TMUAQuestions

Every question from the released TMUA papers, 318 in all. Open one to answer it.

318 questions

Determine the number of stationary points on the curve with equation $y = 3 x^{4} + 4 x^{3} + 6 x^{2} - 5$ TMUA 2022 · Paper 1 · Question 1 · Save
Find the coefficient of the $x^{5}$ term in the expansion of $(1 + x)^{5} \times \sum_{i = 0}^{5} x^{i}$ TMUA 2022 · Paper 1 · Question 2 · Save
Consider the following statement about the positive integer $n$ if $n$ is prime, then $n^{2} + 2$ is not prime Which of the following is a counterexample to this statement? I $n = 2$ II $n = 3$ III $n = 4$ TMUA 2022 · Paper 1 · Question 3 · Save
The point $P$ has coordinates $(p, q)$ , and the equation of a circle is $x^{2} + 2 f x + y^{2} + 2 g y + h = 0$ where $f, g, h, p$ and $q$ are all real constants. Let $L$ be the distance between the centre of the circle and the point $P$ . Which one of the following is sufficient on its own to be able to calculate $L$ ?TMUA 2022 · Paper 1 · Question 4 · Save
A straight line $L$ passes through $(1, 2)$ . Let $P$ be the statement if the $y$ -intercept of $L$ is negative, then the $x$ -intercept of $L$ is positive. Which of the following statements must be true? I $P$ II the converse of $P$ III the contrapositive of $P$ TMUA 2022 · Paper 1 · Question 5 · Save
Consider the following claim: The difference between two consecutive positive cube numbers is always prime. Here is an attempted proof of this claim: I $(x + 1)^{3} = x^{3} + 3 x^{2} + 3 x + 1$ II Taking $x$ to be a positive integer, the difference between two consecutive cube numbers can be expressed as…TMUA 2022 · Paper 1 · Question 7 · Save
A selection, $S$ , of $n$ terms is taken from the arithmetic sequence $1, 4, 7, 10, \dots, 70$ . Consider the following statement: $()$ There are two distinct terms in $S$ whose sum is 74. What is the smallest value of $n$ for which $()$ is necessarily true?TMUA 2022 · Paper 1 · Question 8 · Save
Consider the following statement: $()$ For all real numbers $x$ , if $x < k$ then $x^{2} < k$ What is the complete set of values of $k$ for which $()$ is true?TMUA 2022 · Paper 1 · Question 9 · Save
Which of the following statements is/are true? I For all real numbers $x$ and for all positive integers $n, x < n$ II For all real numbers $x$ , there exists a positive integer $n$ such that $x < n$ III There exists a real number $x$ such that for all positive integers $n, x < n$ TMUA 2022 · Paper 1 · Question 10 · Save
[IMAGE_PLACEHOLDER] The diagram shows a kite $PQRS$ whose diagonals meet at $O$ . $OP = x$ $OQ = y$ $OR = x$ $OS = z$ Which of the following is necessary and sufficient for angle $SPQ$ to be a right angle?TMUA 2022 · Paper 1 · Question 11 · Save
Place the following integrals in order of size, starting with the smallest. $P = \int_{0}^{1} 2^{x} d x$ $Q = \int_{0}^{1} 2^{x} d x$ $R = \int_{0}^{1} (2)^{x} d x$ TMUA 2022 · Paper 1 · Question 12 · Save
Consider the statement $()$ about a real number $x$ : $()$ There exists a real number $y$ such that $x - x y + y$ is negative. For how many real values of $x$ is $()$ true?TMUA 2022 · Paper 1 · Question 13 · Save
Consider the two inequalities: $∣ x + 5∣ < ∣ x + 11∣$ $∣ x + 11∣ < ∣ x + 1∣$ Which one of the following is correct?TMUA 2022 · Paper 1 · Question 14 · Save
The real numbers $x, y$ and $z$ are all greater than 1, and satisfy the equations $lo g_{x} y = z and lo g_{y} z = x$ Which one of the following equations for $lo g_{z} x$ must be true?TMUA 2022 · Paper 1 · Question 15 · Save
In this question, $a_{1}, \dots, a_{100}$ and $b_{1}, \dots, b_{100}$ and $c_{1}, \dots, c_{100}$ are three sequences of integers such that $a_{n} \leq b_{n} + c_{n}$ for each $n$ . Which of the following statements must be true? I…TMUA 2022 · Paper 1 · Question 16 · Save
Consider the following statements about integers $x$ and $y$ : P: $x^{2} + y^{2}$ is divisible by 4. Q: $x$ and $y$ are both even integers. Which one of the following is true?TMUA 2022 · Paper 1 · Question 17 · Save
A triangle has sides $x^{2} - x y + y^{2}, x^{2} + x y + y^{2}$ and $x^{2} + y^{2}$ . Which of the following describes the largest angle of the triangle?TMUA 2022 · Paper 1 · Question 18 · Save
Consider the statement $()$ about a real number $x$ . $()$ $(∣ x ∣ - 1) (∣ x ∣ - 2) \geq 0$ What is the largest possible range of values of $x$ for which $()$ is false?TMUA 2022 · Paper 1 · Question 19 · Save
Let $lo g_{b} x = y$ . Consider the following statements: I $lo g_{x} x^{y} = x$ II $lo g_{x} lo g_{y} y^{x} = x$ III $lo g_{y} lo g_{x} b = - 1$ Which of these statements is/are true for all permissible real numbers $x, y$ and $b$ ?TMUA 2022 · Paper 1 · Question 20 · Save
Determine the number of stationary points on the curve with equation $y = 3 x^{4} + 4 x^{3} + 6 x^{2} - 5$ TMUA 2022 · Paper 2 · Question 1 · Save
Find the coefficient of the $x^{5}$ term in the expansion of $(1 + x)^{5} \times \sum_{i = 0}^{5} x^{i}$ TMUA 2022 · Paper 2 · Question 2 · Save
Consider the following statement about the positive integer $n$ if $n$ is prime, then $n^{2} + 2$ is not prime Which of the following is a counterexample to this statement? I $n = 2$ II $n = 3$ III $n = 4$ TMUA 2022 · Paper 2 · Question 3 · Save
The point $P$ has coordinates $(p, q)$ , and the equation of a circle is $x^{2} + 2 f x + y^{2} + 2 g y + h = 0$ where $f, g, h, p$ and $q$ are all real constants. Let $L$ be the distance between the centre of the circle and the point $P$ . Which one of the following is sufficient on its own to be able to calculate $L$ ?TMUA 2022 · Paper 2 · Question 4 · Save
A straight line $L$ passes through $(1, 2)$ . Let P be the statement if the $y$ -intercept of $L$ is negative, then the $x$ -intercept of $L$ is positive. Which of the following statements must be true? I P II the converse of P III the contrapositive of PTMUA 2022 · Paper 2 · Question 5 · Save
A list consists of $n$ integers. Consider the following statements: P: $n$ is odd. Q: The median of the list is one of the numbers in the list. Which one of the following is true?TMUA 2022 · Paper 2 · Question 6 · Save
Consider the following claim: The difference between two consecutive positive cube numbers is always prime. Here is an attempted proof of this claim: I $(x + 1)^{3} = x^{3} + 3 x^{2} + 3 x + 1$ II Taking $x$ to be a positive integer, the difference between two consecutive cube numbers can be expressed as…TMUA 2022 · Paper 2 · Question 7 · Save
A selection, $S$ , of $n$ terms is taken from the arithmetic sequence $1, 4, 7, 10, \dots, 70$ . Consider the following statement: $()$ There are two distinct terms in $S$ whose sum is 74. What is the smallest value of $n$ for which $()$ is necessarily true?TMUA 2022 · Paper 2 · Question 8 · Save
Consider the following statement: $()$ For all real numbers $x$ , if $x < k$ then $x^{2} < k$ What is the complete set of values of $k$ for which $()$ is true?TMUA 2022 · Paper 2 · Question 9 · Save
Which of the following statements is/are true? I For all real numbers $x$ and for all positive integers $n$ , $x < n$ II For all real numbers $x$ , there exists a positive integer $n$ such that $x < n$ III There exists a real number $x$ such that for all positive integers $n$ , $x < n$ TMUA 2022 · Paper 2 · Question 10 · Save
[IMAGE_PLACEHOLDER] The diagram shows a kite $PQRS$ whose diagonals meet at $O$ . $OP = x$ $OQ = y$ $OR = x$ $OS = z$ Which of the following is necessary and sufficient for angle $SPQ$ to be a right angle?TMUA 2022 · Paper 2 · Question 11 · Save
Place the following integrals in order of size, starting with the smallest. $P = \int_{0}^{1} 2^{x} d x$ $Q = \int_{0}^{1} 2^{x} d x$ $R = \int_{0}^{1} (2)^{x} d x$ TMUA 2022 · Paper 2 · Question 12 · Save
Consider the statement $()$ about a real number $x$ : $()$ There exists a real number $y$ such that $x - x y + y$ is negative. For how many real values of $x$ is $()$ true?TMUA 2022 · Paper 2 · Question 13 · Save
Consider the two inequalities: $∣ x + 5∣ < ∣ x + 11∣$ $∣ x + 11∣ < ∣ x + 1∣$ Which one of the following is correct?TMUA 2022 · Paper 2 · Question 14 · Save
The real numbers $x, y$ and $z$ are all greater than 1, and satisfy the equations $lo g_{x} y = z and lo g_{y} z = x$ Which one of the following equations for $lo g_{z} x$ must be true?TMUA 2022 · Paper 2 · Question 15 · Save
In this question, $a_{1}, \dots, a_{100}$ and $b_{1}, \dots, b_{100}$ and $c_{1}, \dots, c_{100}$ are three sequences of integers such that $a_{n} \leq b_{n} + c_{n}$ for each $n$ . Which of the following statements must be true? I…TMUA 2022 · Paper 2 · Question 16 · Save
A student answered the following question: $a$ and $b$ are non-zero real numbers. Prove that the equation $x^{3} + a x^{2} + b = 0$ has three distinct real roots if $27 b (b + \frac{4 a ^{3}}{27}) < 0$ Here is the student's solution: I We differentiate $y = x^{3} + a x^{2} + b$ to get…TMUA 2022 · Paper 2 · Question 17 · Save
The following graphs (P, Q, R, and S) represent the functions: $y = (cos x)^{c o s x}, y = (sin x)^{s i n x}, y = (cos x)^{s i n x}, and y = (sin x)^{c o s x}$ for $0 < x < \frac{π}{2}$ in some order. [IMAGE_PLACEHOLDER] Which row in the table correctly identifies each graph?TMUA 2022 · Paper 2 · Question 18 · Trigonometry · Save
A polygon has $n$ vertices, where $n \geq 3$ . It has the following properties: Every vertex of the polygon lies on the circumference of a circle $C$ . The centre of the circle $C$ is inside the polygon. The radii from the centre of the circle $C$ to the vertices of the polygon cut the polygon into $n$ triangles of…TMUA 2022 · Paper 2 · Question 19 · Save
The functions $f_{1}$ to $f_{5}$ are defined on the real numbers by $\begin{align} f_1(x) &= \cos x \\ f_2(x) &= \sin(\cos x) \\ f_3(x) &= \cos(\sin(\cos x)) \\ f_4(x) &= \sin(\cos(\sin(\cos x))) \\ f_5(x) &= \cos(\sin(\cos(\sin(\cos x)))) \end{align}$ where all numbers are taken to be in radians. These functions have…TMUA 2022 · Paper 2 · Question 20 · Save
Two circles have the same radius. The centre of one circle is $(- 2, 1)$ . The centre of the other circle is $(3, - 2)$ . The circles intersect at two distinct points. What is the equation of the straight line through the two points at which the circles intersect?TMUA 2021 · Paper 1 · Question 1 · Save
The curve $y = x^{3} - 6 x + 3$ has turning points at $x = α$ and $x = β$ , where $β α$ . Find $\int_{α}^{β} x^{3} - 6 x + 3 d x$ TMUA 2021 · Paper 1 · Question 2 · Save
An arithmetic progression and a convergent geometric progression each have first term $\frac{1}{2}$ The sum of the second terms of the two progressions is $\frac{1}{2}$ The sum of the third terms of the two progressions is $\frac{1}{8}$ What is the sum to infinity of the geometric progression?TMUA 2021 · Paper 1 · Question 3 · Save
Find the minimum value of the function $2^{2 x} - 2^{x + 3} + 4$ TMUA 2021 · Paper 1 · Question 4 · Save
The function $f$ is such that $f (mn) = {f (m) f (n) mn if mn is a multiple of 3 if mn is not a multiple of 3$ for all positive integers $m$ and $n$ . Given that $f (9) + f (16) - f (24) = 0$ , what is the value of $f (3)$ ?TMUA 2021 · Paper 1 · Question 5 · Save
The function $f$ is given by $f (x) = \frac{c o s x + 3}{7 + 5 c o s x - s i n ^{2} x}$ Find the positive difference between the maximum and the minimum values of $f (x)$ .TMUA 2021 · Paper 1 · Question 6 · Save
The function $f$ is such that $f (0) = 0$ , and $x f (x) 0$ for all non-zero values of $x$ . It is given that $\int_{- 2}^{2} f (x) d x = 4$ and $\int_{- 2}^{2} ∣ f (x) ∣ d x = 8$ Evaluate $\int_{- 2}^{0} f (∣ x ∣) d x$ TMUA 2021 · Paper 1 · Question 7 · Save
The line $y = 2 x + 3$ meets the curve $y = x^{2} + b x + c$ at exactly one point. The line $y = 4 x - 2$ also meets the curve $y = x^{2} + b x + c$ at exactly one point. What is the value of $b - c$ ?TMUA 2021 · Paper 1 · Question 8 · Save
Find the area enclosed by the graph of $∣ x ∣ + ∣ y ∣ = 1$ TMUA 2021 · Paper 1 · Question 9 · Save
Use the trapezium rule with 3 strips to estimate $\int_{1}^{2} 2 lo g_{10} x d x$ TMUA 2021 · Paper 1 · Question 10 · Save
The function $f$ is given by $f (x) = x^{\frac{5}{2}} (x^{2} - x + 1)$ Find the fraction of the interval $0 < x < 1$ for which $f (x)$ is decreasing.TMUA 2021 · Paper 1 · Question 11 · Save
The minimum value of the function $x^{4} - p^{2} x^{2}$ is $- 9$ $p$ is a real number. Find the minimum value of the function $x^{2} - p x + 6$ TMUA 2021 · Paper 1 · Question 12 · Save
The function $f$ is such that, for every integer $n$ $\int_{n}^{n + 1} f (x) d x = n + 1$ Evaluate $\sum_{r = 1}^{8} (\int_{0}^{r} f (x) d x)$ TMUA 2021 · Paper 1 · Question 13 · Save
This question uses radians. Find the number of distinct values of $x$ that satisfy the equation $(x + 1) (3 - x) = 2 (1 - cos (π x))$ TMUA 2021 · Paper 1 · Question 14 · Save
The diagram shows the graph of $y = f (x)$ . [IMAGE_PLACEHOLDER] The graph consists of alternating straight-line segments of gradient $1$ and $- 1$ and continues in this way for all values of $x$ . The function $g$ is defined as $g (x) = \sum_{r = 1}^{10} f (2^{r - 1} x)$ Find the value of $\int_{0}^{1} g (x) d x$ TMUA 2021 · Paper 1 · Question 15 · Save
Consider the expansion of $(a + b x)^{n}$ The third term, in ascending powers of $x$ , is $105 x^{2}$ The fourth term, in ascending powers of $x$ , is $210 x^{3}$ The fourth term, in descending powers of $x$ , is $210 x^{3}$ Find the value of $\frac{a ^{2}}{b}$ TMUA 2021 · Paper 1 · Question 16 · Save
Which of the following sketches shows the graph of $sin (x^{2} + y^{2}) = \frac{1}{2}$ where $x^{2} + y^{2} \leq 8 π$ ?TMUA 2021 · Paper 1 · Question 17 · Save
The curve with equation $x = y^{2} - 6 y + 11$ is rotated $9 0^{\circ}$ clockwise about the point $P$ to give the curve $C$ . $P$ has $x$ -coordinate $- 2$ and $y$ -coordinate $3$ . What is the equation of $C$ ?TMUA 2021 · Paper 1 · Question 18 · Save
The equation $sin^{2} (4^{c o s θ} \times 6 0^{\circ}) = \frac{3}{4}$ has exactly three solutions in the range $0^{\circ} \leq θ \leq x^{\circ}$ What is the range of all possible values of $x$ ?TMUA 2021 · Paper 1 · Question 19 · Save
Find the length of the curve with equation $2 lo g_{10} (x - y) = lo g_{10} (2 - 2 x) + lo g_{10} (y + 5)$ TMUA 2021 · Paper 1 · Question 20 · Save
Find the value of $\int_{1}^{4} (3 x + \frac{4}{x ^{2}}) d x$ TMUA 2021 · Paper 2 · Question 1 · Save
$A (0, 2)$ and $C (4, 0)$ are opposite vertices of the square $A BC D$ . What is the equation of the straight line through $B$ and $D$ ?TMUA 2021 · Paper 2 · Question 2 · Save
A student is chosen at random from a class. Each student is equally likely to be chosen. Which of the following conditions is/are necessary for the probability that the student wears glasses to equal $\frac{4}{15}$ ? I Exactly 11 students in the class do not wear glasses. II The number of students in the class is…TMUA 2021 · Paper 2 · Question 3 · Save
Consider the following claim about positive integers $a$ , $b$ and $c$ : if $a$ is a factor of $b c$ , then $a$ is a factor of $b$ or $a$ is a factor of $c$ Which of the following provide(s) a counterexample to this claim? I $a = 5, b = 10, c = 20$ II $a = 8, b = 4, c = 4$ III $a = 6, b = 7, c = 12$ TMUA 2021 · Paper 2 · Question 4 · Save
On which line is the first error in the following argument?TMUA 2021 · Paper 2 · Question 5 · Save
Consider the following two statements about the polynomial $f (x)$ : $P$ : $f (x) = 0$ for exactly three real values of $x$ $Q$ : $f^{'} (x) = 0$ for exactly two real values of $x$ Which one of the following is correct?TMUA 2021 · Paper 2 · Question 6 · Save
A circle has equation $(x - 9)^{2} + (y + 2)^{2} = 4$ A square has vertices at $(1, 0)$ , $(1, 2)$ , $(- 1, 2)$ and $(- 1, 0)$ . A straight line bisects both the area of the circle and the area of the square. What is the $x$ -coordinate of the point where this straight line meets the $x$ -axis?TMUA 2021 · Paper 2 · Question 7 · Save
Consider the following statement about the polynomial $p (x)$ , where $a$ and $b$ are real numbers with $a < b$ : $()$ There exists a number $c$ with $a < c < b$ such that $p^{'} (c) = 0$ . Which one of the following is true?TMUA 2021 · Paper 2 · Question 8 · Save
Consider the following statements about a polynomial $f (x)$ : I $f (x) = p x^{3} + q x^{2} + r x + s$ , where $p \neq = 0$ . II There is a real number $t$ for which $f^{'} (t) = 0$ . III There are real numbers $u$ and $v$ for which $f (u) f (v) < 0$ . Which of these statements is/are sufficient for the equation $f (x) = 0$ to have a real…TMUA 2021 · Paper 2 · Question 9 · Save
The first seven terms of a sequence of positive integers are: $u_{1} = 15$ $u_{2} = 21$ $u_{3} = 30$ $u_{4} = 37$ $u_{5} = 44$ $u_{6} = 51$ $u_{7} = 59$ Consider the following statement about this sequence: $()$ If $n$ is a prime number, then $u_{n}$ is a multiple of 3 or $u_{n}$ is a multiple of 5. What is the smallest value of $n$ …TMUA 2021 · Paper 2 · Question 10 · Save
A student attempts to solve the following problem, where $a$ and $b$ are non-zero real numbers: Show that if $a^{2} - 4 b^{3} \geq 0$ then there exist real numbers $x$ and $y$ such that $a = x y (x + y)$ and $b = x y$ . Consider the following attempt: $(x - y)^{2} \geq 0 (I)$ …TMUA 2021 · Paper 2 · Question 11 · Save
Which of the following statements about polynomials $f$ and $g$ is/are true? I If $f (x) \geq g (x)$ for all $x \geq 0$ , then $\int_{0}^{x} f (t) d t \geq \int_{0}^{x} g (t) d t$ for all $x \geq 0$ . II If $f (x) \geq g (x)$ for all $x \geq 0$ , then $f^{'} (x) \geq g^{'} (x)$ for all $x \geq 0$ . III If $f^{'} (x) \geq g^{'} (x)$ for all…TMUA 2021 · Paper 2 · Question 12 · Save
A region $R$ in the $(x, y)$ -plane is defined by the simultaneous inequalities $y - x < 3$ $y - x^{2} < 1$ Which of the following statements is/are true for every point in $R$ ? I $- 1 < x < 2$ II $(y - x) (y - x^{2}) < 3$ III $y < 5$ TMUA 2021 · Paper 2 · Question 13 · Save
Consider the following simultaneous equations, where $p$ is a real number: $p 2^{x} + lo g_{2} y = 2$ $2^{x} + lo g_{2} y = 1$ What is the complete range of $p$ for which these simultaneous equations have a real solution $(x, y)$ ?TMUA 2021 · Paper 2 · Question 14 · Save
A circle has equation $x^{2} + a x + y^{2} + b y + c = 0$ where $a$ , $b$ and $c$ are non-zero real constants. Which one of the following is a necessary and sufficient condition for the circle to be tangent to the $y$ -axis?TMUA 2021 · Paper 2 · Question 15 · Save
$p$ and $q$ are real numbers, and the equation $x ∣ x ∣ = p x + q$ has exactly $k$ distinct real solutions for $x$ . Which one of the following is the complete list of possible values for $k$ ?TMUA 2021 · Paper 2 · Question 16 · Save
Consider the following functions defined for $x 1$ : $f (x) = lo g_{2} (lo g_{2} x)$ $g (x) = lo g_{2} (lo g_{2} x)$ Which one of the following is true for all values of $x 1$ ?TMUA 2021 · Paper 2 · Question 17 · Save
A student chooses two distinct real numbers $x$ and $y$ with $0 < x < y < 1$ . The student then attempts to draw a triangle $A BC$ with: $A B = 1$ $sin A = x$ $sin B = y$ Which of the following statements is/are correct? I For some choice of $x$ and $y$ , there is exactly one triangle the student could draw. II…TMUA 2021 · Paper 2 · Question 18 · Save
The angle $θ$ can take any of the values $1^{\circ}, 2^{\circ}, 3^{\circ}, \dots, 35 9^{\circ}, 36 0^{\circ}$ . For how many of these values of $θ$ is it true that $sin θ 1 + sin θ 1 - sin θ + cos θ 1 + cos θ 1 - cos θ = 0$ TMUA 2021 · Paper 2 · Question 19 · Save
A sequence of functions $f_{1}, f_{2}, f_{3}, \dots$ is defined by $f_{1} (x) = ∣ x ∣$ $f_{n + 1} (x) = ∣ f_{n} (x) + x ∣ for n \geq 1$ Find the value of $\int_{- 1}^{1} f_{99} (x) d x$ TMUA 2021 · Paper 2 · Question 20 · Save
Which of the following is an expression for the first derivative with respect to $x$ of $\frac{x ^{3} - 5 x ^{2}}{2 x x}$ TMUA 2020 · Paper 1 · Question 1 · Save
$(2 x + 1)$ and $(x - 2)$ are factors of $2 x^{3} + p x^{2} + q$ What is the value of $2 p + q$ ?TMUA 2020 · Paper 1 · Question 2 · Save
Find the complete set of values of $x$ for which $(x + 4) (x + 3) (1 - x) 0 and (x + 2) (x - 2) < 0$ TMUA 2020 · Paper 1 · Question 3 · Save
The $1^{st}$ , $2^{nd}$ and $3^{rd}$ terms of a geometric progression are also the $1^{st}$ , $4^{th}$ and $6^{th}$ terms, respectively, of an arithmetic progression. The sum to infinity of the geometric progression is $12$ . Find the $1^{st}$ term of the geometric…TMUA 2020 · Paper 1 · Question 4 · Save
The curve $S$ has equation $y = p x^{2} + 6 x - q$ where $p$ and $q$ are constants. $S$ has a line of symmetry at $x = - \frac{1}{4}$ and touches the $x$ -axis at exactly one point. What is the value of $p + 8 q$ ?TMUA 2020 · Paper 1 · Question 5 · Save
Find the maximum value of the function $f (x) = \frac{1}{5 ^{2 x} - 4 ( 5 ^{x} ) + 7}$ TMUA 2020 · Paper 1 · Question 6 · Save
Given that $2^{3 x} = 8^{(y + 3)}$ and $4^{(x + 1)} = \frac{1 6 ^{(y + 1)}}{8 ^{(y + 3)}}$ what is the value of $x + y$ ?TMUA 2020 · Paper 1 · Question 7 · Save
The function $f$ is defined for all real $x$ as $f (x) = (p - x) (x + 2)$ Find the complete set of values of $p$ for which the maximum value of $f (x)$ is less than $4$ .TMUA 2020 · Paper 1 · Question 8 · Save
The quadratic expression $x^{2} - 14 x + 9$ factorises as $(x - α) (x - β)$ , where $α$ and $β$ are positive real numbers. Which quadratic expression can be factorised as $(x - α) (x - β)$ ?TMUA 2020 · Paper 1 · Question 9 · Save
The following sequence of transformations is applied to the curve $y = 4 x^{2}$ 1. Translation by $(3 - 5)$ 2. Reflection in the $x$ -axis 3. Stretch parallel to the $x$ -axis with scale factor $2$ What is the equation of the resulting curve?TMUA 2020 · Paper 1 · Question 10 · Save
The quadratic function shown passes through $(2, 0)$ and $(q, 0)$ , where $q 2$ . [IMAGE_PLACEHOLDER] What is the value of $q$ such that the area of region $R$ equals the area of region $S$ ?TMUA 2020 · Paper 1 · Question 11 · Save
How many real solutions are there to the equation $3 cos x = x$ where $x$ is in radians?TMUA 2020 · Paper 1 · Question 12 · Save
Find the coefficient of $x^{2} y^{4}$ in the expansion of $(1 + x + y^{2})^{7}$ TMUA 2020 · Paper 1 · Question 13 · Save
The area enclosed between the line $y = m x$ and the curve $y = x^{3}$ is $6$ . What is the value of $m$ ?TMUA 2020 · Paper 1 · Question 14 · Save
Find the positive difference between the two real values of $x$ for which $(lo g_{2} x)^{4} + 12 (lo g_{2} (\frac{1}{x}))^{2} - 2^{6} = 0$ TMUA 2020 · Paper 1 · Question 15 · Save
The circle $C_{1}$ has equation $(x + 2)^{2} + (y - 1)^{2} = 3$ The circle $C_{2}$ has equation $(x - 4)^{2} + (y - 1)^{2} = 3$ The straight line $l$ is a tangent to both $C_{1}$ and $C_{2}$ and has positive gradient. The acute angle between $l$ and the $x$ -axis is $θ$ Find the value of $tan θ$ TMUA 2020 · Paper 1 · Question 16 · Save
Find the complete set of values of $m$ in terms of $c$ such that the graphs of $y = m x + c$ and $y = x$ have two points of intersection.TMUA 2020 · Paper 1 · Question 17 · Save
Find the number of solutions and the sum of the solutions of the equation $1 - 2 cos^{2} x = ∣ cos x ∣$ where $0 \leq x \leq 18 0^{\circ}$ TMUA 2020 · Paper 1 · Question 18 · Save
Find the lowest positive integer for which $x^{2} - 52 x - 52$ is positive.TMUA 2020 · Paper 1 · Question 19 · Save
For how many values of $a$ is the equation $(x - a) (x^{2} - x + a) = 0$ satisfied by exactly two distinct values of $x$ ?TMUA 2020 · Paper 1 · Question 20 · Save
Find the complete set of values of $k$ for which the line $y = x - 2$ crosses or touches the curve $y = x^{2} + k x + 2$ TMUA 2020 · Paper 2 · Question 1 · Save
Given that $tan θ = 2$ and $18 0^{\circ} < θ < 36 0^{\circ}$ , find the value of $cos θ$ TMUA 2020 · Paper 2 · Question 2 · Save
A student makes the following claim: For all integers $n$ , the expression $4 (\frac{9 n + 1}{2} - \frac{3 n - 1}{2})$ is divisible by 3. Here is the student's argument: $4 (\frac{9 n + 1}{2} - \frac{3 n - 1}{2}) = 2 (2 (\frac{9 n + 1}{2} - \frac{3 n - 1}{2})) (I)$ …TMUA 2020 · Paper 2 · Question 3 · Save
Consider the following statement: Every positive integer $N$ that is greater than 6 can be written as the sum of two non-prime integers that are greater than 1. Which of the following is/are counterexample(s) to this statement? I $N = 5$ II $N = 7$ III $N = 9$ TMUA 2020 · Paper 2 · Question 4 · Save
Which one of the following shows the graph of $y = \frac{2 ^{x}}{1 + 2 ^{x}}$ (Dotted lines indicate asymptotes.)TMUA 2020 · Paper 2 · Question 5 · Save
The function $f (x)$ is defined for all real values of $x$ . Which of the following conditions on $f (x)$ is/are necessary to ensure that $\int_{- 5}^{0} f (x) d x = \int_{0}^{5} f (x) d x$ Condition I: $f (x) = f (- x) for - 5 \leq x \leq 5$ Condition II:…TMUA 2020 · Paper 2 · Question 6 · Save
Consider the following conditions on a parallelogram $PQRS$ , labelled anticlockwise: I length of $PQ = length of QR$ II The diagonal $PR$ intersects the diagonal $QS$ at right angles III $∠ PQR = ∠ QRS$ Which of these conditions is/are individually sufficient for the…TMUA 2020 · Paper 2 · Question 7 · Save
A student is asked to prove whether the following statement () is true or false: () For all real numbers $a$ and $b$ , $∣ a + b ∣ < ∣ a ∣ + ∣ b ∣$ The student's proof is as follows: Statement () is false. A counterexample is $a = 3, b = 4$ , as $∣3 + 4∣ = 7$ and $∣3∣ + ∣4∣ = 7$ , but $7 < 7$ is false. Which of the following best describes…TMUA 2020 · Paper 2 · Question 8 · Save
A student wishes to evaluate the function $f (x) = x sin x$ , where $x$ is in radians, but has a calculator that only works in degrees. What could the student type into their calculator to correctly evaluate $f (4)$ ?TMUA 2020 · Paper 2 · Question 9 · Save
The real numbers $a, b, c$ and $d$ satisfy both $0 < a + b < c + d$ and $0 < a + c < b + d$ Which of the following inequalities must be true? I $a < d$ II $b < c$ III $a + b + c + d 0$ TMUA 2020 · Paper 2 · Question 10 · Save
A spiral line is drawn as shown. [IMAGE_PLACEHOLDER] This spiral pattern continues indefinitely. Which one of the following points is not on the spiral line?TMUA 2020 · Paper 2 · Question 11 · Save
Which one of A–F correctly completes the following statement? Given that $a < b$ , and $f (x) 0$ for all $x$ with $a < x < b$ , the trapezium rule produces an overestimate for $\int_{a}^{b} f (x) d x$ ...TMUA 2020 · Paper 2 · Question 12 · Save
$f (x)$ is a function for which $\int_{0}^{3} (f (x))^{2} d x + \int_{0}^{3} f (x) d x = \int_{0}^{1} f (x) d x$ Which of the following claims about $f (x)$ is/are necessarily true? I $f (x) \leq 0$ for some $x$ with $1 \leq x \leq 3$ II…TMUA 2020 · Paper 2 · Question 13 · Save
An arithmetic sequence $T$ has first term $a$ and common difference $d$ , where $a$ and $d$ are non-zero integers. Property P is: For some positive integer $m$ , the sum of the first $m$ terms of the sequence is equal to the sum of the first $2 m$ terms of the sequence. For example, when $a = 11$ and $d = - 2$ , the sequence…TMUA 2020 · Paper 2 · Question 14 · Save
Which one of the following is a necessary and sufficient condition for $\sum_{k = 1}^{n} sin (\frac{kπ}{3}) = \frac{3}{2}$ to be true?TMUA 2020 · Paper 2 · Question 15 · Save
The Fundamental Theorem of Calculus (FTC) tells us that for any polynomial f : $\frac{d}{d x} (\int_{0}^{x} f (t) d t) = f (x)$ A student calculates $\frac{d}{d x} (\int_{x}^{2 x} t^{2} d t)$ as follows: (I)…TMUA 2020 · Paper 2 · Question 16 · Save
A set of six distinct integers is split into two sets of three. The first set of three integers has a mean of 10 and a median of 8. The second set of three integers has a mean of 12 and a median of 9. What is the smallest possible range of the set of all six integers?TMUA 2020 · Paper 2 · Question 17 · Save
In this question, $f (x) = a x^{3} + b x^{2} + c x + d$ and $g (x) = p x^{3} + q x^{2} + r x + s$ are cubic polynomials. If $f (x) - g (x) 0$ for every real $x$ , which of the following is/are necessarily true? I $a p$ II if $b = q$ then $c = r$ III $d s$ TMUA 2020 · Paper 2 · Question 18 · Save
Nine people are sitting in the squares of a 3 by 3 grid, one in each square, as shown. Two people are called neighbours if they are sitting in squares that share a side. (People in diagonally adjacent squares, which only have a point in common, are not called neighbours.) [IMAGE_PLACEHOLDER] Each of the nine people in…TMUA 2020 · Paper 2 · Question 19 · Save
$x$ is a real number and f is a function. Given that exactly one of the following statements is true, which one is it?TMUA 2020 · Paper 2 · Question 20 · Save
$f (x)$ is a quadratic function in $x$ . The graph of $y = f (x)$ passes through the point $(1, - 1)$ and has a turning point at $(- 1, 3)$ . Find an expression for $f (x)$ .TMUA 2019 · Paper 1 · Question 1 · Save
Find the complete set of values of the real constant $k$ for which the expression $x^{2} + k x + 2 x + 1 - 2 k$ is positive for all real values of $x$ .TMUA 2019 · Paper 1 · Question 2 · Save
Find the coefficient of $x$ in the expression: $(1 + x)^{0} + (1 + x)^{1} + (1 + x)^{2} + (1 + x)^{3} + \dots + (1 + x)^{79} + (1 + x)^{80}$ TMUA 2019 · Paper 1 · Question 3 · Save
The sequence $x_{n}$ is given by: $x_{1} = 10$ $x_{n + 1} = x_{n} for n \geq 1$ What is the value of $x_{100}$ ? [Note that $a^{b^{c}}$ means $a^{(b^{c})}$ ]TMUA 2019 · Paper 1 · Question 4 · Save
S is a geometric sequence. The sum of the first $6$ terms of S is equal to $9$ times the sum of the first $3$ terms of S. The $7^{th}$ term of S is $360$ . Find the $1^{st}$ term of S.TMUA 2019 · Paper 1 · Question 5 · Save
The circles with equations $(x + 4)^{2} + (y + 1)^{2} = 64 and$ $(x - 8)^{2} + (y - 4)^{2} = r^{2} where r 0$ have exactly one point in common. Find the difference between the two possible values of $r$ .TMUA 2019 · Paper 1 · Question 6 · Save
A curve has equation $y = (2 q - x^{2}) (2 q x + 3)$ The gradient of the curve at $x = - 1$ is a function of $q$ . Find the value of $q$ which minimises the gradient of the curve at $x = - 1$ .TMUA 2019 · Paper 1 · Question 7 · Save
The function $f$ is such that $0 < f (x) < 1$ for $0 \leq x \leq 1$ . The trapezium rule with $n$ equal intervals is used to estimate $\int_{0}^{1} f (x) d x$ and produces an underestimate. Using the same number of equal intervals, for which one of the following does the trapezium rule produce…TMUA 2019 · Paper 1 · Question 8 · Save
$p$ is a positive constant. Find the area enclosed between the curves $y = p x$ and $x = p y$ TMUA 2019 · Paper 1 · Question 9 · Save
Evaluate $\int_{- 1}^{3} ∣ x ∣ (1 - x) d x$ TMUA 2019 · Paper 1 · Question 10 · Save
Find the sum of the real values of $x$ that satisfy the simultaneous equations: $lo g_{3} (x y^{2}) = 1$ $(lo g_{3} x) (lo g_{3} y) = - 3$ TMUA 2019 · Paper 1 · Question 11 · Save
It is given that $\frac{d V}{d t} = \frac{24 π ( t - 1 )}{( 1 + t )} for t \geq 1$ and $V = 7$ when $t = 1$ . Find the value of $V$ when $t = 9$ .TMUA 2019 · Paper 1 · Question 12 · Save
Find the maximum value of $4^{s i n x} - 4 \times 2^{s i n x} + \frac{17}{4}$ for real $x$ .TMUA 2019 · Paper 1 · Question 13 · Save
$x$ satisfies the simultaneous equations $sin 2 x + 3 cos 2 x = - 1$ and $3 sin 2 x - cos 2 x = 3$ where $0^{\circ} \leq x \leq 36 0^{\circ}$ . Find the sum of the possible values of $x$ .TMUA 2019 · Paper 1 · Question 14 · Save
Find the real non-zero solution to the equation $\frac{2 ^{(9^{x})}}{8 ^{(3^{x})}} = \frac{1}{4}$ TMUA 2019 · Paper 1 · Question 15 · Save
Given that $2 \int_{0}^{1} f (x) d x + 5 \int_{1}^{2} f (x) d x = 14$ and $\int_{0}^{1} f (x + 1) d x = 6$ find the value of $\int_{0}^{2} f (x) d x$ TMUA 2019 · Paper 1 · Question 16 · Save
Find the fraction of the interval $0 \leq θ \leq π$ for which the inequality $(sin (2 θ) - \frac{1}{2}) (sin θ - cos θ) \geq 0$ is satisfied.TMUA 2019 · Paper 1 · Question 17 · Save
Find the shortest distance between the curve $y = x^{2} + 4$ and the line $y = 2 x - 2$ .TMUA 2019 · Paper 1 · Question 18 · Save
Find the value of $\sum_{k = 0}^{90} sin (10 + 90 k)^{\circ}$ TMUA 2019 · Paper 1 · Question 19 · Save
What is the complete range of values of $k$ for which the curves with equations $y = x^{3} - 12 x$ and $y = k - (x - 2)^{2}$ intersect at three distinct points, of which exactly two have positive $x$ -coordinates?TMUA 2019 · Paper 1 · Question 20 · Save
Find the coefficient of the $x^{4}$ term in the expansion of $x^{2} (2 x + \frac{1}{x})^{6}$ TMUA 2019 · Paper 2 · Question 1 · Save
$(2 x + 1)$ and $(x - 2)$ are factors of $2 x^{3} + p x^{2} + q$ . What is the value of $2 p + q$ ?TMUA 2019 · Paper 2 · Question 2 · Save
$a$ , $b$ and $c$ are real numbers. Given that $ab = a c$ , which of the following statements must be true? I $a = 0$ II $b = 0$ or $c = 0$ III $b = c$ TMUA 2019 · Paper 2 · Question 3 · Save
Consider the following conjecture: If $N$ is a positive integer that consists of the digit $1$ followed by an odd number of $0$ digits and then a final digit $1$ , then $N$ is a prime number. Here are three numbers: I $N = 101$ (which is a prime number) II $N = 1001$ (which equals $7 \times 11 \times 13$ )…TMUA 2019 · Paper 2 · Question 4 · Save
Consider the following statement about the positive integers $a$ , $b$ and $n$ : $()$ : $ab$ is divisible by $n$ The condition 'either $a$ or $b$ is divisible by $n$ ' is:TMUA 2019 · Paper 2 · Question 5 · Save
A student attempts to solve the equation $cos x + sin x tan x = 2 sin x - 1$ in the range $0 \leq x \leq 2 π$ . The student's attempt is as follows: $cos x + sin x tan x = 2 sin x - 1$ So $cos x - sin x + sin x tan x - sin x = - 1$ (I) So $(sin x - cos x) (tan x - 1) = - 1$ …TMUA 2019 · Paper 2 · Question 6 · Save
For which one of the following statements can the fact that $1 2^{2} + 1 6^{2} = 2 0^{2}$ be used to produce a counterexample?TMUA 2019 · Paper 2 · Question 7 · Save
$a$ , $b$ and $c$ are real numbers with $a < b < c < 0$ Which of the following statements must be true? I $a c < ab < a^{2}$ II $b (c + a) 0$ III $\frac{c}{b} \frac{a}{b}$ TMUA 2019 · Paper 2 · Question 8 · Save
A large circular table has $40$ chairs round it. What is the smallest number of people who can be sitting at the table already such that the next person to sit down must sit next to someone?TMUA 2019 · Paper 2 · Question 9 · Save
$PQRS$ is a quadrilateral, labelled anticlockwise. Which one of the following is a necessary but <u>not</u> sufficient condition for $PQRS$ to be a parallelogram?TMUA 2019 · Paper 2 · Question 10 · Save
An arithmetic series has $n$ terms, all of which are integers. The sum of the series is $20$ . Which of the following statements must be true? I The first term of the series is even. II $n$ is even. III The common difference is even.TMUA 2019 · Paper 2 · Question 11 · Save
Most students in a large college study Mathematics. A teacher chooses three different students at random, one after the other. Consider these three probabilities: $R = P (At least one of the students chosen studies Mathematics)$ $S = P (The second student chosen studies Mathematics)$ …TMUA 2019 · Paper 2 · Question 12 · Save
A student approximates the integral $\int_{a}^{b} sin^{2} x d x$ using the trapezium rule with $4$ strips. The resulting approximation is an overestimate. Which of the following is/are necessarily true? I If the student approximates $\int_{- b}^{- a} sin^{2} x d x$ in the same way, the result will be…TMUA 2019 · Paper 2 · Question 13 · Save
Consider the following statements about the polynomial $p (x)$ , where $a < b$ : I $p (a) \leq p (b)$ II $p^{'} (a) \leq p^{'} (b)$ III $p^{''} (a) \leq p^{''} (b)$ Which of these statements is a necessary condition for $p (x)$ to be increasing for $a \leq x \leq b$ ?TMUA 2019 · Paper 2 · Question 14 · Save
The numbers $a$ , $b$ and $c$ are each greater than $1$ . The following logarithms are all to the same base: $lo g (a b^{2} c) = 7$ $lo g (a^{2} b c^{2}) = 11$ $lo g (a^{2} b^{2} c^{3}) = 15$ What is this base?TMUA 2019 · Paper 2 · Question 15 · Save
The graph of the quadratic $y = p x^{2} + q x + p$ where $p 0$ , intersects the $x$ -axis at two distinct points. In which one of the following graphs does the shaded region show the complete set of possible values that $p$ and $q$ could take?TMUA 2019 · Paper 2 · Question 16 · Save
A multiple-choice test question offered the following four options relating to a certain statement: Given that exactly one of these options was correct, which one was it?TMUA 2019 · Paper 2 · Question 17 · Save
Consider the following inequality: $()$ : $a ∣ x ∣ + 1 \leq ∣ x - 2∣$ where $a$ is a real constant. Which one of the following describes the complete set of values of $a$ such that $()$ is true for all real $x$ ?TMUA 2019 · Paper 2 · Question 18 · Save
Find the value of the expression $8 - 42 + 1 + 9 - 122 + 8$ TMUA 2019 · Paper 2 · Question 19 · Save
When the graph of the function $y = f (x)$ , defined on the real numbers, is reflected in the $y$ -axis and then translated by $2$ units in the negative $x$ -direction, the result is the graph of the function $y = g (x)$ . When the graph of the same function $y = f (x)$ is translated by $2$ units in the…TMUA 2019 · Paper 2 · Question 20 · Save
Find the value of $\int_{1}^{4} \frac{3 - 2 x}{x x} d x$ TMUA 2018 · Paper 1 · Question 1 · Save
An arithmetic progression has first term $a$ and common difference $d$ . The sum of the first $5$ terms is equal to the sum of the first $8$ terms. Which one of the following expresses the relationship between $a$ and $d$ ?TMUA 2018 · Paper 1 · Question 2 · Save
Find the shortest distance between the two circles with equations: $(x + 2)^{2} + (y - 3)^{2} = 18$ $(x - 7)^{2} + (y + 6)^{2} = 2$ TMUA 2018 · Paper 1 · Question 3 · Save
Consider the simultaneous equations $3 x^{2} + 2 x y = 4$ $x + y = a$ where $a$ is a real constant. Find the complete set of values of $a$ for which the equations have two distinct real solutions for $x$ .TMUA 2018 · Paper 1 · Question 4 · Save
The function $f$ is defined by $f (x) = x^{3} + a x^{2} + b x + c$ . $a, b$ and $c$ take the values $1, 2$ and $3$ with no two of them being equal and not necessarily in this order. The remainder when $f (x)$ is divided by $(x + 2)$ is $R$ . The remainder when $f (x)$ is divided by $(x + 3)$ is $S$ . What is the largest possible…TMUA 2018 · Paper 1 · Question 5 · Save
Find the number of solutions of the equation $x sin 2 x = cos 2 x$ with $0 \leq x \leq 2 π$ .TMUA 2018 · Paper 1 · Question 6 · Save
The non-zero constant $k$ is chosen so that the coefficients of $x^{6}$ in the expansions of $(1 + k x^{2})^{7}$ and $(k + x)^{10}$ are equal. What is the value of $k$ ?TMUA 2018 · Paper 1 · Question 7 · Save
The sum to infinity of a geometric progression is $6$ . The sum to infinity of the squares of each term in the progression is $12$ . Find the sum to infinity of the cubes of each term in the progression.TMUA 2018 · Paper 1 · Question 8 · Save
Find the complete set of values of the constant $c$ for which the cubic equation $2 x^{3} - 3 x^{2} - 12 x + c = 0$ has three distinct real solutions.TMUA 2018 · Paper 1 · Question 9 · Save
$x$ and $y$ satisfy $∣2 - x ∣ \leq 6$ and $∣ y + 2∣ \leq 4$ . What is the greatest possible value of $∣ x y ∣$ ?TMUA 2018 · Paper 1 · Question 10 · Save
The line $y = m x + 5$ , where $m 0$ , is normal to the curve $y = 10 - x^{2}$ at the point $(p, q)$ . What is the value of $p$ ?TMUA 2018 · Paper 1 · Question 11 · Save
A curve has equation $y = f (x)$ , where $f (x) = x (x - p) (x - q) (r - x)$ with $0 < p < q < r$ . You are given that: $\int_{0}^{r} f (x) d x = 0$ $\int_{0}^{q} f (x) d x = - 2$ $\int_{p}^{r} f (x) d x = - 3$ What is the total area enclosed by the curve and the $x$ -axis for $0 \leq x \leq r$ ?TMUA 2018 · Paper 1 · Question 12 · Save
The line $y = m x + 4$ passes through the points $(3, lo g_{2} p)$ and $(lo g_{2} p, 4)$ . What are the possible values of $p$ ?TMUA 2018 · Paper 1 · Question 14 · Save
Find the sum of the real solutions of the equation: $3^{x} - (3)^{x + 4} + 20 = 0$ TMUA 2018 · Paper 1 · Question 15 · Save
The curve $C$ has equation $y = x^{2} + b x + 2$ , where $b \geq 0$ . Find the value of $b$ that minimises the distance between the origin and the stationary point of the curve $C$ .TMUA 2018 · Paper 1 · Question 16 · Save
There are two sets of data: the mean of the first set is $15$ , and the mean of the second set is $20$ . One of the pieces of data from the first set is exchanged with one of the pieces of data from the second set. As a result, the mean of the first set of data increases from $15$ to $16$ , and the mean of the second set…TMUA 2018 · Paper 1 · Question 17 · Save
What is the smallest positive value of $a$ for which the line $x = a$ is a line of symmetry of the graph of $y = sin (2 x - \frac{4 π}{3})$ ?TMUA 2018 · Paper 1 · Question 18 · Save
A triangle $A BC$ is to be drawn with $A B = 10 cm$ , $BC = 7 cm$ and the angle at $A$ equal to $θ$ , where $θ$ is a certain specified angle. Of the two possible triangles that could be drawn, the larger triangle has three times the area of the smaller one. What is the value of $cos θ$ ?TMUA 2018 · Paper 1 · Question 19 · Save
Find the value of $sin^{2} 0^{\circ} + sin^{2} 1^{\circ} + sin^{2} 2^{\circ} + sin^{2} 3^{\circ} + \dots + sin^{2} 8 7^{\circ} + sin^{2} 8 8^{\circ} + sin^{2} 8 9^{\circ} + sin^{2} 9 0^{\circ}$ TMUA 2018 · Paper 1 · Question 20 · Save
The function f is given, for $x 0$ , by $f (x) = \frac{x ^{3} - 4 x}{2 x}$ Find the value of $f^{'} (4)$ .TMUA 2018 · Paper 2 · Question 1 · Save
Find the value of the constant term in the expansion of $(x^{6} - \frac{1}{x ^{2}})^{12}$ TMUA 2018 · Paper 2 · Question 2 · Save
Consider the following statement: A car journey consists of two parts. In the first part, the average speed is $u km/h$ . In the second part, the average speed is $v km/h$ . Hence the average speed for the whole journey is $\frac{1}{2} (u + v) km/h$ . Which of the following examples of car journeys…TMUA 2018 · Paper 2 · Question 3 · Save
The non-zero real number $c$ is such that the equation $cos x = c$ has two solutions for $0 < x < \frac{3}{2} π$ . How many solutions of the equation $cos^{2} 2 x = c^{2}$ are there in the range $0 < x < \frac{3}{2} π$ ?TMUA 2018 · Paper 2 · Question 4 · Save
The two diagonals of the quadrilateral $Q$ are perpendicular. Consider the following statements: I One of the diagonals of $Q$ is a line of symmetry of $Q$ . II The midpoints of the sides of $Q$ are the vertices of a square. Which of these statements is/are necessarily true for the quadrilateral $Q$ ?TMUA 2018 · Paper 2 · Question 5 · Save
Which one of the following functions provides a counterexample to the statement: if $f^{'} (x) 0$ for all real $x$ , then $f (x) 0$ for all real $x$ .TMUA 2018 · Paper 2 · Question 6 · Save
Sequence 1 is an arithmetic progression with first term $11$ and common difference $3$ . Sequence 2 is an arithmetic progression with first term $2$ and common difference $5$ . Some numbers that appear in Sequence 1 also appear in Sequence 2. Let $N$ be the 20th such number. What is the remainder when $N$ is divided by…TMUA 2018 · Paper 2 · Question 7 · Save
The diagram shows an example of a mountain profile. [IMAGE_PLACEHOLDER] This consists of upstrokes which go upwards from left to right, and downstrokes which go downwards from left to right. The example shown has six upstrokes and six downstrokes. The horizontal line at the bottom is known as sea level. A mountain…TMUA 2018 · Paper 2 · Question 8 · Save
Consider the following attempt to solve the equation $4 x 2 x - 1 = 10 x - 5$ : $4 x 2 x - 1 = 10 x - 5$ $↓$ (I) $4 x 2 x - 1 = 5 (2 x - 1)$ $↓$ (II) $16 x^{2} (2 x - 1) = 25 (2 x - 1)^{2}$ $↓$ (III) $16 x^{2} = 25 (2 x - 1)$ $↓$ (IV) $16 x^{2} - 50 x + 25 = 0$ $↓$ (V)…TMUA 2018 · Paper 2 · Question 9 · Save
The function $f (x)$ is defined for all real numbers. Consider the following three conditions, where $a$ is a real constant: I $f (a - x) = f (a + x)$ for all real $x$ . II $f (2 a - x) = f (x)$ for all real $x$ . III $f (a - x) = f (x)$ for all real $x$ . Which of these conditions is/are necessary and sufficient for the graph…TMUA 2018 · Paper 2 · Question 10 · Save
Consider the equation $2^{x} = m x + c$ , where $m$ and $c$ are real constants. Which of the following statements is/are true? I The equation has a negative real solution only if $c 1$ . II The equation has two distinct real solutions if $c 1$ . III The equation has two distinct positive real solutions if and only if…TMUA 2018 · Paper 2 · Question 11 · Save
Consider the following statement: For any positive integer $N$ there is a positive integer $K$ such that $N (K m + 1) - 1$ is not prime for any positive integer $m$ . Which one of the following is the negation of this statement?TMUA 2018 · Paper 2 · Question 12 · Save
The following is an attempted proof of the conjecture: if $tan θ 0$ , then $sin θ + cos θ 1$ . Suppose $tan θ 0$ , so in particular $cos θ \neq = 0$ . Since $tan θ = \frac{s i n θ}{c o s θ}$ , then $sin θ cos θ = tan θ cos^{2} θ 0$ . (I) It follows that…TMUA 2018 · Paper 2 · Question 13 · Save
In the triangle $PQR$ , $PR = 2$ , $QR = p$ and $∠ RPQ = 3 0^{\circ}$ . What is the set of all the values of $p$ for which this information uniquely determines the length of $PQ$ ?TMUA 2018 · Paper 2 · Question 14 · Save
It is given that $f (x) = x^{3} + 3 q x^{2} + 2$ , where $q$ is a real constant. The equation $f (x) = 0$ has $3$ distinct real roots. Which of the following statements is/are necessarily true? I The equation $f (x) + 1 = 0$ has $3$ distinct real roots. II The equation $f (x + 1) = 0$ has $3$ distinct real roots. III The…TMUA 2018 · Paper 2 · Question 15 · Save
In this question, $x_{1}, x_{2}, x_{3}, \dots$ is an arithmetic progression, all of whose terms are integers. Let $n$ be a positive integer. If the median of the first $n$ terms of the sequence is an integer, which of the following three statements must be true? I The median of the first $n + 2$ terms is an integer. II The…TMUA 2018 · Paper 2 · Question 16 · Save
A positive integer is called a squaresum if and only if it can be written as the sum of the squares of two integers. For example, $61$ and $9$ are both squaresums since $61 = 5^{2} + 6^{2}$ and $9 = 3^{2} + 0^{2}$ . A prime number is called awkward if and only if it has a remainder of $3$ when divided by $4$ . For example, $23$ …TMUA 2018 · Paper 2 · Question 17 · Save
$f (x)$ is a polynomial function defined for all real $x$ . Which of the following is a necessary condition for the inequality $\frac{f ( a ) + f ( b )}{2} \geq f (\frac{a + b}{2})$ to be true for all real numbers $a$ and $b$ with $a < b$ ?TMUA 2018 · Paper 2 · Question 18 · Save
Three real numbers $x$ , $y$ and $z$ satisfy $x yz 1$ . Which one of the following statements must be true?TMUA 2018 · Paper 2 · Question 19 · Save
It is given that the equation $x + p + x = p$ has at least one real solution for $x$ , where $p$ is a real constant. What is the complete set of possible values for $p$ ?TMUA 2018 · Paper 2 · Question 20 · Save
Given that $\frac{d y}{d x} = 3 x^{2} - \frac{2 - 3 x}{x ^{3}}, x \neq = 0$ and $y = 5$ when $x = 1$ , find $y$ in terms of $x$ .TMUA 2017 · Paper 1 · Question 1 · Save
The function $f$ is given by $f (x) = (\frac{2}{x} - \frac{1}{2 x ^{2}})^{2} (x \neq = 0)$ What is the value of $f^{''} (1)$ ?TMUA 2017 · Paper 1 · Question 2 · Save
A line $l$ has equation $y = 6 - 2 x$ A second line is perpendicular to $l$ and passes through the point $(- 6, 0)$ . Find the area of the region enclosed by the two lines and the $x$ -axis.TMUA 2017 · Paper 1 · Question 3 · Save
When $(3 x^{2} + 8 x - 3)$ is multiplied by $(p x - 1)$ and the resulting product is divided by $(x + 1)$ , the remainder is $24$ . What is the value of $p$ ?TMUA 2017 · Paper 1 · Question 4 · Save
$S$ is the complete set of values of $x$ which satisfy both the inequalities $x^{2} - 8 x + 12 < 0 and 2 x + 19$ The set $S$ can also be represented as a single inequality. Which one of the following single inequalities represents the set $S$ ?TMUA 2017 · Paper 1 · Question 5 · Save
A tangent to the circle $x^{2} + y^{2} = 144$ passes through the point $(20, 0)$ and crosses the positive $y$ -axis. What is the value of $y$ at the point where the tangent meets the $y$ -axis?TMUA 2017 · Paper 1 · Question 6 · Save
The first three terms of an arithmetic progression are $p, q$ and $p^{2}$ respectively, where $p < 0$ The first three terms of a geometric progression are $p, p^{2}$ and $q$ respectively. Find the sum of the first $10$ terms of the arithmetic progression.TMUA 2017 · Paper 1 · Question 7 · Save
Find the complete set of values of $x$ , with $0 \leq x \leq π$ , for which $(1 - 2 sin x) cos x \geq 0$ TMUA 2017 · Paper 1 · Question 8 · Save
A circle has equation $x^{2} + y^{2} - 18 x - 22 y + 178 = 0$ 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?TMUA 2017 · Paper 1 · Question 9 · Save
A curve $C$ has equation $y = f (x)$ where $f (x) = p^{3} - 6 p^{2} x + 3 p x^{2} - x^{3}$ and $p$ is real. The gradient of the normal to the curve $C$ at the point where $x = - 1$ is $M$ . What is the greatest possible value of $M$ as $p$ varies?TMUA 2017 · Paper 1 · Question 10 · Save
The sequence $x_{n}$ is defined by the rules $x_{1} = 7$ $x_{n + 1} = \frac{23 x _{n} - 53}{5 x _{n} + 1}$ The first three terms in the sequence are $7, 3, 1$ What is the value of $x_{100}$ ?TMUA 2017 · Paper 1 · Question 11 · Save
The polynomial function $f (x)$ is such that $f (x) 0$ for all values of $x$ . Given $\int_{2}^{4} f (x) d x = A$ , which one of the following statements must be correct?TMUA 2017 · Paper 1 · Question 12 · Save
In the expansion of $(a + b x)^{5}$ the coefficient of $x^{4}$ is $8$ times the coefficient of $x^{2}$ . Given that $a$ and $b$ are non-zero positive integers, what is the smallest possible value of $a + b$ ?TMUA 2017 · Paper 1 · Question 13 · Save
The solution of the simultaneous equations $2^{x} + 3 \times 2^{y} = 3$ $2^{2 x} - 9 \times 2^{2 y} = 6$ is $x = p, y = q$ . Find the value of $p - q$ TMUA 2017 · Paper 1 · Question 14 · Save
It is given that $f (x) = - 2 x^{2} + 10$ Consider the following three curves: (1) $y = f (x)$ (2) $y = f (x + 1)$ (3) the curve $y = f (x + 1)$ reflected in the line $y = 6$ The trapezium rule is used to estimate the area under each of these three curves between $x = 0$ and $x = 1$ . State whether the trapezium rule gives an…TMUA 2017 · Paper 1 · Question 15 · Save
The functions $f$ and $g$ are given by $f (x) = 3 x^{2} + 12 x + 4$ and $g (x) = x^{3} + 6 x^{2} + 9 x - 8$ . What is the complete set of values of $x$ for which one of the functions is increasing and the other decreasing?TMUA 2017 · Paper 1 · Question 16 · Save
The two functions $F (n)$ and $G (n)$ are defined as follows for positive integers $n$ : $F (n) = \frac{1}{n} \int_{0}^{n} (n - x) d x$ $G (n) = \sum_{r = 1}^{n} F (r)$ What is the smallest positive integer $n$ such that $G (n) 150$ ?TMUA 2017 · Paper 1 · Question 17 · Save
The graph of $y = lo g_{10} x$ is translated in the positive $y$ -direction by $2$ units. This translation is equivalent to a stretch of factor $k$ parallel to the $x$ -axis. What is the value of $k$ ?TMUA 2017 · Paper 1 · Question 18 · Save
The set of solutions to the inequality $x^{2} + b x + c < 0$ is the interval $p < x < q$ where $b, c, p$ and $q$ are real constants with $c < 0$ . In terms of $p, q$ and $c$ , what is the set of solutions to the inequality $x^{2} + b c x + c^{3} < 0$ ?TMUA 2017 · Paper 1 · Question 19 · Save
The lengths of the sides $QR, RP$ and $PQ$ in triangle $PQR$ are $a, a + d$ and $a + 2 d$ respectively, where $a$ and $d$ are positive and such that $3 d 2 a$ . What is the full range, in degrees, of possible values for angle $PRQ$ ?TMUA 2017 · Paper 1 · Question 20 · Save
Given that $y = \frac{( 1 - 3 x ) ^{2}}{2 x ^{\frac{3}{2}}}$ , which one of the following is a correct expression for $\frac{d y}{d x}$ ?TMUA 2017 · Paper 2 · Question 1 · Save
$PQRS$ is a rectangle. The coordinates of $P$ and $Q$ are $(0, 6)$ and $(1, 8)$ respectively. The perpendicular to $PQ$ at $Q$ meets the $x$ -axis at $R$ . What is the area of $PQRS$ ?TMUA 2017 · Paper 2 · Question 2 · Save
The first term of a geometric progression is $23$ and the fourth term is $\frac{9}{4}$ What is the sum to infinity of this geometric progression?TMUA 2017 · Paper 2 · Question 3 · Save
The following question appeared in an examination: Given that $tan x = 3$ , find the possible values of $sin 2 x$ . A student gave the following answer: $tan x = 3$ so $x = 6 0^{\circ}$ and $2 x = 12 0^{\circ}$ , therefore $sin 2 x = \frac{3}{2}$ . Which one of the following statements is correct?TMUA 2017 · Paper 2 · Question 4 · Save
Consider the following three statements: 1 $10 p^{2} + 1$ and $10 p^{2} - 1$ are both prime when $p$ is an odd prime. 2 Every prime greater than $5$ is of the form $6 n + 1$ for some integer $n$ . 3 No multiple of $7$ greater than $7$ is prime. The result $91 = 7 \times 13$ can be used to provide a counterexample to which of…TMUA 2017 · Paper 2 · Question 5 · Save
A sequence $u_{0}, u_{1}, u_{2}, \dots$ is defined as follows: $u_{0} = 1$ $u_{n} = \int_{0}^{1} 4 x u_{n - 1} d x for n \geq 1$ What is the value of $u_{1000}$ ?TMUA 2017 · Paper 2 · Question 6 · Save
The graphs of two functions are shown here: $∙ y = a^{x}$ is shown with a solid line, where $a$ is a positive real number $∙ y = f (x)$ is shown with a dashed line [IMAGE_PLACEHOLDER] Which of the following statements (1, 2, 3, 4) could be true? 1 $f (x) = b^{x}$ for some $ba$ 2 $f (x) = b^{x}$ for…TMUA 2017 · Paper 2 · Question 7 · Save
Which one of the following numbers is smallest in value?TMUA 2017 · Paper 2 · Question 8 · Save
Consider the following attempt to prove this true theorem: Theorem: $a^{3} + b^{3} = c^{3}$ has no solutions with $a$ , $b$ and $c$ positive integers. Attempted proof: Suppose that there are positive integers $a$ , $b$ and $c$ such that $a^{3} + b^{3} = c^{3}$ . I We have $a^{3} = c^{3} - b^{3}$ . II Hence $a^{3} = (c - b) (c^{2} + c b + b^{2})$ .…TMUA 2017 · Paper 2 · Question 9 · Save
$f (x)$ is a function defined for all real values of $x$ . Which one of the following is a sufficient condition for $\int_{1}^{3} f (x) d x = 0$ ?TMUA 2017 · Paper 2 · Question 10 · Save
The function $f (x)$ is increasing and $f (0) = 0$ . The positive constants $a$ and $b$ are such that $a < b$ . The area of the region enclosed by the curve $y = f (x)$ , the $x$ -axis and the lines $x = a$ and $x = b$ is denoted by $R$ . The function $g (x)$ is defined by $g (x) = f (x) + 2 f (b)$ . Which of the following is an…TMUA 2017 · Paper 2 · Question 11 · Save
The diagram shows the graphs of $y = sin 2 x$ and $y = cos 2 x$ for $- \frac{π}{2} < x < \frac{π}{2}$ [IMAGE_PLACEHOLDER] Which one of the following is not true?TMUA 2017 · Paper 2 · Question 12 · Save
The positive real numbers $a \times 1 0^{- 3}$ , $b \times 1 0^{- 2}$ and $c \times 1 0^{- 1}$ are each in standard form, and $(a \times 1 0^{- 3}) + (b \times 1 0^{- 2}) = (c \times 1 0^{- 1}) .$ Which of the following statements (I, II, III, IV) must be true? I $a 9$ II $b 9$ III $a < c$ IV $b < c$ TMUA 2017 · Paper 2 · Question 13 · Save
The diagram below shows the graph of $y = x^{2} - 2 b x + c$ . The vertex of this graph is at the point $P$ . [IMAGE_PLACEHOLDER_1] Which one of the following could be the graph of $y = x^{2} - 2 B x + c$ , where $B b$ ?TMUA 2017 · Paper 2 · Question 14 · Save
The function $f$ is defined on the positive integers as follows: $f (1) = 5$ , and for $n \geq 1$ : ${f (n + 1) = 3 f (n) + 1 f (n + 1) = \frac{1}{2} f (n) if f (n) is odd if f (n) is even$ The function $g$ is defined on the positive integers as follows:…TMUA 2017 · Paper 2 · Question 15 · Save
Consider the following statement: $()$ If $f (x)$ is an integer for every integer $x$ , then $f^{'} (x)$ is an integer for every integer $x$ . Which one of the following is a counterexample to $()$ ?TMUA 2017 · Paper 2 · Question 16 · Save
A set $S$ of whole numbers is called stapled if and only if for every whole number $a$ which is in $S$ there exists a prime factor of $a$ which divides at least one other number in $S$ . Let $T$ be a set of whole numbers. Which of the following is true if and only if $T$ is not stapled?TMUA 2017 · Paper 2 · Question 17 · Save
Consider the following problem: Solve the inequality $(\frac{1}{4})^{n} < (\frac{1}{32})^{10}$ , where $n$ is a positive integer. A student produces the following argument: $(\frac{1}{4})^{n} < (\frac{1}{32})^{10}$ $↓$ (I)…TMUA 2017 · Paper 2 · Question 18 · Save
Which one of the following is a sufficient condition for the equation $x^{3} - 3 x^{2} + a = 0$ , where $a$ is a constant, to have exactly one real root?TMUA 2017 · Paper 2 · Question 19 · Save
I have forgotten my 5-character computer password, but I know that it consists of the letters `a`, `b`, `c`, `d`, `e` in some order. When I enter a potential password into the computer, it tells me exactly how many of the letters are in the correct position. When I enter `abcde`, it tells me that none of the letters…TMUA 2017 · Paper 2 · Question 20 · Save
It is given that the expansion of $(a x + b)^{3}$ is $8 x^{3} - p x^{2} + 18 x - 33$ , where $a, b$ and $p$ are real constants. What is the value of $p$ ?TMUA Practice · Paper 1 · Question 1 · Save
The expression $3 x^{3} + 13 x^{2} + 8 x + a$ , where $a$ is a constant, has $(x + 2)$ as a factor. Which one of the following is a complete factorisation of the expression?TMUA Practice · Paper 1 · Question 2 · Save
A line is drawn normal to the curve $y = \frac{2}{x ^{2}}$ at the point on the curve where $x = 1$ . This line cuts the $x$ -axis at $P$ and the $y$ -axis at $Q$ . The length of $PQ$ isTMUA Practice · Paper 1 · Question 3 · Save
The sequence $a_{n}$ is defined by the rule: $a_{n} = (- 1)^{n} - (- 1)^{n - 1} + (- 1)^{n + 2}$ for $n \geq 1$ . Find the value of $\sum_{n = 1}^{39} a_{n}$ TMUA Practice · Paper 1 · Question 4 · Save
What is the total area enclosed between the curve $y = x^{2} - 1$ , the $x$ -axis and the lines $x = - 2$ and $x = 2$ ?TMUA Practice · Paper 1 · Question 5 · Save
P, Q, and R are each mixtures of red and white paint. The percentage by volume of red paint in P is 30%. The percentage by volume of red paint in Q is 20%. The mixtures P, Q, and R are combined in the proportion 12 : 5 : 3 respectively. If the resulting mixture contains 25% by volume of red paint, what percentage by…TMUA Practice · Paper 1 · Question 6 · Save
60% of a sports club's members are women and the remainder are men. This sports club offers the opportunity to play tennis or cricket. Every member plays exactly one of the two sports. $\frac{2}{5}$ of the male members of the club play cricket; $\frac{2}{3}$ of the cricketing members of the club are women. What is the…TMUA Practice · Paper 1 · Question 7 · Save
Find the maximum angle $x$ in the range $0^{\circ} \leq x \leq 36 0^{\circ}$ which satisfies the equation $cos^{2} (2 x) + 3 sin (2 x) - \frac{7}{4} = 0$ TMUA Practice · Paper 1 · Question 8 · Save
The line segment joining the points $(3, 3)$ and $(7, 5)$ is a diameter of a circle. This circle is translated by 3 units in the negative $x$ -direction, then reflected in the $x$ -axis, and then enlarged by a scale factor of 4 about the centre of the resulting circle. The equation of the final circle isTMUA Practice · Paper 1 · Question 9 · Save
How many solutions does the equation $x tan x = 1$ have in the interval $- 2 π \leq x \leq 2 π$ ?TMUA Practice · Paper 1 · Question 10 · Save
The real roots of the equation $4^{2 x} + 12 = 2^{2 x + 3}$ are $p$ and $q$ , where $pq$ . The value of $p - q$ can be expressed asTMUA Practice · Paper 1 · Question 11 · Save
A right circular cylinder is contained within a sphere of radius 5 cm in such a way that the whole of the circumferences of both ends of the cylinder are in contact with the sphere. The diagram shows a planar cross section through the centre of the sphere and cylinder. [IMAGE_PLACEHOLDER] Find, in cubic centimetres,…TMUA Practice · Paper 1 · Question 12 · Save
How many real roots does the equation $3 x^{5} - 10 x^{3} - 120 x + 30 = 0$ have?TMUA Practice · Paper 1 · Question 13 · Save
The terms of an infinite series S are formed by adding together the corresponding terms in two infinite geometric series, T and U. The first term of T and the first term of U are each 4. In order, the first three terms of the combined series S are 8, 3, and $\frac{5}{4}$ . What is the sum to infinity of S?TMUA Practice · Paper 1 · Question 14 · Save
The least possible value of the gradient of the curve $y = (2 x + a) (x - 2 a)^{2}$ at the point where $x = 1$ , as $a$ varies, isTMUA Practice · Paper 1 · Question 15 · Save
Given the simultaneous equations $lo g_{10} 2 + lo g_{10} (y - 1) = 2 lo g_{10} x$ $lo g_{10} (y + 3 - 3 x) = 0$ the values of $y$ areTMUA Practice · Paper 1 · Question 16 · Save
It is given that $y = (1 + 2 cos x) cos 2 x for 0 < x < π$ The complete set of values of $x$ for which $y$ is negative isTMUA Practice · Paper 1 · Question 17 · Save
The function $\frac{1 - x}{3 x ^{2}}$ is defined for all $x \neq = 0$ . The complete set of values of $x$ for which the function is decreasing isTMUA Practice · Paper 1 · Question 18 · Save
The coefficient of $x^{3}$ in the expansion of $(1 + 2 x + 3 x^{2})^{6}$ is equal to twice the coefficient of $x^{4}$ in the expansion of $(1 - a x^{2})^{5}$ . Find all possible values of the constant $a$ .TMUA Practice · Paper 1 · Question 19 · Save
The diagram shows a square-based pyramid with base $PQRS$ and vertex $O$ . All the edges of the pyramid are of length 20 metres. [IMAGE_PLACEHOLDER] Find the shortest distance, in metres, along the outer surface of the pyramid from $P$ to the midpoint of $OR$ .TMUA Practice · Paper 1 · Question 20 · Save
Find the value of $\int_{1}^{2} (x^{2} - \frac{4}{x ^{2}})^{2} d x$ TMUA Practice · Paper 2 · Question 1 · Save
$f (x) = \frac{( x ^{2} + 5 ) ( 2 x )}{4 x ^{3}}, x 0$ Which one of the following is equal to $f^{'} (x)$ ?TMUA Practice · Paper 2 · Question 2 · Save

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