Daniel J Smith Custom TMUA alpha
20 questions · 75 min · Difficulty 36 · by Daniel J Smith
Read the questions freely. Sign in to answer them, start a timed paper and save your progress.
Questions in this paper
- Question 1PQRS is a quadrilateral, labelled anticlockwise. Which one of the following is a sufficient but not necessary condition for PQRS to be a…
- Question 2A student chooses two distinct real numbers x and y with 0<x<y<1. The student then attempts to draw a triangle ABC with AB=1, cos A=x, cos…
- Question 3We say an integer x divides an integer y and write x| y if there exists an integer k such that y=kx Let a,b,n be positive integers. Let (*)…
- Question 4Let n>1 be an integer. Consider the property P(n): For all integers a,b, if n| ab then n| a or n| b. Which of the following is correct?
- Question 5Let P(n) be a statement about integers with the following properties: P(1) and P(2) are true; for all n≥1 the following statement is true…
- Question 6Let real numbers x and y satisfy |x|+|y|=1. What is the maximum possible value of x-2y?
- Question 7Let P and Q be statements. Suppose exactly one of if P then Q and if Q then P is true, and also P or Q is true. Which statement must be…
- Question 8For a>0 with a≠1, consider the equation log_a x=x for x>0. The number of positive solutions is
- Question 9For real parameter t, consider the equation log_3 x+log_x 3=t with x>0 and x≠1. The number of positive solutions x is
- Question 10ABCD is a quadrilateral, labelled anticlockwise. Which one of the following is a sufficient but not necessary condition for ABCD to be a…
- Question 11Given that (x^2)/(4)+y^2=1, what is the greatest possible value of 3x+4y?
- Question 12Fix distinct real numbers a≠ b with a<b. Consider the equation |x-a|+|x-b|=|a-b| Which of the following describes the complete solution set…
- Question 13Consider the statement (*) about a real number x: There exists a real number y such that |x+y|<y. Which one of the following describes all…
- Question 14Let a be a real number for which the equation |x-1|+|x+1|=a has finitely many distinct real solutions for x. Let k be this nonnegative…
- Question 15Consider the following statement about a polynomial p(x) with real coefficients: (*) If p(n) is even for every integer n, then every…
- Question 16Consider the following statement about irrational numbers: (*) If x and y are irrational numbers, then x+y is irrational. Which of the…
- Question 17We say a real function f(x) is continuous at a point x_0∈mathbb R if for all ε>0 there exists δ>0 such that for all x∈mathbb R, if…
- Question 18Let f(x), g(x) and h(x) be real functions and consider the statement (*) If x>0 and f(x)≤0 then g(x)≠1 or h(x)≥-1. Which of the following…
- Question 19We say a sequence of real numbers x_n is a Cauchy sequence if for all ε>0 there exists a positive integer N∈mathbb N such that for all n,m≥…
- Question 20Let f(x), g(x), h(x) and k(x) be real functions and consider the statement (*) If 0 g(x), then h(x)=0 or k(x)≥5. Which of the following…