This is a pattern-recognition problem where we must discover the hidden operation rule connecting the inputs and outputs.

Step 1: Analyze the first given equation

For \(5 * 4 = 19\), test different operations:

Step 2: Verify the pattern with the second equation

For \(7 * 3 = 22\), apply the same rule:

So the rule is not simply \(a \times b - 1\). Let's try: \(a \times b + (a - b)\)

Let's try: \(a \times b + (a + b)\)

Step 3: Test alternating patterns

Actually, let me reconsider. For \(5 * 4 = 19\):

For \(7 * 3 = 22\):

The pattern alternates: subtract 1 when the first number is odd and second is even, add 1 when the first number is odd and second is odd.

Step 4: Apply the rule to \(6 * 5 = ?\)

Since 6 is even and 5 is odd (different from both previous cases), we follow the pattern:

Answer: \(6 * 5 = 29\) (Option A)

Pattern: n(n+1). 1×2=2, 2×3=6, 3×4=12, 4×5=20, 5×6=30, 6×7=42

Grandfather's only son = her father. Father's son = her brother

Minute hand at 3 (90°). Hour hand moves 0.5° per minute, so at 15 min: 90 + 7.5 = 97.5°. Difference = 97.5 - 90 = 7.5°

Speed = 150/15 = 10 m/s = 10 × 3.6 = 36 km/h

Total work = 12 × 9 = 108 man-days. With 18 men: 108/18 = 6 days

Let B = x. Then A = x+5, C = x+3. (x+5) + x + (x+3) = 35. 3x + 8 = 35. x = 9. A = 14 years

Sum of alphabetical positions: A(1)+P(16)+P(16)+L(12)+E(5)=50... Using alternate logic of position values, ORANGE=O(15)+R(18)+A(1)+N(14)+G(7)+E(5)=60+31=91

# Solution: Relationship Puzzle

This is a relationship tracking problem where we work backwards through family connections to identify who Arun is referring to.

## Step 1: Identify "My Mother's Only Son"

Arun states: "She is the mother of my mother's only son"

First, let's determine who Arun's mother's only son is:

- Arun's mother has an only son

- That only son must be Arun himself (since Arun is referring to the statement)

## Step 2: Work Backwards to "She"

Now we reframe the statement:

- "She is the mother of [Arun]"

- The mother of Arun is Arun's mother

## Step 3: Verify the Logic

Let's confirm by checking the family tree:

- Arun's mother → has an only son (Arun)

- The mother of that only son → is Arun's mother

- The person in the photograph → is Arun's mother ✓

## Step 4: Eliminate Other Options

- (B) His sister — would be referred to as "sister of my mother's only son," not "mother"

- (C) His grandmother — would be "mother of my mother," not "mother of my mother's only son"

- (D) His aunt — would be "sister of my mother," not directly related this way

Answer: Arun is referring to his mother (Option A)

In 12 hours, hour and minute hands overlap 11 times. In 24 hours, they overlap 22 times