TMUA 2022 Paper 2

Determine the number of stationary points on the curve with equation \(y = 3x^4 + 4x^3 + 6x^2 - 5\)

Find the coefficient of the \(x^5\) term in the expansion of \((1+x)^5 \times \displaystyle\sum_{i=0}^5 x^i\)

Consider the following statement about the positive integer \(n\) if \(n\) is prime, then \(n^2 + 2\) is *not* prime Which of the following is a *counterexample* to this statement?
I. \(n = 2\)
II. \(n = 3\)
III. \(n = 4\)

The point \(P\) has coordinates \((p, q)\), and the equation of a circle is \(x^2 + 2f x + y^2 + 2g y + h = 0\) where \(f\), \(g\), \(h\), \(p\) and \(q\) are all real constants. Let \(L\) be the distance between the centre of the circle and the point \(P\). Which one of the following is *sufficient* on its own to be able to calculate \(L\)?

A straight line \(L\) passes through \((1, 2)\). Let P be the statement *if* the \(y\)-intercept of \(L\) is negative, *then* the \(x\)-intercept of \(L\) is positive. Which of the following statements *must* be true?
I. P
II. the converse of P
III. the contrapositive of P

A list consists of \(n\) integers. Consider the following statements: P: \(n\) is odd. Q: The median of the list is one of the numbers in the list. Which one of the following is true?

Consider the following claim: The difference between two consecutive positive cube numbers is always prime. Here is an attempted proof of this claim:
I. \((x+1)^3 = x^3 + 3x^2 + 3x + 1\)
II. Taking \(x\) to be a positive integer, the difference between two consecutive cube numbers can be expressed as \((x+1)^3 - x^3 = 3x^2 + 3x + 1\)
III. It is impossible to factorise \(3x^2 + 3x + 1\) into two linear factors with integer coefficients because its discriminant is negative.
IV. Therefore for every positive integer value of \(x\) the integer \(3x^2 + 3x + 1\) cannot be factorised.
V. Hence, the difference between two consecutive cube numbers will always be prime. Which of the following best describes this proof?

A selection, \(S\), of \(n\) terms is taken from the arithmetic sequence \(1, 4, 7, 10, ..., 70\). Consider the following statement: \((*)\) There are two distinct terms in \(S\) whose sum is \(74\). What is the smallest value of \(n\) for which \((*)\) is *necessarily* true?

Consider the following statement: \((*)\) For all real numbers \(x\), if \(x < k\) then \(x^2 < k\) What is the complete set of values of \(k\) for which \((*)\) is true?

Which of the following statements is/are true?
I. *For all* real numbers \(x\) and *for all* positive integers \(n\), \(x < n\)
II. *For all* real numbers \(x\), *there exists* a positive integer \(n\) such that \(x < n\)
III. *There exists* a real number \(x\) such that *for all* positive integers \(n\), \(x < n\)

The diagram shows a kite \(P Q R S\) whose diagonals meet at \(O\). \(O P = x\) \(O Q = y\) \(O R = x\) \(O S = z\) Which of the following is *necessary and sufficient* for angle \(S P Q\) to be a right angle?

Place the following integrals in order of size, starting with the smallest. \(P = \displaystyle\int_{0}^{1} 2^{\sqrt{x}} d x\) \(Q = \displaystyle\int_{0}^{1} 2^x d x\) \(R = \displaystyle\int_{0}^{1} (\sqrt{2})^x d x\)

Consider the statement \((*)\) about a real number \(x\): \((*)\) *There exists* a real number \(y\) such that \(x - x y + y\) is negative. For how many real values of \(x\) is \((*)\) true?

Consider the two inequalities: \(|x+5| < |x+11|\) \(|x+11| < |x+1|\) Which one of the following is correct?

The real numbers \(x\), \(y\) and \(z\) are all greater than 1, and satisfy the equations \(\log_x y = z\) and \(\log_y z = x\) Which one of the following equations for \(\log_z x\) must be true?

In this question, \(a_1, ..., a_100\) and \(b_1, ..., b_100\) and \(c_1, ..., c_100\) are three sequences of integers such that \(a_n \leq b_n + c_n\) for each \(n\). Which of the following statements *must* be true?
I. \((\text{minimum of } a_1, ..., a_100) \leq (\text{minimum of } b_1, ..., b_100) + (\text{minimum of } c_1, ..., c_100)\)
II. \((\text{minimum of } a_1, ..., a_100) \geq (\text{minimum of } b_1, ..., b_100) + (\text{minimum of } c_1, ..., c_100)\)
III. \((\text{maximum of } a_1, ..., a_100) \leq (\text{maximum of } b_1, ..., b_100) + (\text{maximum of } c_1, ..., c_100)\)

A student answered the following question: \(a\) and \(b\) are non-zero real numbers. Prove that the equation \(x^3 + a x^2 + b = 0\) has three distinct real roots if \(27 b \left(b + \dfrac{4 a^3}{27}\right) < 0\) Here is the student's solution:
I. We differentiate \(y = x^3 + a x^2 + b\) to get \(\dfrac{d y}{d x} = 3x^2 + 2a x = x(3x + 2a)\). Solving \(\dfrac{d y}{d x} = 0\) shows that the stationary points are at \((0, b)\) and \(\left(-\dfrac{2a}{3}, b + \dfrac{4 a^3}{27}\right)\)
II. If \(27 b \left(b + \dfrac{4 a^3}{27}\right) < 0\), then \(b\) and \(b + \dfrac{4 a^3}{27}\) must have opposite signs, and so one of the stationary points is above the \(x\)-axis and one is below.
III. If the cubic has three distinct real roots, then one of the stationary points is above the \(x\)-axis and one is below.
IV. Hence if \(27 b \left(b + \dfrac{4 a^3}{27}\right) < 0\), then the equation has three distinct real roots. Which one of the following options best describes the student's solution?

P, Q, R and S show the graphs of \(y = (\cos x)^{\cos x}\), \(y = (\sin x)^{\sin x}\), \(y = (\cos x)^{\sin x}\) and \(y = (\sin x)^{\cos x}\) for \(0 < x < \dfrac{\pi}{2}\) in some order. Which row in the following table correctly identifies the graphs?

A polygon has \(n\) vertices, where \(n \geq 3\). It has the following properties: - Every vertex of the polygon lies on the circumference of a circle \(C\). - The centre of the circle \(C\) is inside the polygon. - The radii from the centre of the circle \(C\) to the vertices of the polygon cut the polygon into \(n\) triangles of equal area. For which values of \(n\) are these properties *sufficient* to deduce that the polygon is regular?

The functions \(f_1\) to \(f_5\) are defined on the real numbers by \(f_1(x) = \cos x\) \(f_2(x) = \sin(\cos x)\) \(f_3(x) = \cos(\sin(\cos x))\) \(f_4(x) = \sin(\cos(\sin(\cos x)))\) \(f_5(x) = \cos(\sin(\cos(\sin(\cos x))))\) where all numbers are taken to be in radians. These functions have maximum values \(m_1\), \(m_2\), \(m_3\), \(m_4\) and \(m_5\), respectively. Which one of the following statements is true?