Euclid Sample Paper 1
20 questions · 75 min · Difficulty 57 · by Euclid
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Questions in this paper
- Question 1Let P(x)=x^4-4x^3+6x^2-4x+k. Exactly one real value of x satisfies P(x)=0. What is the value of k?
- Question 2What is the coefficient of x^9 in the expansion of (1-x)^7(1+x+x^2)^7?
- Question 3The triangle with vertices (1,3), (3,1) and (2,4) is reflected in the line x+2y=2. Recall that the centroid of a triangle with vertices…
- Question 4For each positive integer n, define a_n=√ n(√(n+1)-√ n). Which of the following is true for every positive integer n?
- Question 5Which of the following describes all real solutions of |x^2-5|<2|x|?
- Question 6The real numbers a,b,c satisfy a+b+c=6 ab+bc+ca=9. What is the greatest possible value of c?
- Question 7The parabola y=x^2+px-3 meets the x-axis at Q and R, and the y-axis at P. A circle with centre (-1,-1) passes through P, Q, and R. What is…
- Question 8For how many integers n is (n^2+9)/(n+3) a positive integer?
- Question 9For real x, let lfloor xrfloor be the greatest integer less than or equal to x. What is the value of Σ_(n=1)^(100)lfloor√ nrfloor?
- Question 10A sequence (x_n) is defined by x_1=2, x_(n+1)=(x_n)/(1+x_n). What is the value of Σ_(n=1)^(20)x_nx_(n+1)?
- Question 11The trapezium rule, using any positive number of equal-width strips, is applied to ∫_0^2(x^4-4x^3+6x^2) mathrm dx. Which statement is true?
- Question 12For real θ, define T=sinθ+cosθ+2sinθcosθ. Let M be the maximum possible value of T, and let m be the minimum possible value of T. What is…
- Question 13A sequence of squares S_1,S_2,S_3,… is arranged as shown. Their horizontal diagonals lie consecutively on the positive x-axis. The…
- Question 14For which values of k does |x^2-1|=kx+1 have exactly three distinct real solutions?
- Question 15For each integer k satisfying -4≤ k≤4, consider the two curves y=2^x y=(-1)^kx+k. Let N_k be the number of points of intersection of these…
- Question 16A differentiable function f:(0,∞)→mathbb R satisfies f'(x)=-frac4x^2. For each t>0, the tangent to the curve y=f(x) at x=t meets the…
- Question 17A monic cubic polynomial f has three distinct real roots. The two bounded regions enclosed by the curve y=f(x) and the x-axis have equal…
- Question 18A quartic polynomial P has leading coefficient 4 and satisfies P(x)≥0 for every real x. It has exactly two distinct real roots and also…
- Question 19For a real parameter p, consider the two curves C_p: (|x|-1)^2+y^2=(p+1)^2 and D_p: x^2+(|y|-1)^2=(p+1)^2. The curves intersect at eight…
- Question 20For which values of k does the equation x^6-12x^4-kx^3-12x^2+1=0 have exactly four distinct real solutions?