Section 1 (3 POINTS FOR EACH ANSWER)
QUESTION 01
If set \(𝐴 = \{1,2,4 \}\), \(𝐵 = \{2,4,5 \}\), then \((𝐴 \cap 𝐵) = \bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(\{2, 4\}\)
(B) \(\{1, 2, 4, 5\}\)
(C) \(\{1\}\)
(D) \(\{1, 5\}\)
QUESTION 02
The domain of the function \((x – 1)^{\frac 12} + (4 – x)^{- \frac 32}\) is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \([1,4)\)
(B) \((- \sim, -1) \)
(C) \(4, + \sim)\)
(D) \((1, 4] \)
QUESTION 03
In the following formulae, tthe one that must be correct is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(\cos 2x = 2 \cos^2 x – 1 \)
(B) \(\cos 2x = 2 \sin^2 x – 1\)
(C) \(\sin 2x = 2 \cos^2 x – 1\)
(D) \(\sin 2x = 2 \sin^2 x – 1\)
QUESTION 04
Let \(𝑚\) be a real number. If line \(𝑙_1: 2𝑥 + 𝑚𝑦 + 1 = 0\) is parallel to line \(𝑙_2 : 𝑦 = 3𝑥 – 1\), then \(𝑚 = \bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(- \dfrac 23 \)
(B) \(\dfrac 23\)
(C) \(6 \)
(D) \(-6 \)
QUESTION 05
If the distance from moving point \(𝑃\) to point \(𝐹(2,0)\) equals to the distance from \(𝑃\) to the straight line \(𝑥 + 2 = 0\), then the trajectory equation of \(𝑃\) is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(y^2 = 8x\)
(B) \(y^2 = -8x\)
(C) \(x^2 = 8y\)
(D) \(x^2 = -8y\)
QUESTION 06
In the following functions, the one that is decreasing in its domain is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(y = 2^{-x}\)
(B) \(y = \cot x\)
(C) \(y = \dfrac {1}{x^2 + 1}\)
(D) \(y = x\)
QUESTION 07
Among the following options, tthe correct one is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\).
(A) \(y = x^3 + \dfrac 1x\) and \(y = \log_2 (x + \sqrt{x^2 + 1})\) are both odd functions
(B) \(y = x^3 + \dfrac 1x\) is an odd function but \(y = \log_2 (x + \sqrt{x^2 + 1})\) is not an odd function
(C) \(y = \log_2 (x + \sqrt{x^2 + 1})\) is an odd function but \(y = x^3 + \dfrac 1x\) is not an odd function
(D) Neither \(y = x^3 + \dfrac 1x\) nor \(y = \log_2 (x + \sqrt{x^2 + 1})\) is an odd functions
QUESTION 08
Given that \(\{a_n\}\) is an arithmetic sequence, and \(a_2 = 12\), \(𝑎_8 = 18\), then \(a_5 = \bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
Which of the following lines is parallel to \(3x – y + 1 = 0\). \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(15 \)
(B) \(6 \sqrt{6}\)
(C) \(30 \)
(D) \(216 \)
QUESTION 09
If the inverse function of \(f(x)\) is \(f^{-1} (x) = x^2 \: (x > 0)\), then \(f(4) = \bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(2\)
(B) \(-2 \)
(C) \(16\)
(D) \(-16 \)
QUESTION 10
\(\arctan \left(\dfrac {5 \pi}{6}\right) = \bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \( – \dfrac {\pi}{6}\)
(B) \(\dfrac {\pi}{6} \)
(C) \(- \dfrac {5 \pi}{6} \)
(D) \(\dfrac {5 \pi}{6} \)
QUESTION 11
Let \((-2 \sqrt{3}, 0)\) be one focus of the ellipse \(\dfrac {x^2}{a^2} + \dfrac {y^2}{b^2} = 1\). If \(a = 2b\), then \(b = \bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(2 \)
(B) \(\dfrac {\sqrt{60}}{5} \)
(C) \(4 \)
(D) \(\dfrac {12}{5} \)
QUESTION 12
Let \(a,b\) be positive real numbers, then statement “\(a + b \leq 2\)” is a \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\) for statement “\(a^2 + b^2 \leq 2\)”.
(A) necessary but not sufficient condition
(B) sufficient but not necessary condition
(C) neither sufficient nor necessary condition
(D) sufficient and necessary condition
QUESTION 13
The derivative of function \(\sin (2x + 3)\) is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(2 \cos (2x + 3) \)
(B) \(\cos (2x + 3) \)
(C) \(-2 \cos (2x + 3) \)
(D) \(-\cos (2x + 3) \)
QUESTION 14
The tangential equation of curve \(y = x^3 + 3x + 1\) at point \(P(0,1)\) is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(3x – y + 1 = 0 \)
(B) \(x – 3y + 3 = 0\)
(C) \(3x + y – 1 = 0\)
(D) \(x + 3y – 3 = 0 \)
Section 2 (4 POINTS FOR EACH ANSWER)
QUESTION 15
The solution set of the inequality \(\dfrac {1}{x – 2} < \dfrac 1x\) is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \((0,2) \)
(B) \((1,3) \)
(C) \(-\sim, -1) \)
(D) \((2,+ \sim) \)
(E) \(-1,0)\)
QUESTION 16
If the first term \(a_1\) of the geometric sequence \(\{a_n\}\) is \(1\), and the common quotient \(q = 2\), then the sum \(s_8\) of the first 8 terms of \(\{a_n\}\) is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) 255
(B) 127
(C) 63
(D) 511
(E) 512
QUESTION 17
If real numbers \(a,b\) satisfy \(2^{2a – b} = 4^{a + b} = 3\), then \(a = \bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(\log_4 3\)
(B) \(\log_2 3 \)
(C) \(\log_3 2 \)
(D) \(\log_3 4 \)
(E) \(\log_4 2 \)
QUESTION 18
Let the median of \(a, b, 1, 2\) be \(3\), the average of \(a, b, 1, 2\) be \(4\), than \(ab = \bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) 36
(B) 22
(C) 30
(D) 40
(E) 42
QUESTION 19
If point \(P\) is on the curve \(C: 2 x^2 – 4x + 2y^2 – 12y = 5\), and point \(Q\) is on the line \(x + y + 3 = 0\), then the minimum distance between point \(P\) and \(Q\) is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(\sqrt{2} \)
(B) \(\dfrac {\sqrt{2}}{2} \)
(C) \(1 \)
(D) \(0\)
(E) \(2\)
QUESTION 20
Let complex number \(z\) be defined \(z = \sqrt{3} + 1\), where \(i\) is the unit of imaginart numbers, then \(z^5 = \bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A)
(B)
(C)
(D)
(E)
QUESTION 21
In all two-digit numbers, the number of those which the sum of its ones digit and its tens digit is an even number is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) 45
(B) 25
(C) 40
(D) 20
(E) 50
QUESTION 22
If real numbers \(x,y\) satisfy \(\sin x \cos y = \dfrac 45\), \(\sin y \cos x = \dfrac 15\), then \(\cos 2x = \bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(- \dfrac 35\)
(B) \(\dfrac 35\)
(C) \(\dfrac 45\)
(D) \(- \dfrac 45\)
(E) \(\dfrac 25\)
QUESTION 23
Three students participated in high jump, long jump and shot put competitions. If everyone chooses two of these three items, then the probability that there are exactly two people choosing same items is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(\dfrac 23\)
(B) \(\dfrac 13\)
(C) \(\dfrac 29\)
(D) \(\dfrac 19\)
(E) \(\dfrac 16\)
QUESTION 24
Let the foci of the hyperbola \(C: x^2 – \dfrac {y^2}{3} = 1\) be \(F_1\) and \(F_2\) respectively. If point \(P\) is on \(C\) such that \(\dfrac {\sin (PF_2F_1)}{\sin (PF_1F_2)} = 2\), then \(\cos (F_1PF_2) = \bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(\dfrac 14\)
(B) \(-\dfrac 12\)
(C) \(\dfrac 16\)
(D) \(-\dfrac 15\)
(E) \(\dfrac 13\)
QUESTION 25
Let \(PABC\) be a tetrahedron whose edges have a length of 6 . If points \(D, E, F\) are the barycenter of triangles \(\Delta PAB\), \(\Delta PBC\), \(\Delta PAC\) respectively, then the volume of the trigonal pyramid \(PDEF\) is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(\dfrac {4\sqrt{2}}{3}\)
(B) \(\dfrac {\sqrt{2}}{3}\)
(C) \(\dfrac {3\sqrt{2}}{2}\)
(D) \(2\sqrt{2}\)
(E) \(4\sqrt{2}\)
QUESTION 26
Let sequences \(\{a_n\}\), \(\{b_n\}\) and \(\{c_n\}\) satisfy:
for an arbitrary integer \(n\), \(a_{n + 1} = (-1)^n \:.\: (a_n^2 + 1)\)
\(b_n = a_n \:.\: a_{n +1}\)
\(c_n = \cos a_n\)
Then the one that must be correct in the following statements is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(\{b_n\}\) is a decreasing sequence
(B) \(\{b_n\}\) is a increasing sequence
(C) \(\{c_n\}\) is a increasing sequence
(D) \(\{a_n\}\) is a increasingg sequence
(E) \(\{c_n\}\) is a decreasing sequence
QUESTION 27
Let points \(A, B, C\) be the three vertices of triangle \(\Delta ABC\), such that for an arbitrary real number \(\lambda\), the following always holds true:
\(| \overrightarrow {BA} – \lambda \overrightarrow {BC} | \geq 2 | \overrightarrow {BC} |\).
If \(| \overrightarrow {BC} | = 1\) then the minimum value of the perimeter of the triangle \(\Delta ABC\) is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(1 + \sqrt{17}\)
(B) \(1 + \sqrt{12}\)
(C) \(1 + \sqrt{7}\)
(D) \(1 + \sqrt{22}\)
(E) \(1 + \sqrt{27}\)
QUESTION 28
If \(x, y, z\) are all real numbers, and \(x^2 + y^2 + z^2 = 1\), then the maximum value of \(xy + 2yz\) is \(\bbox[10px, border: 2px solid red]{(\: \dotso \: )}\)
(A) \(\dfrac {\sqrt{5}}{2}\)
(B) \(\dfrac {\sqrt{2}}{2}\)
(C) \(\dfrac {\sqrt{3}}{2}\)
(D) \(2\)
(E) \(\dfrac {1}{2}\)