Worked TMUA problems
60 TMUA-calibre problems, free to read.
Each page carries one TMUA-calibre question, the full worked solution, and a video walkthrough where one has been produced. Pitched at the harder end of TMUA to give a real sense of the bank's calibre.
TMUA Sequences and Series
5 problemsRead the topic guide · Arithmetic, geometric, sum to infinity, and the binomial expansion for positive integer powers.
- LEMMA-ARITHMETIC-SERIES-01
The internal angles of quadrilateral (···) form an arithmetic progression. Triangles (···) and (···) are similar with (···) and (···) . Moreover, the angles in each of these two…
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- LEMMA-ARITHMETIC-SERIES-02
The numbers, in order, of each row and the numbers, in order, of each column of a (···) array of integers form an arithmetic progression of length (···) The numbers in positions…
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- LEMMA-ARITHMETIC-SERIES-03
What is the smallest positive integer (···) such that the sum of the first (···) positive integers equals (···) ?
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- LEMMA-ARITHMETIC-SERIES-04
Three non-zero real numbers form an arithmetic progression (in that order); their squares, taken in the same order, form a geometric progression. What are all possible values of…
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- LEMMA-ARITHMETIC-SERIES-05
The sum of the first (···) terms of an arithmetic progression, whose first term is an integer (not necessarily positive) and whose common difference is (···) , is known to be…
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TMUA Graph Sketching and Transformations
5 problemsRead the topic guide · Where curves go, how they shift, and how to read a transformed graph back into algebra.
- LEMMA-GRAPH-TRANSFORMATIONS-02
The function (···) is even, that is (···) for all real (···) . Which one of the following statements about the graph of (···) must be true?
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- LEMMA-GRAPH-TRANSFORMATIONS-01
The graph of (···) passes through the two points (···) and (···) . Which one of the following pairs of points must lie on the graph of (···) ?
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- LEMMA-GRAPH-TRANSFORMATIONS-03
Let (···) and (···) for all real (···) . Which one of the following statements about (···) in relation to (···) is true?
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- LEMMA-GRAPH-TRANSFORMATIONS-04
Which one of the following statements about the graph of (···) , relative to the graph of (···) , is correct?
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- LEMMA-GRAPH-TRANSFORMATIONS-05
Starting from the graph of (···) , the following three transformations are applied in order: first, a reflection in the (···) -axis; then, a translation by (···) units upward…
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TMUA Inequalities
4 problemsRead the topic guide · Sign analysis, denominator traps, and the rules you can't apply to inequalities without checking the sign.
- LEMMA-INEQUALITIES-01
Real numbers (···) and (···) are chosen with (···) such that no triangle with positive area has side lengths (···) , (···) , and (···) or (···) , (···) , and (···) . What is the…
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- LEMMA-INEQUALITIES-03
Consider the statement: (···) for all (···) with (···) . The statement is true
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- LEMMA-INEQUALITIES-04
The set of all real numbers (···) satisfying (···) is
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- LEMMA-INEQUALITIES-05
For every positive integer (···) , let (···) denote the integer closest to (···) , and let (···) . The number of elements in (···) is
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TMUA Integration with the Power Rule
4 problemsRead the topic guide · Area between curves, definite integrals, recovering a function from its derivative — the calculus core of Paper 1.
- LEMMA-INTEGRATION-POWER-RULE-01
The function (···) satisfies (···) for every real (···) , and (···) . What is the value of (···) ?
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- LEMMA-INTEGRATION-POWER-RULE-03
Which of the following is equal to (···) ?
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- LEMMA-INTEGRATION-POWER-RULE-02
For which positive value of (···) is (···) ?
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- LEMMA-INTEGRATION-POWER-RULE-04
Suppose (···) and (···) . What is the value of (···) ?
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TMUA Exponentials and Logarithms
5 problemsRead the topic guide · Substitute, factor, solve — quadratic-in-disguise is the most common pattern on Paper 1.
- LEMMA-EXPONENTIALS-LOGS-01
A right rectangular prism whose surface area and volume are numerically equal has edge lengths (···) and (···) What is (···)
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- LEMMA-EXPONENTIALS-LOGS-02
Without using a calculator, which one of the following statements about (···) and (···) is correct?
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- LEMMA-EXPONENTIALS-LOGS-03
Find the set of all real solutions of the equation (···) .
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- LEMMA-EXPONENTIALS-LOGS-04
What is the sum of all the solutions of (···) in the interval (···) ?
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- LEMMA-EXPONENTIALS-LOGS-05
What is the precise interval on which the function (···) is monotonically decreasing?
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TMUA Polynomials
6 problemsRead the topic guide · Factor theorem, remainder theorem, and the moves you make on a polynomial when one root is given for free.
- LEMMA-POLYNOMIALS-01
The roots of (···) are (···) and (···) What is the value of (···)
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- LEMMA-POLYNOMIALS-02
Let (···) be a (monic) polynomial with real coefficients satisfying (···) . What is the value of (···) ?
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- LEMMA-POLYNOMIALS-04
The polynomial equation (···) has at least one real root if and only if
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- LEMMA-POLYNOMIALS-05
How many different real values of (···) make the equation (···) have two identical (repeated) real roots?
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- LEMMA-STATIONARY-POINTS-04
The polynomial equation (···) has at least one real root if and only if
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- LEMMA-STATIONARY-POINTS-05
How many different real values of (···) make the equation (···) have two identical (repeated) real roots?
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TMUA Quadratics
6 problemsRead the topic guide · Discriminants, vertex form, parameter conditions — Paper 1's most-tested algebra family.
- LEMMA-NECESSARY-SUFFICIENT-05
The quadratic (···) has two distinct real roots, and the difference between the roots is greater than (···) and less than (···) — call this statement (···) . Which one of the…
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- LEMMA-POLYNOMIALS-03
If (···) , (···) , (···) are distinct odd natural numbers, then the number of rational roots of the quadratic (···)
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- LEMMA-QUADRATICS-02
Consider the quadratic equation (···) . The number of pairs (···) for which the equation has solutions of the form (···) and (···) for some (···) is
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- LEMMA-QUADRATICS-01
Consider the function (···) , (···) . Then the equation (···) has
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- LEMMA-QUADRATICS-03
Consider a quadratic equation (···) , where (···) , (···) and (···) are positive real numbers. If the equation has no real roots, then which of the following is true?
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- LEMMA-QUADRATICS-05
If (···) , (···) , (···) are distinct odd natural numbers, then the number of rational roots of the quadratic (···)
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TMUA Stationary Points
3 problemsRead the topic guide · Differentiate, set to zero, classify with the second derivative. Then read the question.
- LEMMA-STATIONARY-POINTS-01
The curve (···) has stationary points at (···) and (···) . Which one of the following correctly classifies them?
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- LEMMA-STATIONARY-POINTS-02
What is the unique stationary point of the curve (···) on its domain?
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- LEMMA-STATIONARY-POINTS-03
For which value of the real constant (···) does the curve (···) have its local minimum value equal to (···) ?
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TMUA Coordinate Geometry
5 problemsRead the topic guide · Lines, circles, gradients, distances, and the algebra of where two shapes meet.
- LEMMA-STRAIGHT-LINES-01
Square (···) lies in the first quadrant. Points (···) and (···) lie on lines (···) , and (···) , respectively. What is the sum of the coordinates of the center of the square (···)…
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- LEMMA-STRAIGHT-LINES-04
Let (···) and (···) . Let (···) and (···) be points on the (···) -axis, with (···) below (···) and (···) . Let (···) be the point of intersection of the lines (···) and (···) .…
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- LEMMA-STRAIGHT-LINES-02
A straight line is drawn through the point (···) making an angle (···) , with (···) , with the positive direction of the (···) -axis, meeting the line (···) at a point (···) such…
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- LEMMA-STRAIGHT-LINES-05
A straight line segment (···) of length (···) moves with end (···) on the (···) -axis and end (···) on the (···) -axis. The locus of the point (···) on the segment for which (···)…
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- LEMMA-TRIG-EQUATIONS-05
A straight line is drawn through the point (···) making an angle (···) , with (···) , with the positive direction of the (···) -axis, meeting the line (···) at a point (···) such…
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TMUA Trigonometric Identities
5 problemsRead the topic guide · Pythagorean, double-angle, compound-angle, and the small set of moves that collapses a long expression into one line.
- LEMMA-TRIG-IDENTITIES-01
Let (···) What is the value of (···)
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- LEMMA-TRIG-IDENTITIES-02
Let (···) . What is the mean of (···) ?
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- LEMMA-TRIG-IDENTITIES-03
If the area of the circumcircle of a regular polygon with (···) sides is (···) , then the area of the circle inscribed in the polygon is
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- LEMMA-TRIG-IDENTITIES-04
If (···) , then (···) equals
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- LEMMA-TRIG-IDENTITIES-05
Let (···) be an angle in the second quadrant ( (···) ) with (···) . Then the value of (···) is
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TMUA Trigonometric Equations
5 problemsRead the topic guide · Multiple-angle substitution, the unit circle, and counting solutions in a fixed interval.
- LEMMA-QUADRATICS-04
For which values of (···) with (···) does the quadratic in (···) given by (···) have repeated roots?
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- LEMMA-TRIG-EQUATIONS-02
How many values of (···) in the interval (···) satisfy (···)
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- LEMMA-TRIG-EQUATIONS-01
How many solutions does the equation (···) have on the interval (···)
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- LEMMA-TRIG-EQUATIONS-03
Suppose (···) is a real number such that the equation (···) has more than one solution in the interval (···) . The set of all such (···) that can be written in the form (···)…
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- LEMMA-TRIG-EQUATIONS-04
How many angles (···) with (···) satisfy (···) ?
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combinations-permutations
1 problemarea-under-curve
1 problemlogic-of-arguments
3 problems- LEMMA-NECESSARY-SUFFICIENT-01
Let (···) be positive real numbers satisfying:
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- LEMMA-NECESSARY-SUFFICIENT-03
Let (···) be statements such that: if (···) is true then (···) is true; if (···) is true then (···) is true; and if (···) is true then at least one of (···) and (···) is false.…
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- LEMMA-NECESSARY-SUFFICIENT-02
Let (···) be statements such that: if (···) is true then both (···) and (···) are true; and if both (···) and (···) are true then (···) is false. Which of the following must hold?
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