TMUA_Mock_Exam_Paper_1.pdf

TMUA Mock Exam Paper 1 AdmissionsEdge You have 75 minutes to answer all 20 Questions - Good Luck! 1 Question 1 Consider the parallelogram formed by the lines y = mx, y = mx + 1, y = nx, and y = nx + 1. The area of this parallelogram equa...

TMUA Mock Exam Paper 1 AdmissionsEdge You have 75 minutes to answer all 20 Questions - Good Luck! 1 Question 1 Consider the parallelogram formed by the lines y = mx, y = mx + 1, y = nx, and y = nx + 1. The area of this parallelogram equals: 1 A |m−n| 2 B |m+n| 1 C |m+n| |m+n| D (m−n)2 1 E |m−n|3 F |m + n|3 2 Question 2 Positive real numbers x,y not equal to 1 satisfy log2 x = logy 16 and xy = 64. What is (log2 xy )2 ? 25 A 2 B 20 45 C 2 D 25 E 32 3 Question 3 What is the inverse of the function f (x) = 2x(x−1) , with domain and range the set of real values greater than or equal to 1? A 0.5x(x−1) p
B 0.5 1 + 1 + 4 log2 x p
C 0.5 1 − 1 + 4 log2 x D 2x(x−1) E Undefined 4 Question 4 Given A = (sin x)2 + (cos x)4 , then for all real values of x, we have what? A 1≤A≤2 3 B 4 ≤A≤1 13 C 16 ≤A≤1 3 13 D 4 ≤A≤ 16 3 E 0≤A≤ 4 F − 13 16 ≤ A ≤ 0 5 Question 5 Circles with radii 1, 2, and 3 are mutually externally tangent. What is the area of the triangle determined by the points of tangency? 3 A 5 4 B 5 C 1 6 D 5 4 E 3 5 F 3 6 Question 6 The equations representing two pairs of opposite sides of a parallelogram are x2 − 5x + 6 = 0 and y 2 − 6y + 5 = 0. What are the equations of its diagonals? A x + 4y = 13 and y = 4x − 7 B 4x + y = 13 and 4y = x − 7 C 4x + y = 13 and y = 4x − 7 D y − 4x = 13 and y + 4x = 7 E y = 5 and x = 2 7 Question 7 The value of  v   u v s u u  !  1 u 1 u 1 1   3√2 t4 −  3√2 4 − 6 + log 32  √ 4− √ ··· u  t  3 2 3 2  is equal to what? A 2 B 3 C 4 D 5 E 6 F 7 8 Question 8 How many solutions of the equation tan x + sec x = 2 cos x lie in the interval [0, 2π]? (Define sec(x) as 1/cos(x)) A 0 B 1 C 2 D 3 E 4 F 5 G 6 9 Question 9 What is the area of the region enclosed by the graph of the equation x2 + y 2 = |x| + |y|? A π √ B π+ 2 C π+2 √ D π+2 2 E 2π √ F 2π + 2 √ G 2π + 2 2 10 Question 10 What is the number of√ terms√with rational coefficients among the 1001 terms in the expansion of (x 3 2 + y 2 3)1000 A 0 B 100 C 166 D 167 E 500 F 501 G 667 11 Question 11 Consider sequences of positive real numbers of the form x, 2000, y... in which each term after the first is 1 less than the product of its two immediate neighbors. For how many different values of x does the term 2001 appear somewhere in the sequence? Hint: Consider periodicity. A 0 B 1 C 2 D 3 E 4 F 5 G 6 12 Question 12 Suppose that |x + y| + |x − y| = 2. What is the maximum possible value of x2 − 6x + y 2 ? A 10 B 9 C 8 D 7 E 6 F 5 G 4 13 Question 13 (n2 −1)! For how many integers n between 1 and 50, inclusive, is (n!)n an integer? (Recall that 0! = 1.) A 35 B 34 C 33 D 32 E 30 F 29 14 Question 14 Let D(n) denote the number of ways of writing the positive integer n as a product n = f1 f2 · · · fk , where k ≥ 1, and the f ’s are integers strictly greater than 1, and the order in which the factors are listed matters (that is, two representations differing only in the order of the factors are considered distinct). For example, 6 can be written as 6, 2×3, and 3×2 so D(6) = 3. What is D(96)? A 112 B 128 C 144 D 172 E 184 F 196 15 Question 15 √ √ Let f1 (x) = 1 − x, and for integers n ≥ 2, let fn (x) = fn−1 ( n2 − x). If N is the largest value of n for which the domain of fn is nonempty, the domain of fN is [c]. What is N + c? A -226 B -144 C -20 D 20 E 144 F 226 16 Question 16 Define {x} as the fraction part of a positive real number x. For example {1.5} = 0.5 and {2.36} = 0.36. Calculate the integral Z 1   1 √ dx 0 x You may need to use the following result: for any positive integer k define ∞ X 1 = ζ(k) n=1 nk A 2 B 2 − ζ(2) C 2 − 2ζ(2) D 2 − ζ(4) E 2 − 2ζ(4) F 2 − 4ζ(4) G 1 17 Question 17 Let S be the circle x2 + y 2 = 4 and let P be a point on S lying in the top right quadrant. Let the tangent to S at P intersect the coordinate axes at M and N. Then on which of the following curves does the midpoint of M N lie? A (x + y)2 = 3xy B x2/3 + y 2/3 = 24/3 C x2 + y 2 = 2xy D x2 + y 2 = x2 y 2 18 Question 18 For 135◦ <
x < 180◦ , points P =
cos x, cos2 x , Q = cot x, cot2 x , R =

tan x, tan2 x and S = sin x, sin2 x are the vertices of a trapezoid with P in the top-left corner, labelled clockwise. What is sin(2x)? You may use the facts that sin(2x) = 2sin(x)cos(x) and that cot(x) = 1/tanx A 10 B 15 C 20 D 25 E 30 F 35 G 40 19 Question 19 At a competition with N players, the number of players given elite status is equal to 21+⌊log2 (N −1)⌋ − N. Suppose that 19 players are given elite status. What is the sum of the two smallest possible values of N ? Note that ⌊ x⌋ is the greatest integer less than or equal to x. A 38 B 90 C 154 D 406 E 1024 F 2048 G 512 20 Question 20 √ Let ABCD be a square of side length 3+1. Point P is on AC such that AP = √ 2. The square region bounded by ABCD is rotated 90◦ counterclockwise with center P , sweeping out a region whose area is 1c (aπ + b), where a, b, and c are positive integers and (a,b,c) are co-prime. What is a + b + c? A 15 B 17 C 19 D 21 E 23 F 25 21