tmua.me Extreme Set A Paper 1
20 questions · 75 min · Difficulty 82 · by tmua.me
Read the questions freely. Sign in to answer them, start a timed paper and save your progress.
Questions in this paper
- Question 1A polynomial P(x) of degree 3 has three distinct real roots a, b and c. It is given that P(a+b)=P(b+c)=P(c+a)=4 and that P(1)=-1. What is…
- Question 2The circles C and D have equations C: x^2+y^2=1, D: (x-3)^2+y^2=1. A point P moves once around D. The two tangents from P to C meet C at A…
- Question 3A positive integer N has exactly six positive divisors that are perfect squares and exactly four positive divisors that are perfect cubes…
- Question 4The point A lies on the x-axis with non-negative x-coordinate, and the point B lies on the y-axis with non-negative y-coordinate. The…
- Question 5A number is initially 0. A fair coin is tossed eight times, with independent results. After each toss, the current number is doubled if the…
- Question 6The curve C has equation y=x+frac1x, x>0. The tangent and the normal to C at a point P meet the y-axis at A and B, respectively. The origin…
- Question 7For which values of the real constant k does the equation log_2(x^2-2x-k)=1+log_2(2x-k) have exactly one real solution for x?
- Question 8Four distinct integers are chosen from 1,2,3,…,12. How many different selections have a product that is a perfect square? Selections which…
- Question 9A cubic polynomial f(x) has coefficient 1 for x^3 and satisfies ∫_0^1 f(x) dx= ∫_1^2 f(x) dx= ∫_2^3 f(x) dx=0. What is the greatest…
- Question 10Six distinct real numbers have mean 7, median 8 and range 9. Their smallest value is a. Which of the following is the complete set of…
- Question 11A real number r satisfies 0<r<1. For each integer n≥0, the point P_n has coordinates P_n=(r^n,r^(2n)). A broken line joins P_0 to P_1, P_1…
- Question 12The angles α and β satisfy 0<α<fracπ2, 0<β<fracπ2, tanα+tanβ=√2. What is the least possible value of sin^2α+sin^2β ?
- Question 13In a triangle ABC, the point D lies on the side BC with BD:DC=3:2, and the point E lies on the side CA with CE:EA=2:1. The line segments AD…
- Question 14The real constants a,b,c have the property that both x^2-2ax+b≥0 x^2-2cx+a≥0 hold for every real number x. What is the greatest possible…
- Question 15For r=0,1,…,8, let a_r be the coefficient of x^r in the expansion of (1-x)^8. What is the value of Σ_(r=0)^8(a_r)/((r+1)(r+2)(r+3)) ?
- Question 16The positive real numbers x and y satisfy x+y=frac92, |log_2 x|+|log_2 y|=2. What is the product of all the distinct values of x for which…
- Question 17A particle starts at the centre P of a rectangle with side lengths 3 and 2. It moves in straight lines inside the rectangle. Whenever it…
- Question 18Six cards are numbered 1,2,3,4,5,6. Four cards are drawn successively without replacement, with each remaining card equally likely at each…
- Question 19A cube has side length 1. Two distinct parallel planes, Π_1 and Π_2, are each perpendicular to the same body diagonal VW. Each plane cuts…
- Question 20For each real number k, let E(k) be the greatest value of |x^3-kx| for -1≤ x≤1. What is the least possible value of E(k)?