Viluxent Set 1 Paper 1
20 questions · 75 min · Difficulty 61 · by Viluxent
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Questions in this paper
- Question 1Real numbers x,y,z satisfy x+y+z=6, xy+yz+xz=9. Find the number of ordered triples (x,y,z) of real numbers for which xyz attains its…
- Question 2How many values of x∈[0,2π] satisfy both 4sin^2x-4sin xcos x+cos^2x=0 and 2sin^2x+3sin xcos x-2cos^2x=0 simultaneously?
- Question 3How many real solutions does the equation 2^x+2^(-x)=x^2+2 have?
- Question 4Find the number of ordered pairs of positive integers (x,y) with x>y such that x^2-y^2=14400 and x-y is a perfect square.
- Question 5Let p be a real number, and let n denote the number of points of intersection of the curves y=|x^3-p^3| y=p^3|x-1|. n cannot take which of…
- Question 6The circle C has equation (x-5)^2+(y-4)^2=9. The two tangent lines from the origin O to C meet at O, making an angle θ between them. Find…
- Question 7In triangle ABC, the side lengths a,b,c (opposite A,B,C respectively) are in arithmetic progression, and B=60°. Given that the area of the…
- Question 8A differentiable function f is defined for all x>0 and satisfies f(xy)=f(x)+f(y) all x,y>0, together with f(2)=1. Find f'(8).
- Question 9Let P(x)=1+2x+3x^2+4x^3+…+3000x^(2999). Find the remainder when P(x) is divided by x^2-x+1.
- Question 10Let lfloor yrfloor denote the greatest integer less than or equal to y. Find the value of ∫_(50)^(1/3)log_5((lfloor3xrfl…
- Question 11Let g_0(x)=x for x∈[0,1), and for n≥0 define g_(n+1)(x)=2g_n(x)-lfloor2g_n(x)rfloor (that is, g_(n+1)(x) is the fractional part of…
- Question 12For a positive integer n, let d(n) denote the number of digits in the decimal representation of n (so d(7)=1, d(58)=2, d(100)=3, and so…
- Question 13A particle travels from A(0,0) to a point on the line 3x+4y=24, and then on to B(5,-4). Find the minimum possible total length of its path.
- Question 14Positive integers a and b satisfy (a+b+1)(a-b+1)=7200 and (a+1)/(b)+frac ba+1=(130)/(63). Find the value of a.
- Question 15Three circles, each of radius 2, are mutually tangent to one another (each pair touches at exactly one point). Find the exact area of the…
- Question 16Let f(x)=(x)/(x-1) for x≠1, and let g be the inverse function of f. How many real solutions does the equation f(x)=g(x) have?
- Question 17Let f(x)=(3x-1)/(x+2). How many real solutions does the equation f(f(x))=x have?
- Question 18Let p be a non-zero real constant, and consider the curve y=x^4-4px^3+6p^2x^2+1. For how many values of p does this curve have a point of…
- Question 19Which of the following is true?
- Question 20Triangle PQR has PQ=5, QR=8, and ∠ PQR=60°. Point M lies on QR with QM=3. An ant starts at P at time t=0 and walks in a straight line…