Euclid R2Drew2 Edition Paper 1
20 questions · 75 min · Difficulty 64 · by Euclid
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Questions in this paper
- Question 1The roots of x^2+px+q=0 are a and b. The roots of x^2+qx+p=0 are a+1 and b+1. What is p+q?
- Question 2An integer n≥3 has the following property: In the expansion of (1+x)^n(1-x), the coefficient of x^2 is equal to the coefficient of x^3…
- Question 3The equation 9^x-10·6^x+9·4^x=0 has solutions p and q. What is (frac32)^(p+q)?
- Question 4The function f is defined by f(x)=x^2-6x+7. Distinct real numbers a and b satisfy f(a)=f(b) f(a+1)=f(b-2). What is ab?
- Question 5An integer n>3 satisfies n^2-9=2^k, where k is a non-negative integer. What is k?
- Question 6The circles x^2+y^2=25 and (x-8)^2+y^2=r^2 intersect in a common chord of length 8. There are two possible values of r^2. What is their sum?
- Question 7When the polynomial P(x) is divided by (x-1)^2, the remainder is 4x-1. What is the remainder when P(x^2-3x+3) is divided by (x-2)^2?
- Question 8For real x, define F(x)=|x^2-5x|+|x^2-5x+4|. The set of values of x for which F(x) is least consists of two closed intervals. What is the…
- Question 9In triangle ABC, AB=12, AC=7, ∠ ABC=30°. There are two non-congruent triangles satisfying these conditions. Their areas are K_1 and K_2…
- Question 10In triangle OAB, the point P lies on OA and the point Q lies on OB, where OP:PA=2:3, OQ:QB=3:4. The lines AQ and BP meet at R. The line OR…
- Question 11The equation log_2(x^2-5x+8)+log_2(8-x^2+5x)=4 has two real solutions p and q. What is pq?
- Question 12Evaluate ∫_0^1fracx^6+x^3+1x^3+x^(3/2)+1 mathrm dx.
- Question 13For x>0, define F(x)=(x^4+4x^2+1)/(x^4+2x^3+3x^2-2x+1). What is the least possible value of F(x)?
- Question 14A point P lies inside triangle ABC. Through P, a line is drawn parallel to BC. This line cuts off a smaller triangle at A with area 4…
- Question 15What is the sum of all values of θ satisfying 0≤θ<2π and |sinθ|+|cosθ|=(1+√3)/(2)?
- Question 16At three distinct points A, B and C on the parabola y=x^2, the normals to the parabola all meet at the same point P. The point A has…
- Question 17A line is called a GP-secant of a curve if it meets the curve at three distinct points whose x-coordinates are positive and form a…
- Question 18For a real number a, consider the graphs y=|x^2-1| and y=|(x-a)^2-1|. The graphs intersect at exactly three distinct points. Which of the…
- Question 19A rectangle ABCD has area 3. The points P, Q and R lie on the sides BC, CD and DA, respectively. Among all possible rectangles ABCD of area…
- Question 20Let f(x)=x^2-2. Define f_1(x)=f(x) and, for n≥1, f_(n+1)(x)=fbigl(f_n(x)bigr). How many distinct real solutions does the equation f_8(x)=2…