KCET 2025 Math Paper

Given:
‌

→

a

=

^

i

+2

^

j

+

^

k

and ‌‌

→

b

=

^

i

−

^

j

+4

^

k

‌

→

c

=

^

i

+

^

j

+

^

k

We need the expression

→

a

+λ

→

b

to be perpendicular to

→

c

:
(

→

a

+λ

→

b

)⋅

→

c

=0
First, calculate

→

a

+λ

→

b

:
Expanding this, we get:
‌=

^

i

+2

^

j

+

^

k

+λ

^

i

−λ

^

j

+4λ

^

k

‌=(1+λ)

^

i

+(2−λ)

^

j

+(1+4λ)

^

k

Next, calculate the dot product with

→

c

:
Performing the dot product:
‌=(1+λ)⋅1+(2−λ)⋅1+(1+4λ)⋅1
‌=(1+λ)+(2−λ)+(1+4λ)
‌=1+λ+2−λ+1+4λ
‌=4+4λ
Set the expression to zero for perpendicularity:
4+4λ=0
Solving for λ :
‌4λ=−4
‌λ=−1
Thus, the correct value of λ is -1 .