tmua.me Extreme Set B Paper 2
20 questions · 75 min · Difficulty 87 · by tmua.me
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Questions in this paper
- Question 1The points A=(-1,0) and B=(1,0) are fixed. The line L has equation y=mx+c, where m and c are real constants. Consider the statements - S…
- Question 2The quadratic polynomial q(x)=ax^2+bx+c has integer coefficients, with a>0. It satisfies q(n)>0 for every integer n, but q(x)<0 for at…
- Question 3Let k≥0. Consider the statements - S: 0≤ k 1, b>1, c>1, the three numbers a+(k)/(a), b+(k)/(b), c+(k)/(c) can also be used as the side…
- Question 4The vertices of a convex quadrilateral are A,B,C,D, in that order. The areas of the four triangles ABC, ABD, ACD, BCD are 1, 2, 3 and 4, in…
- Question 5The points A=(5/3,0) and B=(0,5/3) are fixed. A point P=(x,y) lies on the circle x^2+y^2=1, with x≠0. The gradients of AP and BP are m and…
- Question 6Three coins A, B and C are tossed independently. Their probabilities of landing heads are p, q and r, respectively, where 0<p<1, 0<q<1…
- Question 7Initially the six numbers 1,2,3,4,5,6 are written on a board. In each move, one of the current entries is chosen uniformly at random and…
- Question 8Eight distinct integers are selected from 1,2,3,…,16. No selected integer is a factor of any other selected integer. What is the least…
- Question 9A real-valued function f, defined for every real number, satisfies f(x+1)=f(x)+2x+1, f(2x)=4f(x) for every real x. Also, 0≤ f(x)≤1 whenever…
- Question 10An array has 5 labelled rows and 4 labelled columns. Each of its 20 cells is coloured red or blue. Whenever two distinct rows and two…
- Question 11A polynomial P(x) has degree 4 and coefficient 1 for x^4. Two of its real roots are a and b, where b-a=2. Consider the statements - S: The…
- Question 12Each use of a machine produces one of the numbers 1, 2 and 3. Each number has a fixed positive probability of being produced, and the…
- Question 13For a real constant k with 0≤ k≤2, define a_n=n+((-1)^n k)/(n) (n=1,2,3,…). Consider the statement: For every real number ε>0, there exist…
- Question 14Triangle ABC is right-angled at A, with AB=AC=5. Three circles have centres A, B and C and radii r, 2r and √2 r, respectively, where r>0…
- Question 15Let f(x)=x^4-frac32x^2, and let k be a real constant. Consider the statements - S: For every real number h>0, the equation f(x+h)-f(x)=kh…
- Question 16Nine players have distinct ranks 1,2,…,9, with rank 1 the strongest. Whenever two players play, the player with the smaller rank wins. At…
- Question 17A point P lies on the parabola y=x^2. Its reflections in the three lines x=0, y=0, 3x+2y=12 are three distinct collinear points. What is…
- Question 18For a real constant k, define f_k(x)=x^3-kx^2+10x. Consider the statements - S: k≥13/2. - T: For every pair of distinct positive real…
- Question 19The eight distinct integers 1, 2, 3, 4, 6, 8, 12, 24 are placed at the eight vertices of a fixed cube, with every placement equally likely…
- Question 20The four numbers sinα, cosα, sinβ, cosβ, when arranged in increasing order, form an arithmetic progression with positive common difference…