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

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