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JEE Advanced 2021 Paper 2
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Section:
Mathematics
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© examsnet.com
Question : 43
Total: 57
Let
O
be the origin and
O
A
=
2
∧
i
+
2
∧
j
+
∧
k
,
O
B
=
∧
i
−
2
∧
j
+
2
∧
k
and
→
O
C
=
‌
1
2
(
→
O
B
−
λ
→
O
A
)
for some
λ
>
0
. If
|
→
O
B
×
→
O
C
|
=
‌
9
2
, then which of the following statements is (are) TRUE?
[JEE Adv 2021 P2]
Projection of
→
O
C
on
O
A
is
−
‌
3
2
Area of the triangle
O
A
B
is
‌
9
2
Area of the triangle
A
B
C
is
‌
9
2
The acute angle between the diagonals of the parallelogram with adjacent sides
O
A
and
O
C
is
‌
Ï€
3
Validate
Solution:
O
B
×
O
C
=
‌
1
2
O
B
×
(
O
B
−
λ
O
A
)
=
‌
λ
2
(
O
A
×
O
B
)
....(i)
Since,
O
A
â‹…
O
B
=
0
therfore
O
A
⟂
r
O
B
∴
|
O
B
×
O
C
|
=
|
‌
λ
2
(
O
A
×
O
B
)
|
  from(i)
=
‌
|
λ
|
2
|
O
A
|
O
B
|
sin
‌
Ï€
2
⇒
‌
9
2
=
‌
|
λ
|
2
×
3
×
3
⇒
λ
=
1
(
given
λ
>
0
)
So,
→
O
C
=
‌
→
O
B
−
→
O
A
2
=
‌
1
2
(
−
∧
i
−
4
∧
j
+
∧
k
)
(a) Projection
O
C
on
O
A
=
‌
O
C
â‹…
O
A
|
O
A
|
=
‌
‌
1
2
(
−
2
−
8
+
1
)
3
=
−
‌
3
2
(b) Area of
â–³
D
A
B
=
‌
1
2
|
O
A
×
O
B
|
=
‌
9
2
(b) Area of
â–³
D
A
B
=
‌
1
2
|
O
A
×
O
B
|
=
‌
9
2
(c) Area of
â–³
A
B
C
=
‌
1
2
|
→
A
B
×
→
A
C
|
=
|
∧
i
∧
j
∧
k
−
1
−
4
1
−
5
2
−
4
−
1
2
|
=
‌
1
2
|
6
∧
i
−
3
∧
j
−
6
∧
k
|
=
‌
9
2
Let acute angle between diagonals is
θ
then
cos
‌
θ
=
‌
(
O
A
+
O
C
)
â‹…
(
O
A
−
O
C
)
(
O
A
+
O
C
)
(
O
A
−
O
C
)
=
‌
18
3
√
2
√
90
∴
‌
‌
θ
â‰
‌
Ï€
3
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