tmua.me Extreme Set C Paper 2
20 questions · 75 min · Difficulty 90 · by tmua.me
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Questions in this paper
- Question 1Let a and b be real constants, and define q(x)=x^2+ax+b. Consider the following statement. For every real number x, at least one of q(x)…
- Question 2For a real constant k, the equation 2^(-|x|)+3 2^(-|x-2|)=k has four distinct real solutions x_1,x_2,x_3,x_4. What is the value of…
- Question 3Each use of either of two machines produces one of the numbers 0, 1 and 2, each with positive probability. The fixed probabilities may be…
- Question 4Two squares of side length 1 have the same centre O. One is obtained by rotating the other through an angle θ, where 0<θ<fracπ4. Their…
- Question 5Let m be a real constant. Consider the following statement. For every list of five distinct real numbers with mean m and median 1, the…
- Question 6Initially, tank A contains 1 litre of pure dye, and tank B contains 2 litres of water. A cycle consists of transferring t litres from A to…
- Question 7A real-valued function f, defined for every real number, satisfies f(x)+f(-x)=2x^2+2, f(x)f(-x)=x^4+x^2+1 for every real x. Consider the…
- Question 8Triangle ABC has AB=√(10), BC=√(17), CA=√5. A point P lies strictly inside the triangle and satisfies ∠ APB=90°, ∠ BPC=∠ CPA=135°. What is…
- Question 9Let p and q be relatively prime positive integers, and let b,c,d be rational numbers. Define P(x)=frac pq x^3+bx^2+cx+d. Consider the…
- Question 10A point P is chosen at random inside an equilateral triangle ABC of side length 2. The probability that P lies in a region is proportional…
- Question 11Three events A, B and C each have probability 2/3. Each pair of these events is independent. A student writes the following argument, where…
- Question 12A quadratic polynomial f with real coefficients satisfies ∫_0^1 f(x)^2 mathrm dx=1, ∫_0^1 f(x)f(1-x) mathrm dx=0. What is the greatest…
- Question 13For 0<x<1 and each positive integer n, define F_n(x)=nx^n(1-x). Consider the following statements. 1. For every real x with 0<x<1, there…
- Question 14Let θ_1,…,θ_m be all the distinct real solutions with 0≤θ<2π of sin(fracπ2(2sinθ+cosθ)) = cos(fracπ2(sinθ-2cosθ)). What is the value of…
- Question 15The distinct cubic polynomials P and Q both have coefficient 1 for x^3. Neither polynomial has a repeated real root, and P(x)Q(x)≥0 every…
- Question 16Four coins A,B,C,D are tossed independently. Each coin has a fixed probability of heads strictly between 0 and 1, and coin A has…
- Question 17The positive real numbers a,b,c,d satisfy ab=1, bc=2, cd=6, da=3. Consider the two conditions - S: a+b+c+d is rational. - T: ac+bd is…
- Question 18A particle moves in a plane. At time t≥0, its coordinates are x(t) and y(t), where x(0)=y(0)=0, (mathrm dx)/(mathrm dt)=1-t^2, (mathrm…
- Question 19Let n be a positive integer. Consider the following statement. There exists a polynomial P with real coefficients and degree exactly 3 such…
- Question 20The vertices of a rectangle are A,B,C,D, in that order, with AB=2√3, BC=2. A point P in the plane of the rectangle satisfies PA^2+PC^2=12…