Daniel J Smith Custom TMUA Set A Paper 2
20 questions · 75 min · Difficulty 53 · by Daniel J Smith
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Questions in this paper
- Question 1For each positive integer n, define E_n=n^4+4 For which one of the following statements is n=3 a counterexample?
- Question 2A student is asked to prove the statement: if u and v are real numbers such that u^2+v^2=0, then u=0 and v=0 They write the following…
- Question 3We 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 m,n≥…
- Question 4Let S be the statement about integers n: (*) For every integer n there exists an integer m such that m>n and m is even Which of the…
- Question 5Consider the statement about integers n: (*) For every integer n, if n is even then n^2+2n is divisible by 4 Which of the following…
- Question 6A statement S is given: For all real numbers x, if x^2≥1 then x^4≥ x^2 A student wishes to prove S by contradiction and writes the…
- Question 7For real numbers a and b, consider the statement S(a,b): if ab 0 Which of the following descriptions is correct?
- Question 8Consider the statement: For every positive integer n there exists a positive integer k such that for all positive integers m, the number…
- Question 9Let f(x)=ax^2+bx+c be a quadratic function with real coefficients such that f(x)≥0 for all real x Suppose that for every real x…
- Question 10Let C be the circle x^2+y^2=9 Let P be the point (a,0) where a>3 The two tangents from P to C touch the circle at T_1 and T_2 What is the…
- Question 11Let lfloor xrfloor denote the greatest integer less than or equal to x Define f(x)=lfloor2xrfloor-lfloor xrfloor for all real x The value…
- Question 12A sequence (a_n) is defined by a_1=1 and, for n≥1, a_(n+1)=fraca_n+4√(a_n+1) Which of the following statements is true?
- Question 13Let P and Q be statements Suppose exactly one of (if P then Q) and (if Q then P) is true Suppose further that (P or Q) is true Which…
- Question 14Let x be a real number Which one of the following statements is a sufficient condition for exactly two of the other four statements?
- Question 15A non-empty finite set S of whole numbers is called *linked* if and only if for every a in S, there exists b in S with b≠ a such that…
- Question 16Consider the statement: if 0≤ f(x)≤1 for all real x with 0≤ x≤1, then (*) ∫_0^1 f(x) dx≤∫_0^1(f(x))^2 dx Which of the following functions…
- Question 17Consider the claim: For all integers a, b, c, if ab=ac then b=c A student's argument is as follows: - I. Suppose ab=ac - II. Then ab-ac=0 -…
- Question 18Let f be a function defined for all real x, and assume the integral exists Which one of the following is a sufficient condition for…
- Question 19Consider the two statements about a real-valued function f: (A) For all real numbers x there exists a real number y such that y>x and…
- Question 20The positive real numbers p×10^(-4), q×10^(-2) and r×10^(-1) are each in standard form, and (p×10^(-4))+(q×10^(-2))=(r×10^(-1)) How many of…