UPSC NDA I 2025 Mathematics Paper

Given,
Points A(k,1,−1),B(2k,0,2), and C(2+2k,k,1) are the vertices of the triangle.
The vectors AB and BC are given by:
‌

→

AB

=B−A=(2k−k,0−1,2−(−1))=(k,−1,3)
‌

→

BC

=C−B=(2+2k−2k,k−0,1−2)=(2,k,−1)
The dot product of

→

AB

and

→

BC

is:

→

AB

⋅

→

BC

=(k)(2)+(−1)(k)+(3)(−1)=2k−k−3=k−3
Since the vectors are perpendicular, their dot product must be zero:
k−3=0
Thus, k=3.
∴ The value of k is 3 .