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Question bank SMO 2015

Exam papers

110 questions

Question 1

Find the exact value of $\dfrac{\sqrt{50}+7}{\sqrt{50}-7}+\dfrac{\sqrt{50}-7}{\sqrt{50}+7}$.
  1. 197
  2. 198
  3. 199
  4. 200
  5. 201

Question 2

Simplify $\dfrac{3^x+63}{21^{x-2}+7^{x-1}}$.
  1. $\frac3{7^x}$
  2. $\frac{9\times49}{7^x}$
  3. $\frac9{7^x}$
  4. $\frac{9\times7}{7^x}$
  5. $\frac{3\times7}{7^x}$

Question 3

Suppose $m$ and $n$ are real numbers such that the roots of the equation $2x^2-mx+8=0$ are $\alpha$ and $\beta$ while the roots of the equation $5x^2-10x+5n=0$ are $\dfrac1\alpha$ and $\dfrac1\beta$. Find the value of $mn$.
  1. $\frac14$
  2. 4
  3. 8
  4. 12
  5. 16

Question 4

Find the largest number among the following numbers:
  1. $\sqrt8+\sqrt8$
  2. $\sqrt7+\sqrt9$
  3. $\sqrt6+\sqrt{10}$
  4. $\sqrt5+\sqrt{11}$
  5. $\sqrt4+\sqrt{12}$

Question 5

Which of the following is true?
  1. $\cos221^\circ\gt \sin319^\circ$
  2. $\cos139^\circ\gt \sin221^\circ$
  3. $\sin139^\circ\gt \cos41^\circ$
  4. $\sin221^\circ\gt \cos139^\circ$
  5. $\sin41^\circ\gt \cos319^\circ$

Question 6

Find the smallest number among the following numbers:
  1. $\log_{2015}2016$
  2. $\log_{2016}2017$
  3. $\log_{2017}2018$
  4. $\log_{2018}2019$
  5. $\log_{2019}2020$

Question 7

If $x$ and $y$ are non-zero numbers such that $x\gt y$, which of the following is always true?
  1. $\frac1x\lt \frac1y$
  2. $\frac xy\gt 1$
  3. $|x|\gt |y|$
  4. $\frac1{xy^2}\gt \frac1{x^2y}$
  5. $\frac xy\gt \frac yx$

Question 8

If $f(x)=2|x^2-1|-x-1$, where $0\le x\le\dfrac32$, find the range of $f(x)$.
  1. $-3\le f(x)\le1$
  2. $-3\le f(x)\le\frac32$
  3. $-2\le f(x)\le1$
  4. $-2\le f(x)\le\frac32$
  5. $-2\le f(x)\le2$

Question 9

How many prime numbers $p$ satisfy the inequality $$\left|2+\log_{3/p}(3p^2)\right|\gt \frac32?$$
  1. 6
  2. 7
  3. 8
  4. 9
  5. 10

Question 10

When a polynomial $f(x)$ is divided by $(x-1)$ and $(x+5)$, the remainders are −6 and 6 respectively. Let $r(x)$ be the remainder when $f(x)$ is divided by $x^2+4x-5$. Find the value of $r(-2)$.
  1. 0
  2. 1
  3. 2
  4. 3
  5. 5

Question 11

Find the value of $$\frac{2+\log_2 3}{1+\log_2 3}+\frac{3+\log_3 4}{1+\log_3 2}.$$

Question 12

Find the absolute value of the coefficient of $\dfrac1x$ in the expansion of $$\left(2x^2-\frac1x\right)^{10}.$$

Question 13

If $a+b=\dfrac{25}4$ and $(1+\sqrt a)(1+\sqrt b)=\dfrac{15}2$, find the value of $ab$.

Question 14

Find the value of $p$ if there is a unique value of $x$ satisfying the equation $p^{2x+1}+1=\sqrt{44}\,e^{x\ln p}$.

Question 15

Suppose $x$ and $y$ are real numbers such that $x^2$ and $y^2$ are positive integers. Find the maximum value of $x^2-xy$ if $$(3x^2-y^2)^2+(x^2+y^2)^2=72.$$

Question 16

Find the smallest positive integer $x$ (measured in degrees) such that $$\tan(x-160^\circ)=\frac{\cos50^\circ}{1-\sin50^\circ}.$$

Question 17

Find the smallest positive integer $k$ such that $$\frac1{\log_{3^k}2015!}+\frac1{\log_{4^k}2015!}+\cdots+\frac1{\log_{2015^k}2015!}\gt 2015.$$

Question 18

Let $a\lt b\lt c$ be the three real roots of the equation $$3x^3-35x^2+500=0.$$ Find $\dfrac{500^2}{(bc)^2}+\dfrac{500^2}{(ac)^2}+\dfrac{500^2}{(ab)^2}$.

Question 19

A function $f$ satisfies $f(x)+f(3x)=x^2+1$ for all real numbers $x$. If $f(2)+f(18)=6$, determine the value of $f(6)$.

Question 20

It is given that $n$ consecutive positive even integers, the smallest of which is $x$, have a sum of 154. Find the smallest possible value of $x$.