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TMUA algebra and functions questions
All 39 algebra and functions questions from the TMUA past papers, 2016 to 2023, newest first, each with a worked solution.
2023
- P1 Q2 · The graphs of y = x² + 5x + 6 and y = mx - 3, where m is a constant, are plotted on the same set…
- P2 Q1 · Given that (1)/(√(x) - 6) - (1)/(√(x) + 6) = (3)/(11) what is the value of x?
- P2 Q18 · The equation x⁴ + bx² + c = 0 has four distinct real roots if and only if which of the following…
2022
- P1 Q12 · A family of quadratic curves is given by y_k = 2(x - (k)/(2))² + (k²)/(2) + 4k + 3 where k is any…
- P1 Q13 · Given that (a³ + (2)/(b³))((2)/(a³) - b³) = √(2) where a and b are real numbers, what is the least… hard
- P1 Q16 · The solutions to 7x⁴ - 6x² + 1 = 0 are ±cosθ and ±cosβ. Which one of the following equations has…
2021
- P1 Q5 · The function f is such that, for all positive integers m and n: f(mn) = f(m)f(n) if mn is a…
- P1 Q8 · The line y = 2x + 3 meets the curve y = x² + bx + c at exactly one point. The line y = 4x - 2 also…
- P1 Q12 · The minimum value of the function x⁴ - p²x² is -9, where p is a real number. Find the minimum…
- P2 Q16 · p and q are real numbers, and the equation x|x| = px + q has exactly k distinct real solutions for… hard
2020
- P1 Q2 · (2x + 1) and (x - 2) are factors of 2x³ + px² + q. What is the value of 2p + q?
- P1 Q3 · Find the complete set of values of x for which (x+4)(x+3)(1-x) > 0 and (x+2)(x-2) < 0
- P1 Q5 · The curve S has equation y = px² + 6x - q where p and q are constants. S has a line of symmetry at…
- P1 Q8 · The function f is defined for all real x as f(x) = (p - x)(x + 2) Find the complete set of values…
- P1 Q9 · The quadratic expression x² - 14x + 9 factorises as (x - α)(x - β), where α and β are positive…
- P1 Q17 · Find the complete set of values of m in terms of c such that the graphs of y = mx + c and y = √(x)…
- P1 Q19 · Find the lowest positive integer for which x² - 52x - 52 is positive.
- P1 Q20 · For how many values of a is the equation (x - a)(x² - x + a) = 0 satisfied by exactly two distinct… hard
- P2 Q1 · Find the complete set of values of k for which the line y = x - 2 crosses or touches the curve y =…
- P2 Q18 · In this question, f(x) = ax³ + bx² + cx + d and g(x) = px³ + qx² + rx + s are cubic polynomials.…
2019
- P1 Q1 · f(x) is a quadratic function in x. The graph of y = f(x) passes through the point (1, -1) and has…
- P1 Q2 · Find the complete set of values of the real constant k for which the expression x² + kx + 2x + 1 -…
- P2 Q2 · (2x + 1) and (x - 2) are factors of 2x³ + px² + q. What is the value of 2p + q?
- P2 Q3 · a, b and c are real numbers. Given that ab = ac, which of the following statements must be true?…
- P2 Q16 · The graph of the quadratic y = px² + qx + p where p > 0, intersects the x-axis at two distinct…
- P2 Q19 · Find the value of the expression √(8 - 4√(2) + 1) + √(9 - 12√(2) + 8)
2018
- P1 Q4 · Consider the simultaneous equations 3x² + 2xy = 4 x + y = a where a is a real constant. Find the…
- P1 Q5 · The function f is defined by f(x) = x³ + ax² + bx + c. a, b and c take the values 1, 2 and 3 with…
- P1 Q10 · x and y satisfy |2 - x| ≤ 6 and |y + 2| ≤ 4. What is the greatest possible value of |xy|?
- P1 Q17 · There are two sets of data: the mean of the first set is 15, and the mean of the second set is 20.…
- P2 Q20 · It is given that the equation √(x+p) + √(x) = p has at least one real solution for x, where p is a…
2017
- P1 Q4 · When (3x² + 8x - 3) is multiplied by (px - 1) and the resulting product is divided by (x + 1), the…
- P1 Q5 · S is the complete set of values of x which satisfy both the inequalities x² - 8x + 12 9 The set S…
- P1 Q19 · The set of solutions to the inequality x² + bx + c In terms of p, q and c, what is the set of… hard
2016
- P1 Q1 · It is given that the expansion of (ax + b)³ is 8x³ - px² + 18x - 3√(3), where a, b and p are real…
- P1 Q2 · The expression 3x³ + 13x² + 8x + a, where a is a constant, has (x + 2) as a factor. Which one of…
- P1 Q6 · P, Q, and R are each mixtures of red and white paint. The percentage by volume of red paint in P…
- P1 Q7 · 60% of a sports club's members are women and the remainder are men. This sports club offers the…
- P2 Q15 · Consider the quadratic f(x) = x² - 2px + q and the statement: (*) f(x) = 0 has two real roots…