IKMC 2027 - Grades 11-12 - Self-study SET 11 - Thematic Batch 3

Question 1

A geometric sequence starts with $a_1=\sqrt[4]{11}$ and $a_2=\sqrt[6]{11}$. What is its fourth term?

1. 11$\sqrt{9}$
2. 10$\sqrt{11}$
3. 12$\sqrt{11}$
4. 24$\sqrt{11}$
5. 1

Question 2

For integers x, y, z, what is the minimum of $\sqrt{x^{2}+1}$+$\sqrt{(x-y)^{2}+4}$+$\sqrt{(y-z)^{2}+1}$+$\sqrt{(z-10)^{2}+9}$?

Original printed choices: A) 7; B) 13; C) 4 + $\sqrt{109}$; D) $\sqrt{149}$; E) $\sqrt{125}$

Editorial task: Resolve the missing assumption or inconsistent choices. Derive the valid literal conclusion and distinguish it from the source’s intended answer.

Question 3

Alex lists all three-digit numbers that can be formed using odd digits. What is the sum of these numbers?

1. 69375
2. 19375
3. 244130625
4. 78125
5. 125

Question 4

A quadrilateral is circumscribed about a circle. Its perimeter is in the ratio 4: 3 to the circle’s circumference. What is the ratio of their areas?

1. 4 : π
2. 3$\sqrt{2}$ : π
3. 16 : 9
4. π : 3
5. 4 : 3

Question 5

Sina chooses and writes down a number from 1 to 2018. Zera also chooses a number from 1 to 2018. What is the probability that Zera’s number is larger?

Original printed choices: A) 1009/2018; B) 2019/4036; C) 2017/4036; D) 2017/4038; E) 2019/4038

Editorial task: Resolve the missing assumption or inconsistent choices. Derive the valid literal conclusion and distinguish it from the source’s intended answer.

Question 6

A 100×200 rectangular sheet has a unit-square grid. Tom cuts a zigzag border around it in the pattern shown. What is the perimeter of the resulting shape?

1. 600
2. 600$\sqrt{2}$
3. 592$\sqrt{2}$
4. 592
5. 596$\sqrt{2}$

Question 7

In $\overline{FOUR}=\overline{TWO}+\overline{TWO}$, different letters represent different digits and the same letter always represents the same digit. What is the greatest possible value of $\overline{FOUR}$?

1. 2468
2. 1974
3. 1876
4. 1846
5. 1682

Question 8

Fifteen numbers are arranged around the wheel as shown, and one is 10. Every seven consecutive entries have the same sum. Which listed value could be the sum of all fifteen numbers?

1. 100
2. 120
3. 160
4. 180
5. All the other choices are wrong

Question 9

There are 2021 balls in a row, each blue or red, with at least one of each colour. Every block of ten consecutive balls has the same number of blue balls, and every block of twelve consecutive balls has the same number of red balls. If the leftmost ball is blue, how many are red?

1. 1009
2. 1010
3. 1011
4. 2020
5. 2021

Question 10

How many positive integers p make p, p+2, p+4 all prime?

1. 1
2. 2
3. 3
4. From 4 to 10
5. More than 100

Question 11

What is the minimum of $\sqrt{1+x^{2}}$+$\sqrt{1+(3-x)^{2}}$?

Original printed choices: A) 1 + $\sqrt{10}$; B) 2$\sqrt{1 + 9/4}$; C) $\sqrt{13}$; D) $\sqrt{2}$ + $\sqrt{5}$; E) 5

Editorial task: Resolve the missing assumption or inconsistent choices. Derive the valid literal conclusion and distinguish it from the source’s intended answer.

Question 12

Hasan cuts his hollow conical hat horizontally at half its height, turns the upper part upside down and inserts it into the lower part, as shown. The original cone has radius 2 and height 8. What volume does the resulting shape enclose?

Model note: “Volume” means the enclosed capacity of the folded hollow shape, as in the source, rather than the material volume of the hat.

1. 7π
2. 8π
3. 11π
4. 9π
5. 7π

Question 13

An ant normally travels 5 metres along AB. Two obstacles, each adding a vertical height of 1 metre as shown, are built on the route. The ant must now climb vertically up and down both obstacles. How long is its new route?

1. 7
2. 9
3. 5 + 4$\sqrt{2}$
4. 9 − 2$\sqrt{2}$
5. It depends on the orientation of the obstacles.

Question 14

A circle is inscribed in a quarter of a larger disk, as shown. What is the ratio of the grey area to the white area?

1. 2 : 1
2. (1 + $\sqrt{2}$) : 1
3. (3 + 2$\sqrt{2}$) : 4
4. (2 + $\sqrt{2}$) : 2
5. 4 : (2$\sqrt{2}$ − 1)

Question 15

A rectangular sheet has length x and width y, with x>y. It can be rolled in two ways to form the curved surface of a cylinder. What is the volume ratio of the taller cylinder to the shorter cylinder?

1. y² : x²
2. y : x
3. 1:1
4. x : y
5. x² : y²

Question 16

In triangle ABC, the two circles through B and C meet at D and E as shown. Which statement about angle GEF must be true?

1. $\widehat{GEF}$ = 2$\widehat{A}$
2. $\widehat{GEF}$ = $\widehat{B}$ + $\widehat{C}$
3. $\widehat{GEF}$ = 180° − $\widehat{B}$ − $\widehat{C}$
4. $\widehat{GEF}$ = $\widehat{B}$ + $\widehat{C}$ − $\widehat{A}$
5. The size of $\widehat{GEF}$ depends on the dimensions of triangle ABC.

Question 17

In the diagram, ∠CAD=11° and AB equals the radius of the circle centred at O. What is ∠COD?

1. 60°
2. 44°
3. 33°
4. 66°
5. 22,5°

Question 18

Square ABCD has side 1. Two disks have centres A and C and both have radius 1. What is their common area?

1. 1/2
2. (π − 1)/2
3. (π − 2)/2
4. π − 3
5. π/4

Question 19

A footballer takes three consecutive penalties. He says the probability of scoring at least once is 0.999. What is his probability of scoring a single penalty?

Original printed choices: A) 0,7; B) 0,8; C) 0,9; D) 0,99; E) 0,999

Editorial task: Resolve the missing assumption or inconsistent choices. Derive the valid literal conclusion and distinguish it from the source’s intended answer.

Question 20

All animals of the same species have the same mass. D wolves weigh K kilograms, and E elephants weigh as much as M wolves. What is the mass of one elephant?

1. DKEM
2. DK/(EM)
3. KE/(DM)
4. KM/(DE)
5. None of the other four choices is correct.