NDA 2026 Mathematics PYQ — Let two lines of regression be and for some data. What is the val… | Mathem Solvex | Mathem Solvex
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NDA 2026 — Mathematics PYQ
NDA | Mathematics | 2026
Let two lines of regression be x+y+11=0 and 2x+3y+4=0 for some data. What is the value of the correlation coefficient between x and y?
मान लीजिए किन्हीं आंकड़ों के लिए दो समाश्रयण-रेखाएं (lines of regression) x + y + 11 = 0 और 2x + 3y + 4 = 0 हैं। x और y के बीच सहसंबंध गुणांक का मान क्या है ?
Choose the correct answer:
A.
−32
(Correct Answer)
B.
−61
C.
32
D.
61
Correct Answer:
−32
Explanation
1. Express the lines in slope-intercept form (y=mx+c):
Line 1:x+y+11=0
y=−x−11
The slope is m1=−1.
Line 2:2x+3y+4=0
3y=−2x−4
y=−32x−34
The slope is m2=−32.
2. Relate the slopes to regression coefficients (byx and bxy):
The regression coefficients are related to the slopes of these lines. We know the property r2=byx⋅bxy. For the regression lines, the slopes are byx and bxy1.
We test the two possible assignments for the regression coefficients to ensure the resulting ∣r∣≤1:
Case 1: Assume byx=−1 and bxy1=−32⟹bxy=−23.
r2=byx⋅bxy=(−1)⋅(−23)=23=1.5
Since r2 cannot be greater than 1, this assignment is incorrect.
Case 2: Assume byx=−32 and bxy1=−1⟹bxy=−1.
r2=byx⋅bxy=(−32)⋅(−1)=32
This satisfies ∣r∣≤1.
3. Calculate r:
Since both regression coefficients are negative, the correlation coefficient r must also be negative:
r=−32
Correct Option:
(a) −32
Explanation
1. Express the lines in slope-intercept form (y=mx+c):
Line 1:x+y+11=0
y=−x−11
The slope is m1=−1.
Line 2:2x+3y+4=0
3y=−2x−4
y=−32x−34
The slope is m2=−32.
2. Relate the slopes to regression coefficients (byx and bxy):
The regression coefficients are related to the slopes of these lines. We know the property r2=byx⋅bxy. For the regression lines, the slopes are byx and bxy1.
We test the two possible assignments for the regression coefficients to ensure the resulting ∣r∣≤1:
Case 1: Assume byx=−1 and bxy1=−32⟹bxy=−23.
r2=byx⋅bxy=(−1)⋅(−23)=23=1.5
Since r2 cannot be greater than 1, this assignment is incorrect.
Case 2: Assume byx=−32 and bxy1=−1⟹bxy=−1.
r2=byx⋅bxy=(−32)⋅(−1)=32
This satisfies ∣r∣≤1.
3. Calculate r:
Since both regression coefficients are negative, the correlation coefficient r must also be negative:
