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TMUA differentiation and integration questions
All 68 differentiation and integration questions from the TMUA past papers, 2016 to 2023, newest first, each with a worked solution.
2023
- P1 Q1 · Given that ∫₀¹ (ax + b),dx = 1 and ∫₀¹ x(ax + b),dx = 1 find the value of a + b.
- P1 Q3 · For any integer n ≥ 0, ∫_nⁿ⁺¹ f(x),dx = n + 1 Evaluate ∫₀³ f(x),dx + ∫₁³ f(x),dx + ∫₂³ f(x),dx +…
- P1 Q10 · The trapezium rule with 4 strips is used to estimate the integral ∫_-2²√(4 - x²),dx What is the…
- P1 Q14 · The function f(x) = (2)/(3)x³ + 2mx² + n, m > 0 has three distinct real roots. What is the… hard
- P1 Q19 · The solution to the differential equation (dy)/(dx) = |-6x| for all x is y = f(x) + c, where c is…
- P2 Q2 · Evaluate ∫₉¹⁶((1)/(√(x)) + √(x))² dx - ∫₉¹⁶((1)/(√(x)) - √(x))² dx
- P2 Q17 · The ceiling of x, written ⌈ x ⌉, is defined to be the value of x rounded up to the nearest…
- P2 Q20 · Let f be a polynomial with real coefficients. The integral I_p,q where p Which of the following… hard
2022
- P1 Q3 · Given the following statements about a function f f''(x) = a for all x f(0) = 1, f(1) = 2 ∫₀¹…
- P1 Q6 · Given that ∫_log_2 5^log₂ 20 x,dx = log₂ M what is the value of M?
- P1 Q7 · Find the finite area enclosed between the line y = 0 and the curve y = x² - 4|x| - 12.
- P1 Q15 · A rectangle is drawn in the region enclosed by the curves p and q, where p(x) = 8 - 2x² q(x) = x²… hard
- P2 Q1 · Determine the number of stationary points on the curve with equation y = 3x⁴ + 4x³ + 6x² - 5
- P2 Q12 · Place the following integrals in order of size, starting with the smallest. P = ∫₀¹ 2^√(x),dx Q =…
2021
- P1 Q2 · The curve y = x³ - 6x + 3 has turning points at x = α and x = β, where β > α. Find ∫_α^β x³ - 6x +…
- P1 Q7 · The function f is such that f(0) = 0, and x,f(x) > 0 for all non-zero values of x. It is given… hard
- P1 Q10 · Use the trapezium rule with 3 strips to estimate ∫_(1)/(2)² 2log₁₀ x,dx
- P1 Q11 · The function f is given by f(x) = x^(1)/(7)(x² - x + 1) Find the fraction of the interval 0 < x <…
- P1 Q13 · The function f is such that, for every integer n ∫_nⁿ⁺¹ f(x),dx = n + 1 Evaluate ∑_r=1⁸(∫₀^r…
- P1 Q15 · The diagram shows the graph of y = f(x). It consists of alternating straight-line segments of… hard
- P2 Q1 · Find the value of ∫₁⁴(3√(x) + (4)/(x²))dx
- P2 Q8 · Consider the following statement about the polynomial p(x), where a and b are real numbers with a…
- P2 Q12 · Which of the following statements about polynomials f and g is/are true? I If…
- P2 Q20 · A sequence of functions f₁, f₂, f₃, … is defined by f₁(x) = |x| f_n+1(x) = |f_n(x) + x| for n ≥ 1… hard
2020
- P1 Q1 · Which of the following is an expression for the first derivative with respect to x of (x³ -…
- P1 Q11 · The diagram shows a quadratic function passing through (2, 0) and (q, 0), where q > 2. The curve…
- P1 Q14 · The area enclosed between the line y = mx and the curve y = x³ is 6. What is the value of m?
- P2 Q6 · The function f(x) is defined for all real values of x. Which of the following conditions on f(x)…
- P2 Q12 · Which one of A – F correctly completes the following statement? Given that a 0 for all x with a
- P2 Q13 · f(x) is a function for which ∫₀³ (f(x))² dx + ∫₀³ f(x),dx = ∫₀¹ f(x),dx Which of the following…
- P2 Q16 · The Fundamental Theorem of Calculus (FTC) tells us that for any polynomial f: (d)/(dx)(∫₀^x…
2019
- P1 Q7 · A curve has equation y = (2q - x²)(2qx + 3) The gradient of the curve at x = -1 is a function of…
- P1 Q8 · The function f is such that 0 The trapezium rule with n equal intervals is used to estimate ∫₀¹…
- P1 Q9 · p is a positive constant. Find the area enclosed between the curves y = p√(x) and x = p√(y).
- P1 Q10 · Evaluate ∫_-1³ |x|(1-x),dx
- P1 Q12 · It is given that (dV)/(dt) = (24π(t-1))/(1 + √(t)) for t ≥ 1 and V = 7 when t = 1. Find the value…
- P1 Q16 · Given that 2∫₀¹ f(x),dx + 5∫₁² f(x),dx = 14 and ∫₀¹ f(x+1),dx = 6 find the value of ∫₀² f(x),dx
- P2 Q13 · A student approximates the integral ∫_a^b sin² x,dx using the trapezium rule with 4 strips. The…
- P2 Q14 · Consider the following statements about the polynomial p(x), where a I p(a) ≤…
2018
- P1 Q1 · Find the value of ∫₁⁴ (3 - 2x)/(x√(x)),dx
- P1 Q9 · Find the complete set of values of the constant c for which the cubic equation 2x³ - 3x² - 12x + c…
- P1 Q11 · The line y = mx + 5, where m > 0, is normal to the curve y = 10 - x² at the point (p, q). What is…
- P1 Q12 · A curve has equation y = f(x), where f(x) = x(x - p)(x - q)(r - x) with 0 You are given that: ∫₀^r…
- P1 Q13 · The function f(x) has derivative f'(x). The diagram shows the graph of y = f'(x). Coming in from…
- P2 Q1 · The function f is given, for x > 0, by f(x) = (x³ - 4x)/(2√(x)) Find the value of f'(4).
- P2 Q18 · f(x) is a polynomial function defined for all real x. Which of the following is a necessary…
2017
- P1 Q1 · Given that (dy)/(dx) = 3x² - (2 - 3x)/(x³), x ≠ 0 and y = 5 when x = 1, find y in terms of x.
- P1 Q2 · The function f is given by f(x) = ((2)/(x) - (1)/(2x²))² (x ≠ 0) What is the value of f''(1)?
- P1 Q10 · A curve C has equation y = f(x) where f(x) = p³ - 6p²x + 3px² - x³ and p is real. The gradient of… hard
- P1 Q12 · The polynomial function f(x) is such that f(x) > 0 for all values of x. Given ∫₂⁴ f(x),dx = A,…
- P1 Q15 · It is given that f(x) = -2x² + 10 Consider the following three curves: (1) y = f(x)…
- P1 Q16 · The functions f and g are given by f(x) = 3x² + 12x + 4 and g(x) = x³ + 6x² + 9x - 8. What is the…
- P2 Q1 · Given that y = (1-3x)²2x^(3)/(2) which one of the following is a correct expression for (dy)/(dx)?
- P2 Q6 · A sequence u₀, u₁, u₂, … is defined as follows: u₀ = 1 u_n = ∫₀¹ 4x,u_n-1,dx for n ≥ 1 What is the…
- P2 Q10 · f(x) is a function defined for all real values of x. Which one of the following is a sufficient…
- P2 Q11 · The function f(x) is increasing and f(0) = 0. The positive constants a and b are such that a The…
- P2 Q16 · Consider the following statement: (*) If f(x) is an integer for every integer x, then f'(x) is an…
- P2 Q19 · Which one of the following is a sufficient condition for the equation x³ - 3x² + a = 0, where a is…
2016
- P1 Q3 · A line is drawn normal to the curve y = (2)/(x²) at the point on the curve where x = 1. This line…
- P1 Q5 · What is the total area enclosed between the curve y = x² - 1, the x-axis and the lines x = -2 and…
- P1 Q12 · A right circular cylinder is contained within a sphere of radius 5 cm in such a way that the whole…
- P1 Q13 · How many real roots does the equation 3x⁵ - 10x³ - 120x + 30 = 0 have?
- P1 Q15 · The least possible value of the gradient of the curve y = (2x + a)(x - 2a)² at the point where x =…
- P1 Q18 · The function 1-x[3]x² is defined for all x ≠ 0. The complete set of values of x for which the…
- P2 Q1 · Find the value of ∫₁² (x² - (4)/(x²))² dx
- P2 Q2 · f(x) = (x²+5)(2x)[4]x³, x > 0 Which one of the following is equal to f'(x)?
- P2 Q11 · f(x) is a polynomial with real coefficients. The equation f(x) = 0 has exactly two real roots, x =…
- P2 Q18 · Consider this statement about a function f(x): (*) If (f(x))² ≤ 1 for all -1 ≤ x ≤ 1 then…