Test Index

UPSC NDA 2 2016 Mathematics Solved Paper

Question Numbers: 65-66

Direction: Consider the following for the next two (02) items that follow:

Let

{\rm{\vec a}} = {\rm{\hat i}} + {\rm{\hat j}},{\rm{\;\vec b}} = 3{\rm{\hat i}} + 4{\rm{\hat k}}

and

{\rm{\vec b}} = {\rm{\vec c}} + {\rm{\vec d}}

where

{\rm{\vec c}}

is parallel to

{\rm{\vec a}}

and

{\rm{\vec d}}

is perpendicular to

{\rm{\vec a}}

Question : 66 of 120

Marks: +1, -0

{\rm{\vec d}} = {\rm{x\hat i}} + {\rm{y\hat j}} + {\rm{z\hat k}}

1. y - x = 4

2z - 3 = 0

Select the correct answer using the code given below:

Solution:

Concept:

Vector addition equates components to solve for unknowns.

Explanation:

Given

{\rm{\vec d}} = x{\rm{\hat i}} + y{\rm{\hat j}} + z{\rm{\hat k}}

From the context,

{\rm{\vec b}} = {\rm{\vec c}} + {\rm{\vec d}}

with

{\rm{\vec b}} = 3{\rm{\hat i}} + 4{\rm{\hat k}}

and

{\rm{\vec c}} = \frac{3}{2}({\rm{\hat i}} + {\rm{\hat j}})

Substitute:

3{\rm{\hat i}} + 4{\rm{\hat k}} = \frac{3}{2}{\rm{\hat i}} + \frac{3}{2}{\rm{\hat j}} + x{\rm{\hat i}} + y{\rm{\hat j}} + z{\rm{\hat k}}

Combine like terms:

3{\rm{\hat i}} + 0{\rm{\hat j}} + 4{\rm{\hat k}} = \left(\frac{3}{2} + x\right){\rm{\hat i}} + \left(\frac{3}{2} + y\right){\rm{\hat j}} + z{\rm{\hat k}}

Equate coefficients:

3 = \frac{3}{2} + x

⇒

x = \frac{3}{2}

0 = \frac{3}{2} + y

⇒

y = -\frac{3}{2}

4 = z

⇒

z = 4

Check statement 1:

y - x = -\frac{3}{2} - \frac{3}{2} = -3 \neq 4

. So statement 1 is false.

Check statement 2:

2z - 3 = 2(4) - 3 = 8 - 3 = 5 \neq 0

. So statement 2 is false.

Answer:

Neither statement 1 nor statement 2 is correct. The correct option is D.

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