TMUA 2017 Paper 1

Given that \( \dfrac{d y}{d x} = 3x^2 - \dfrac{2 - 3x}{x^3}, \quad x \neq 0 \) and \(y = 5\) when \(x = 1\), find \(y\) in terms of \(x\).

The function \(f\) is given by \( f(x) = \left(\dfrac{2}{x} - \dfrac{1}{2 x^2}\right)^2 \quad (x \neq 0) \) What is the value of \(f''(1)\)?

A line \(l\) has equation \(y = 6 - 2x\) A second line is perpendicular to \(l\) and passes through the point \((-6, 0)\). Find the area of the region enclosed by the two lines and the \(x\)-axis.

When \((3 x^2 + 8 x - 3)\) is multiplied by \((p x - 1)\) and the resulting product is divided by \((x + 1)\), the remainder is \(24\). What is the value of \(p\)?

Let \(S\) be the set of all values of \(x\) for which \( x^2 - 8 x + 12 < 0 \quad \text{and} \quad 2 x + 1 > 9 \) Which one of the following is equivalent to the statement that \(x\) is in \(S\)?

A tangent to the circle \(x^2 + y^2 = 144\) passes through the point \((20, 0)\) and crosses the positive \(y\)-axis. What is the value of \(y\) at the point where the tangent meets the \(y\)-axis?

The first three terms of an arithmetic progression are \(p\), \(q\) and \(p^2\) respectively, where \(p < 0\) The first three terms of a geometric progression are \(p\), \(p^2\) and \(q\) respectively. Find the sum of the first \(10\) terms of the arithmetic progression.

Find the complete set of values of \(x\), with \(0 \leq x \leq \pi\), for which \( (1 - 2 \sin x) \cos x \geq 0 \)

A circle has equation \(x^2 + y^2 - 18 x - 22 y + 178 = 0\) A regular hexagon is drawn inside this circle so that the vertices of the hexagon touch the circle. What is the area of the hexagon?

A curve \(C\) has equation \(y = f(x)\) where \( f(x) = p^3 - 6 p^2 x + 3 p x^2 - x^3 \) and \(p\) is real. The gradient of the normal to the curve \(C\) at the point where \(x = -1\) is \(M\). What is the greatest possible value of \(M\) as \(p\) varies?

The sequence \(x_n\) is defined by the rules \( x_1 = 7 \) \( x_{n+1} = \dfrac{23 x_n - 53}{5 x_n + 1} \) The first three terms in the sequence are \(7, 3, 1\) What is the value of \(x_100\)?

The function \(f(x)\) is defined for \(x \geq 0\). Given that \( \displaystyle\int_{2}^{4} f(x) d x = A, \) which one of the following is true?

In the expansion of \((a + b x)^5\) the coefficient of \(x^4\) is \(8\) times the coefficient of \(x^2\). Given that \(a\) and \(b\) are non-zero *positive* integers, what is the smallest possible value of \(a + b\)?

The solution of the simultaneous equations \( 2^x + 3 \times 2^y = 3 \) \( 2^{2 x} - 9 \times 2^{2 y} = 6 \) is \(x = p\), \(y = q\). Find the value of \(p - q\)

It is given that \(f(x) = -2 x^2 + 10\) Consider the following three curves: (1) \(y = f(x)\) (2) \(y = f(x + 1)\) (3) the curve \(y = f(x + 1)\) reflected in the line \(y = 6\) The trapezium rule is used to estimate the area under each of these three curves between \(x = 0\) and \(x = 1\). State whether the trapezium rule gives an overestimate or underestimate for each of these areas.

The functions \(f\) and \(g\) are given by \(f(x) = 3 x^2 + 12 x + 4\) and \(g(x) = x^3 + 6 x^2 + 9 x - 8\). What is the complete set of values of \(x\) for which one of the functions is increasing and the other decreasing?

The two functions \(F(n)\) and \(G(n)\) are defined as follows for positive integers \(n\): \( F(n) = \dfrac{1}{n} \displaystyle\int_{0}^{n} (n - x) d x \) \( G(n) = \displaystyle\sum_{r=1}^n F(r) \) What is the smallest positive integer \(n\) such that \(G(n) > 150\)?

The graph of \(y = \log_10 x\) is translated in the positive \(y\)-direction by \(2\) units. This translation is equivalent to a stretch of factor \(k\) parallel to the \(x\)-axis. What is the value of \(k\)?

The set of solutions to the inequality \(x^2 + b x + c < 0\) is the interval \(p < x < q\) where \(b\), \(c\), \(p\) and \(q\) are real constants with \(c < 0\). In terms of \(p\), \(q\) and \(c\), what is the set of solutions to the inequality \(x^2 + b c x + c^3 < 0\)?

The lengths of the sides \(Q R\), \(R P\) and \(P Q\) in triangle \(P Q R\) are \(a\), \(a + d\) and \(a + 2 d\) respectively, where \(a\) and \(d\) are positive and such that \(3 d > 2 a\). What is the full range, in degrees, of possible values for angle \(P R Q\)?