ITMC 2018 Grade 11 National Preliminary set01 Next → Printed positive marks are preserved per question. The source uses different blank-answer and incorrect-answer rules by section, including an optional bonus penalty; the import does not reproduce that complete official scoring scheme.
$f_1(x)=(1/x)^{13}$, $x\ne0$
$f_2(x)=0^x$, $x\ne0$
$f_3(x)=1^{2x}$, $x\in\mathbb R$
$f_4(x)=(\sqrt3-\sqrt2)^{-x}$, $x\in\mathbb R$
$f_5(x)=(\log_2 3)^{-x}$, $x\in\mathbb R$
Given $\log_{72}2=a$, find $\log_{72}3$.
$\frac2{1+3a}$
$\frac{1+3a}2$
$\frac{1-3a}2$
$\frac{3a-1}2$
$\frac1{3a}-2$
Let $f(x)=\log_{2561}x$. Which set is the domain of $f\circ f$?
$[1,\infty)$
$(1,\infty)$
$(0,\infty)$
$[0,\infty)$
$(0,1)\cup(1,\infty)$
$ABCD$ is a rectangle. Construct square $ABEF$ with $C$ on $BE$ and $D$ on $AF$, and square $AGHD$ with $G$ on $AB$ and $H$ on $CD$. Their areas are $\log_4 27$ and $\log_9 8$, respectively. Find the area of $ABCD$.
$\frac32\log_2 3$
$\frac94\log_3 2$
$\frac32$
$\frac94$
None of these
Which of these values is the largest negative real number? Angles are in radians.
$\cos2$
$\cos3$
$\cos4$
$\cos5$
$\cos6$
For real $x$, simplify $\sin(x-1)\cos1+\cos(x-1)\sin1$.
$\sin x$
$\cos x$
$1$
$\sin x\cos2$
$\cos x\cos2$
In triangle $ABC$, $\cos A=5/13$ and $\cos B=3/5$. Find $\cos C$.
$-\frac{63}{65}$
$-\frac{33}{65}$
$1$
$\frac{33}{65}$
$\frac{63}{65}$
Let $\theta$ be acute and $\csc\theta+\cot\theta=3+2\sqrt2$. Find $\cos^2\theta$.
$\frac18$
$\frac78$
$\frac{17}{72}$
$\frac19$
$\frac89$
Find $\sqrt3\,\sqrt[3]2$.
$\sqrt2\,\sqrt[3]3$
$\sqrt[6]{108}$
$\sqrt[5]6$
$\sqrt[6]6$
None of these
Given $37^x=\sqrt[3]{20}+\sqrt[3]{17}$ and $37^y=\sqrt[3]{400}-\sqrt[3]{340}+\sqrt[3]{289}$, which relation is correct?
$y=-x+1$
$y=-x+\log37$
$y=1/x$
$y=(\log37)/x$
None of these
Real $x,y$ satisfy $(x+1)^2+(y+2)^2=0$. Find $(x-1)^2+(y-2)^2$.
$0$
$5$
$10$
$20$
$25$
Let $n$ be the greatest integer solution of $\left|\left||x-2018|+2018\right|-2018\right|=543$. Which listed number divides $n$?
$5$
$7$
$11$
$13$
$17$
Let $A$ be the solution set of $\log_{x+1}(x^2-3x+1)=1$. Consider (1) $A$ has two elements; (2) the product of all elements of $A$ is 0; (3) the sum of all elements of $A$ is 4. Which is correct?
Fewer than two statements are correct
Only (1) and (2)
Only (1) and (3)
Only (2) and (3)
All three
Use $\log_{10}3=0.4771$ and $\log_{10}5=0.6990$ to choose the correct ordering.
$\frac43\lt \log_3 4\lt \sqrt[3]4$
$\log_3 4\lt \frac43\lt \sqrt[3]4$
$\sqrt[3]4\lt \log_3 4\lt \frac43$
$\log_3 4\lt \sqrt[3]4\lt \frac43$
$\frac43\lt \sqrt[3]4\lt \log_3 4$
An integer $x\in[0,90]$ satisfies $\tan x^\circ=(\cos10^\circ+\cos70^\circ)/(\sin10^\circ+\sin70^\circ)$. Find $x$.
$30$
$40$
$50$
$60$
$70$
Real $x\in[0,2\pi]$ satisfies $\sin(x-4)=-1$. How many statements are correct? (1) $\sin x\lt 0$; (2) $\cos x\gt 0$; (3) $\sin2x\gt 0$; (4) $\cos2x\lt 0$.
$0$
$1$
$2$
$3$
$4$
Using $\pi=3.14$ as instructed, evaluate $\arcsin(\sin(-5))$.
$-5$
$-1.28$
$1.28$
$-1.86$
$1.86$
With universal set $\mathbb R$, solve $(3-2\sqrt2)^{x^2-5x-6}=(3+2\sqrt2)^{6+5x-x^2}$.
$\varnothing$
$[-1,6]$
$(-\infty,-1]\cup[6,\infty)$
$\mathbb R\setminus\{-1,6\}$
$\mathbb R$
The polynomial $x^4+5x^3+px^2+qx+r$ is divisible by $2+5x+6x^2+x^3$, where $p,q,r$ are real. Find $(p+r)q$.
$-8$
$3$
$4$
$8$
$9$
Real $x$ satisfies $(x-1)(x-2)(x-3)-(x-1)(x-2)+x-4=0$. Find the sum of all possible $x$.
$-7$
$1$
$3$
$4$
$7$
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