tmua.me Extreme Set C Paper 1
20 questions · 75 min · Difficulty 85 · by tmua.me
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Questions in this paper
- Question 1The positive integers occurring in either of the sequences 1,4,7,10,… 1,2,4,8,… are combined into one increasing sequence a_1,a_2,a_3,…. An…
- Question 2A sequence of real numbers satisfies u_(n+1)=u_n^2-3 (n=1,2,3,…). Its first three terms form an arithmetic progression with nonzero common…
- Question 3Five distinct positive integers are written in increasing order. When the smallest is removed, the mean of the four remaining integers…
- Question 4A real number t>2 and integers a,b satisfy t^3-2t=a, t^2-at+b=2. What is the smallest possible value of a?
- Question 5For nonzero real x, define f(x)=x^2+frac1x^2, g(x)=x^2-3x. What is the sum of the squares of all the distinct nonzero real numbers a for…
- Question 6For real x with x≠0 and x≠-1, what is the least possible value of x^2+(x+1)^2+frac12x^2+frac12(x+1)^2 ?
- Question 7For r=0,1,…,49, let a_r be the coefficient of x^r in (1+x)^(47)(1-x)^2. These coefficients satisfy the identity a_r=bino…
- Question 8A rectangular sheet ABCD is folded along the diagonal AC, so that the two triangular parts lie in different planes. In the folded position…
- Question 9For a real constant k, the polynomial p(t)=t^3-kt^2+(k-1)t+(k)/(32) has sinθ and cosθ as roots, where 0<θ<dfracπ4. What is the value of k?
- Question 10A circle meets the parabola y=x^2 at exactly three distinct points. Their x-coordinates, in increasing order, form an arithmetic…
- Question 11Three points P,Q,R start at the vertices A,B,C, respectively, of a triangle. They travel along AB,BC,CA, respectively, at constant speeds…
- Question 12For each real angle θ, let m and M be the smallest and largest, respectively, of the four numbers |sinθ|, |cosθ|, |sinθ+cosθ|, |sinθ-cosθ|…
- Question 13Three consecutive vertices A,B,C of a parallelogram ABCD lie on the circle x^2+y^2=1. The fourth vertex satisfies OD=√5, where O is the…
- Question 14A rectangle has sides parallel to the coordinate axes. One vertex is (-9,-5), and the opposite vertex lies on the circle x^2+y^2=25. What…
- Question 15A cubic polynomial f satisfies f(0)=0, f(1)=1, f'(0)f'(1)=4. It is also given that f'(x)≥0 whenever 0≤ x≤1. What is the greatest possible…
- Question 16For a real constant h, consider the curves C_1: y=(x^3)/(3)-(x)/(2), C_2: y=((x-2)^3)/(3)-(x-2)/(2)+h. For each value of h for which the…
- Question 17Six counters initially carry the numbers 1, 2, 3, 4, 5, 7. At each stage, one pair of the counters currently present is chosen uniformly at…
- Question 18Nine cards are numbered 1,2,…,9. Three are drawn without replacement, with every selection of three cards equally likely. After seeing…
- Question 19Five cards are drawn successively without replacement from nine cards numbered 1,2,…,9. Each remaining card is equally likely at each draw…
- Question 20For a≥0, let f_a(x)=x^3+a x(1-x). The trapezium rule is used to estimate ∫_0^1 f_a(x) mathrm dx with two strips whose boundaries are x=0…