TMUA Practice Paper 2

Evaluate \(\displaystyle\int_{1}^{2} \left(x^2 - \dfrac{4}{x^2}\right)^2 d x\).

Given \(f(x) = \dfrac{2x^3 + 10x}{x^{\dfrac{3}{4}}}\), find \(f'(x)\).

For \(0 < x < 2\pi\), the equation \(8 \sin^2 x + 4 \cos^2 x = 7\) has several solutions. What is the largest such solution?

Five urns P, Q, R, S and T each make a statement about how many balls are in each urn. Exactly one of these statements is true. Which urn makes the true statement?

Consider the statement (∗): every whole number \(n\) that is 1 less or 5 less than a multiple of 6 is prime. How many counterexamples to (∗) are there with \(1 \leq n \leq 49\)?

A sequence of functions is defined by \(f_1(x) = x^{10}\) and \(f_{n+1}(x) = x f_n'(x)\). Evaluate \(\displaystyle\sum_{n=1}^{20} f_n(x)\).

A proof that if \(\log_c d = (\log_a b)^2\) then \(d = b^{x y}\) (where \(x = \log_a b\) and \(y = \log_a c\)) is to be assembled from the given numbered lines. Which ordering of the lines gives a correct proof?

A point \((x, y)\) satisfies both \(x + y > 6\) and \(x - y < 4\). Which of the following conditions must then hold for every such point? (1) \(x > 1\); (2) \(y > 5\); (3) \((x + y)(x - y) > -24\).

Triangles \(A B C\) and \(X Y Z\) have \(A B = X Y\), \(B C = X Y\) given as equal corresponding pairs and equal areas. Which of the following extra conditions guarantee that the two triangles are congruent? (1) the areas are equal; (2) an angle at the end of the given side is equal; (3) two pairs of corresponding angles are equal.

Let \(x\) and \(y\) be real numbers, which may be positive or negative. Which one of the following conditions is sufficient to guarantee that \(x < y\)?

A polynomial \(y = f(x)\) meets the \(x\)-axis only at \(x = -p\) and \(x = p\). Which of the following statements must be true? (1) \(f\) has exactly one stationary point between \(-p\) and \(p\); (2) \(\displaystyle\int_{-p}^p f(x) d x = 2 \displaystyle\int_{0}^{p} f(x) d x\); (3) \(y = -f(-x)\) also meets the \(x\)-axis only at \(x = -p\) and \(x = p\).

For an arithmetic series with first term \(a\) and common difference \(d\), the sum of the first \(n\) terms is \(S_n\). Given that \(S_8 > 3 S_6\), what can be deduced about the signs of \(a\) and \(d\)?

In this question \(a\), \(b\) and \(c\) are positive integers. The following is an attempted proof of the false statement: If \(a\) divides \(b c\), then \(a\) divides \(b\) or \(a\) divides \(c\). ['\(a\) divides \(b c\)' means '\(a\) is a factor of \(b c\)'] Which line contains the error in this proof? 1. The statement is equivalent to 'if \(a\) does not divide \(b\) and \(a\) does not divide \(c\) then \(a\) does not divide \(b c\)'. 2. Suppose \(a\) does not divide \(b\) and \(a\) does not divide \(c\). Then the remainder when dividing \(b\) by \(a\) is \(r\), where \(0 < r < a\), and the remainder when dividing \(c\) by \(a\) is \(s\), where \(0 < s < a\). 3. So \(b = a x + r\) and \(c = a y + s\) for some integers \(x\) and \(y\). 4. Thus \(b c = a(a x y + x s + y r) + r s\). 5. So the remainder when dividing \(b c\) by \(a\) is \(r s\). 6. Since \(r > 0\) and \(s > 0\), it follows that \(r s > 0\). 7. Hence \(a\) does not divide \(b c\).

For a quartic \(y = f(x)\), the equation \(f(x) = 1\) has \(p\) solutions, \(f(x) = 2\) has \(q\) solutions, \(f(x) = 3\) has \(r\) solutions and \(f(x) = 4\) has \(s\) solutions. For which of the following sets of values is it impossible to draw such a quartic?

The quadratic \(f(x) = x^2 - 2 p x + q\) has two real roots whose difference \(r_2 - r_1\) satisfies \(2 < r_2 - r_1 < 4\) (condition ∗). This holds if and only if which of the following is true?

In the diagram, \(S R = 3\) and \(Q P = 12\), with the various triangles formed being similar. Find the length \(U T\).

Consider the graphs \(y = 3 \sin x + 2\) and \(y = x + c\). Which of the following statements are true for a suitable choice of \(c\)? (1) there is exactly one solution with \(0 \leq x \leq \pi\) and at least one solution with \(-\pi < x < 0\); (2) there is exactly one solution with \(0 \leq x \leq \pi\) and no solutions with \(x < 0\); (3) there is exactly one solution with \(0 \leq x \leq \pi\) and no solutions with \(x > 2\pi\).

Which of the following functions is a counterexample to the statement: 'If \((f(x))^2 \leq 1\) for all \(-1 \leq x \leq 1\), then \(\displaystyle\int_{-1}^1 (f(x))^2 d x \geq \displaystyle\int_{-1}^1 f(x) d x\)'?

The plan view, front elevation and side elevation of a solid object made of unit cubes are shown. How many unit cubes could the object contain?

The interior angle of a regular \(n\)-gon is \(\dfrac{3}{4}\) of the interior angle of a regular \(m\)-gon. How many pairs of integers \((n, m)\) with \(n, m \geq 3\) satisfy this condition?