Cryptokyo TMUA Problem Set
19 questions · 71 min · Difficulty 39 · by Cryptokyo
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Questions in this paper
- Question 1Evaluate the integral ∫_0^1(1-x)/(1-√ x) d x.
- Question 2Rationalise the denominator of frac11+√3+√5.
- Question 3Find the coefficient of x^6 in the expansion of (1+x)(1+x^2)(1+x^3)(1+x^4)(1+x^5)(1+x^6).
- Question 4A quadratic polynomial f(x) has two distinct real roots α and β. We are given that αβ=1 and that α+β=γ. What is the possible range of…
- Question 5The polynomial x^2+bx+c has roots α and β. Find α^4+β^4 in terms of b and c.
- Question 6Find f'(2) for f(x)=(x^3+8x^2+11x-20)/(x^2+4x-5).
- Question 7Without using a calculator, compute the exact value of √[3]19.683.
- Question 8How many positive integers are there less than 1000 that are not divisible by 2 or 3?
- Question 9For which values of a does the equation 3^(2x)-a×3^(x+1)+a^2=0 have two distinct real solutions? I. a 0 III. No such values of a IV. a=0
- Question 10The logarithm function log(x) is defined for x>0. For which values of a does the equation log(x+a)=log(x)+log(a) have a solution?
- Question 11Find a counter-example to the statement: if p is prime then 2^p-1 is prime.
- Question 12We say a function is multiplicatively bounded if and only if f(xy)≤ f(x)f(y) and f(0)>0. Is f(x)≥1 a necessary condition for a…
- Question 13For this question we assume our given function f(x) is differentiable and thus is (Riemann) integrable. We say f(x) is odd if and only if…
- Question 14How many solutions are there to the equation tan(2x-xsin x)=1 for 0<x<π?
- Question 15A discrete random variable X takes non-negative values with distribution mathbb P(X=k)=ab^(-k), where k≥0 is an integer and a>0 and b>1 are…
- Question 16Find the value of log(frac12)+log(frac43)+log(frac94)+log(frac165)+…+log((n^2)/(n+1)) in the form log f(n) where f(n) is some function of n.
- Question 17We define the floor function lfloor xrfloor to be equal to the largest integer less than or equal to x, so that lfloor2rfloor=2 and…
- Question 18A sequence u_k is defined such that u_1=a, u_2=b and u_(n+1)=u_n-u_(n-1) for n≥2. Find the value of Σ_(k=1)^(62)u_k in terms of a and b.
- Question 19A sequence of functions f_k(x) is defined such that f_1(x)=α+β x and f_(n+1)(x)=frac14(d^2)/(d x^2)(∫_0^x xf_n(x) d x) for n≥1. What is the…