ITMC 2018 Grade 10 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.
$(A-B)^{\prime}$
$A^{\prime}\cup B$
$(B^{\prime}-A^{\prime})^{\prime}$
$B-A$
$(B^{\prime}\cap A)^{\prime}$
Finite set $A$ has no proper subset, and $A\in B$. What is the least number of elements of $A\cup\mathcal P(B)$? Here $\mathcal P$ denotes the power set.
0
1
2
3
4
$A=\{1,3,4\}$, $B=\{1,\{3\},\{4\},\{1,3,4\}\}$ and $C=\{\{4\},A,B\}$. How many elements are in $A\cup B\cup C$?
9
8
7
6
5
Let $S=\{a,b,\{a\},\{a,b\}\}$ and $A=\{x\mid x\subset S\text{ and }\{a\}\subset x\}$. What is $n(A)$? The source uses the symbol $\subset$.
15
14
8
7
4
$p\to(q\to r)$ is false. Consider (1) $p\leftrightarrow(q\leftrightarrow r)$; (2) $p\land(q\land r)$; (3) $p\lor(q\lor r)$. Which are false?
(1) and (2) only
(1) and (3) only
(2) and (3) only
All three
Fewer than two are false
Which proposition is not logically equivalent to the others?
$(p\lor\neg p)\land q$
$q\lor(p\land\neg p)$
$(p\leftrightarrow\neg p)\leftrightarrow\neg q$
$(p\to\neg p)\to\neg q$
$q\land q$
Consider (1) $\neg p\to(q\lor r)$; (2) $(\neg q\land\neg r)\to p$; (3) $p\lor q\lor r$. Which statement is true?
Only (1),(2) are equivalent.
Only (1),(3) are equivalent.
Only (2),(3) are equivalent.
No pair is equivalent.
All three are equivalent.
Find the negation of $\forall x\lt 0\ \exists y\lt 0\ [x\lt y\to x^2>y^2]$.
$\exists x\lt 0\ \forall y\lt 0\ [x\ge y\to x^2\le y^2]$
$\exists x\lt 0\ \forall y\lt 0\ [x\lt y\land x^2\le y^2]$
$\exists x\ge0\ \forall y\ge0\ [x\ge y\to x^2\le y^2]$
$\exists x\ge0\ \forall y\ge0\ [x\lt y\land x^2\le y^2]$
$\exists x\ge0\ \forall y\ge0\ [x\ge y\lor x^2>y^2]$
How many real solutions satisfy $\left|\left||x-2561|+2561\right|-2561\right|=543$?
1
2
3
4
More than 4
Find the solution set of $x^2(x-2)\le4(x-2)$.
$(-\infty,-2]$
$[-2,\infty)$
$[-2,2]$
$(-\infty,2]$
None of these.
Given $a-b>a$ and $a+b\lt b$, which statement must be true?
$a\lt b$
$b\lt a$
$a\lt b\lt 0$
$a\lt 0$ but $b>0$
$a\lt 0$ and $b\lt 0$
$2^x=\sqrt[3]{20}-\sqrt[3]{18}$ and $2^y=\sqrt[3]{400}+\sqrt[3]{360}+\sqrt[3]{324}$. Which relation holds?
$y=1-x$
$y=2-x$
$y=1/x$
$y=2/x$
None of these.
Of 40 surveyed students,36 entered both the fifth and sixth Thailand Mathematics Contests. Among these 36,24 won gold at the fifth and 25 at the sixth. What is the least number who won gold at both?
11
13
17
24
25
For propositions $a,b,c$, which choice for $P$ makes $((a\land b)\land(b\to c))\to P$ a tautology?
$\neg b$
$a\land\neg c$
$c\to\neg b$
$\neg a\lor(b\land c)$
$(a\lor c)\land\neg b$
$\forall x\ \exists y\ [(x-2)(2-y)>0]$ is true. Which set can be the common domain of x and y?
$\{0\}$
$\{1\}$
$\{3\}$
$\{0,1\}$
$\{1,3\}$
Real $x$ satisfies $1/x>-5$ and $1/x\lt 2$. Which statement describes the solutions?
$-1/5\lt x\lt 1/2$
$-1/2\lt x\lt 5$
$x>1/2$
$x>1/2\text{ or }-1/5\lt x\lt 0$
$x>1/2\text{ or }x\lt -1/5$
Which ordering is true?
$\frac{12}7\lt \frac{\sqrt6+1}2\lt \sqrt3$
$\frac{12}7\lt \sqrt3\lt \frac{\sqrt6+1}2$
$\sqrt3\lt \frac{\sqrt6+1}2\lt \frac{12}7$
$\frac{\sqrt6+1}2\lt \sqrt3\lt \frac{12}7$
$\frac{\sqrt6+1}2\lt \frac{12}7\lt \sqrt3$
Suppose $3x^2+9x^2+kx-12$ is divisible by $x+3$, with real k. Which polynomial divides $3x^2-9x^2+kx-12$? Both leading exponents are printed as 2.
$3x^2-4$
$3x^2+4$
$3x^2-x+4$
$3x-4$
$3x+4$
Real $x$ satisfies $(x+1)(x+2)(x+3)+(x+1)(x+2)+x+4=0$. Find the sum of all possible real x.
$-7$
$-6$
$-4$
$-3$
$-1$
Real $a,b$ satisfy $a^2+25a+61=0$ and $b^2+25b+61=0$. Find $(a+3)(b+3)$.
$-149$
$-5$
$145$
$167$
$217$
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