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1

### WB JEE 2022

English
Bengali

If $$\overrightarrow a = \widehat i + \widehat j - \widehat k$$, $$\overrightarrow b = \widehat i - \widehat j + \widehat k$$ and $$\overrightarrow c$$ is unit vector perpendicular to $$\overrightarrow a$$ and coplanar with $$\overrightarrow a$$ and $$\overrightarrow b$$, then unit vector $$\overrightarrow d$$ perpendicular to both $$\overrightarrow a$$ and $$\overrightarrow c$$ is

A
$$\pm {1 \over {\sqrt 6 }}\left( {2\widehat i - \widehat j + \widehat k} \right)$$
B
$$\pm {1 \over {\sqrt 2 }}\left( {\widehat j + \widehat k} \right)$$
C
$$\pm {1 \over {\sqrt 6 }}\left( {\widehat i - 2\widehat j + \widehat k} \right)$$
D
$$\pm {1 \over {\sqrt 2 }}\left( {\widehat j - \widehat k} \right)$$

দেওয়া আছে $$\overrightarrow a = \widehat i + \widehat j - \widehat k$$, $$\overrightarrow b = \widehat i - \widehat j + \widehat k$$, $$\overrightarrow c$$ একটি একক ভেক্টর $$\overrightarrow a$$ এর উপর লম্ব এবং $$\overrightarrow a$$ ও $$\overrightarrow b$$ এর সঙ্গে একতলীয়। সেক্ষেত্রে $$\overrightarrow a$$ ও $$\overrightarrow c$$ উভয়ের উপর লম্ব ও একক ভেক্টর $$\overrightarrow d$$ হবে

A
$$\pm {1 \over {\sqrt 6 }}\left( {2\widehat i - \widehat j + \widehat k} \right)$$
B
$$\pm {1 \over {\sqrt 2 }}\left( {\widehat j + \widehat k} \right)$$
C
$$\pm {1 \over {\sqrt 6 }}\left( {\widehat i - 2\widehat j + \widehat k} \right)$$
D
$$\pm {1 \over {\sqrt 2 }}\left( {\widehat j - \widehat k} \right)$$
2

### WB JEE 2021

English
Bengali
If a($$\alpha$$ $$\times$$ $$\beta$$) + b($$\beta$$ $$\times$$ $$\gamma$$) + c($$\gamma$$ + $$\alpha$$) = 0, where a, b, c are non-zero scalars, then the vectors $$\alpha$$, $$\beta$$, $$\gamma$$ are
A
parallel
B
non-coplanar
C
coplanar
D
mutually perpendicular

## Explanation

We have,

a($$\alpha$$ $$\times$$ $$\beta$$) + b($$\beta$$ $$\times$$ $$\gamma$$) + c($$\gamma$$ + $$\alpha$$) = 0

Clearly a($$\alpha$$ $$\times$$ $$\beta$$), ($$\beta$$ $$\times$$ $$\gamma$$) and ($$\gamma$$ + $$\alpha$$) are coplanar vector $$\therefore$$ $$\alpha$$, $$\beta$$ and $$\gamma$$ are also coplanar.
দেওয়া আছে a($$\alpha$$ $$\times$$ $$\beta$$) + b($$\beta$$ $$\times$$ $$\gamma$$) + c($$\gamma$$ + $$\alpha$$) = 0, যেখানে a, b, c অশূণ্য স্কেলার। সেক্ষেত্রে $$\alpha$$, $$\beta$$, $$\gamma$$ ভেক্টরত্রয়
A
পরস্পর সমান্তরাল
B
সমতলীয় নয়
C
সমতলীয়
D
পরস্পর লম্ব

## Explanation

আমাদের কাছে,

a($$\alpha$$ $$\times$$ $$\beta$$) + b($$\beta$$ $$\times$$ $$\gamma$$) + c($$\gamma$$ + $$\alpha$$) = 0

স্পষ্টতই a($$\alpha$$ $$\times$$ $$\beta$$), ($$\beta$$ $$\times$$ $$\gamma$$) এবং ($$\gamma$$ + $$\alpha$$) হল সমতলীয় ভেক্টর

$$\therefore$$ এছাড়াও $$\alpha$$, $$\beta$$ and $$\gamma$$ হল সমতলীয়.
3

### WB JEE 2021

English
Bengali
let $$\alpha$$, $$\beta$$, $$\gamma$$ be three non-zero vectors which are pairwise non-collinear. if $$\alpha$$ + 3$$\beta$$ is collinear with $$\gamma$$ and $$\beta$$ + 2$$\gamma$$ is collinear with $$\alpha$$ then $$\alpha$$ + 3$$\beta$$ + 6$$\gamma$$ is
A
$$\gamma$$
B
0
C
$$\gamma$$ + $$\gamma$$
D
$$\alpha$$

## Explanation

Given, $$\alpha$$ + 3$$\beta$$ is collinear with $$\gamma$$

$$\therefore$$ $$\alpha$$ + 3$$\beta$$ = $$\lambda$$$$\gamma$$ ...... (1)

and $$\beta$$ + 2$$\gamma$$ is collinear with $$\alpha$$

$$\therefore$$ $$\beta$$ + 2$$\gamma$$ = $$\mu$$$$\alpha$$ ...... (2)

Add 6$$\gamma$$ with equation (1), we get

$$\alpha$$ + 3$$\beta$$ + 6$$\gamma$$ = (6 + $$\lambda$$)$$\gamma$$ ...... (3)

Multiply equation (2) with (3) and add $$\alpha$$, we get

$$\alpha$$ + 3$$\beta$$ + 6$$\gamma$$ = (3$$\mu$$ + 1)$$\alpha$$ ..... (4)

From (3) and (4), we get

(6 + $$\lambda$$)$$\gamma$$ = (3$$\mu$$ + 1)$$\alpha$$ ..... (5)

As, $$\alpha$$, $$\beta$$, $$\gamma$$ are not pairwise collinear so in equation (5) coefficient of $$\gamma$$ and $$\alpha$$ must be zero.

$$\therefore$$ 6 + $$\lambda$$ = 0 and 3$$\mu$$ + 1 = 0

$$\therefore$$ From equation (3),

$$\lambda$$ + 3$$\beta$$ + 6$$\gamma$$ = (6 + $$\lambda$$)$$\gamma$$ = 0 $$\times$$ $$\gamma$$ = 0

মনে কর $$\alpha$$, $$\beta$$, $$\gamma$$ তিনটি অ-শূণ্য ভেক্টর যাদের মধ্যে কোন দুটি একযােগে সমরেখ নয়। যদি $$\alpha$$ + 3$$\beta$$ ও $$\gamma$$ সমরেখ হয় এবং $$\beta$$ + 2$$\gamma$$ ও $$\alpha$$ সমরেখ হয়, তবে $$\alpha$$ + 3$$\beta$$ + 6$$\gamma$$ হবে
A
$$\gamma$$
B
0
C
$$\gamma$$ + $$\gamma$$
D
$$\alpha$$

## Explanation

প্রদত্ত, $$\alpha$$ + 3$$\beta$$, $$\gamma$$-এর সাথে সমরেখ

$$\therefore$$ $$\alpha$$ + 3$$\beta$$ = $$\lambda$$1$$\gamma$$

$$\Rightarrow$$ $$\beta = {{{\lambda _1}} \over 3}\gamma - {\alpha \over 3}$$ .... (i)

এবং $$\beta$$ + 2$$\gamma$$ হল $$\alpha$$-এর সাথে সমরেখ

$$\therefore$$ $$\beta$$ + 2$$\gamma = \lambda$$2$$\alpha$$

$$\Rightarrow$$ $$\beta$$ = $$\lambda$$2$$\alpha$$ $$-$$ 2$$\gamma$$ ...... (ii)

সমীকরণ (i) এবং (ii) থেকে, আমরা পাই

$${{{\lambda _1}} \over 3}\gamma - {\alpha \over 3} = {\lambda _2}\alpha - 2\gamma$$

$$\Rightarrow \alpha \left( {{\lambda _2} + {1 \over 3}} \right) = \gamma \left( {{{{\lambda _1}} \over 3} + 2} \right)$$

$$\Rightarrow {\lambda _2} + {1 \over 3} = 0$$ এবং $${{{\lambda _1}} \over 3} + 2 = 0$$

$$\Rightarrow {\lambda _2} = - {1 \over 3}$$ এবং $${{{\lambda _1}} \over 3} = - 2$$

$$\Rightarrow {\lambda _1} = - 6$$ এবং $${\lambda _2} = - {1 \over 3}$$

সমীকরণ (i) এবং (ii) থেকে, $$\beta = - 2\gamma - {\alpha \over 3}$$

$$\therefore$$ $$\alpha + 3\beta + 6\gamma = \alpha + 3\left( { - 2\gamma - {\alpha \over 3}} \right) + 6\gamma$$

$$= 0$$
4

### WB JEE 2020

English
Bengali
The unit vector in ZOX plane, making angles $$45^\circ$$ and $$60^\circ$$ respectively with $$\alpha = 2\widehat i + 2\widehat j - \widehat k$$ and $$\beta = \widehat j - \widehat k$$ is
A
$${1 \over {\sqrt 2 }}\widehat i + {1 \over {\sqrt 2 }}\widehat j$$
B
$${1 \over {\sqrt 2 }}\widehat i - {1 \over {\sqrt 2 }}\widehat k$$
C
$${1 \over {\sqrt 2 }}\widehat i - {1 \over {\sqrt 2 }}\widehat j$$
D
$${1 \over {\sqrt 2 }}\widehat i + {1 \over {\sqrt 2 }}\widehat k$$

## Explanation

Let the unit vector in ZOX plane be

$$a = x\widehat i + z\widehat k,\left| a \right| = 1$$

$$a.\alpha = \left| a \right|\left| \alpha \right|\cos 45^\circ$$

$$\Rightarrow$$ $$(x\widehat i + z\widehat k).(2\widehat i + 2\widehat j - \widehat k) = 1 \times 3 \times {1 \over {\sqrt 2 }}$$

[$$\because$$ $$\alpha = 2\widehat i + 2\widehat j - \widehat k$$]

$$2x - z = {3 \over {\sqrt 2 }}$$

and $$a.\beta = \left| a \right|\left| \beta \right|\cos 60^\circ$$

$$\Rightarrow (x\widehat i + z\widehat k).(\widehat j - \widehat k) = 1 \times \sqrt 2 \times {1 \over 2}$$ [$$\because$$ $$\beta = \widehat j - \widehat k$$]

$$- z = {1 \over {\sqrt 2 }}$$

$$z = - {1 \over {\sqrt 2 }}$$ and $$x = {1 \over {\sqrt 2 }}$$

$$\therefore$$ $$a = {1 \over {\sqrt 2 }}\widehat i - {1 \over {\sqrt 2 }}\widehat k$$

ZOX তলে একক ভেক্টর যথাক্রমে 45$$^\circ$$ ও 60$$^\circ$$ কোণ উৎপন্ন করে $$\overrightarrow \alpha$$ ও $$\overrightarrow \beta$$ এর সঙ্গে যেখানে $$\overrightarrow \alpha = 2\widehat i + 2\widehat j - \widehat k$$ এবং $$\overrightarrow \beta = \widehat j - \widehat k$$। সেক্ষেত্রে উক্ত একক ভেক্টরটি হবে

A
$${1 \over {\sqrt 2 }}\widehat i + {1 \over {\sqrt 2 }}\widehat j$$
B
$${1 \over {\sqrt 2 }}\widehat i - {1 \over {\sqrt 2 }}\widehat k$$
C
$${1 \over {\sqrt 2 }}\widehat i - {1 \over {\sqrt 2 }}\widehat j$$
D
$${1 \over {\sqrt 2 }}\widehat i + {1 \over {\sqrt 2 }}\widehat k$$

## Explanation

একক ভেক্টরটি $$\overrightarrow A = a\widehat i + b\widehat k$$ হলে $$\left| {\overrightarrow A } \right| = \sqrt {{a^2} + {b^2}} = 1$$

$$\overrightarrow \alpha \,.\,\overrightarrow A = 2a - b = \left| {\overrightarrow \alpha } \right|\left| {\overrightarrow A } \right|\cos 45^\circ = {3 \over {\sqrt 2 }}$$

$$\overrightarrow \beta \,.\,\overrightarrow A = - b = \left| {\overrightarrow \beta } \right|\left| {\overrightarrow A } \right|\cos 60^\circ = \sqrt 2 .{1 \over 2} = {1 \over {\sqrt 2 }}$$

$$\therefore$$ $$b = - {1 \over {\sqrt 2 }}$$ এবং $$2a = {3 \over {\sqrt 2 }} + b = \sqrt 2 \Rightarrow a = {1 \over {\sqrt 2 }}$$

$$\therefore$$ $$\overrightarrow A = {1 \over {\sqrt 2 }}\widehat i - {1 \over {\sqrt 2 }}\widehat k$$

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