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TMUAmockz Set 1 Paper 1

20 questions · 75 min · Difficulty 52 · by TMUAmockz

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Questions in this paper

  1. Question 1The curves C_n are defined by |x|+|y|=2^(2-n) for non-negative integers n. The region between C_n and C_(n+1) is shaded when n is even, and…
  2. Question 2Let f(x)=x^3-3px-p(1+p), where p>0 and p≠1. How many times does the curve y=f(x) cross the x-axis?
  3. Question 3Find the area of the region of the coordinate plane satisfying 0≤ x≤fracπ2, 0≤ y≤fracπ2, sin(x+y)≤frac12.
  4. Question 4For x^2+2x-3>0, how many distinct real solutions are there to the equation (x^2+2x-3)^(x^2-4)=(x^2+2x-3)^(2x)?
  5. Question 5Evaluate ∫_(-2)^4|x-1|(x^2-2x-3)d x
  6. Question 6For how many values of the constant k does the equation (kx^2-7x+6)(x-k)=0 have exactly two distinct real roots?
  7. Question 7For some integer n>1, the infinite series dfrac2n+dfrac3n^2+dfrac2n^3+dfrac3n^4+…=dfrac751. Find the value of n.
  8. Question 8The coefficient of x^2 in the expansion of (1+2x+2x^2)^3 is equal to one third of the coefficient of x^2 in the expansion of (1+mx)^4. Find…
  9. Question 9A circle centred at A with radius 1 and a circle centred at B with radius 9 are externally tangent to one another. A third circle is…
  10. Question 10Two lines y=x and y=-x cross the vertical line x=4. A circle sits in the region to the right of x=4, touching all three lines. Its centre…
  11. Question 11A triangle has vertices A=(1,2), B=(3,3) and C=(3m+1,2), where m≠0. The line y-2=m(x-1) divides the triangle into two regions of equal…
  12. Question 12Find the sum of all values of θ in the interval 0°<θ<180°that satisfy cosθ=sin5θ.
  13. Question 13An arithmetic progression has first term 25 and common difference d. The numbers obtained by adding 10 to the second term and adding 29 to…
  14. Question 14Triangle PQR has fixed vertices Q=(0,3) and R=(4,0). The third vertex P=(a,b) has positive integer coordinates satisfying a+2b=14. What is…
  15. Question 15Evaluate Σ_(k=1)^(49)sin^2((kπ)/(50)).
  16. Question 16The function g(x)=lfloor xrfloor·lceil xrceil is defined for all real x, where lfloor xrfloor denotes the floor of x and lceil xrceil the…
  17. Question 17A regular hexagon PQRSTU has vertex P at (1,1) and vertex R at (7,5). Find the area of the hexagon.
  18. Question 18Which of the following graphs best represents the curve with equation |y|=-tan x in the range -π<x<π (excluding points where tan x is…
  19. Question 19Positive integers p,q and r satisfy p<q<r. The simultaneous equations 2x+y=12 y=|x-p|+|x-q|+|x-r| have exactly one solution. Find the…
  20. Question 20The equation 16x^4-12x^2+1=0 has four real solutions. The second largest solution can be written in the form fracp+q√54 where p and q are…