11. In \(ΔABC, D\) is the midpoint of \(BC, AM\) and \(MN\) are straight lines. If \(AB = m.AM\) and \(AC = n.AN\), find \((m + n)\).
(A) \(1\)
(B) \(2\)
(C) \(\frac{5}{2}\)
(D) \(\frac{3}{2}\)
(E) \(5\)
12. Given \(25^x=2000, 8^y=2000\), the value of \(\frac{1}{x}+\frac{1}{y}\) is ____
(A). \(\frac{1}{2}\)
(B). \(\frac{1}{3}\)
(C). \(\frac{3}{2}\)
(D). \(1\)
(E). \(2\)
13. Evaluate
\((1-\frac{1}{2^2}).(1-\frac{1}{3^2}).(1-\frac{1}{4^2})×…×(1-\frac{1}{2015^2}).(1-\frac{1}{2016^2})\)
(A). \(\frac{2015}{4032}\)
(B). \(\frac{2017}{4032}\)
(C). \(\frac{1}{2}\)
(D). \(1\)
(E). \(2\)
14. \(x\) and \(y\) are both real numbers that satisfy \(a^x<a^y\) for \(0<a<1\). Which of the following is true?
(A) \(\frac{1}{x^2+1}>\frac{1}{y^2+1}\)
(B) \(\ln{(x^2+1)}>\ln{(y^2+1)}\)
(C) \(\sin x>\sin y\)
(D) \(x^3 > y^3\)
(E) None of the above
15. Suppose \(A=(2+1)(2^2+1)(2^4+1)(2^8+1)(2^{16}+1)(2^{32}+1)\), find the ones digit in \(A-2016\)
(A) 3
(B) 2
(C) 1
(D) 0
(E) -1
16. Find the value of
\(\cos^2 α + \cos^2(α+\frac{2π}{3})+\cos^2(α-\frac{2π}{3})\)
(A) \(\frac{1}{2}\)
(B) \(1\)
(C) \(\frac{3}{2}\)
(D) \(\frac{\sqrt 5}{2}\)
(E) None of the above
17. \(a, b, c\) and \(d\) are all natural numbers, such that \(a^5 = b^4, c^3 = d^2\) and \(a-c=31\). Find the value of \(a\)
(A) 64
(B) 125
(C) 243
(D) 256
(E) None of the above
18. Given that \(\sqrt x + \sqrt y=\sqrt{2016}\), for \(0<x<y\). Find the number of pairs of \((x,y)\), that satisfy the equation.
(A) 3
(B) 5
(C) 7
(D) 9
(E) None of the above
19. The altitude \(OA\) of the cone is given as \(3\sqrt 3\) cm. The area of the lateral surface is a semicircle. Find the area of the lateral surface in π cm².
(A) 18
(B) 20
(C) 22
(D) 24
(E) 26
20. Find the value of \(m^4 +\frac{1}{m^4}\), given that \(m^2 – 2m +1 = 0\), for \(m∈R\)
(A) 20016
(B) 20160
(C) 20164
(D) 20169
(E) None of the above