COMEDK 2024 Shift 2 Paper

To determine the new coordinates of point B after it has been rotated about point A through an angle of 15∘ in the anticlockwise direction, we can use rotation formulas. First, let's denote the original coordinates as:
A(2,0)
B(3,1)
Let's transform point B to the origin by translating point A to the origin:
In vector notation, point B relative to A is:

→

AB

=(3−2,1−0)=(1,1)
After the rotation, the new coordinates C can be determined using the rotation matrix. The rotation matrix for a counterclockwise rotation by angle θ is:
(

cos‌θ	−sin‌θ
sin‌θ	cos‌θ

)
For an angle of 15∘, we have:
θ=15∘
The cosine and sine of 15∘ are:
‌cos‌15∘=‌

√3+1

2√2

‌sin‌15∘=‌

√3−1

2√2

Using the rotation matrix:
(

cos‌15∘	−sin‌15∘
sin‌15∘	cos‌15∘

)(

1

1

)
This multiplication gives us:
(

‌ √3+1 2√2 ⋅1−‌ √3−1 2√2 ⋅1

‌ √3−1 2√2 ⋅1+‌ √3+1 2√2 ⋅1

)
This simplifies to:
Now, we revert the translation back by adding A(2,0) to get the coordinates of C :
C=(2+‌

1

√2

,√‌

3

2

)
The correct option is:
Option D
(2+‌

1

√2

,√‌

3

2

)