tmua.co.uk TMUA Mock Paper 1
20 questions · 75 min · Difficulty 56 · by tmua.co.uk
Read the questions freely. Sign in to answer them, start a timed paper and save your progress.
Questions in this paper
- Question 1The sum of the first n terms of a sequence is given by the formula S_n=5n^2+kn, where k is a constant. The 6th term of the sequence is 52…
- Question 2Let P(x) be defined by: P(x)=x^(2027)-4x^(2026)+6. When P(x) is divided by x^2-1, the remainder is R(x)=ax+b. What is R(x)?
- Question 3Find the value of the constant k such that the equation (x^2-4x)^2-2k(x^2-4x)+k^2-9=0 has exactly three distinct real solutions for x.
- Question 4A sequence u_n is defined by u_0=3 and the recurrence relation u_(n+1)=1-frac1u_n for n≥0. Find the value of the sum S=Σ_(n=0)^(25)u_n.
- Question 5Consider the infinite series S=frac919+(99)/(19^2)+(999)/(19^3)+(9999)/(19^4)+…. In each term, the numerator is formed by writing the digit…
- Question 6In how many ways can 729 be written as a sum of two or more consecutive positive integers?
- Question 7Let P(x)=(2x^2-3x+2)^3. What is the sum of the coefficients of the even powers of x in the expansion of P(x)? (Count the constant term as…
- Question 8For integers n with 0≤ n≤20, consider the expression log_2((n-9)/(n-2)). For how many such integers n is this expression defined as a real…
- Question 9The curve C has equation y=x^3-6x^2+4x. The tangent to C at the point where x=1 intersects C again at the point P. The tangent to C at P…
- Question 10What is the shortest distance between the graphs of the following two equations in the xy-plane? |x|+|y|=6, |x-1|+|y-4|=13.
- Question 11The sum S_n is defined by S_n=Σ_(r=1)^nlog_2(1+(2r+3)/(r(r+2))). Find the smallest integer n such that S_n≥4+log_2 5.
- Question 12Consider the equation: 4sin xcos x+2√2cos x=k(2sin x+√2). Find the sum of the squares of all possible real values for the constant k such…
- Question 13Let f(y)=y^2-2. Find the number of distinct real solutions x to the equation f(f(f(|x|)))=0.
- Question 14Let a and b be positive real numbers. For every real x the following inequality holds: a≤frac41+e^x+frac11+e^(-x)+1≤ b. What is the minimum…
- Question 15For a real number x, let [x] denote the largest integer less than or equal to x. Find the value of ∫_(-4)^4(6x^2+[x^3]) dx.
- Question 16Evaluate the sum frac1log_2(64!)+frac1log_3(64!)+frac1log_4(64!)+…+frac1log_(64)(64!).
- Question 17Let S=Σ_(r=0)^(19)(-1)^rbinom32rbinom3213+r. Evaluate S.
- Question 18Let θ be a real number. Define two infinite series: S_1(θ)=sin^2θ+sin^4θ+sin^6θ+… S_2(θ)=cos^2θ+cos^4θ+cos^6θ+…. Both series converge…
- Question 19Find the sum of all distinct real solutions to the following equation: (x^2-7x+11)^(x^2-2x-2)=1.
- Question 20Let the “bimode” of a data set be equal to the regular mode if there is a singular most common value, but equal to the mean of the modes if…