\(\frac{\sqrt{2}}{2}\)
\(\frac{5\sqrt{2}}{12}\)
\(-\frac{121}{72}\)
\(\frac{1}{3}\)
\(-\frac{1}{3}\)
\(\frac{55}{72}\)
2 / 20
A baker believes that the amount of bread \( B \) and the number of muffins \( M \) produced are related by the equation \( M = aB^n \). She collects the data from a day at the bakery and creates a scatter graph with \( \log B \) as the \( x \)-axis, and \( \log M \) on the \( y \)-axis. The results have a straight line gradient of \( 5 \) and \( y \)-intercept \( -1 \). What are the exact values for \( a \) and \( n \)?
\(a = -1,\ n = \log(5)\)
\(a = -1,\ n = -5\)
\(a = 1,\ n = \log(5)\)
\(a = 100,\ n = 5\)
\(a = -0.1,\ n = -5\)
\(a = 0.1,\ n = 5\)
\(a = 10,\ n = 5\)
\(a = 10,\ n = \log(5)\)
3 / 20
A circle has the equation \( x^2 – 12x + y^2 – 10y + 12 = 0 \), and is tangent to two sides of the triangle. Some coordinates on the triangle are labelled. Find the exact value of the shaded area.
\(528 – 49\pi\)
\(552 – 49^2\pi\)
\(4290 – 36^2\pi\)
\(276 – 49\pi\)
\(169 – 36\pi\)
4 / 20
The graph of \( y = \sqrt{5x – 2} \) undergoes the below transformations in the given order:
I quad Translated horizontally left by \( 4 \) II quad Translated vertically down by \( 6 \) III quad Vertical reflection in the axis \( y = 0 \) IV quad Stretch factor \( \frac{1}{3} \) in the \( y \)-axis V quad Stretch factor \( 2 \) in the \( x \)-axis
Which of the following equations describes the transformed graph?
\(y = \sqrt{2 – \frac{5x}{18}} – 2\)
\(y = \frac{\sqrt{-10x – 6 – 4}}{3}\)
\(y = -\frac{\sqrt{\frac{5}{2}x – 22 – 6}}{3}\)
\(y = -\frac{\sqrt{\frac{5}{2}x + 18 – 6}}{3}\)
\(y = \frac{\sqrt{\frac{5}{2}x – 22 – 6}}{3}\)
\(y = -\frac{\sqrt{\frac{5}{2}x + 22 – 6}}{3}\)
\(y = \frac{\sqrt{\frac{5}{2}x + 14}}{3}\)
5 / 20
Solve the differential equation, given that when \( y = 5 \) and \( x = 3 \). What is the value of the constant of integration? \[ \frac{dy}{dx} = \frac{(7x + 3)y^{2/5}}{x}. \]
\(8(5)^{2/5}\)
\(\frac{1}{3}(5)^{8/5} – 21 – 3\ln 3\)
\(-\frac{1}{8}(5)^{4/5} – 7 + \ln 3\)
\(\frac{1}{8}(5)^{4/5} – 21 – 3\ln 3\)
\(\frac{5}{3}(5)^{3/5} + 21 + \ln 3\)
\(-8(5)^{2/5}\)
6 / 20
On a cheese farm, the mass of cheese produced and measured on the scales T minutes after the cheese machine has started, is G grams. For any time, the rate of cheese production is proportional to the mass of cheese formed. However, cheese is removed from scales at a constant rate of 10 grams per minute. When the mass of cheese on the scales was 150g, and the rate of change of cheese weight on the scales was 90 g/minute. When the rate of change of mass of cheese was 50g/min, what was the mass of cheese formed? (You may find it beneficial to create a differential equation to show this.)
7 / 20
Find the coefficient of \( x^3 \) in the binomial expansion of \[ y = (2x – 1)^9. \]
8 / 20
The function \( f(x) \) has a stationary point at \( (5, 12) \) and \( f”(x) = 18x + 12 \). Find \( f(x) \).
\(3x^3 + 3x^2 – 316x + 1142\)
\(6x^3 + 3x^2 – 189x + 132\)
\(3x^3 + 6x^2 – 285x + 912\)
\(x^3 + 12x^2 – 128x + 227\)
\(2x^3 + 14x^2 – 336x + 1242\)
\(9x^2 + 12x – 273\)
9 / 20
Find the equation of the normal to the curve with the inverse function \[ y = \frac{\sqrt{x – 10}}{\sqrt{5}} + 7 \] at the point where \( x = 1 \).
\(y – 190 = \frac{1}{60}x – \frac{1}{60}\)
\(7y = 3x + 17\)
\(2y = \frac{1}{30}(x – 1) – 190\)
\(y = 190x\)
\(30y = \frac{1}{2}x + \frac{379}{2}\)
10 / 20
Calculate the exact solution to the equation: \[ \log_{6}(4x + 3) = \log_{6}(9x – 5) + 2. \]
\(\frac{578}{693}\)
\(\frac{653}{855}\)
\(-\frac{896}{146}\)
\(-\frac{203}{5}\)
\(\frac{183}{320}\)
\(\frac{219}{34}\)
11 / 20
How many values satisfy the following equation for \( -2\pi \leq x \leq 2\pi \)? \[ 2 \sin x \cos x = \cos x. \]
12 / 20
A semi-circle shares its diameter with the unique edge of an isosceles triangle, as shown. The equal sides of the triangle are both \( 7 \,\text{cm} \) long, and the angle between them is \( 30^\circ \).
Calculate the perimeter of the composite shape.
(Hint: It is given that \( \sin 75^\circ = \frac{\sqrt{6} + \sqrt{2}}{4} \).)
\(14 + \frac{7\pi}{12}(\sqrt{6} – \sqrt{3})\)
\(7 + \frac{7\pi}{12}(\sqrt{6} – \sqrt{2})\)
\(14 + \frac{7\pi}{4}(\sqrt{6} – \sqrt{2})\)
\(\frac{7\pi}{24}(\sqrt{6} – \sqrt{2})\)
\(\frac{7\pi}{12}(\sqrt{6} – \sqrt{2})\)
\(\frac{7\pi}{12}(\sqrt{6} – \sqrt{3})\)
\(14 + \frac{3\pi}{8}(3\sqrt{2} – \sqrt{3})\)
\(\frac{3\pi}{8}(3\sqrt{2} – \sqrt{3})\)
13 / 20
Calculate the sum of the real solutions of the following: \[ 9^{2x} + 6 = 3^{2x + 2}. \]
\(\log_{3} 12\)
\(\log_{3} 6\)
\(\frac{1}{2} \log_{2} 6\)
14 / 20
Two circles \( A \) and \( B \) have the respective equations: \[ (x – 2)^2 + (y – 4)^2 = 4 \] \[ x^2 – 6x + y^2 + 8x = 0. \] Find the \( x \)-coordinate of the point on \( A \) that is closest to circle \( B \).
\(\frac{120 + \sqrt{20 – (4 \times 56 \times 225)}}{112}\)
\(\frac{230 + \sqrt{900 – (4 \times 26 \times 126)}}{52}\)
\(\frac{160 + \sqrt{100 – (4 \times 80 \times 220)}}{160}\)
\(\frac{260 + \sqrt{(-260)^2 – (4 \times 65 \times 256)}}{130}\)
\(\frac{16 + \sqrt{256 – (4 \times 10 \times 255)}}{20}\)
\(\frac{16 + \sqrt{256 – (4 \times 13 \times 256)}}{26}\)
15 / 20
A circle, of diameter 4cm, contains a square with two vertices on the diameter of the circle, and 2 vertices on the circumference of the circle, as shown below.
Calculate the area of the square.
\(\frac{4}{5}\)
\(\frac{16}{5}\)
\(4\)
\(\frac{7}{2}\)
\(\frac{9}{2}\)
\(\frac{17}{4}\)
\(\frac{64}{5}\)
16 / 20
An arithmetic series has first term a and a common difference d.
The sum of the first 10 terms is 1075. The third term of the sequence is 15.
Find the values of a and d
\(a = 5,\ d = -4\)
\(a = 7,\ d = 3\)
\(a = -24,\ d = 19.5\)
\(a = -59,\ d = 37\)
\(a = 3,\ d = 6\)
\(a = -2.5,\ d = 8.75\)
17 / 20
Solve: \[ 2^{x+1} – 2^{-x} = 0. \]
\(x = -\frac{1}{2}\)
\(x = \frac{1}{2}\)
18 / 20
A cubic polynomial satisfies the following conditions: \[ f(3) = 0. \] It can be expressed in the form \( (x – a)(x – 1)(x + 1) \). Find the total area under the curve between \( x = -1 \) and \( x = 1 \).
\(-\frac{1}{12}\)
\(\frac{1}{2}\)
19 / 20
State the minimum and maximum points of the graph of \[ y = -2 \sin\left(-3x + \frac{3\pi}{2}\right) + 7, \quad 0 \leq x < \pi. \]
\((-\pi/3, 8), (0, 6), (\pi/3, 8)\)
\((0, 9), (\pi/3, 5), (2\pi/3, 9)\)
\((-\pi/3, 9), (0, 5), (\pi/3, 9)\)
\((0, 8), (\pi/3, 6), (2\pi/3, 8)\)
\((0, 8), (\pi/6, 6), (2\pi/6, 8)\)
\((0, 9), (\pi/6, 6), (2\pi/6, 9)\)
20 / 20
A swimming pool is filled with water such that the volume of the pool in gallons \( G \) over time in minutes \( t \) is given by \[ G = 30t – 12t^3 + 9. \] At what fraction of time \( t \) is the water flowing at the maximum rate?
\(\sqrt{\frac{30}{12}}\)
\(\frac{1}{12}\)
\(\sqrt{\frac{30}{36}}\)
\(\sqrt{\frac{6}{6}}\)
\(\sqrt{\frac{30}{16}}\)
\(\sqrt{\frac{6}{36}}\)
Your score is
The average score is 0%
Restart quiz
Added to cart
Check out our shop to see what's available
