Skip to content / Đến nội dung
ThichHocToan.com

SASMO 2019 Grade 10

Question 1

Scoring for this platform: initial 0; Q1–15 correct +2, Q16–25 correct +4; wrong or blank 0; maximum 70. Official SASMO rules (reference only): initial 15; Section A correct +2, wrong −1, blank 0; Section B correct +4, wrong or blank 0; maximum 85.

One-fourth of $16^{40}$ is equal to
  1. $4^{40}$
  2. $16^{10}$
  3. $4^{79}$
  4. $4^{41}$
  5. None of the above

Question 2

The following calculations lead to the conclusion that $\pi\gt{}2019$.

(1) Let $x=\pi=3.14159\ldots$.

(2) $\frac{2x+38}{x-1000}\lt{}4$.

(3) $2x+38\lt{}4(x-1000)=4x-4000$.

(4) $2x\gt{}4038\Rightarrow x\gt{}2019$.

(5) Therefore, $\pi\gt{}2019$.

Which transition is the first that contains an error?
  1. (1) to (2)
  2. (2) to (3)
  3. (3) to (4)
  4. (4) to (5)
  5. None of the above

Question 3

The moment of inertia measures an object’s tendency to resist rotational acceleration. For a hollow sphere rotating about an axis through its centre, $I=\frac23mr^2$, where $m$ is its mass and $r$ its radius. If its moment of inertia becomes half the original and its mass is doubled, what happens to its radius?
  1. Halved
  2. Doubled
  3. Reduced to one-quarter of the original
  4. Reduced to one-eighth of the original
  5. None of the above

Question 4

Which of the following statements is not true?
  1. $\tan\theta\gt{}0$ for $0^\circ\lt{}\theta\lt{}90^\circ$
  2. $\sin\theta\lt{}0$ for $270^\circ\lt{}\theta\lt{}360^\circ$
  3. $\cos\theta\gt{}\sin\theta$ for $0^\circ\lt{}\theta\lt{}90^\circ$
  4. $\cos\theta\lt{}0$ for $90^\circ\lt{}\theta\lt{}180^\circ$
  5. None of the above

Question 5

How many prime numbers $x$ satisfy $\frac{x+15}{2}\ge\frac{3x-7}{4}+1\gt{}5\frac12$?
  1. 25
  2. 7
  3. 8
  4. 12
  5. None of the above

Question 6

Let $m$ be a positive integer such that the following sequence is an arithmetic progression. How many terms are in the list $\{6,13,20,27,\ldots,m+7\}$?
  1. $\frac{m+8}{7}$
  2. $\frac{m+1}{7}$
  3. $m+1$
  4. $\frac{m+7}{7}$
  5. None of the above

Question 7

A positive integer $n$ leaves remainder $3$ when divided by $5$. Which of the following numbers is divisible by $5$?
  1. $n+3$
  2. $n-2$
  3. $5n+3$
  4. $5n+2$
  5. $2n+4$

Question 8

The value of $\sqrt{18}+\sqrt{27}$ lies between two consecutive positive integers. Which are they?
  1. 9 and 10
  2. 8 and 9
  3. 10 and 11
  4. 7 and 8
  5. None of the above

Question 9

The coefficients in $(x+y)^n$ are related to Pascal’s triangle shown below. For example, $(x+y)^3=x^3+3x^2y+3xy^2+y^3$, whose coefficients $1,3,3,1$ form the fourth row. What is the sum of the coefficients of $(x+y)^8$?

Question illustration
  1. 128
  2. 256
  3. 512
  4. 127
  5. None of the above

Question 10

In a particular week this year, the probability of rain on any given day in Singapore is $50\%$. What is the probability that it rains on exactly two days of that week?

Interpretation: Assume rain events on different days are independent, as required by the intended source solution.
  1. $\frac14$
  2. $\frac3{64}$
  3. $\frac{21}{64}$
  4. $\frac{21}{128}$
  5. None of the above

Question 11

Find the remainder when $19^5-19$ is divided by $30$.
  1. 0
  2. 1
  3. 9
  4. 18
  5. None of the above

Question 12

In which figure is the number of black circles $111$ more than the number of white circles in the same figure?

Interpretation: Continue the displayed construction: add one row at each step, keeping the boundary white and the interior black.

Question illustration
  1. Figure 18
  2. Figure 19
  3. Figure 20
  4. Figure 12
  5. None of the above

Question 13

What is the last digit of $549^{123}\times549^{123}$?
  1. 1
  2. 9
  3. 3
  4. 6
  5. None of the above

Question 14

What is the $2019$th digit after the decimal point in the decimal form of $\frac{2019}{7}$?
  1. 5
  2. 8
  3. 7
  4. 1
  5. None of the above

Question 15

Which net below can be folded to form the solid shown?

Interpretation: Interpret the hidden lower rear position as filled: the solid has a complete 2 by 2 lower layer, with two diagonally opposite unit cubes above it. Fold only along the drawn panel edges. The hidden position is not fully specified by the perspective drawing; this convention makes the intended solid precise.

Question illustration

Question illustration
  1. Net A
  2. Net B
  3. Net C
  4. Net D
  5. Net E

Question 16

A number $A=20192019\ldots2019$ is formed by writing $2019$ next to itself $n$ times. What is the least positive $n$ for which $A$ is divisible by $9$?

Answer format: enter the integer answer without units. Leading zeros are optional; the original paper uses four-digit boxes.

Question 17

A list contains five positive integers. Their average is $86$, their median is $99$, and the largest is $12$ more than $20$ times the smallest. What is the maximum possible value of the largest number?

Answer format: enter the integer answer without units. Leading zeros are optional; the original paper uses four-digit boxes.

Question 18

How many four-digit perfect squares start with two 9s?

Answer format: enter the integer answer without units. Leading zeros are optional; the original paper uses four-digit boxes.

Question 19

How many digits are in $3^3\times8^{14}\times25^{18}$?

Answer format: enter the integer answer without units. Leading zeros are optional; the original paper uses four-digit boxes.

Question 20

In the diagram, $ABCD$ is a square and $H$ is the intersection of its diagonals. Points $B,E,F,G$ are collinear, with $G$ on $AD$ and $E$ on $AC$. Also, $AF$ bisects $\angle CAD$, and $\angle AFB=90^\circ$. If $EH=13\sqrt2$, find $AG$.

Answer format: enter the integer answer without units. Leading zeros are optional; the original paper uses four-digit boxes.

Question illustration