Y and A Tutoring Paper 1 with Answers and Hints
20 questions · 75 min · Difficulty 40 · by Y and A Tutoring
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Questions in this paper
- Question 1Consider the points (x,y) that lie on the circle with equation (x-2)^2+(y-3)^2=4. What is the largest possible value that x+y can take?
- Question 2The coefficient of x^5 in the binomial expansion of (kx^2-2x+1)^4 is -32. Find all possible real values of the constant k.
- Question 3What is the shortest possible distance between the circle (x-3)^2+(y+2)^2=8 and the straight line y=x+3?
- Question 4A student uses the trapezium rule with 2 strips of equal width to estimate the value of the definite integral: ∫_0^2√(4-x^2) d x. By…
- Question 5A sequence is defined by u_n=2^(3n-1) for integers n≥1. Which of the following statements correctly describes the sequence defined by…
- Question 6How many solutions does the equation cos(πsin x)=0 have in the interval 0≤ x≤2π?
- Question 7Let f(x)=2x^3-9x^2+12x and g(x)=c, where c is a real constant. Let m be the number of distinct real solutions to the equation f(x)=g(x)…
- Question 8Let P_1(x), P_2(x) and P_3(x) be three quadratic polynomials with real coefficients. The polynomial Q(x)=P_1(x)P_2(x)P_3(x) has exactly 5…
- Question 9A quadratic function f(x) has a minimum value of 4 at x=2. Another quadratic function g(x) has a maximum value of 4 at x=2. How many…
- Question 10Three fair six-sided dice are rolled. What is the probability that the three numbers rolled can form the side lengths of a non-degenerate…
- Question 11In triangle ABC, the side lengths are BC=a, CA=b and AB=c. It is given that the side lengths satisfy the algebraic relationship…
- Question 12Find the number of real pairs (x,y) that satisfy the following simultaneous equations: 3^x+9^y=10, 3^y+9^x=10.
- Question 13Four straight lines are drawn on a 2D plane. None of the lines are parallel to each other, and no three lines intersect at a single point…
- Question 14Find the sum of all valid solutions to the following equation in the interval 0≤ x≤2π: sin^3 x=(sin^2 x-1)cos x.
- Question 15Let f(x)=x^3-6x^2+12x-5. What is the exact value of f(2-√[3]5)+f(2+√[3]5)?
- Question 16A function f is defined for all real numbers and has the property that f(x+y)=f(x)f(y) for all real numbers x and y. It is given that there…
- Question 17A bag contains exactly 3 red marbles and n blue marbles (where n≥2). Three marbles are drawn at random without replacement. Given that the…
- Question 18Find the sum of all positive integers n such that the expression: (n^2+7n+22)/(n+3) evaluates to an integer.
- Question 19Find the complete set of values of x for which the following inequality holds: (2x-1)/(x+2)>x.
- Question 20Let f(x)=fracx√(1+x^2). Let f_n(x) denote the n-th iteration of f(x) such that f_2(x)=f(f(x)), f_3(x)=f(f(f(x))), and so on. Evaluate…