Question

The $y$-coordinate of $A_8$ is

• A. 13
• B. 10
• C. 7
• D. 5.5
• E. None of the above
• A. 1
• B. 2
• C. 3
• D. 7
• J. None of the above

Answer 1

The Correct Option is C

Step 1: Represent the $x$-coordinates.
Given that $x_1, x_2, \ldots, x_n$ are in Arithmetic Progression (A.P). Let the first term be $a$ and common difference $d$. We know: \[ x_2 = a+d = 2, \quad x_{24} = a+23d = 68 \] Solving: \[ a+d=2 \quad \text{and} \quad a+23d=68 \] Subtracting: $(a+23d)-(a+d)=66 \Rightarrow 22d=66 \Rightarrow d=3$. Then $a=2-d=2-3=-1$. Step 2: Find $x_8$.
\[ x_8 = a+7d = -1+7(3)=20 \] Step 3: Equation of the line $y=px+q$.
We know $A_2(2,-2)$ and $A_{24}(68,31)$ lie on the line. So: \[ -2=2p+q, \quad 31=68p+q \] Subtracting: $(31-(-2))=33=(68-2)p=66p \Rightarrow p=\tfrac{1}{2}$. Substitute back: $-2=2(\tfrac{1}{2})+q \Rightarrow -2=1+q \Rightarrow q=-3$. Hence line equation: \[ y=\tfrac{1}{2}x-3 \] Step 4: Find $y_8$.
At $x_8=20$: \[ y_8=\tfrac{1}{2}(20)-3=10-3=7 \] \[ \boxed{7} \]

Answer 2

Step 1: Recall x-coordinates in A.P.
From Q74, we found a = -1 and d = 3.
So x-coordinates are:
x₁ = -1, x₂ = 2, x₃ = 5, x₄ = 8, x₅ = 11, …

Step 2: Equation of line.
From Q74, the line is y = (1/2)x − 3.

Step 3: Find y-coordinates.
- For x₁ = -1: y₁ = (1/2)(-1) − 3 = -0.5 − 3 = -3.5
- For x₂ = 2: y₂ = (1/2)(2) − 3 = 1 − 3 = -2
- For x₃ = 5: y₃ = (1/2)(5) − 3 = 2.5 − 3 = -0.5
- For x₄ = 8: y₄ = (1/2)(8) − 3 = 4 − 3 = 1
- For x₅ = 11: y₅ = (1/2)(11) − 3 = 5.5 − 3 = 2.5

Thus, initial coordinates are:
A₁(-1, -3.5), A₂(2, -2), A₃(5, -0.5), A₄(8, 1), A₅(11, 2.5), …

Step 4: Condition for being in the first quadrant.
Both x > 0 and y > 0.

- A₁(-1, -3.5): x < 0, so not in first quadrant.
- A₂(2, -2): y < 0, so not in first quadrant.
- A₃(5, -0.5): y < 0, so not in first quadrant.
- A₄(8, 1): both positive ⇒ first quadrant.
- A₅(11, 2.5): both positive ⇒ first quadrant.
- Beyond this, x keeps increasing, and y = (1/2)x − 3 keeps increasing. So all further points will be in the first quadrant.

Step 5: Count.
Only A₁, A₂, and A₃ are not in the first quadrant.

So, total = 3 points.

Answer: 3