TMUA 2021 Paper 2

Find the value of \(\displaystyle\int_{1}^{4} \left(3 \sqrt{x} + \dfrac{4}{x^2}\right) d x\)

\(A(0, 2)\) and \(C(4, 0)\) are opposite vertices of the square \(A B C D\). What is the equation of the straight line through \(B\) and \(D\)?

A student is chosen at random from a class. Each student is equally likely to be chosen. Which of the following conditions is/are *necessary* for the probability that the student wears glasses to equal \(\dfrac{4}{15}\)? I Exactly 11 students in the class do not wear glasses. II The number of students in the class is divisible by 3. III The class contains 30 students, and 8 of them wear glasses.

Consider the following claim about positive integers \(a\), \(b\) and \(c\): *if* \(a\) is a factor of \(b c\), *then* \(a\) is a factor of \(b\) *or* \(a\) is a factor of \(c\) Which of the following provide(s) a *counterexample* to this claim? I \(a = 5, b = 10, c = 20\) II \(a = 8, b = 4, c = 4\) III \(a = 6, b = 7, c = 12\)

On which line is the first error in the following argument?

Consider the following two statements about the polynomial \(f(x)\): P: \(f(x) = 0\) for exactly three real values of \(x\) Q: \(f'(x) = 0\) for exactly two real values of \(x\) Which one of the following is correct?

A circle has equation \((x - 9)^2 + (y + 2)^2 = 4\) A square has vertices at \((1, 0)\), \((1, 2)\), \((-1, 2)\) and \((-1, 0)\). A straight line bisects both the area of the circle and the area of the square. What is the \(x\)-coordinate of the point where this straight line meets the \(x\)-axis?

Consider the following statement about the polynomial \(p(x)\), where \(a\) and \(b\) are real numbers with \(a < b\): \((*)\) There exists a number \(c\) with \(a < c < b\) such that \(p'(c) = 0\). Which one of the following is true?

Consider the following statements about a polynomial \(f(x)\): I \(f(x) = p x^3 + q x^2 + r x + s\), where \(p \neq 0\). II There is a real number \(t\) for which \(f'(t) = 0\). III There are real numbers \(u\) and \(v\) for which \(f(u) f(v) < 0\). Which of these statements is/are *sufficient* for the equation \(f(x) = 0\) to have a real solution?

The first seven terms of a sequence of positive integers are: \(u_1 = 15\), \(u_2 = 21\), \(u_3 = 30\), \(u_4 = 37\), \(u_5 = 44\), \(u_6 = 51\), \(u_7 = 59\) Consider the following statement about this sequence: \((*)\) *If* \(n\) is a prime number, *then* \(u_n\) is a multiple of 3 *or* \(u_n\) is a multiple of 5. What is the smallest value of \(n\) that provides a *counterexample* to \((*)\)?

A student attempts to solve the following problem, where \(a\) and \(b\) are non-zero real numbers: Show that *if* \(a^2 - 4 b^3 \geq 0\) *then* there exist real numbers \(x\) and \(y\) such that \(a = x y (x + y)\) and \(b = x y\). Consider the following attempt: \((x - y)^2 \geq 0\) (I) so \(x^2 + y^2 - 2 x y \geq 0\) (II) so \((x + y)^2 - 4 x y \geq 0\) (III) so \(x^2 y^2 (x + y)^2 - 4 x^3 y^3 \geq 0\) (IV) so \(a^2 - 4 b^3 \geq 0\) (V) Which of the following best describes this attempt?

Which of the following statements about polynomials \(f\) and \(g\) is/are true? I If \(f(x) \geq g(x)\) for all \(x \geq 0\), then \(\displaystyle\int_{0}^{x} f(t) d t \geq \displaystyle\int_{0}^{x} g(t) d t\) for all \(x \geq 0\). II If \(f(x) \geq g(x)\) for all \(x \geq 0\), then \(f'(x) \geq g'(x)\) for all \(x \geq 0\). III If \(f'(x) \geq g'(x)\) for all \(x \geq 0\), then \(f(x) \geq g(x)\) for all \(x \geq 0\).

A region \(R\) in the \((x, y)\)-plane is defined by the simultaneous inequalities \(y - x < 3\) \(y - x^2 < 1\) Which of the following statements is/are true for *every* point in \(R\)? I \(-1 < x < 2\) II \((y - x)(y - x^2) < 3\) III \(y < 5\)

Consider the following simultaneous equations, where \(p\) is a real number: \(p 2^x + \log_2 y = 2\) \(2^x + \log_2 y = 1\) What is the complete range of \(p\) for which these simultaneous equations have a real solution \((x, y)\)?

A circle has equation \(x^2 + a x + y^2 + b y + c = 0\) where \(a\), \(b\) and \(c\) are non-zero real constants. Which one of the following is a *necessary and sufficient* condition for the circle to be tangent to the \(y\)-axis?

\(p\) and \(q\) are real numbers, and the equation \(x |x| = p x + q\) has exactly \(k\) distinct real solutions for \(x\). Which one of the following is the complete list of possible values for \(k\)?

Consider the following functions defined for \(x > 1\): \(f(x) = \log_2(\log_2 \sqrt{x})\) \(g(x) = \log_2(\sqrt{\log_2 x})\) Which one of the following is true for all values of \(x > 1\)?

A student chooses two distinct real numbers \(x\) and \(y\) with \(0 < x < y < 1\). The student then attempts to draw a triangle \(A B C\) with: \(A B = 1\), \(\sin A = x\), \(\sin B = y\) Which of the following statements is/are correct? I For some choice of \(x\) and \(y\), there is exactly *one* triangle the student could draw. II For some choice of \(x\) and \(y\), there are exactly *two* different triangles the student could draw. III For some choice of \(x\) and \(y\), there are exactly *three* different triangles the student could draw. (Note that congruent triangles are considered to be the same.)

The angle \(\theta\) can take any of the values \(1^{\circ}, 2^{\circ}, 3^{\circ}, ..., 359^{\circ}, 360^{\circ}\). For how many of these values of \(\theta\) is it true that \(\sin \theta \sqrt{1 + \sin \theta} \sqrt{1 - \sin \theta} + \cos \theta \sqrt{1 + \cos \theta} \sqrt{1 - \cos \theta} = 0\)

A sequence of functions \(f_1, f_2, f_3, ...\) is defined by \(f_1(x) = |x|\) \(f_{n+1}(x) = |f_n(x) + x|\) for \(n \geq 1\) Find the value of \(\displaystyle\int_{-1}^1 f_99(x) d x\)