Viluxent Set 1 Paper 2
20 questions · 75 min · Difficulty 51 · by Viluxent
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Questions in this paper
- Question 1Let n be a positive integer, and consider the statements P:n^2-1 is divisible by 24. Q:gcd(n,6)=1. Which one of the following is true?
- Question 2You have n stacks of coins, each containing n coins weighing n grams each. However, one stack is defective, and every coin in that stack…
- Question 3How many solutions does 3sin^2 x+5sin xcos x-2cos^2 x=0 have for 0≤ x<2π?
- Question 4Let b and c be real numbers with c≠0. P: The equation x^2+bx+c=0 has two distinct positive real roots. Q: The equation cx^2+bx+1=0 has two…
- Question 5Find Σ_(k=0)^∞lfloorfrac1+√((800,000,000,000,000)/(4^k))2rfloor.
- Question 6Quadratic Chase is a single-player game. To start the game, the player chooses two distinct nonzero integers a and b and writes the…
- Question 7A sequence is defined by a_1=k, for some real number k, and a_(n+1)=a_n^2-2 (n≥1). P: The sequence (a_n) is bounded. Q: -2≤ k≤2. Which one…
- Question 8Find the value of Σ_(k=1)^(1680)frac1k√(k+1)+(k+1)√(k).
- Question 9A ray of light travels from A(1,3) to the y-axis, reflects, travels to the x-axis, reflects again, and then arrives at B(4,7). Find the…
- Question 10Consider the following attempt to solve the equation |x-3|=2x+1: (I) &|x-3|=2x+1 (II) &(x-3)^2=(2x+1)^2 (III) &x^2-6x+9=4x^2+4x+1 (IV)…
- Question 11Let f(x)=(2x+1)/(x-3), for x≠3. How many real solutions does the equation f(f(x))=x have?
- Question 12Two circles of radii 4 and 9 are externally tangent to one another. A common external tangent line touches the two circles at points A and…
- Question 13Consider the statement: “For every real number x, there exists a real number y such that x+y>5 and xy<1.” Which of the following is the…
- Question 14How many real numbers x satisfy 9^x+25^x=15^(x+1)?
- Question 15In the expansion of (1+x+x^4)^(50) as a polynomial in x, find the sum of the coefficients of the odd-power terms (i.e. the terms…
- Question 16A sequence satisfies a_1=a_2=1 and a_(n+2)=a_(n+1)+a_n+1 for n≥1. (I) a_n is odd for every positive integer n. (II) a_n+1 is a power of 2…
- Question 17In triangle ABC, AB=6 and AC=10. The median from A meets BC at its midpoint M, and AM=7. Find the exact area of triangle ABC.
- Question 18A cone is inscribed in a sphere of fixed radius R (the apex and the entire circular base rim lie on the sphere), so that its volume is as…
- Question 19Let k be a positive real constant. For each nonzero real number m, define the line L_m:y=mx+k/m. (I) Every line L_m is tangent to the…
- Question 20Let f be differentiable on mathbb R with f'(x)>0 for all real x. Define g(x)=∫_0^x f(t) d t. P &: g is strictly increasing on mathbb R. Q…