Euclid Sample Paper 2
20 questions · 75 min · Difficulty 54 · by Euclid
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Questions in this paper
- Question 1Let x,y>0 and x+y=1. The following statements may be used to prove frac1x+frac1y≥4. P:& (x-y)^2≥0, Q:& x^2+y^2≥2xy, R:& (x+y)^2≥4xy, S:&…
- Question 2A sequence (a_n) satisfies a_(n+2)=a_(n+1)+a_n for every positive integer n. A student claims: If a_1 is even, then every term of the…
- Question 3A sequence satisfies a_(n+1)=1-a_n for every positive integer n. Consider P: The sequence has least period 2, Q: a_1≠frac12. Which option…
- Question 4Consider the statement: For every real number x, there exists a positive integer n such that x<n≤ x+2. Which is its logical negation?
- Question 5A function f is continuous on [0,1]. Consider P: f(c)=0 for some c∈(0,1), Q: f(0)f(1)<0. Which option correctly describes the relationship…
- Question 6A student calculates n^2+n+11 for n=0,1,2,3 and obtains four prime numbers. The student conjectures that the expression is prime for every…
- Question 7A real sequence (a_n) satisfies a_(n+2)=a_(n+1)-a_n for every positive integer n. Which statements must be true? - I. a_(n+3)=-a_n for…
- Question 8Let n be a positive integer. Consider P: 24| n^2, Q: 6| n. Which option correctly describes the relationship between P and Q?
- Question 9In triangle ABC, let M be the midpoint of BC. Define P:& ∠ BAC=90°, Q:& AM=frac12BC, R:& MA=MB=MC. Which statements are true? - I. P if and…
- Question 10The positive real numbers a<b<c are the side lengths of a non-degenerate triangle. Define x=b+c-a, y=c+a-b, z=a+b-c. Which statements must…
- Question 11For how many integers n with 1≤ n≤500 is n^2+3n+2 divisible by 35?
- Question 12The following lines can be arranged to prove that there are no positive integers a and b satisfying a^2=2b^2. - P. Suppose that such pairs…
- Question 13Let f be twice differentiable on mathbb R. Define P: f has at least four distinct real roots, Q: f' has at least three distinct real roots…
- Question 14A differentiable function f satisfies f'(x)=x^2-4 for every real x, and f(0)=0. Which statements must be true? - I. The equation f(x)=k has…
- Question 15For each n=1,2,…,100, the statement S_n is: At least n of the integers m with 1≤ m≤100 have m^2-1 divisible by 24. Exactly how many of the…
- Question 16A differentiable function f:mathbb R→mathbb R is defined as *wobbly* if there exists a real number x such that f'(x)=0. The function f is…
- Question 17Define f(x)=x^2-2x. For a real number k, the equation f(f(x))=k has exactly four distinct real solutions. Which of the following statements…
- Question 18A differentiable function f satisfies f'(x)=(x^2-1)(x^2-4) for every real x. Which of the following statements must be true? - I…
- Question 19For real k, consider |x^2-4x+3|=k|x-2|+frac32. For which values of k does the equation have exactly six distinct real solutions?
- Question 20For a real number a, consider the equation x^2+2xy+ay^2=1. Define P: For every real x, there exists a real y satisfying the equation, Q…