Daniel J Smith Custom TMUA Set A Paper 1
20 questions · 75 min · Difficulty 50 · by Daniel J Smith
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Questions in this paper
- Question 1Let f(x)=a^(cos x)+a^(-cos x) with a>0 and real x The difference between the maximum and minimum values of f(x) is 4 Find the sum of the…
- Question 2For non-zero real numbers a and b, consider the equation log_3(x-a)=log_3x+b The number of real solutions x depends on a and b. Which of…
- Question 3Fix a>0 with a≠1 For x>0 consider the equation log_a x=x The number of positive solutions of this equation is
- Question 4Define functions g_1, g_2, g_3, g_4 on the real numbers by g_1(x)&=sin x g_2(x)&=sin(sin x) g_3(x)&=sin(sin(sin x)) g_4(x)&=sin(sin(sin(sin…
- Question 5Consider P(x)=(1+x)^8(1-2x)^5 The coefficient of x^3 in P(x) is equal to k times the coefficient of x^2 in P(x) What is the value of k?
- Question 6Let f be a polynomial defined for all real x, and define J=∫_(-2)^3(f(x)-f(|x|)) dx Which of the following expressions is equal to J for…
- Question 7Let a>0 and k be real numbers Consider the cubic g(x)=x^3-3ax+k For which complete range of values of k does the equation g(x)=0 have three…
- Question 8Consider the curve defined by (|x|-4)^2+(y-1)^2=1 How many distinct straight lines passing through the origin are tangents to this curve?
- Question 9For a real constant k, define A(k)=∫_(-1)^2|x-k| dx For which value of k is A(k) minimised?
- Question 10Let a and b be real numbers with a+b=1 What is the least possible value of (a^2+b)(b^2+a) ?
- Question 11Let k be an integer with 0≤ k≤10 Consider the quadratic equation x^2-2kx+(k^2-3k+2)=0 For how many values of k does the equation have two…
- Question 12The interior angle of a regular polygon with n sides exceeds the interior angle of a regular polygon with m sides by exactly 18°How many…
- Question 13Let c be a real constant. Consider the equation |sin x|+|cos x|=c for 0≤ x<2π Which of the following gives the number of distinct solutions…
- Question 14Let a, b, c be real constants with a>0 and c≠0. Suppose the inequality ax^2+bx+c>0 has solution set x s where r 0 ?
- Question 15A sequence (u_n) is defined by u_1>0 and u_(n+1)=(u_n)/(1+u_n) for all integers n≥1 Which of the following is equal to Σ_(n=1)^∞u_n?
- Question 16Let f(x)=x^3-3x+1 How many distinct real solutions does the equation f(x)=0 have?
- Question 17Let a and b be positive integers and define a sequence v_n by v_1=a, v_2=b, v_(n+2)=2v_(n+1)-v_n n≥1 Suppose the highest common factor of a…
- Question 18How many solutions are there to the equation 2sin x=|cos x| for 0°≤ x<360°, and what is the sum of all the solutions?
- Question 19A circle has centre O and radius 5 Points A and B lie on the circumference The area of triangle AOB is 10 What is the greatest possible…
- Question 20Let f(x)=x^3-3x Consider the following three curves: - (1) y=f(x) - (2) y=f(x)+2 - (3) the curve y=f(x) reflected in the x-axis The…