NIMCET 2020 Mathematics PYQ — If , , are three non-zero vectors with no two of which are collin… | Mathem Solvex | Mathem Solvex
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NIMCET 2020 — Mathematics PYQ
NIMCET | Mathematics | 2020
If a, b, c are three non-zero vectors with no two of which are collinear, a+2b is collinear with c and b+3c is collinear with a, then ∣a+2b+6c∣ will be equal to:
Choose the correct answer:
A.
Zero
(Correct Answer)
B.
9
C.
1
D.
None
Correct Answer:
Zero
Explanation
Step 1: Use the first condition of collinearity
Since a+2b is collinear with c, it can be written as a scalar multiple of c. Let λ be a scalar constant:
a+2b=λc
Rearranging the terms to express a, we get:
a=λc−2b— (Equation 1)
Step 2: Use the second condition of collinearity
Similarly, since b+3c is collinear with a, it can be written as a scalar multiple of a. Let μ be another scalar constant:
b+3c=μa— (Equation 2)
Step 3: Substitute Equation 1 into Equation 2
Substitute the value of a from Equation 1 into Equation 2:
b+3c=μ(λc−2b)
Expanding the right-hand side:
b+3c=μλc−2μb
Step 4: Group the vector components
Bring all the terms containing b to one side and c to the other:
b+2μb=μλc−3c
(1+2μ)b=(μλ−3)c
Step 5: Compare the coefficients
We are given that b and c are non-zero and non-collinear vectors. Therefore, the linear combination (1+2μ)b−(μλ−3)c=0 can only hold true if the individual scalar coefficients are both equal to zero:
For the b coefficient:
1+2μ=0⟹μ=−21
For the c coefficient:
μλ−3=0⟹μλ=3
Substitute the value of μ=−21 into the second condition:
(−21)λ=3⟹λ=−6
Step 6: Evaluate the target expression
Now substitute the value of λ=−6 back into Equation 1:
a+2b=−6c
Bring −6c to the left-hand side of the equation:
a+2b+6c=0
Finally, taking the magnitude on both sides:
∣a+2b+6c∣=∣0∣=0
Correct Answer:
A) Zero
Explanation
Step 1: Use the first condition of collinearity
Since a+2b is collinear with c, it can be written as a scalar multiple of c. Let λ be a scalar constant:
a+2b=λc
Rearranging the terms to express a, we get:
a=λc−2b— (Equation 1)
Step 2: Use the second condition of collinearity
Similarly, since b+3c is collinear with a, it can be written as a scalar multiple of a. Let μ be another scalar constant:
b+3c=μa— (Equation 2)
Step 3: Substitute Equation 1 into Equation 2
Substitute the value of a from Equation 1 into Equation 2:
b+3c=μ(λc−2b)
Expanding the right-hand side:
b+3c=μλc−2μb
Step 4: Group the vector components
Bring all the terms containing b to one side and c to the other:
b+2μb=μλc−3c
(1+2μ)b=(μλ−3)c
Step 5: Compare the coefficients
We are given that b and c are non-zero and non-collinear vectors. Therefore, the linear combination (1+2μ)b−(μλ−3)c=0 can only hold true if the individual scalar coefficients are both equal to zero:
For the b coefficient:
1+2μ=0⟹μ=−21
For the c coefficient:
μλ−3=0⟹μλ=3
Substitute the value of μ=−21 into the second condition:
(−21)λ=3⟹λ=−6
Step 6: Evaluate the target expression
Now substitute the value of λ=−6 back into Equation 1:
