NIMCET 2025 Paper

Section: Mathematics

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Question : 26 of 120

Marks: +1, -0

Let

{a}↖{→}

and

{c}↖{→}

be unit vectors such that the angle between them is

\cos ^{-1} (\;{1}/{4})

If

{b}↖{→}=2 {c}↖{∧}+λ {a}↖{∧},

where

λ>0

and

|{b}↖{→}|=4

, then

λ

is equal to:

$\;{1}/{3}$
$\;{1}/{4}$
4
3

Solution:

Since

{a}↖{→}

and

{c}↖{→}

are unit vectors and

{b}↖{→}=2 {c}↖{→}+λ {a}↖{→},

we use:

|{b}↖{→}|^2=|2 {c}↖{→}+λ {a}↖{→}|^2=16 .

Compute:

|b|^2=4+λ^2+2(2)(λ)({a}↖{→} ⋅ {c}↖{→})

Given:

{a}↖{→} ⋅ {c}↖{→}=\;{1}/{4}

So:

4+λ^2+4 λ (\;{1}/{4}) =16

4+λ^2+λ=16

λ^2+λ-12=0

Factor:

(λ-3)(λ+4)=0

λ=3 \;\; (λ>0)

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