138 Reasoning question on Symbol notations

Here are some reasoning questions involving symbol notations:

1. Mathematical Symbol-Based Reasoning

Question:
If A⋆B=A2+B2A \star B = A^2 + B^2A⋆B=A2+B2, what is the value of 3⋆43 \star 43⋆4?

Solution:
3⋆4=32+42=9+16=253 \star 4 = 3^2 + 4^2 = 9 + 16 = 253⋆4=32+42=9+16=25.

2. Logical Symbol Replacement

Question:
In a certain code, △\triangle△ means addition, □\square□ means subtraction, and ∘\circ∘ means multiplication. What is the value of:5△3∘2□45 \triangle 3 \circ 2 \square 45△3∘2□4

Solution:(5+3)×2−4=8×2−4=16−4=12(5 + 3) \times 2 – 4 = 8 \times 2 – 4 = 16 – 4 = 12(5+3)×2−4=8×2−4=16−4=12

3. Pattern-Based Symbol Reasoning

Question:
If α\alphaα represents a prime number and β\betaβ represents an even number, which of the following expressions is always true?

α+β\alpha + \betaα+β is even.
α×β\alpha \times \betaα×β is odd.
α−β\alpha – \betaα−β is odd.

Solution:

Prime numbers (except 2) are odd.
Even numbers (β\betaβ) are multiples of 2.
Odd + Even = Odd (so statement 1 is false).
Odd × Even = Even (so statement 2 is false).
Odd – Even = Odd (so statement 3 is true).

Correct answer: Option 3

4. Symbol-Based Coding Reasoning

Question:
If ♡\heartsuit♡ stands for “greater than,” ♠\spadesuit♠ stands for “less than,” and ♣\clubsuit♣ stands for “equal to,” which of the following is correct?

A) 5♡3♣2♠15 \heartsuit 3 \clubsuit 2 \spadesuit 15♡3♣2♠1
B) 7♠9♣97 \spadesuit 9 \clubsuit 97♠9♣9
C) 4♡2♣24 \heartsuit 2 \clubsuit 24♡2♣2

Solution:

A) 5>3=2<15 > 3 = 2 < 15>3=2<1 → False
B) 7<9=97 < 9 = 97<9=9 → True
C) 4>2=24 > 2 = 24>2=2 → True

Correct answers: B and C

2. Symbol Substitution in Equations

Question:
If P@Q=P2−2PQ+Q2P @ Q = P^2 – 2PQ + Q^2P@Q=P2−2PQ+Q2, then what is the value of 7@37 @ 37@3?

Solution:7@3=72−2(7)(3)+32=49−42+9=167 @ 3 = 7^2 – 2(7)(3) + 3^2 = 49 – 42 + 9 = 167@3=72−2(7)(3)+32=49−42+9=16

6. Custom Arithmetic Operation

Question:
A new operation α\alphaα is defined as:AαB=A2−B2+2ABA \alpha B = A^2 – B^2 + 2ABAαB=A2−B2+2AB

Find the value of 5α35 \alpha 35α3.

Solution:5α3=52−32+2(5)(3)5 \alpha 3 = 5^2 – 3^2 + 2(5)(3)5α3=52−32+2(5)(3) =25−9+30=46= 25 – 9 + 30 = 46=25−9+30=46

7. Symbol-Based Logical Deduction

Question:
In a certain code language:

A⋆BA \star BA⋆B means “A is greater than B.”
A△BA \triangle BA△B means “A is less than B.”
A⋄BA \diamond BA⋄B means “A is equal to B.”

Which of the following is always true?
A) 7⋆5△67 \star 5 \triangle 67⋆5△6
B) 8△10⋄108 \triangle 10 \diamond 108△10⋄10
C) 9⋄9⋆89 \diamond 9 \star 89⋄9⋆8

Solution:

A) 7>5<67 > 5 < 67>5<6 → False
B) 8<10=108 < 10 = 108<10=10 → True
C) 9=9>89 = 9 > 89=9>8 → True

Correct answers: B and C

8. Symbol Pattern Identification

Question:
A pattern is defined as follows:A△B=(A+B)−(A×B)A \triangle B = (A + B) – (A \times B)A△B=(A+B)−(A×B)

Find the missing number:4△3=−5,6△2=−6,5△?=−44 \triangle 3 = -5, \quad 6 \triangle 2 = -6, \quad 5 \triangle ? = -44△3=−5,6△2=−6,5△?=−4

Solution:
Using the given operation:A△B=(A+B)−(A×B)A \triangle B = (A + B) – (A \times B)A△B=(A+B)−(A×B)

For 5△B=−45 \triangle B = -45△B=−4:(5+B)−(5×B)=−4(5 + B) – (5 \times B) = -4(5+B)−(5×B)=−4 5+B−5B=−45 + B – 5B = -45+B−5B=−4 5+B=5B−45 + B = 5B – 45+B=5B−4 5+4=5B−B5 + 4 = 5B – B5+4=5B−B 9=4B9 = 4B9=4B B=94B = \frac{9}{4}B=49

Since BBB should be an integer, there is no valid integer solution. This suggests an issue with the question setup. If you’d like, I can refine it to ensure integer values.

9. Symbol-Based Coding Problem

Question:
If #\## stands for addition, \@\@\@ stands for subtraction, and &\&& stands for multiplication, what is the value of:6&2#8\@46 \& 2 \# 8 \@ 46&2#8\@4

Solution:(6×2)+8−4=12+8−4=16(6 \times 2) + 8 – 4 = 12 + 8 – 4 = 16(6×2)+8−4=12+8−4=16

Correct answer: 16

10. Symbol-Based Inequality Reasoning

Question:
In a certain notation:

A⋆BA \star BA⋆B means “A is greater than B.”
A#BA \# BA#B means “A is equal to B.”
A@BA @ BA@B means “A is less than B.”

Which of the following is always true?
A) 5⋆3#3@75 \star 3 \# 3 @ 75⋆3#3@7
B) 8@10⋆48 @ 10 \star 48@10⋆4
C) 9#9@109 \# 9 @ 109#9@10

Solution:

A) 5>3=3<75 > 3 = 3 < 75>3=3<7 → True
B) 8<10>48 < 10 > 48<10>4 → False
C) 9=9<109 = 9 < 109=9<10 → True

Correct answers: A and C

11. Custom Mathematical Operation

Question:
A new operation ⋄\diamond⋄ is defined as:A⋄B=A2+2AB+B2A \diamond B = A^2 + 2AB + B^2A⋄B=A2+2AB+B2

Find the value of 4⋄24 \diamond 24⋄2.

Solution:4⋄2=42+2(4)(2)+224 \diamond 2 = 4^2 + 2(4)(2) + 2^24⋄2=42+2(4)(2)+22 =16+16+4=36= 16 + 16 + 4 = 36=16+16+4=36

Correct answer: 36

12. Symbol-Based Logical Statement

Question:
In a given symbolic system:

P♡QP \heartsuit QP♡Q means “P is the father of Q.”
P♠QP \spadesuit QP♠Q means “P is the brother of Q.”
P♣QP \clubsuit QP♣Q means “P is the son of Q.”

Which of the following statements is logically possible?
A) A♡B♠CA \heartsuit B \spadesuit CA♡B♠C
B) X♣Y♡ZX \clubsuit Y \heartsuit ZX♣Y♡Z
C) M♠N♣OM \spadesuit N \clubsuit OM♠N♣O

Solution:

A) AAA is the father of BBB, and BBB is the brother of CCC → Possible
B) XXX is the son of YYY, and YYY is the father of ZZZ → Possible
C) MMM is the brother of NNN, and NNN is the son of OOO → Possible

Correct answer: All are possible

13. Symbolic Number Series

Question:
A sequence is defined as follows:A△B=(A+B)−(A×B)A \triangle B = (A + B) – (A \times B)A△B=(A+B)−(A×B)

If 3△2=−13 \triangle 2 = -13△2=−1 and 5△4=−55 \triangle 4 = -55△4=−5, find 6△36 \triangle 36△3.

Solution:
Using the given operation:A△B=(A+B)−(A×B)A \triangle B = (A + B) – (A \times B)A△B=(A+B)−(A×B)

For 6△36 \triangle 36△3:(6+3)−(6×3)=9−18=−9(6 + 3) – (6 \times 3) = 9 – 18 = -9(6+3)−(6×3)=9−18=−9

Correct answer: -9

14. Symbolic Arithmetic Puzzle

Question:
If ⋆\star⋆ means square of the number, #\## means double the number, and \@\@\@ means add 5 to the number, what is the value of:4⋆#\@34 \star \# \@ 34⋆#\@3

Solution:4⋆=42=164 \star = 4^2 = 164⋆=42=16 16#=16×2=3216 \# = 16 \times 2 = 3216#=16×2=32 32\@=32+5=3732 \@ = 32 + 5 = 3732\@=32+5=37

Correct answer: 37

15. Symbolic Inequality Challenge

Question:
In a symbolic notation:

A#BA \# BA#B means “A is greater than or equal to B.”
A\@BA \@ BA\@B means “A is less than B.”
A&BA \& BA&B means “A is equal to B.”

Which of the following is always true?
A) 6#5\@46 \# 5 \@ 46#5\@4
B) 8\@9&98 \@ 9 \& 98\@9&9
C) 10&10#910 \& 10 \# 910&10#9

Solution:

A) 6≥5<46 \geq 5 < 46≥5<4 → False
B) 8<9=98 < 9 = 98<9=9 → True
C) 10=10≥910 = 10 \geq 910=10≥9 → True

Correct answers: B and C

16. Custom Arithmetic Operation

Question:
A new operation ⊕\oplus⊕ is defined as:A⊕B=A2−B2+ABA \oplus B = A^2 – B^2 + ABA⊕B=A2−B2+AB

Find the value of 7⊕37 \oplus 37⊕3.

Solution:7⊕3=72−32+(7×3)7 \oplus 3 = 7^2 – 3^2 + (7 \times 3)7⊕3=72−32+(7×3) =49−9+21=61= 49 – 9 + 21 = 61=49−9+21=61

Correct answer: 61

17. Symbol-Based Logical Deduction

Question:
In a certain code language:

A♡BA \heartsuit BA♡B means “A is the mother of B.”
A♠BA \spadesuit BA♠B means “A is the sister of B.”
A♣BA \clubsuit BA♣B means “A is the daughter of B.”

Which of the following is logically correct?
A) X♡Y♠ZX \heartsuit Y \spadesuit ZX♡Y♠Z
B) M♣N♡OM \clubsuit N \heartsuit OM♣N♡O
C) P♠Q♣RP \spadesuit Q \clubsuit RP♠Q♣R

Solution:

A) XXX is the mother of YYY, and YYY is the sister of ZZZ → Possible
B) MMM is the daughter of NNN, and NNN is the mother of OOO → Possible
C) PPP is the sister of QQQ, and QQQ is the daughter of RRR → Possible

Correct answer: All are possible

18. Symbolic Number Series

Question:
A sequence is defined as follows:A⋄B=(A×B)+(A+B)A \diamond B = (A \times B) + (A + B)A⋄B=(A×B)+(A+B)

If 3⋄2=113 \diamond 2 = 113⋄2=11 and 5⋄4=295 \diamond 4 = 295⋄4=29, find 6⋄36 \diamond 36⋄3.

Solution:
Using the given operation:A⋄B=(A×B)+(A+B)A \diamond B = (A \times B) + (A + B)A⋄B=(A×B)+(A+B)

For 6⋄36 \diamond 36⋄3:(6×3)+(6+3)=18+9=27(6 \times 3) + (6 + 3) = 18 + 9 = 27(6×3)+(6+3)=18+9=27

Correct answer: 27

19. Symbolic Arithmetic Puzzle

Question:
If ⋆\star⋆ means cube of the number, #\## means half the number, and \@\@\@ means subtract 4 from the number, what is the value of:4⋆#\@34 \star \# \@ 34⋆#\@3

Solution:4⋆=43=644 \star = 4^3 = 644⋆=43=64 64#=64÷2=3264 \# = 64 \div 2 = 3264#=64÷2=32 32\@=32−4=2832 \@ = 32 – 4 = 2832\@=32−4=28

Correct answer: 28

20. Symbolic Inequality Challenge

Question:
In a symbolic notation:

A#BA \# BA#B means “A is greater than or equal to B.”
A\@BA \@ BA\@B means “A is less than B.”
A&BA \& BA&B means “A is equal to B.”

Which of the following is always true?
A) 7#5\@47 \# 5 \@ 47#5\@4
B) 10\@12&1210 \@ 12 \& 1210\@12&12
C) 8&8#78 \& 8 \# 78&8#7

Solution:

A) 7≥5<47 \geq 5 < 47≥5<4 → False
B) 10<12=1210 < 12 = 1210<12=12 → True
C) 8=8≥78 = 8 \geq 78=8≥7 → True

Correct answers: B and C

21. Symbol-Based Logical Puzzle

Question:
If:

P△QP \triangle QP△Q means “P is the father of Q.”
P□QP \square QP□Q means “P is the brother of Q.”
P\hexagonQP \hexagon QP\hexagonQ means “P is the son of Q.”

Which of the following is definitely incorrect?
A) X△Y□ZX \triangle Y \square ZX△Y□Z
B) A\hexagonB△CA \hexagon B \triangle CA\hexagonB△C
C) P□Q\hexagonRP \square Q \hexagon RP□Q\hexagonR

Solution:

A) XXX is the father of YYY, and YYY is the brother of ZZZ → Possible
B) AAA is the son of BBB, and BBB is the father of CCC → Possible
C) PPP is the brother of QQQ, and QQQ is the son of RRR → Incorrect, because a brother cannot directly relate to someone being a son.

Correct answer: C

22. Symbolic Equation Challenge

Question:
If:A□B=A2+B2−2ABA \square B = A^2 + B^2 – 2ABA□B=A2+B2−2AB

Find the value of 6□26 \square 26□2.

Solution:6□2=62+22−2(6)(2)6 \square 2 = 6^2 + 2^2 – 2(6)(2)6□2=62+22−2(6)(2) =36+4−24=16= 36 + 4 – 24 = 16=36+4−24=16

Correct answer: 16

23. Symbol-Based Logical Condition

Question:
If:

P♡QP \heartsuit QP♡Q means “P is taller than Q.”
P♠QP \spadesuit QP♠Q means “P is shorter than Q.”
P♣QP \clubsuit QP♣Q means “P is as tall as Q.”

Which of the following is always true?
A) 6♡5♠46 \heartsuit 5 \spadesuit 46♡5♠4
B) 8♠10♣108 \spadesuit 10 \clubsuit 108♠10♣10
C) 9♣9♡89 \clubsuit 9 \heartsuit 89♣9♡8

Solution:

A) 6>5<46 > 5 < 46>5<4 → False
B) 8<10=108 < 10 = 108<10=10 → True
C) 9=9>89 = 9 > 89=9>8 → True

Correct answers: B and C

24. Custom Arithmetic Operation

Question:
A new operation ⊗\otimes⊗ is defined as:A⊗B=(A+B)×(A−B)A \otimes B = (A + B) \times (A – B)A⊗B=(A+B)×(A−B)

Find the value of 8⊗38 \otimes 38⊗3.

Solution:8⊗3=(8+3)×(8−3)8 \otimes 3 = (8 + 3) \times (8 – 3)8⊗3=(8+3)×(8−3) =11×5=55= 11 \times 5 = 55=11×5=55

Correct answer: 55

25. Symbolic Relationship Reasoning

Question:
In a certain notation system:

A♡BA \heartsuit BA♡B means “A is the sister of B.”
A♠BA \spadesuit BA♠B means “A is the father of B.”
A♢BA \diamondsuit BA♢B means “A is the uncle of B.”

Which of the following is definitely incorrect?
A) X♠Y♡ZX \spadesuit Y \heartsuit ZX♠Y♡Z
B) A♢B♠CA \diamondsuit B \spadesuit CA♢B♠C
C) P♡Q♢RP \heartsuit Q \diamondsuit RP♡Q♢R

Solution:

A) XXX is the father of YYY, and YYY is the sister of ZZZ → Possible
B) AAA is the uncle of BBB, and BBB is the father of CCC → Possible
C) PPP is the sister of QQQ, and QQQ is the uncle of RRR → Incorrect (because a sister cannot be directly related to an uncle).

Correct answer: C

26. Symbol-Based Number Puzzle

Question:
A mathematical operation △\triangle△ is defined as:A△B=A2+B2−ABA \triangle B = A^2 + B^2 – ABA△B=A2+B2−AB

If 5△2=215 \triangle 2 = 215△2=21 and 6△3=336 \triangle 3 = 336△3=33, find 7△47 \triangle 47△4.

Solution:
Using the operation:A△B=A2+B2−ABA \triangle B = A^2 + B^2 – ABA△B=A2+B2−AB

For 7△47 \triangle 47△4:72+42−(7×4)=49+16−28=377^2 + 4^2 – (7 \times 4) = 49 + 16 – 28 = 3772+42−(7×4)=49+16−28=37

Correct answer: 37

27. Symbolic Arithmetic Challenge

Question:
If ⋆\star⋆ means cube of the number, #\## means triple the number, and \@\@\@ means subtract 6 from the number, find the value of:3⋆#\@23 \star \# \@ 23⋆#\@2

Solution:3⋆=33=273 \star = 3^3 = 273⋆=33=27 27#=27×3=8127 \# = 27 \times 3 = 8127#=27×3=81 81\@=81−6=7581 \@ = 81 – 6 = 7581\@=81−6=75

Correct answer: 75

28. Symbolic Inequality Puzzle

Question:
In a symbolic notation:

A#BA \# BA#B means “A is greater than or equal to B.”
A\@BA \@ BA\@B means “A is less than B.”
A&BA \& BA&B means “A is equal to B.”

Which of the following is always true?
A) 9#8\@79 \# 8 \@ 79#8\@7
B) 11\@12&1211 \@ 12 \& 1211\@12&12
C) 10&10#910 \& 10 \# 910&10#9

Solution:

A) 9≥8<79 \geq 8 < 79≥8<7 → False
B) 11<12=1211 < 12 = 1211<12=12 → True
C) 10=10≥910 = 10 \geq 910=10≥9 → True

Correct answers: B and C

29. Symbolic Relationship Puzzle

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the brother of B.”
A□BA \square BA□B means “A is the father of B.”
A\hexagonBA \hexagon BA\hexagonB means “A is the grandfather of B.”

Which of the following is definitely incorrect?
A) X□Y△ZX \square Y \triangle ZX□Y△Z
B) A\hexagonB□CA \hexagon B \square CA\hexagonB□C
C) P△Q\hexagonRP \triangle Q \hexagon RP△Q\hexagonR

Solution:

A) XXX is the father of YYY, and YYY is the brother of ZZZ → Possible
B) AAA is the grandfather of BBB, and BBB is the father of CCC → Possible
C) PPP is the brother of QQQ, and QQQ is the grandfather of RRR → Incorrect (a brother cannot directly be linked to a grandfather).

Correct answer: C

30. Symbol-Based Mathematical Equation

Question:
If:A⋄B=(A×B)+(A−B)A \diamond B = (A \times B) + (A – B)A⋄B=(A×B)+(A−B)

Find the value of 7⋄47 \diamond 47⋄4.

Solution:7⋄4=(7×4)+(7−4)7 \diamond 4 = (7 \times 4) + (7 – 4)7⋄4=(7×4)+(7−4) =28+3=31= 28 + 3 = 31=28+3=31

Correct answer: 31

31. Custom Arithmetic Operation

Question:
A new operation ⊕\oplus⊕ is defined as:A⊕B=A2+B2+2AB−(A+B)A \oplus B = A^2 + B^2 + 2AB – (A + B)A⊕B=A2+B2+2AB−(A+B)

Find the value of 5⊕35 \oplus 35⊕3.

Solution:5⊕3=52+32+2(5)(3)−(5+3)5 \oplus 3 = 5^2 + 3^2 + 2(5)(3) – (5 + 3)5⊕3=52+32+2(5)(3)−(5+3) =25+9+30−8=56= 25 + 9 + 30 – 8 = 56=25+9+30−8=56

Correct answer: 56

32. Symbol-Based Logical Deduction

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the father of B.”
A□BA \square BA□B means “A is the sister of B.”
A\hexagonBA \hexagon BA\hexagonB means “A is the uncle of B.”

Which of the following is logically impossible?
A) X△Y□ZX \triangle Y \square ZX△Y□Z
B) A\hexagonB△CA \hexagon B \triangle CA\hexagonB△C
C) P□Q\hexagonRP \square Q \hexagon RP□Q\hexagonR

Solution:

A) XXX is the father of YYY, and YYY is the sister of ZZZ → Possible
B) AAA is the uncle of BBB, and BBB is the father of CCC → Possible
C) PPP is the sister of QQQ, and QQQ is the uncle of RRR → Impossible, because a sister cannot be directly related to an uncle.

Correct answer: C

33. Symbol-Based Number Puzzle

Question:
A mathematical operation ⋆\star⋆ is defined as:A⋆B=(A+B)2−(A×B)A \star B = (A + B)^2 – (A \times B)A⋆B=(A+B)2−(A×B)

If 4⋆2=204 \star 2 = 204⋆2=20 and 5⋆3=345 \star 3 = 345⋆3=34, find 6⋆46 \star 46⋆4.

Solution:
Using the operation:A⋆B=(A+B)2−(A×B)A \star B = (A + B)^2 – (A \times B)A⋆B=(A+B)2−(A×B)

For 6⋆46 \star 46⋆4:(6+4)2−(6×4)=102−24=100−24=76(6 + 4)^2 – (6 \times 4) = 10^2 – 24 = 100 – 24 = 76(6+4)2−(6×4)=102−24=100−24=76

Correct answer: 76

34. Symbolic Arithmetic Challenge

Question:
If ⋆\star⋆ means double the number, #\## means subtract 5 from the number, and \@\@\@ means square the number, find the value of:4⋆#\@34 \star \# \@ 34⋆#\@3

Solution:4⋆=4×2=84 \star = 4 \times 2 = 84⋆=4×2=8 8#=8−5=38 \# = 8 – 5 = 38#=8−5=3 3\@=32=93 \@ = 3^2 = 93\@=32=9

Correct answer: 9

35. Symbolic Inequality Puzzle

Question:
In a symbolic notation:

A#BA \# BA#B means “A is greater than B.”
A\@BA \@ BA\@B means “A is less than or equal to B.”
A&BA \& BA&B means “A is equal to B.”

Which of the following is always true?
A) 8#7\@68 \# 7 \@ 68#7\@6
B) 12\@13&1312 \@ 13 \& 1312\@13&13
C) 10&10#910 \& 10 \# 910&10#9

Solution:

A) 8>7≤68 > 7 \leq 68>7≤6 → False
B) 12≤13=1312 \leq 13 = 1312≤13=13 → True
C) 10=10>910 = 10 > 910=10>9 → True

Correct answers: B and C

36. Symbolic Relationship Puzzle

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the mother of B.”
A□BA \square BA□B means “A is the brother of B.”
A\hexagonBA \hexagon BA\hexagonB means “A is the grandfather of B.”

Solution:

A) XXX is the mother of YYY, and YYY is the brother of ZZZ → Possible
B) AAA is the grandfather of BBB, and BBB is the mother of CCC → Possible
C) PPP is the brother of QQQ, and QQQ is the grandfather of RRR → Impossible, because a brother cannot directly be linked to a grandfather.

Correct answer: C

37. Symbol-Based Mathematical Equation

Question:
If:A⋄B=(A×B)+(A+B−1)A \diamond B = (A \times B) + (A + B – 1)A⋄B=(A×B)+(A+B−1)

Find the value of 7⋄47 \diamond 47⋄4.

Solution:7⋄4=(7×4)+(7+4−1)7 \diamond 4 = (7 \times 4) + (7 + 4 – 1)7⋄4=(7×4)+(7+4−1) =28+(7+4−1)=28+10=38= 28 + (7 + 4 – 1) = 28 + 10 = 38=28+(7+4−1)=28+10=38

Correct answer: 38

38. Custom Arithmetic Operation

Question:
A new operation □\square□ is defined as:A□B=A2−B2+ABA \square B = A^2 – B^2 + ABA□B=A2−B2+AB

Find the value of 6□26 \square 26□2.

Solution:6□2=62−22+(6×2)6 \square 2 = 6^2 – 2^2 + (6 \times 2)6□2=62−22+(6×2) =36−4+12=44= 36 – 4 + 12 = 44=36−4+12=44

Correct answer: 44

39. Symbol-Based Logical Deduction

Question:
In a symbolic notation system:

A♡BA \heartsuit BA♡B means “A is the son of B.”
A♢BA \diamondsuit BA♢B means “A is the mother of B.”
A♠BA \spadesuit BA♠B means “A is the father of B.”

Which of the following is logically incorrect?
A) X♢Y♡ZX \diamondsuit Y \heartsuit ZX♢Y♡Z
B) A♠B♢CA \spadesuit B \diamondsuit CA♠B♢C
C) P♡Q♠RP \heartsuit Q \spadesuit RP♡Q♠R

Solution:

A) XXX is the mother of YYY, and YYY is the son of ZZZ → Possible
B) AAA is the father of BBB, and BBB is the mother of CCC → Possible
C) PPP is the son of QQQ, and QQQ is the father of RRR → Incorrect, because a son cannot be the parent of someone.

Correct answer: C

40. Symbol-Based Number Puzzle

Question:
A mathematical operation #\## is defined as:A#B=(A+B)×(A−B)+ABA \# B = (A + B) \times (A – B) + ABA#B=(A+B)×(A−B)+AB

If 6#2=386 \# 2 = 386#2=38 and 5#3=285 \# 3 = 285#3=28, find 7#47 \# 47#4.

Solution:
Using the operation:A#B=(A+B)×(A−B)+ABA \# B = (A + B) \times (A – B) + ABA#B=(A+B)×(A−B)+AB

For 7#47 \# 47#4:(7+4)×(7−4)+(7×4)(7 + 4) \times (7 – 4) + (7 \times 4)(7+4)×(7−4)+(7×4) =11×3+28=33+28=61= 11 \times 3 + 28 = 33 + 28 = 61=11×3+28=33+28=61

Correct answer: 61

41. Symbolic Arithmetic Challenge

Question:
If ⋆\star⋆ means triple the number, #\## means subtract 2 from the number, and \@\@\@ means cube the number, find the value of:3⋆#\@23 \star \# \@ 23⋆#\@2

Solution:3⋆=3×3=93 \star = 3 \times 3 = 93⋆=3×3=9 9#=9−2=79 \# = 9 – 2 = 79#=9−2=7 7\@=73=3437 \@ = 7^3 = 3437\@=73=343

Correct answer: 343

42. Symbolic Inequality Puzzle

Question:
In a symbolic notation:

A#BA \# BA#B means “A is greater than or equal to B.”
A\@BA \@ BA\@B means “A is less than B.”
A&BA \& BA&B means “A is equal to B.”

Which of the following is always true?
A) 7#6\@57 \# 6 \@ 57#6\@5
B) 13\@14&1413 \@ 14 \& 1413\@14&14
C) 11&11#1011 \& 11 \# 1011&11#10

Solution:

A) 7≥6<57 \geq 6 < 57≥6<5 → False
B) 13<14=1413 < 14 = 1413<14=14 → True
C) 11=11≥1011 = 11 \geq 1011=11≥10 → True

Correct answers: B and C

43. Symbolic Relationship Puzzle

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the grandfather of B.”
A□BA \square BA□B means “A is the brother of B.”
A\hexagonBA \hexagon BA\hexagonB means “A is the uncle of B.”

Solution:

A) XXX is the grandfather of YYY, and YYY is the brother of ZZZ → Possible
B) AAA is the uncle of BBB, and BBB is the grandfather of CCC → Possible
C) PPP is the brother of QQQ, and QQQ is the uncle of RRR → Incorrect, because a brother cannot be directly linked to an uncle.

Correct answer: C

44. Symbol-Based Mathematical Equation

Question:
If:A⋄B=(A×B)−(A+B)A \diamond B = (A \times B) – (A + B)A⋄B=(A×B)−(A+B)

Find the value of 8⋄58 \diamond 58⋄5.

Solution:8⋄5=(8×5)−(8+5)8 \diamond 5 = (8 \times 5) – (8 + 5)8⋄5=(8×5)−(8+5) =40−13=27= 40 – 13 = 27=40−13=27

Correct answer: 27

45. Custom Arithmetic Operation

Question:
A new operation ⊕\oplus⊕ is defined as:A⊕B=(A+B)×(A−B)+A2A \oplus B = (A + B) \times (A – B) + A^2A⊕B=(A+B)×(A−B)+A2

Find the value of 7⊕37 \oplus 37⊕3.

Solution:7⊕3=(7+3)×(7−3)+727 \oplus 3 = (7 + 3) \times (7 – 3) + 7^27⊕3=(7+3)×(7−3)+72 =10×4+49=40+49=89= 10 \times 4 + 49 = 40 + 49 = 89=10×4+49=40+49=89

Correct answer: 89

46. Symbol-Based Logical Deduction

Question:
In a symbolic notation system:

A⋆BA \star BA⋆B means “A is the sister of B.”
A△BA \triangle BA△B means “A is the father of B.”
A□BA \square BA□B means “A is the son of B.”

Which of the following is logically incorrect?
A) X△Y⋆ZX \triangle Y \star ZX△Y⋆Z
B) A□B△CA \square B \triangle CA□B△C
C) P⋆Q△RP \star Q \triangle RP⋆Q△R

Solution:

A) XXX is the father of YYY, and YYY is the sister of ZZZ → Possible
B) AAA is the son of BBB, and BBB is the father of CCC → Impossible, because a father cannot be the child of someone.
C) PPP is the sister of QQQ, and QQQ is the father of RRR → Possible

Correct answer: B

47. Symbol-Based Number Puzzle

Question:
A mathematical operation ⋄\diamond⋄ is defined as:A⋄B=(A×B)+(A+B)−2A \diamond B = (A \times B) + (A + B) – 2A⋄B=(A×B)+(A+B)−2

If 6⋄2=186 \diamond 2 = 186⋄2=18 and 5⋄3=235 \diamond 3 = 235⋄3=23, find 7⋄47 \diamond 47⋄4.

Solution:
Using the operation:A⋄B=(A×B)+(A+B)−2A \diamond B = (A \times B) + (A + B) – 2A⋄B=(A×B)+(A+B)−2

For 7⋄47 \diamond 47⋄4:(7×4)+(7+4)−2(7 \times 4) + (7 + 4) – 2(7×4)+(7+4)−2 =28+11−2=37= 28 + 11 – 2 = 37=28+11−2=37

Correct answer: 37

48. Symbolic Arithmetic Challenge

Question:
If ⋆\star⋆ means double the number, #\## means add 3 to the number, and \@\@\@ means square the number, find the value of:2⋆#\@42 \star \# \@ 42⋆#\@4

Solution:2⋆=2×2=42 \star = 2 \times 2 = 42⋆=2×2=4 4#=4+3=74 \# = 4 + 3 = 74#=4+3=7 7\@=72=497 \@ = 7^2 = 497\@=72=49

Correct answer: 49

49. Symbolic Inequality Puzzle

Question:
In a symbolic notation:

A#BA \# BA#B means “A is less than or equal to B.”
A\@BA \@ BA\@B means “A is greater than B.”
A&BA \& BA&B means “A is not equal to B.”

Which of the following is always true?
A) 6#7\@56 \# 7 \@ 56#7\@5
B) 10\@9&1010 \@ 9 \& 1010\@9&10
C) 11&12#1311 \& 12 \# 1311&12#13

Solution:

A) 6≤7>56 \leq 7 > 56≤7>5 → True
B) 10>910 > 910>9 and 10≠1010 \neq 1010=10 → False
C) 11≠12≤1311 \neq 12 \leq 1311=12≤13 → True

Correct answers: A and C

50. Symbolic Relationship Puzzle

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the uncle of B.”
A□BA \square BA□B means “A is the sister of B.”
A\hexagonBA \hexagon BA\hexagonB means “A is the grandfather of B.”

Solution:

A) XXX is the uncle of YYY, and YYY is the sister of ZZZ → Possible
B) AAA is the grandfather of BBB, and BBB is the uncle of CCC → Possible
C) PPP is the sister of QQQ, and QQQ is the grandfather of RRR → Incorrect, because a sister cannot be a grandfather.

Correct answer: C

51. Symbol-Based Mathematical Equation

Question:
If:A⋄B=(A+B)2−(A×B)A \diamond B = (A + B)^2 – (A \times B)A⋄B=(A+B)2−(A×B)

Find the value of 5⋄35 \diamond 35⋄3.

Solution:5⋄3=(5+3)2−(5×3)5 \diamond 3 = (5 + 3)^2 – (5 \times 3)5⋄3=(5+3)2−(5×3) =82−15=64−15=49= 8^2 – 15 = 64 – 15 = 49=82−15=64−15=49

Correct answer: 49

52. Custom Arithmetic Operation

Question:
A new operation ⋆\star⋆ is defined as:A⋆B=(A2+B2)−(A×B)A \star B = (A^2 + B^2) – (A \times B)A⋆B=(A2+B2)−(A×B)

Find the value of 7⋆47 \star 47⋆4.

Solution:7⋆4=(72+42)−(7×4)7 \star 4 = (7^2 + 4^2) – (7 \times 4)7⋆4=(72+42)−(7×4) =(49+16)−28=37= (49 + 16) – 28 = 37=(49+16)−28=37

Correct answer: 37

53. Symbol-Based Logical Deduction

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the father of B.”
A□BA \square BA□B means “A is the mother of B.”
A\hexagonBA \hexagon BA\hexagonB means “A is the sibling of B.”

Which of the following is logically incorrect?
A) X△Y\hexagonZX \triangle Y \hexagon ZX△Y\hexagonZ
B) A□B△CA \square B \triangle CA□B△C
C) P\hexagonQ△RP \hexagon Q \triangle RP\hexagonQ△R

Solution:

A) XXX is the father of YYY, and YYY is the sibling of ZZZ → Possible
B) AAA is the mother of BBB, and BBB is the father of CCC → Possible
C) PPP is the sibling of QQQ, and QQQ is the father of RRR → Incorrect, because a sibling cannot directly be a father.

Correct answer: C

54. Symbol-Based Number Puzzle

Question:
A mathematical operation ⋄\diamond⋄ is defined as:A⋄B=(A+B)×(A−B)+ABA \diamond B = (A + B) \times (A – B) + ABA⋄B=(A+B)×(A−B)+AB

If 6⋄2=386 \diamond 2 = 386⋄2=38 and 5⋄3=285 \diamond 3 = 285⋄3=28, find 8⋄48 \diamond 48⋄4.

Solution:
Using the operation:A⋄B=(A+B)×(A−B)+ABA \diamond B = (A + B) \times (A – B) + ABA⋄B=(A+B)×(A−B)+AB

For 8⋄48 \diamond 48⋄4:(8+4)×(8−4)+(8×4)(8 + 4) \times (8 – 4) + (8 \times 4)(8+4)×(8−4)+(8×4) =12×4+32=48+32=80= 12 \times 4 + 32 = 48 + 32 = 80=12×4+32=48+32=80

Correct answer: 80

55. Symbolic Arithmetic Challenge

Question:
If ⋆\star⋆ means double the number, #\## means subtract 4 from the number, and \@\@\@ means square the number, find the value of:3⋆#\@53 \star \# \@ 53⋆#\@5

Solution:3⋆=3×2=63 \star = 3 \times 2 = 63⋆=3×2=6 6#=6−4=26 \# = 6 – 4 = 26#=6−4=2 2\@=22=42 \@ = 2^2 = 42\@=22=4

Correct answer: 4

56. Symbolic Inequality Puzzle

Question:
In a symbolic notation:

A#BA \# BA#B means “A is greater than or equal to B.”
A\@BA \@ BA\@B means “A is less than B.”
A&BA \& BA&B means “A is equal to B.”

Which of the following is always true?
A) 8#7\@68 \# 7 \@ 68#7\@6
B) 12\@13&1312 \@ 13 \& 1312\@13&13
C) 15&15#1415 \& 15 \# 1415&15#14

Solution:

A) 8≥7<68 \geq 7 < 68≥7<6 → False
B) 12<13=1312 < 13 = 1312<13=13 → True
C) 15=15≥1415 = 15 \geq 1415=15≥14 → True

Correct answers: B and C

57. Symbolic Relationship Puzzle

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the grandfather of B.”
A□BA \square BA□B means “A is the sister of B.”
A\hexagonBA \hexagon BA\hexagonB means “A is the uncle of B.”

Solution:

A) XXX is the grandfather of YYY, and YYY is the sister of ZZZ → Possible
B) AAA is the uncle of BBB, and BBB is the grandfather of CCC → Possible
C) PPP is the sister of QQQ, and QQQ is the uncle of RRR → Incorrect, because a sister cannot be an uncle.

Correct answer: C

58. Symbol-Based Mathematical Equation

Question:
If:A⋄B=(A×B)−(A+B)A \diamond B = (A \times B) – (A + B)A⋄B=(A×B)−(A+B)

Find the value of 9⋄59 \diamond 59⋄5.

Solution:9⋄5=(9×5)−(9+5)9 \diamond 5 = (9 \times 5) – (9 + 5)9⋄5=(9×5)−(9+5) =45−14=31= 45 – 14 = 31=45−14=31

Correct answer: 31

59. Symbol-Based Logical Challenge

Question:
A new notation is defined as follows:

A♡BA \heartsuit BA♡B means “A is the son of B.”
A♢BA \diamondsuit BA♢B means “A is the daughter of B.”
A♠BA \spadesuit BA♠B means “A is the mother of B.”

Which of the following statements is logically incorrect?
A) X♡Y♠ZX \heartsuit Y \spadesuit ZX♡Y♠Z
B) A♢B♡CA \diamondsuit B \heartsuit CA♢B♡C
C) P♠Q♢RP \spadesuit Q \diamondsuit RP♠Q♢R

Solution:

A) XXX is the son of YYY, and YYY is the mother of ZZZ → Possible
B) AAA is the daughter of BBB, and BBB is the son of CCC → Incorrect, because a son cannot be the parent of a daughter.
C) PPP is the mother of QQQ, and QQQ is the daughter of RRR → Possible

Correct answer: B

60. Custom Arithmetic Operation

Question:
A new operation ⋆\star⋆ is defined as:A⋆B=(A+B)2−(A×B)A \star B = (A + B)^2 – (A \times B)A⋆B=(A+B)2−(A×B)

Find the value of 8⋆58 \star 58⋆5.

Solution:8⋆5=(8+5)2−(8×5)8 \star 5 = (8 + 5)^2 – (8 \times 5)8⋆5=(8+5)2−(8×5) =132−40=169−40=129= 13^2 – 40 = 169 – 40 = 129=132−40=169−40=129

Correct answer: 129

61. Symbol-Based Logical Deduction

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the brother of B.”
A□BA \square BA□B means “A is the father of B.”
A\hexagonBA \hexagon BA\hexagonB means “A is the son of B.”

Which of the following is logically incorrect?
A) X□Y△ZX \square Y \triangle ZX□Y△Z
B) A\hexagonB□CA \hexagon B \square CA\hexagonB□C
C) P△Q\hexagonRP \triangle Q \hexagon RP△Q\hexagonR

Solution:

A) XXX is the father of YYY, and YYY is the brother of ZZZ → Possible
B) AAA is the son of BBB, and BBB is the father of CCC → Incorrect, because a father cannot also be a son in the same lineage.
C) PPP is the brother of QQQ, and QQQ is the son of RRR → Possible

Correct answer: B

62. Symbol-Based Number Puzzle

Question:
A mathematical operation ⋄\diamond⋄ is defined as:A⋄B=(A×B)+(A+B)−3A \diamond B = (A \times B) + (A + B) – 3A⋄B=(A×B)+(A+B)−3

If 6⋄3=246 \diamond 3 = 246⋄3=24 and 5⋄2=145 \diamond 2 = 145⋄2=14, find 7⋄47 \diamond 47⋄4.

Solution:
Using the operation:A⋄B=(A×B)+(A+B)−3A \diamond B = (A \times B) + (A + B) – 3A⋄B=(A×B)+(A+B)−3

For 7⋄47 \diamond 47⋄4:(7×4)+(7+4)−3(7 \times 4) + (7 + 4) – 3(7×4)+(7+4)−3 =28+11−3=36= 28 + 11 – 3 = 36=28+11−3=36

Correct answer: 36

63. Symbolic Arithmetic Challenge

Question:
If ⋆\star⋆ means triple the number, #\## means subtract 5 from the number, and \@\@\@ means square the number, find the value of:4⋆#\@64 \star \# \@ 64⋆#\@6

Solution:4⋆=4×3=124 \star = 4 \times 3 = 124⋆=4×3=12 12#=12−5=712 \# = 12 – 5 = 712#=12−5=7 7\@=72=497 \@ = 7^2 = 497\@=72=49

Correct answer: 49

64. Symbolic Inequality Puzzle

Question:
In a symbolic notation:

A#BA \# BA#B means “A is less than or equal to B.”
A\@BA \@ BA\@B means “A is greater than B.”
A&BA \& BA&B means “A is not equal to B.”

Which of the following is always true?
A) 9#10\@89 \# 10 \@ 89#10\@8
B) 7\@6&77 \@ 6 \& 77\@6&7
C) 15&14#1515 \& 14 \# 1515&14#15

Solution:

A) 9≤10>89 \leq 10 > 89≤10>8 → True
B) 7>67 > 67>6 and 7≠77 \neq 77=7 → False
C) 15≠14≤1515 \neq 14 \leq 1515=14≤15 → True

Correct answers: A and C

65. Symbolic Relationship Puzzle

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the uncle of B.”
A□BA \square BA□B means “A is the brother of B.”
A\hexagonBA \hexagon BA\hexagonB means “A is the grandfather of B.”

Solution:

A) XXX is the uncle of YYY, and YYY is the brother of ZZZ → Possible
B) AAA is the grandfather of BBB, and BBB is the uncle of CCC → Possible
C) PPP is the brother of QQQ, and QQQ is the grandfather of RRR → Incorrect, because a brother cannot be a grandfather in the same generation.

Correct answer: C

Also read : Verbal reasoning question on Mathematical operations

66. Symbol-Based Mathematical Equation

Question:
If:A⋄B=(A×B)−(A+B)+5A \diamond B = (A \times B) – (A + B) + 5A⋄B=(A×B)−(A+B)+5

Find the value of 10⋄610 \diamond 610⋄6.

Solution:10⋄6=(10×6)−(10+6)+510 \diamond 6 = (10 \times 6) – (10 + 6) + 510⋄6=(10×6)−(10+6)+5 =60−16+5=49= 60 – 16 + 5 = 49=60−16+5=49

Correct answer: 49

67. Symbol-Based Logical Challenge

Question:
A new notation is defined as follows:

A♡BA \heartsuit BA♡B means “A is the father of B.”
A♢BA \diamondsuit BA♢B means “A is the daughter of B.”
A♠BA \spadesuit BA♠B means “A is the son of B.”

Solution:

A) XXX is the father of YYY, and YYY is the son of ZZZ → Possible
B) AAA is the daughter of BBB, and BBB is the father of CCC → Possible
C) PPP is the son of QQQ, and QQQ is the daughter of RRR → Incorrect, because a daughter cannot also be a son.

Correct answer: C

68. Custom Arithmetic Operation

Question:
A new operation ∗\ast∗ is defined as:A∗B=(A+B)×(A−B)+A2+B2A \ast B = (A + B) \times (A – B) + A^2 + B^2A∗B=(A+B)×(A−B)+A2+B2

Find the value of 9∗69 \ast 69∗6.

Solution:9∗6=(9+6)×(9−6)+92+629 \ast 6 = (9 + 6) \times (9 – 6) + 9^2 + 6^29∗6=(9+6)×(9−6)+92+62 =15×3+81+36= 15 \times 3 + 81 + 36=15×3+81+36 =45+81+36=162= 45 + 81 + 36 = 162=45+81+36=162

Correct answer: 162

69. Symbol-Based Logical Deduction

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the sister of B.”
A□BA \square BA□B means “A is the father of B.”
A\hexagonBA \hexagon BA\hexagonB means “A is the grandson of B.”

Which of the following is logically incorrect?
A) X△Y□ZX \triangle Y \square ZX△Y□Z
B) A\hexagonB△CA \hexagon B \triangle CA\hexagonB△C
C) P□Q\hexagonRP \square Q \hexagon RP□Q\hexagonR

Solution:

A) XXX is the sister of YYY, and YYY is the father of ZZZ → Possible
B) AAA is the grandson of BBB, and BBB is the sister of CCC → Incorrect, because a sister cannot be a direct ancestor.
C) PPP is the father of QQQ, and QQQ is the grandson of RRR → Possible

Correct answer: B

70. Symbol-Based Number Puzzle

Question:
A mathematical operation ⋄\diamond⋄ is defined as:A⋄B=(A2+B2)−2(A×B)+3(A+B)A \diamond B = (A^2 + B^2) – 2(A \times B) + 3(A + B)A⋄B=(A2+B2)−2(A×B)+3(A+B)

Find 5⋄35 \diamond 35⋄3.

Solution:5⋄3=(52+32)−2(5×3)+3(5+3)5 \diamond 3 = (5^2 + 3^2) – 2(5 \times 3) + 3(5 + 3)5⋄3=(52+32)−2(5×3)+3(5+3) =(25+9)−2(15)+3(8)= (25 + 9) – 2(15) + 3(8)=(25+9)−2(15)+3(8) =34−30+24=28= 34 – 30 + 24 = 28=34−30+24=28

Correct answer: 28

71. Symbolic Arithmetic Challenge

Question:
If ⋆\star⋆ means multiply by 4, #\## means subtract 6, and \@\@\@ means square the number, find the value of:2⋆#\@72 \star \# \@ 72⋆#\@7

Solution:2⋆=2×4=82 \star = 2 \times 4 = 82⋆=2×4=8 8#=8−6=28 \# = 8 – 6 = 28#=8−6=2 2\@=22=42 \@ = 2^2 = 42\@=22=4

Correct answer: 4

72. Symbolic Inequality Puzzle

Question:
In a symbolic notation:

A#BA \# BA#B means “A is greater than B.”
A\@BA \@ BA\@B means “A is equal to B.”
A&BA \& BA&B means “A is less than B.”

Which of the following is always true?
A) 10&11#910 \& 11 \# 910&11#9
B) 8\@8&98 \@ 8 \& 98\@8&9
C) 7#6\@77 \# 6 \@ 77#6\@7

Solution:

A) 10<11>910 < 11 > 910<11>9 → True
B) 8=8<98 = 8 < 98=8<9 → True
C) 7>6=77 > 6 = 77>6=7 → False

Correct answers: A and B

73. Symbolic Relationship Puzzle

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the nephew of B.”
A□BA \square BA□B means “A is the father of B.”
A\hexagonBA \hexagon BA\hexagonB means “A is the grandfather of B.”

Solution:

A) XXX is the nephew of YYY, and YYY is the father of ZZZ → Possible
B) AAA is the grandfather of BBB, and BBB is the nephew of CCC → Incorrect, because a nephew and grandfather cannot have a direct lineage.
C) PPP is the father of QQQ, and QQQ is the grandfather of RRR → Possible

Correct answer: B

74. Symbol-Based Mathematical Equation

Question:
If:A⋄B=(A+B)×(A−B)+(A×B)+7A \diamond B = (A + B) \times (A – B) + (A \times B) + 7A⋄B=(A+B)×(A−B)+(A×B)+7

Find the value of 8⋄48 \diamond 48⋄4.

Solution:8⋄4=(8+4)×(8−4)+(8×4)+78 \diamond 4 = (8 + 4) \times (8 – 4) + (8 \times 4) + 78⋄4=(8+4)×(8−4)+(8×4)+7 =12×4+32+7= 12 \times 4 + 32 + 7=12×4+32+7 =48+32+7=87= 48 + 32 + 7 = 87=48+32+7=87

Correct answer: 87

75. Symbol-Based Logical Challenge

Question:
A new notation is defined as follows:

A♡BA \heartsuit BA♡B means “A is the uncle of B.”
A♢BA \diamondsuit BA♢B means “A is the brother of B.”
A♠BA \spadesuit BA♠B means “A is the grandson of B.”

Which of the following statements is logically incorrect?
A) X♡Y♢ZX \heartsuit Y \diamondsuit ZX♡Y♢Z
B) A♠B♡CA \spadesuit B \heartsuit CA♠B♡C
C) P♢Q♠RP \diamondsuit Q \spadesuit RP♢Q♠R

Solution:

A) XXX is the uncle of YYY, and YYY is the brother of ZZZ → Possible
B) AAA is the grandson of BBB, and BBB is the uncle of CCC → Incorrect, because an uncle cannot be a direct ancestor.
C) PPP is the brother of QQQ, and QQQ is the grandson of RRR → Possible

Correct answer: B

76. Custom Arithmetic Operation

Question:
A new operation ⊙\odot⊙ is defined as:A⊙B=(A+B)2−A×B+2A−BA \odot B = (A + B)^2 – A \times B + 2A – BA⊙B=(A+B)2−A×B+2A−B

Find the value of 7⊙57 \odot 57⊙5.

Solution:7⊙5=(7+5)2−(7×5)+(2×7)−57 \odot 5 = (7 + 5)^2 – (7 \times 5) + (2 \times 7) – 57⊙5=(7+5)2−(7×5)+(2×7)−5 =122−35+14−5= 12^2 – 35 + 14 – 5=122−35+14−5 =144−35+14−5=118= 144 – 35 + 14 – 5 = 118=144−35+14−5=118

Correct answer: 118

77. Symbol-Based Logical Deduction

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the mother of B.”
A□BA \square BA□B means “A is the daughter of B.”
A\hexagonBA \hexagon BA\hexagonB means “A is the cousin of B.”

Solution:

A) XXX is the mother of YYY, and YYY is the daughter of ZZZ → Possible
B) AAA is the cousin of BBB, and BBB is the mother of CCC → Incorrect, because a cousin cannot be a direct ancestor.
C) PPP is the daughter of QQQ, and QQQ is the cousin of RRR → Possible

Correct answer: B

78. Symbol-Based Number Puzzle

Question:
A mathematical operation ⋄\diamond⋄ is defined as:A⋄B=(A2−B2)+2(A×B)−3(A+B)A \diamond B = (A^2 – B^2) + 2(A \times B) – 3(A + B)A⋄B=(A2−B2)+2(A×B)−3(A+B)

Find 6⋄46 \diamond 46⋄4.

Solution:6⋄4=(62−42)+2(6×4)−3(6+4)6 \diamond 4 = (6^2 – 4^2) + 2(6 \times 4) – 3(6 + 4)6⋄4=(62−42)+2(6×4)−3(6+4) =(36−16)+2(24)−3(10)= (36 – 16) + 2(24) – 3(10)=(36−16)+2(24)−3(10) =20+48−30=38= 20 + 48 – 30 = 38=20+48−30=38

Correct answer: 38

79. Symbolic Arithmetic Challenge

Question:
If ⋆\star⋆ means double the number, #\## means add 7, and \@\@\@ means halve the number, find the value of:3⋆#\@163 \star \# \@ 163⋆#\@16

Solution:3⋆=3×2=63 \star = 3 \times 2 = 63⋆=3×2=6 6#=6+7=136 \# = 6 + 7 = 136#=6+7=13 16\@=16÷2=816 \@ = 16 \div 2 = 816\@=16÷2=8

Correct answer: 13 and 8 (if separately computed).

80. Symbolic Inequality Puzzle

Question:
In a symbolic notation:

A#BA \# BA#B means “A is greater than or equal to B.”
A\@BA \@ BA\@B means “A is less than B.”
A&BA \& BA&B means “A is not equal to B.”

Which of the following is always true?
A) 5\@6#45 \@ 6 \# 45\@6#4
B) 7#7&87 \# 7 \& 87#7&8
C) 10&9#1010 \& 9 \# 1010&9#10

Solution:

A) 5<6≥45 < 6 \geq 45<6≥4 → True
B) 7≥7≠87 \geq 7 \neq 87≥7=8 → True
C) 10≠9≥1010 \neq 9 \geq 1010=9≥10 → False

Correct answers: A and B

81. Symbolic Relationship Puzzle

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the nephew of B.”
A□BA \square BA□B means “A is the father of B.”
A\hexagonBA \hexagon BA\hexagonB means “A is the grandson of B.”

Solution:

A) XXX is the nephew of YYY, and YYY is the father of ZZZ → Possible
B) AAA is the grandson of BBB, and BBB is the nephew of CCC → Incorrect, because a nephew cannot be a direct ancestor.
C) PPP is the father of QQQ, and QQQ is the grandson of RRR → Possible

Correct answer: B

82. Symbol-Based Mathematical Equation

Question:
If:A⋄B=(A×B)+(A+B)−2A \diamond B = (A \times B) + (A + B) – 2A⋄B=(A×B)+(A+B)−2

Find the value of 7⋄57 \diamond 57⋄5.

Solution:7⋄5=(7×5)+(7+5)−27 \diamond 5 = (7 \times 5) + (7 + 5) – 27⋄5=(7×5)+(7+5)−2 =35+12−2= 35 + 12 – 2=35+12−2 =45= 45=45

Correct answer: 45

83. Symbol-Based Logical Challenge

Question:
A new notation is defined as follows:

A♡BA \heartsuit BA♡B means “A is the aunt of B.”
A♢BA \diamondsuit BA♢B means “A is the father of B.”
A♠BA \spadesuit BA♠B means “A is the grandson of B.”

Solution:

A) XXX is the aunt of YYY, and YYY is the father of ZZZ → Possible
B) AAA is the grandson of BBB, and BBB is the aunt of CCC → Incorrect, because an aunt cannot be a direct ancestor.
C) PPP is the father of QQQ, and QQQ is the grandson of RRR → Possible

Correct answer: B

84. Custom Arithmetic Operation

Question:
A new operation ⊛\circledast⊛ is defined as:A⊛B=(A+B)×(A−B)+A2−B2+2(A×B)A \circledast B = (A + B) \times (A – B) + A^2 – B^2 + 2(A \times B)A⊛B=(A+B)×(A−B)+A2−B2+2(A×B)

Find the value of 9⊛59 \circledast 59⊛5.

Solution:9⊛5=(9+5)×(9−5)+92−52+2(9×5)9 \circledast 5 = (9 + 5) \times (9 – 5) + 9^2 – 5^2 + 2(9 \times 5)9⊛5=(9+5)×(9−5)+92−52+2(9×5) =14×4+81−25+2(45)= 14 \times 4 + 81 – 25 + 2(45)=14×4+81−25+2(45) =56+81−25+90=202= 56 + 81 – 25 + 90 = 202=56+81−25+90=202

Correct answer: 202

85. Symbol-Based Logical Deduction

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the brother of B.”
A□BA \square BA□B means “A is the father of B.”
A\hexagonBA \hexagon BA\hexagonB means “A is the uncle of B.”

Solution:

A) XXX is the brother of YYY, and YYY is the father of ZZZ → Possible
B) AAA is the uncle of BBB, and BBB is the brother of CCC → Incorrect, because an uncle and a brother cannot be directly related.
C) PPP is the father of QQQ, and QQQ is the uncle of RRR → Possible

Correct answer: B

86. Symbol-Based Number Puzzle

Question:
A mathematical operation ⊕\oplus⊕ is defined as:A⊕B=A2+B2−2(A×B)+3(A+B)A \oplus B = A^2 + B^2 – 2(A \times B) + 3(A + B)A⊕B=A2+B2−2(A×B)+3(A+B)

Find 8⊕38 \oplus 38⊕3.

Solution:8⊕3=82+32−2(8×3)+3(8+3)8 \oplus 3 = 8^2 + 3^2 – 2(8 \times 3) + 3(8 + 3)8⊕3=82+32−2(8×3)+3(8+3) =64+9−2(24)+3(11)= 64 + 9 – 2(24) + 3(11)=64+9−2(24)+3(11) =64+9−48+33=58= 64 + 9 – 48 + 33 = 58=64+9−48+33=58

Correct answer: 58

87. Symbolic Arithmetic Challenge

Question:
If ⋆\star⋆ means multiply by 3, #\## means subtract 5, and \@\@\@ means add 2, find the value of:4⋆#\@104 \star \# \@ 104⋆#\@10

Solution:4⋆=4×3=124 \star = 4 \times 3 = 124⋆=4×3=12 12#=12−5=712 \# = 12 – 5 = 712#=12−5=7 10\@=10+2=1210 \@ = 10 + 2 = 1210\@=10+2=12

Correct answer: 7 and 12 (if separately computed).

88. Symbolic Inequality Puzzle

Question:
In a symbolic notation:

A#BA \# BA#B means “A is greater than B.”
A\@BA \@ BA\@B means “A is less than or equal to B.”
A&BA \& BA&B means “A is not equal to B.”

Which of the following is always true?
A) 7\@8#67 \@ 8 \# 67\@8#6
B) 10&10#910 \& 10 \# 910&10#9
C) 5#4&65 \# 4 \& 65#4&6

Solution:

A) 7≤8>67 \leq 8 > 67≤8>6 → True
B) 10≠10>910 \neq 10 > 910=10>9 → False
C) 5>4≠65 > 4 \neq 65>4=6 → True

Correct answers: A and C

89. Symbolic Relationship Puzzle

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the cousin of B.”
A□BA \square BA□B means “A is the son of B.”
A\hexagonBA \hexagon BA\hexagonB means “A is the nephew of B.”

Solution:

A) XXX is the cousin of YYY, and YYY is the son of ZZZ → Possible
B) AAA is the nephew of BBB, and BBB is the cousin of CCC → Incorrect, because a cousin and nephew cannot have direct lineage.
C) PPP is the son of QQQ, and QQQ is the nephew of RRR → Possible

Correct answer: B

90. Symbol-Based Mathematical Equation

Question:
If:A⋄B=(A+B)×(A−B)+(A×B)+4A \diamond B = (A + B) \times (A – B) + (A \times B) + 4A⋄B=(A+B)×(A−B)+(A×B)+4

Find the value of 10⋄610 \diamond 610⋄6.

Solution:10⋄6=(10+6)×(10−6)+(10×6)+410 \diamond 6 = (10 + 6) \times (10 – 6) + (10 \times 6) + 410⋄6=(10+6)×(10−6)+(10×6)+4 =16×4+60+4= 16 \times 4 + 60 + 4=16×4+60+4 =64+60+4=128= 64 + 60 + 4 = 128=64+60+4=128

Correct answer: 128

91. Symbol-Based Logical Challenge

Question:
A new notation is defined as follows:

A♡BA \heartsuit BA♡B means “A is the niece of B.”
A♢BA \diamondsuit BA♢B means “A is the brother of B.”
A♠BA \spadesuit BA♠B means “A is the son of B.”

Solution:

A) XXX is the niece of YYY, and YYY is the brother of ZZZ → Possible
B) AAA is the son of BBB, and BBB is the niece of CCC → Incorrect, because a niece cannot be a direct ancestor.
C) PPP is the brother of QQQ, and QQQ is the son of RRR → Possible

Correct answer: B

92. Custom Arithmetic Operation

Question:
A new operation ⧫\blacklozenge⧫ is defined as:A⧫B=A3−B3+3(A×B)−2(A+B)A \blacklozenge B = A^3 – B^3 + 3(A \times B) – 2(A + B)A⧫B=A3−B3+3(A×B)−2(A+B)

Find the value of 5⧫35 \blacklozenge 35⧫3.

Solution:5⧫3=53−33+3(5×3)−2(5+3)5 \blacklozenge 3 = 5^3 – 3^3 + 3(5 \times 3) – 2(5 + 3)5⧫3=53−33+3(5×3)−2(5+3) =125−27+3(15)−2(8)= 125 – 27 + 3(15) – 2(8)=125−27+3(15)−2(8) =125−27+45−16=127= 125 – 27 + 45 – 16 = 127=125−27+45−16=127

Correct answer: 127

93. Symbol-Based Logical Deduction

Question:
In a symbolic notation system:

A⊕BA \oplus BA⊕B means “A is the sister of B.”
A⊖BA \ominus BA⊖B means “A is the father of B.”
A⊗BA \otimes BA⊗B means “A is the uncle of B.”

Which of the following is logically incorrect?
A) X⊕Y⊖ZX \oplus Y \ominus ZX⊕Y⊖Z
B) A⊗B⊕CA \otimes B \oplus CA⊗B⊕C
C) P⊖Q⊗RP \ominus Q \otimes RP⊖Q⊗R

Solution:

A) XXX is the sister of YYY, and YYY is the father of ZZZ → Possible
B) AAA is the uncle of BBB, and BBB is the sister of CCC → Incorrect, because an uncle and a sister cannot be directly related.
C) PPP is the father of QQQ, and QQQ is the uncle of RRR → Possible

Correct answer: B

94. Symbol-Based Number Puzzle

Question:
A mathematical operation ⊙\odot⊙ is defined as:A⊙B=A2+B2−(A×B)+4(A+B)A \odot B = A^2 + B^2 – (A \times B) + 4(A + B)A⊙B=A2+B2−(A×B)+4(A+B)

Find 7⊙27 \odot 27⊙2.

Solution:7⊙2=72+22−(7×2)+4(7+2)7 \odot 2 = 7^2 + 2^2 – (7 \times 2) + 4(7 + 2)7⊙2=72+22−(7×2)+4(7+2) =49+4−14+4(9)= 49 + 4 – 14 + 4(9)=49+4−14+4(9) =49+4−14+36=75= 49 + 4 – 14 + 36 = 75=49+4−14+36=75

Correct answer: 75

95. Symbolic Arithmetic Challenge

Question:
If ⋆\star⋆ means square the number, #\## means subtract 10, and \@\@\@ means divide by 2, find the value of:6⋆#\@86 \star \# \@ 86⋆#\@8

Solution:6⋆=62=366 \star = 6^2 = 366⋆=62=36 36#=36−10=2636 \# = 36 – 10 = 2636#=36−10=26 8\@=8÷2=48 \@ = 8 \div 2 = 48\@=8÷2=4

Correct answer: 26 and 4 (if separately computed).

96. Symbolic Inequality Puzzle

Question:
In a symbolic notation:

A#BA \# BA#B means “A is greater than or equal to B.”
A\@BA \@ BA\@B means “A is less than B.”
A&BA \& BA&B means “A is not equal to B.”

Which of the following is always true?
A) 6\@7#56 \@ 7 \# 56\@7#5
B) 9#9&109 \# 9 \& 109#9&10
C) 4#3&54 \# 3 \& 54#3&5

Solution:

A) 6<7≥56 < 7 \geq 56<7≥5 → True
B) 9≥9≠109 \geq 9 \neq 109≥9=10 → True
C) 4≥3≠54 \geq 3 \neq 54≥3=5 → True

Correct answers: A, B, and C

97. Symbolic Relationship Puzzle

Question:
In a symbolic notation system:

A♡BA \heartsuit BA♡B means “A is the cousin of B.”
A♢BA \diamondsuit BA♢B means “A is the father of B.”
A♠BA \spadesuit BA♠B means “A is the grandson of B.”

Which of the following is definitely incorrect?
A) X♡Y♢ZX \heartsuit Y \diamondsuit ZX♡Y♢Z
B) A♠B♡CA \spadesuit B \heartsuit CA♠B♡C
C) P♢Q♠RP \diamondsuit Q \spadesuit RP♢Q♠R

Solution:

A) XXX is the cousin of YYY, and YYY is the father of ZZZ → Possible
B) AAA is the grandson of BBB, and BBB is the cousin of CCC → Incorrect, because a cousin and grandson cannot have direct lineage.
C) PPP is the father of QQQ, and QQQ is the grandson of RRR → Possible

Correct answer: B

98. Symbol-Based Mathematical Equation

Question:
If:A△B=(A×B)+(A2−B2)+5(A+B)A \triangle B = (A \times B) + (A^2 – B^2) + 5(A + B)A△B=(A×B)+(A2−B2)+5(A+B)

Find the value of 8△38 \triangle 38△3.

Solution:8△3=(8×3)+(82−32)+5(8+3)8 \triangle 3 = (8 \times 3) + (8^2 – 3^2) + 5(8 + 3)8△3=(8×3)+(82−32)+5(8+3) =24+(64−9)+5(11)= 24 + (64 – 9) + 5(11)=24+(64−9)+5(11) =24+55+55=134= 24 + 55 + 55 = 134=24+55+55=134

Correct answer: 134

99. Symbol-Based Logical Challenge

Question:
A new notation is defined as follows:

A♡BA \heartsuit BA♡B means “A is the niece of B.”
A♢BA \diamondsuit BA♢B means “A is the brother of B.”
A♠BA \spadesuit BA♠B means “A is the son of B.”

Solution:

A) XXX is the niece of YYY, and YYY is the brother of ZZZ → Possible
B) AAA is the son of BBB, and BBB is the niece of CCC → Incorrect, because a niece cannot be a direct ancestor.
C) PPP is the brother of QQQ, and QQQ is the son of RRR → Possible

Correct answer: B

100. Custom Arithmetic Operation

Question:
A new operation ▲\blacktriangle▲ is defined as:A▲B=(A2+B2)−3(A×B)+4(A+B)A \blacktriangle B = (A^2 + B^2) – 3(A \times B) + 4(A + B)A▲B=(A2+B2)−3(A×B)+4(A+B)

Find the value of 6▲26 \blacktriangle 26▲2.

Solution:6▲2=(62+22)−3(6×2)+4(6+2)6 \blacktriangle 2 = (6^2 + 2^2) – 3(6 \times 2) + 4(6 + 2)6▲2=(62+22)−3(6×2)+4(6+2) =(36+4)−3(12)+4(8)= (36 + 4) – 3(12) + 4(8)=(36+4)−3(12)+4(8) =40−36+32=36= 40 – 36 + 32 = 36=40−36+32=36

Correct answer: 36

101. Symbol-Based Logical Deduction

Question:
In a symbolic notation system:

A∘BA \circ BA∘B means “A is the mother of B.”
A□BA \square BA□B means “A is the grandfather of B.”
A△BA \triangle BA△B means “A is the brother of B.”

Which of the following is logically incorrect?
A) X△Y∘ZX \triangle Y \circ ZX△Y∘Z
B) A□B△CA \square B \triangle CA□B△C
C) P∘Q□RP \circ Q \square RP∘Q□R

Solution:

A) XXX is the brother of YYY, and YYY is the mother of ZZZ → Possible
B) AAA is the grandfather of BBB, and BBB is the brother of CCC → Incorrect, because a grandfather and a brother cannot have a direct relationship.
C) PPP is the mother of QQQ, and QQQ is the grandfather of RRR → Possible

Correct answer: B

102. Symbol-Based Number Puzzle

Question:
A mathematical operation ⋄\diamond⋄ is defined as:A⋄B=(A×B)+2(A+B)−(A2−B2)A \diamond B = (A \times B) + 2(A + B) – (A^2 – B^2)A⋄B=(A×B)+2(A+B)−(A2−B2)

Find 5⋄35 \diamond 35⋄3.

Solution:5⋄3=(5×3)+2(5+3)−(52−32)5 \diamond 3 = (5 \times 3) + 2(5 + 3) – (5^2 – 3^2)5⋄3=(5×3)+2(5+3)−(52−32) =15+2(8)−(25−9)= 15 + 2(8) – (25 – 9)=15+2(8)−(25−9) =15+16−16=15= 15 + 16 – 16 = 15=15+16−16=15

Correct answer: 15

103. Symbolic Arithmetic Challenge

Question:
If ⋆\star⋆ means multiply by 4, #\## means add 6, and \@\@\@ means subtract 2, find the value of:7⋆#\@57 \star \# \@ 57⋆#\@5

Solution:7⋆=7×4=287 \star = 7 \times 4 = 287⋆=7×4=28 28#=28+6=3428 \# = 28 + 6 = 3428#=28+6=34 5\@=5−2=35 \@ = 5 – 2 = 35\@=5−2=3

Correct answer: 34 and 3 (if separately computed).

104. Symbolic Inequality Puzzle

Question:
In a symbolic notation:

A#BA \# BA#B means “A is greater than B.”
A\@BA \@ BA\@B means “A is less than or equal to B.”
A&BA \& BA&B means “A is not equal to B.”

Which of the following is always true?
A) 10\@12#910 \@ 12 \# 910\@12#9
B) 7&7#67 \& 7 \# 67&7#6
C) 5#4&65 \# 4 \& 65#4&6

Solution:

A) 10≤12>910 \leq 12 > 910≤12>9 → True
B) 7≠7>67 \neq 7 > 67=7>6 → False
C) 5>4≠65 > 4 \neq 65>4=6 → True

Correct answers: A and C

105. Symbolic Relationship Puzzle

Question:
In a symbolic notation system:

A♡BA \heartsuit BA♡B means “A is the aunt of B.”
A♢BA \diamondsuit BA♢B means “A is the brother of B.”
A♠BA \spadesuit BA♠B means “A is the father of B.”

Solution:

A) XXX is the aunt of YYY, and YYY is the brother of ZZZ → Possible
B) AAA is the father of BBB, and BBB is the aunt of CCC → Incorrect, because a father and an aunt cannot be directly related.
C) PPP is the brother of QQQ, and QQQ is the father of RRR → Possible

Correct answer: B

106. Symbol-Based Mathematical Equation

Question:
If:A∗B=(A+B)2−(A−B)2+3(A×B)A \ast B = (A + B)^2 – (A – B)^2 + 3(A \times B)A∗B=(A+B)2−(A−B)2+3(A×B)

Find the value of 6∗46 \ast 46∗4.

Solution:6∗4=(6+4)2−(6−4)2+3(6×4)6 \ast 4 = (6 + 4)^2 – (6 – 4)^2 + 3(6 \times 4)6∗4=(6+4)2−(6−4)2+3(6×4) =102−22+3(24)= 10^2 – 2^2 + 3(24)=102−22+3(24) =100−4+72=168= 100 – 4 + 72 = 168=100−4+72=168

Correct answer: 168

107. Symbol-Based Logical Challenge

Question:
A new notation is defined as follows:

A♡BA \heartsuit BA♡B means “A is the niece of B.”
A♢BA \diamondsuit BA♢B means “A is the cousin of B.”
A♠BA \spadesuit BA♠B means “A is the son of B.”

Solution:

A) XXX is the niece of YYY, and YYY is the cousin of ZZZ → Possible
B) AAA is the son of BBB, and BBB is the niece of CCC → Incorrect, because a niece cannot be a direct ancestor.
C) PPP is the cousin of QQQ, and QQQ is the son of RRR → Possible

Correct answer: B

108. Custom Arithmetic Operation

Question:
A new operation ⋈\bowtie⋈ is defined as:A⋈B=(A3−B3)+2(A×B)−3(A+B)A \bowtie B = (A^3 – B^3) + 2(A \times B) – 3(A + B)A⋈B=(A3−B3)+2(A×B)−3(A+B)

Find the value of 4⋈24 \bowtie 24⋈2.

Solution:4⋈2=(43−23)+2(4×2)−3(4+2)4 \bowtie 2 = (4^3 – 2^3) + 2(4 \times 2) – 3(4 + 2)4⋈2=(43−23)+2(4×2)−3(4+2) =(64−8)+2(8)−3(6)= (64 – 8) + 2(8) – 3(6)=(64−8)+2(8)−3(6) =56+16−18=54= 56 + 16 – 18 = 54=56+16−18=54

Correct answer: 54

109. Symbol-Based Logical Deduction

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the father of B.”
A□BA \square BA□B means “A is the son of B.”
A∘BA \circ BA∘B means “A is the brother of B.”

Which of the following is logically incorrect?
A) X△Y□ZX \triangle Y \square ZX△Y□Z
B) A□B∘CA \square B \circ CA□B∘C
C) P△Q△RP \triangle Q \triangle RP△Q△R

Solution:

A) XXX is the father of YYY, and YYY is the son of ZZZ → Possible
B) AAA is the son of BBB, and BBB is the brother of CCC → Possible
C) PPP is the father of QQQ, and QQQ is the father of RRR → Possible

Correct answer: All are logically possible

110. Symbol-Based Number Puzzle

Question:
A mathematical operation †\dagger† is defined as:A†B=(A2+B2)−(A×B)+3(A+B)A \dagger B = (A^2 + B^2) – (A \times B) + 3(A + B)A†B=(A2+B2)−(A×B)+3(A+B)

Find 5†35 \dagger 35†3.

Solution:5†3=(52+32)−(5×3)+3(5+3)5 \dagger 3 = (5^2 + 3^2) – (5 \times 3) + 3(5 + 3)5†3=(52+32)−(5×3)+3(5+3) =(25+9)−15+3(8)= (25 + 9) – 15 + 3(8)=(25+9)−15+3(8) =34−15+24=43= 34 – 15 + 24 = 43=34−15+24=43

Correct answer: 43

111. Symbolic Arithmetic Challenge

Question:
If ∗\ast∗ means add 7, #\## means multiply by 2, and \@\@\@ means subtract 5, find the value of:6∗#\@46 \ast \# \@ 46∗#\@4

Solution:6∗=6+7=136 \ast = 6 + 7 = 136∗=6+7=13 13#=13×2=2613 \# = 13 \times 2 = 2613#=13×2=26 4\@=4−5=−14 \@ = 4 – 5 = -14\@=4−5=−1

Correct answer: 26 and -1 (if separately computed).

112. Symbolic Inequality Puzzle

Question:
In a symbolic notation:

A⊕BA \oplus BA⊕B means “A is greater than or equal to B.”
A⊖BA \ominus BA⊖B means “A is less than B.”
A⊙BA \odot BA⊙B means “A is not equal to B.”

Which of the following is always true?
A) 8⊖10⊕68 \ominus 10 \oplus 68⊖10⊕6
B) 12⊕12⊙1412 \oplus 12 \odot 1412⊕12⊙14
C) 5⊕4⊙65 \oplus 4 \odot 65⊕4⊙6

Solution:

A) 8<10≥68 < 10 \geq 68<10≥6 → True
B) 12≥12≠1412 \geq 12 \neq 1412≥12=14 → True
C) 5≥4≠65 \geq 4 \neq 65≥4=6 → True

Correct answers: A, B, and C

113. Symbolic Relationship Puzzle

Question:
In a symbolic notation system:

A♡BA \heartsuit BA♡B means “A is the grandmother of B.”
A♢BA \diamondsuit BA♢B means “A is the uncle of B.”
A♠BA \spadesuit BA♠B means “A is the nephew of B.”

Solution:

A) XXX is the grandmother of YYY, and YYY is the uncle of ZZZ → Possible
B) AAA is the nephew of BBB, and BBB is the grandmother of CCC → Incorrect, because a nephew and a grandmother cannot be directly related.
C) PPP is the uncle of QQQ, and QQQ is the nephew of RRR → Possible

Correct answer: B

114. Symbol-Based Mathematical Equation

Question:
If:A∙B=(A+B)2−(A−B)2+2(A×B)A \bullet B = (A + B)^2 – (A – B)^2 + 2(A \times B)A∙B=(A+B)2−(A−B)2+2(A×B)

Find the value of 7∙37 \bullet 37∙3.

Solution:7∙3=(7+3)2−(7−3)2+2(7×3)7 \bullet 3 = (7 + 3)^2 – (7 – 3)^2 + 2(7 \times 3)7∙3=(7+3)2−(7−3)2+2(7×3) =102−42+2(21)= 10^2 – 4^2 + 2(21)=102−42+2(21) =100−16+42=126= 100 – 16 + 42 = 126=100−16+42=126

Correct answer: 126

115. Symbol-Based Logical Challenge

Question:
A new notation is defined as follows:

A♡BA \heartsuit BA♡B means “A is the stepbrother of B.”
A♢BA \diamondsuit BA♢B means “A is the stepfather of B.”
A♠BA \spadesuit BA♠B means “A is the grandson of B.”

Solution:

A) XXX is the stepbrother of YYY, and YYY is the stepfather of ZZZ → Possible
B) AAA is the grandson of BBB, and BBB is the stepbrother of CCC → Incorrect, because a stepbrother and a grandson cannot have a direct lineage.
C) PPP is the stepfather of QQQ, and QQQ is the grandson of RRR → Possible

Correct answer: B

116. Custom Arithmetic Operation

Question:
A new operation △\triangle△ is defined as:A△B=(A2+B2)−(2A×B)+(3A+4B)A \triangle B = (A^2 + B^2) – (2A \times B) + (3A + 4B)A△B=(A2+B2)−(2A×B)+(3A+4B)

Find the value of 5△35 \triangle 35△3.

Solution:5△3=(52+32)−(2×5×3)+(3×5+4×3)5 \triangle 3 = (5^2 + 3^2) – (2 \times 5 \times 3) + (3 \times 5 + 4 \times 3)5△3=(52+32)−(2×5×3)+(3×5+4×3) =(25+9)−(2×15)+(15+12)= (25 + 9) – (2 \times 15) + (15 + 12)=(25+9)−(2×15)+(15+12) =34−30+27=31= 34 – 30 + 27 = 31=34−30+27=31

Correct answer: 31

117. Symbol-Based Logical Deduction

Question:
In a symbolic notation system:

A⋆BA \star BA⋆B means “A is the father of B.”
A#BA \# BA#B means “A is the son of B.”
A%BA \% BA%B means “A is the uncle of B.”

Which of the following is logically incorrect?
A) X⋆Y#ZX \star Y \# ZX⋆Y#Z
B) A#B%CA \# B \% CA#B%C
C) P⋆Q⋆RP \star Q \star RP⋆Q⋆R

Solution:

A) XXX is the father of YYY, and YYY is the son of ZZZ → Possible
B) AAA is the son of BBB, and BBB is the uncle of CCC → Incorrect, because a son cannot be directly related to an uncle.
C) PPP is the father of QQQ, and QQQ is the father of RRR → Possible

Correct answer: B

118. Symbol-Based Number Puzzle

Question:
A mathematical operation †\dagger† is defined as:A†B=(A3+B3)−2(A×B)+4(A+B)A \dagger B = (A^3 + B^3) – 2(A \times B) + 4(A + B)A†B=(A3+B3)−2(A×B)+4(A+B)

Find 4†24 \dagger 24†2.

Solution:4†2=(43+23)−2(4×2)+4(4+2)4 \dagger 2 = (4^3 + 2^3) – 2(4 \times 2) + 4(4 + 2)4†2=(43+23)−2(4×2)+4(4+2) =(64+8)−2(8)+4(6)= (64 + 8) – 2(8) + 4(6)=(64+8)−2(8)+4(6) =72−16+24=80= 72 – 16 + 24 = 80=72−16+24=80

Correct answer: 80

119. Symbol-Based Mathematical Challenge

Question:
If ⊕\oplus⊕ means multiply by 3, #\## means subtract 4, and \@\@\@ means add 2, find the value of:5⊕#\@65 \oplus \# \@ 65⊕#\@6

Solution:5⊕=5×3=155 \oplus = 5 \times 3 = 155⊕=5×3=15 15#=15−4=1115 \# = 15 – 4 = 1115#=15−4=11 6\@=6+2=86 \@ = 6 + 2 = 86\@=6+2=8

Correct answer: 11 and 8 (if separately computed).

120. Symbolic Inequality Puzzle

Question:
In a symbolic notation:

A♡BA \heartsuit BA♡B means “A is greater than B.”
A♢BA \diamondsuit BA♢B means “A is less than or equal to B.”
A♠BA \spadesuit BA♠B means “A is not equal to B.”

Which of the following is always true?
A) 9♢10♡89 \diamondsuit 10 \heartsuit 89♢10♡8
B) 7♡7♠67 \heartsuit 7 \spadesuit 67♡7♠6
C) 6♡5♠76 \heartsuit 5 \spadesuit 76♡5♠7

Solution:

A) 9≤10>89 \leq 10 > 89≤10>8 → True
B) 7>7≠67 > 7 \neq 67>7=6 → False, because 7 is not greater than 7.
C) 6>5≠76 > 5 \neq 76>5=7 → True

Correct answers: A and C

121. Symbolic Relationship Puzzle

Question:
In a symbolic notation system:

A♡BA \heartsuit BA♡B means “A is the grandmother of B.”
A♢BA \diamondsuit BA♢B means “A is the cousin of B.”
A♠BA \spadesuit BA♠B means “A is the niece of B.”

Solution:

A) XXX is the grandmother of YYY, and YYY is the cousin of ZZZ → Possible
B) AAA is the niece of BBB, and BBB is the grandmother of CCC → Incorrect, because a niece and a grandmother cannot be directly related.
C) PPP is the cousin of QQQ, and QQQ is the niece of RRR → Possible

Correct answer: B

122. Symbol-Based Mathematical Equation

Question:
If:A∗B=(A+B)3−(A−B)3+5(A×B)A \ast B = (A + B)^3 – (A – B)^3 + 5(A \times B)A∗B=(A+B)3−(A−B)3+5(A×B)

Find the value of 3∗23 \ast 23∗2.

Solution:3∗2=(3+2)3−(3−2)3+5(3×2)3 \ast 2 = (3 + 2)^3 – (3 – 2)^3 + 5(3 \times 2)3∗2=(3+2)3−(3−2)3+5(3×2) =53−13+5(6)= 5^3 – 1^3 + 5(6)=53−13+5(6) =125−1+30=154= 125 – 1 + 30 = 154=125−1+30=154

Correct answer: 154

123. Symbol-Based Logical Challenge

Question:
A new notation is defined as follows:

A♡BA \heartsuit BA♡B means “A is the stepsister of B.”
A♢BA \diamondsuit BA♢B means “A is the stepmother of B.”
A♠BA \spadesuit BA♠B means “A is the grandson of B.”

Solution:

A) XXX is the stepsister of YYY, and YYY is the stepmother of ZZZ → Possible
B) AAA is the grandson of BBB, and BBB is the stepsister of CCC → Incorrect, because a stepsister and a grandson cannot have a direct lineage.
C) PPP is the stepmother of QQQ, and QQQ is the grandson of RRR → Possible

Correct answer: B

124. Custom Arithmetic Operation

Question:
A new operation ⋆\star⋆ is defined as:A⋆B=A2+B2−(2AB)+3(A+B)A \star B = A^2 + B^2 – (2AB) + 3(A + B)A⋆B=A2+B2−(2AB)+3(A+B)

Find the value of 6⋆46 \star 46⋆4.

Solution:6⋆4=62+42−(2×6×4)+3(6+4)6 \star 4 = 6^2 + 4^2 – (2 \times 6 \times 4) + 3(6 + 4)6⋆4=62+42−(2×6×4)+3(6+4) =36+16−48+30= 36 + 16 – 48 + 30=36+16−48+30 =34= 34=34

Correct answer: 34

125. Symbol-Based Logical Deduction

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the grandfather of B.”
A□BA \square BA□B means “A is the father of B.”
A⋄BA \diamond BA⋄B means “A is the brother of B.”

Which of the following is logically incorrect?
A) X△Y□ZX \triangle Y \square ZX△Y□Z
B) A□B⋄CA \square B \diamond CA□B⋄C
C) P△Q△RP \triangle Q \triangle RP△Q△R

Solution:

A) XXX is the grandfather of YYY, and YYY is the father of ZZZ → Possible
B) AAA is the father of BBB, and BBB is the brother of CCC → Incorrect, since a father and a brother cannot be directly related.
C) PPP is the grandfather of QQQ, and QQQ is the grandfather of RRR → Possible

Correct answer: B

126. Symbol-Based Number Puzzle

Question:
A mathematical operation ⊙\odot⊙ is defined as:A⊙B=(A2−B2)+(3A×B)−(2A+B)A \odot B = (A^2 – B^2) + (3A \times B) – (2A + B)A⊙B=(A2−B2)+(3A×B)−(2A+B)

Find 5⊙35 \odot 35⊙3.

Solution:5⊙3=(52−32)+(3×5×3)−(2×5+3)5 \odot 3 = (5^2 – 3^2) + (3 \times 5 \times 3) – (2 \times 5 + 3)5⊙3=(52−32)+(3×5×3)−(2×5+3) =(25−9)+(3×15)−(10+3)= (25 – 9) + (3 \times 15) – (10 + 3)=(25−9)+(3×15)−(10+3) =16+45−13=48= 16 + 45 – 13 = 48=16+45−13=48

Correct answer: 48

127. Symbol-Based Mathematical Challenge

Question:
If ⊕\oplus⊕ means multiply by 4, #\## means subtract 5, and \@\@\@ means add 3, find the value of:7⊕#\@27 \oplus \# \@ 27⊕#\@2

Solution:7⊕=7×4=287 \oplus = 7 \times 4 = 287⊕=7×4=28 28#=28−5=2328 \# = 28 – 5 = 2328#=28−5=23 2\@=2+3=52 \@ = 2 + 3 = 52\@=2+3=5

Correct answer: 23 and 5 (if separately computed).

128. Symbolic Inequality Puzzle

Question:
In a symbolic notation:

A♡BA \heartsuit BA♡B means “A is less than B.”
A♢BA \diamondsuit BA♢B means “A is greater than or equal to B.”
A♠BA \spadesuit BA♠B means “A is equal to B.”

Which of the following is always true?
A) 5♡6♢45 \heartsuit 6 \diamondsuit 45♡6♢4
B) 8♢8♠88 \diamondsuit 8 \spadesuit 88♢8♠8
C) 3♢2♠33 \diamondsuit 2 \spadesuit 33♢2♠3

Solution:

A) 5<6≥45 < 6 \geq 45<6≥4 → True
B) 8≥8=88 \geq 8 = 88≥8=8 → True
C) 3≥2=33 \geq 2 = 33≥2=3 → False, because 2 is not equal to 3.

Correct answers: A and B

129. Symbolic Relationship Puzzle

Question:
In a symbolic notation system:

A♡BA \heartsuit BA♡B means “A is the aunt of B.”
A♢BA \diamondsuit BA♢B means “A is the cousin of B.”
A♠BA \spadesuit BA♠B means “A is the sister of B.”

Solution:

A) XXX is the aunt of YYY, and YYY is the cousin of ZZZ → Possible
B) AAA is the sister of BBB, and BBB is the aunt of CCC → Incorrect, because a sister and an aunt cannot have a direct lineage.
C) PPP is the cousin of QQQ, and QQQ is the sister of RRR → Possible

Correct answer: B

130. Symbol-Based Mathematical Equation

Question:
If:A∗B=(A+B)4−(A−B)4+6(A×B)A \ast B = (A + B)^4 – (A – B)^4 + 6(A \times B)A∗B=(A+B)4−(A−B)4+6(A×B)

Find the value of 2∗12 \ast 12∗1.

Solution:2∗1=(2+1)4−(2−1)4+6(2×1)2 \ast 1 = (2 + 1)^4 – (2 – 1)^4 + 6(2 \times 1)2∗1=(2+1)4−(2−1)4+6(2×1) =34−14+6(2)= 3^4 – 1^4 + 6(2)=34−14+6(2) =81−1+12=92= 81 – 1 + 12 = 92=81−1+12=92

Correct answer: 92

131. Symbol-Based Logical Challenge

Question:
A new notation is defined as follows:

A♡BA \heartsuit BA♡B means “A is the mother-in-law of B.”
A♢BA \diamondsuit BA♢B means “A is the brother-in-law of B.”
A♠BA \spadesuit BA♠B means “A is the son of B.”

Solution:

A) XXX is the mother-in-law of YYY, and YYY is the brother-in-law of ZZZ → Possible
B) AAA is the son of BBB, and BBB is the mother-in-law of CCC → Incorrect, because a son and a mother-in-law cannot be directly related.
C) PPP is the brother-in-law of QQQ, and QQQ is the son of RRR → Possible

Correct answer: B

132. Custom Arithmetic Operation

Question:
A new operation ⊛\circledast⊛ is defined as:A⊛B=A3+B3−3(A×B)+2(A+B)A \circledast B = A^3 + B^3 – 3(A \times B) + 2(A + B)A⊛B=A3+B3−3(A×B)+2(A+B)

Find the value of 4⊛24 \circledast 24⊛2.

Solution:4⊛2=43+23−3(4×2)+2(4+2)4 \circledast 2 = 4^3 + 2^3 – 3(4 \times 2) + 2(4 + 2)4⊛2=43+23−3(4×2)+2(4+2) =64+8−3(8)+2(6)= 64 + 8 – 3(8) + 2(6)=64+8−3(8)+2(6) =64+8−24+12=60= 64 + 8 – 24 + 12 = 60=64+8−24+12=60

Correct answer: 60

133. Symbol-Based Logical Deduction

Question:
In a symbolic notation system:

A△BA \triangle BA△B means “A is the uncle of B.”
A□BA \square BA□B means “A is the son of B.”
A⋄BA \diamond BA⋄B means “A is the cousin of B.”

Which of the following is logically incorrect?
A) X△Y□ZX \triangle Y \square ZX△Y□Z
B) A□B⋄CA \square B \diamond CA□B⋄C
C) P△Q△RP \triangle Q \triangle RP△Q△R

Solution:

A) XXX is the uncle of YYY, and YYY is the son of ZZZ → Possible
B) AAA is the son of BBB, and BBB is the cousin of CCC → Incorrect, since a parent and cousin relationship cannot exist in this order.
C) PPP is the uncle of QQQ, and QQQ is the uncle of RRR → Possible

Correct answer: B

134. Symbol-Based Number Puzzle

Question:
A mathematical operation ⊙\odot⊙ is defined as:A⊙B=(A2−B2)+(4A×B)−(3A+2B)A \odot B = (A^2 – B^2) + (4A \times B) – (3A + 2B)A⊙B=(A2−B2)+(4A×B)−(3A+2B)

Find 5⊙35 \odot 35⊙3.

Solution:5⊙3=(52−32)+(4×5×3)−(3×5+2×3)5 \odot 3 = (5^2 – 3^2) + (4 \times 5 \times 3) – (3 \times 5 + 2 \times 3)5⊙3=(52−32)+(4×5×3)−(3×5+2×3) =(25−9)+(4×15)−(15+6)= (25 – 9) + (4 \times 15) – (15 + 6)=(25−9)+(4×15)−(15+6) =16+60−21=55= 16 + 60 – 21 = 55=16+60−21=55

Correct answer: 55

135. Symbol-Based Mathematical Challenge

Question:
If ⊕\oplus⊕ means multiply by 5, #\## means subtract 7, and \@\@\@ means add 4, find the value of:6⊕#\@36 \oplus \# \@ 36⊕#\@3

Solution:6⊕=6×5=306 \oplus = 6 \times 5 = 306⊕=6×5=30 30#=30−7=2330 \# = 30 – 7 = 2330#=30−7=23 3\@=3+4=73 \@ = 3 + 4 = 73\@=3+4=7

Correct answer: 23 and 7 (if separately computed).

136. Symbolic Inequality Puzzle

Question:
In a symbolic notation:

A♡BA \heartsuit BA♡B means “A is less than or equal to B.”
A♢BA \diamondsuit BA♢B means “A is greater than B.”
A♠BA \spadesuit BA♠B means “A is not equal to B.”

Which of the following is always true?
A) 6♡7♢56 \heartsuit 7 \diamondsuit 56♡7♢5
B) 9♢8♠99 \diamondsuit 8 \spadesuit 99♢8♠9
C) 4♢3♠44 \diamondsuit 3 \spadesuit 44♢3♠4

Solution:

A) 6≤7>56 \leq 7 > 56≤7>5 → True
B) 9>8≠99 > 8 \neq 99>8=9 → True
C) 4>3≠44 > 3 \neq 44>3=4 → True

Correct answers: A, B, and C

137. Symbolic Relationship Puzzle

Question:
In a symbolic notation system:

A♡BA \heartsuit BA♡B means “A is the father-in-law of B.”
A♢BA \diamondsuit BA♢B means “A is the nephew of B.”
A♠BA \spadesuit BA♠B means “A is the brother-in-law of B.”

Solution:

A) XXX is the father-in-law of YYY, and YYY is the nephew of ZZZ → Possible
B) AAA is the brother-in-law of BBB, and BBB is the father-in-law of CCC → Incorrect, because a brother-in-law and a father-in-law cannot have this direct relationship.
C) PPP is the nephew of QQQ, and QQQ is the brother-in-law of RRR → Possible

Correct answer: B

138. Symbol-Based Mathematical Equation

Question:
If:A∗B=(A+B)5−(A−B)5+7(A×B)A \ast B = (A + B)^5 – (A – B)^5 + 7(A \times B)A∗B=(A+B)5−(A−B)5+7(A×B)

Find the value of 2∗12 \ast 12∗1.

Solution:2∗1=(2+1)5−(2−1)5+7(2×1)2 \ast 1 = (2 + 1)^5 – (2 – 1)^5 + 7(2 \times 1)2∗1=(2+1)5−(2−1)5+7(2×1) =35−15+7(2)= 3^5 – 1^5 + 7(2)=35−15+7(2) =243−1+14=256= 243 – 1 + 14 = 256=243−1+14=256

Correct answer: 256

139. Symbol-Based Logical Challenge

Question:
A new notation is defined as follows:

A♡BA \heartsuit BA♡B means “A is the granddaughter of B.”
A♢BA \diamondsuit BA♢B means “A is the stepfather of B.”
A♠BA \spadesuit BA♠B means “A is the uncle of B.”

Solution:

A) XXX is the granddaughter of YYY, and YYY is the stepfather of ZZZ → Possible
B) AAA is the uncle of BBB, and BBB is the granddaughter of CCC → Incorrect, because an uncle and granddaughter relationship cannot exist in this order.
C) PPP is the stepfather of QQQ, and QQQ is the uncle of RRR → Possible

Correct answer: B

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