TS EAMCET 18-July-2022 Shift 1 Solved Paper

Let a=a

^

i

+b

^

j

+c

^

k

a be a vector in the plane containing vectors
b=

^

i

+2

^

j

+

^

k

‌ and ‌c=2

^

i

−

^

j

+

^

k

a⋅(b×c)=0
|

a	b	c
1	2	1
2	−1	1

|=0
⇒‌‌3a+b−5c=0‌‌‌⋅⋅⋅⋅⋅⋅⋅(i)
⇒a⟂(i+j+3k)⇒a⋅(i+j+3k)=0
a+b+3c=0‌‌‌⋅⋅⋅⋅⋅⋅⋅(ii)
Projection of a on b is 3√6 .
‌‌

a⋅b

|b|

=3√6
‌‌

a+2b+c

√1+4+1

=3√6
‌a+2b+c=18‌‌‌⋅⋅⋅⋅⋅⋅⋅(iii)
On subtracting Eq. (ii) from Eq. (i),
. 2a−8c=0⇒a=4c‌‌‌⋅⋅⋅⋅⋅⋅⋅(iv)
On multiplying 2 in Eq. (ii) and subtracting
Eq. (iii), we get
a+5c=−18‌‌‌⋅⋅⋅⋅⋅⋅⋅(v)
From Eqs. (iv) and (v), c=−2 and a=−8
On putting in Eq. (iii), b=14
a‌=−8

^

i

+14

^

j

−2

^

k

|a|2‌=82+142+22=64+196+4=264