Maths Made Easy TMUA Paper 1
18 questions · 68 min · Difficulty 49 · by Maths Made Easy
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2 questions have been withdrawn and are left out of practice sessions.
Question paper ↗Mark scheme ↗Original question paper (before corrections) ↗Original mark scheme (before corrections) ↗
Questions in this paper
- Question 1Real numbers x and y satisfy x^2+xy+y^2=7. Find the complete set of values that x+y can take.
- Question 2A geometric progression has first term a and common ratio r. The 1st, 2nd, and 4th terms form an arithmetic progression (in that order)…
- Question 3How many solutions does the equation 4sin^4x-3sin^2x+sin^2xcos^2x=0 have in the interval 0≤ x≤2π?
- Question 4Find the smallest positive integer n for which Σ_(k=1)^nlog_3(1+frac1k)>3.
- Question 5Withdrawn
- Question 6For real numbers x and y with xy=1, find the maximum value of ((x+y)^2)/(x^2+y^2).
- Question 7Let u_(n+1)=u_n^2-2 with u_1=t. For how many integer values of t does there exist some positive integer N such that u_N=2?
- Question 8Positive real numbers a,b,c satisfy ab+bc+ca=1. What is the least upper bound of (a+b+c)/((1+a)(1+b)(1+c)) ?
- Question 9For positive real numbers x,y define S=frac xy+frac yx+x+y. Which is the greatest lower bound of S?
- Question 10Let a_1=1 and a_(n+1)=a_n+dfrac1a_n. Which statement is true?
- Question 11The polynomial f(x)=x^3+cx^2+dx-6 satisfies both of the following: - (x+2) is a factor of f(x); - f(x) leaves a remainder of 3 when divided…
- Question 12Find the coefficient of x^3 in the expansion of (2-x)^3(1+x)^5.
- Question 13What is the smallest integer n such that every placement of n distinct points strictly inside a unit square guarantees two points at…
- Question 14Find the number of integer solutions to |x|+|y|≤400.
- Question 15The lengths of a triangle's sides follow a geometric progression (common ratio r). It follows that:
- Question 16What is the area of the polygon formed by all points (x,y) in the plane satisfying the inequality bigl||x|-2bigr|+bigl||y|-2bigr|≤4?
- Question 17For real x, find the number of solutions to sin(πcos x)=cos(πsin x).
- Question 18A pyramid has a square base of side length s and four equilateral triangular faces. A sphere of radius r is inscribed in the pyramid…
- Question 19A rectangle is inscribed in a circle of radius R, so that all four vertices lie on the circle. Find the maximum possible area of the…
- Question 20Withdrawn