Juan Set 2 Paper 1
26 questions · 98 min · Difficulty 48 · by Juan
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Questions in this paper
- Question 1Let m and n be real numbers. In the coordinate plane, the circle passing through the point A(4,1) is given by x^2+y^2-2x+6y=n. The line…
- Question 2For digits A, D, T, and Y, the following addition is given: r@ r &TYT &AYT +&YDT hline &2024 where TYT<AYT. What is the product Y· D· T?
- Question 3Let a, x, y be positive real numbers such that -2x^2+y^2=2a, 3x^2-2y^2=-6a. What is the ratio (y)/(x)?
- Question 4Let a be a digit such that the remainder when 25! is divided by (23!-a) is 60^2. What is a? (Note that n!=n·(n-1)·(n-2)…3·2·1)
- Question 5Let x, y, z be distinct prime numbers such that x+y· z=21. What is x+y+z?
- Question 6Let a be an integer such that the inequality 0<|x^2-2x+2|-x^2-x<a is satisfied by exactly 4 integer values of x. What is the sum of the…
- Question 7Let a and b be positive real numbers such that for each of the equations ax^2-2x+b=0, bx^2-3bx+a=0 the sum of the roots is 1 more than the…
- Question 8Let a and b be real numbers, and let the function f, defined on the set of real numbers by f(x)=x^3+9x^2+ax+b take positive values for…
- Question 9For an arithmetic sequence (a_n) with common difference r, it is given that a_(30)-a_(25) 53. What is the sum of the possible integer…
- Question 10Let a>1, and let x, y be positive real numbers such that, arranged from smallest to largest, log_a x, log_a y, log_a(x+y) are consecutive…
- Question 11Melisa is going to stack, on a table, 10 circular pieces of cardboard whose radii are 1 cm, 2 cm, 3 cm, …, 10 cm (each a different natural…
- Question 12A certain bus arrives at the stop near Duru's house at exactly 09:02 with probability (7)/(10), and at exactly 09:03 with probability…
- Question 13Let m and n be natural numbers such that the constant term in the expansion of (x+(5)/(x^m))^n is 60. What is m+n?
- Question 14Let a and b be real numbers such that, in the coordinate plane, the parabola y=x^2+ax+b is tangent to the x-axis and to the line y=x. What…
- Question 15Let a and b be real numbers, and let the function f, defined by f(x)=ax^3+bx^2+x+7 be strictly increasing everywhere. Given that f(-1)=0…
- Question 16Let k and m be real numbers, and let the functions f and g be defined on the set of real numbers by f(x)=2x^3-9x^2-mx-k, g(x)=x^3· f(x)…
- Question 17Let f and g be differentiable functions defined on the set of real numbers satisfying ∫_1^2f'(3x)d x=4, ∫ f(2x)d x=g(x)+C (C constant)…
- Question 18Let a=sin(40°), b=sec(40°), c=tan(40°). Which of the following is the correct ordering of a, b, and c?
- Question 19Which of the following is the simplified form of the expression (cos(2x+y)+sin(2x-y))/(cos(2x)+sin(2x))
- Question 20Let 0<a<(π)/(2) satisfy cos^2a-cos(2a)=sin(2a). Which of the following is true for this value of a?
- Question 21Bonus question. Let 0≤ x≤π satisfy √(2)sin(4x)-cos(8x)=1. What is the sum of the values of x that satisfy this equation?
- Question 22Bonus question. The altitude of a triangle is a straight line drawn from a corner vertex down to the opposite side so that it hits that…
- Question 23Bonus question. The centroid of a triangle can be calculated by taking the mean of the x-coordinates and the y-coordinates of its vertices…
- Question 24Bonus question. In the coordinate plane, what is the area, in square units, of the triangular region bounded by the lines 2x-y=0, x+2y=0…
- Question 25Bonus question. In the coordinate plane, let C be the reflection of the point A(a,b) about the point B(3,0), and let D be the reflection of…
- Question 26Bonus question. In the coordinate plane, the point A(11,9) lies in the interior of a circle that is tangent to the line y=x at the point…