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1

WB JEE 2022

MCQ (Single Correct Answer)
English
Bengali

If $$p = \left[ {\matrix{ 1 & \alpha & 3 \cr 1 & 3 & 3 \cr 2 & 4 & 4 \cr } } \right]$$ is the adjoint of the $$3 \times 3$$ matrix A and det A = 4, then $$\alpha$$ is equal to

A
4
B
11
C
5
D
0

যদি $$p = \left[ {\matrix{ 1 & \alpha & 3 \cr 1 & 3 & 3 \cr 2 & 4 & 4 \cr } } \right]$$, $$3 \times 3$$ ম্যাট্রিক্স A এর adjoint ম্যাট্রিক্স হয় এবং det A = 4 হয় তবে $$\alpha$$ এর মান হবে

A
4
B
11
C
5
D
0
2

WB JEE 2022

MCQ (Single Correct Answer)
English
Bengali

If $$\Delta (x) = \left| {\matrix{ {x - 2} & {{{(x - 1)}^2}} & {{x^3}} \cr {x - 1} & {{x^2}} & {{{(x + 1)}^3}} \cr x & {{{(x + 1)}^2}} & {{{(x + 2)}^3}} \cr } } \right|$$, then coefficient of x in $$\Delta$$x is

A
2
B
$$-$$2
C
3
D
$$-$$4

যদি $$\Delta (x) = \left| {\matrix{ {x - 2} & {{{(x - 1)}^2}} & {{x^3}} \cr {x - 1} & {{x^2}} & {{{(x + 1)}^3}} \cr x & {{{(x + 1)}^2}} & {{{(x + 2)}^3}} \cr } } \right|$$ হয়, তবে $$\Delta$$x -এ x পদের সহগ হবে

A
2
B
$$-$$2
C
3
D
$$-$$4
3

WB JEE 2022

MCQ (Single Correct Answer)
English
Bengali

Under which of the following condition(s) does(do) the system of equations $$\left( {\matrix{ 1 & 2 & 4 \cr 2 & 1 & 2 \cr 1 & 2 & {(a - 4)} \cr } } \right)\left( {\matrix{ x \cr y \cr z \cr } } \right) = \left( {\matrix{ 6 \cr 4 \cr a \cr } } \right)$$ possesses(possess) unique solution ?

A
$$\forall$$ a $$\in$$ R
B
a = 8
C
for all integral values of a
D
a $$\ne$$ 8

নিম্নলিখিত কোন শর্তাবলীর অধীনে

$$\left( {\matrix{ 1 & 2 & 4 \cr 2 & 1 & 2 \cr 1 & 2 & {(a - 4)} \cr } } \right)\left( {\matrix{ x \cr y \cr z \cr } } \right) = \left( {\matrix{ 6 \cr 4 \cr a \cr } } \right)$$ সমীকরণগুচ্ছের অনন্য সমাধান থাকবে ?

A
$$\forall$$a $$\in$$ R
B
a = 8
C
a-এর সকল পূর্ণসংখ্যা মানের জন্য
D
a $$\ne$$ 8
4

WB JEE 2021

MCQ (Single Correct Answer)
English
Bengali
The determinant $$\left| {\matrix{ {{a^2} + 10} & {ab} & {ac} \cr {ab} & {{b^2} + 10} & {bc} \cr {ac} & {bc} & {{c^2} + 10} \cr } } \right|$$ is
A
divisible by 10 but not by 100
B
divisible by 100
C
not divisible by 100
D
not divisible by 10

Explanation

We have,

$$\left| {\matrix{ {{a^2} + 10} & {ab} & {ac} \cr {ab} & {{b^2} + 10} & {bc} \cr {ac} & {bc} & {{c^2} + 10} \cr } } \right|$$

$$ = {1 \over {abc}}\left| {\matrix{ {a({a^2} + 10)} & {a{b^2}} & {a{c^2}} \cr {{a^2}b} & {b({b^2} + 0)} & {b{c^2}} \cr {{a^2}c} & {{b^2}c} & {c({c^2} + 10)} \cr } } \right|$$

Taking common a, b, c from R1, R2 and R3 respectively

$$ = {{abc} \over {abc}}\left| {\matrix{ {{a^2} + 10} & {{b^2}} & {{c^2}} \cr {{a^2}} & {{b^2} + 10} & {{c^2}} \cr {{a^2}} & {{b^2}} & {{c^2} + 10} \cr } } \right|$$

Applying C1 $$\to$$ C1 + C2 + C3, we get

$$ = \left| {\matrix{ {{a^2} + {b^2} + {c^2} + 10} & {{b^2}} & {{c^2}} \cr {{a^2} + {b^2} + {c^2} + 10} & {{b^2} + 10} & {{c^2}} \cr {{a^2} + {b^2} + {c^2} + 0} & {{b^2}} & {{c^2} + 10} \cr } } \right|$$

$$ = ({a^2} + {b^2} + {c^2} + 10)\left| {\matrix{ 1 & {{b^2}} & {{c^2}} \cr 1 & {{b^2} + 10} & {{c^2}} \cr 1 & {{b^2}} & {{c^2} + 10} \cr } } \right|$$

R2 $$\to$$ R2 $$-$$ R1 and R3 $$\to$$ R3 $$-$$ R1

$$ = ({a^2} + {b^2} + {c^2} + 10)\left| {\matrix{ 1 & {{b^2}} & {{c^2}} \cr 0 & {10} & 0 \cr 0 & 0 & {10} \cr } } \right|$$

$$ = ({a^2} + {b^2} + {c^2} + 10)\,(100)$$

$$\because$$ If is divisible by 100.
$$\left| {\matrix{ {{a^2} + 10} & {ab} & {ac} \cr {ab} & {{b^2} + 10} & {bc} \cr {ac} & {bc} & {{c^2} + 10} \cr } } \right|$$ নির্ণায়কটি
A
10 দ্বারা বিভাজ্য কিন্তু 100 দ্বারা বিভাজ্য নয়
B
100 দ্বারা বিভাজ্য
C
100 দ্বারা বিভাজ্য নয়
D
10 দ্বারা বিভাজ্য নয়

Explanation

আমাদের কাছে,

$$\left| {\matrix{ {{a^2} + 10} & {ab} & {ac} \cr {ab} & {{b^2} + 10} & {bc} \cr {ac} & {bc} & {{c^2} + 10} \cr } } \right|$$

$$ = {1 \over {abc}}\left| {\matrix{ {a({a^2} + 10)} & {a{b^2}} & {a{c^2}} \cr {{a^2}b} & {b({b^2} + 0)} & {b{c^2}} \cr {{a^2}c} & {{b^2}c} & {c({c^2} + 10)} \cr } } \right|$$

যথাক্রমে R1, R2 এবং R3 থেকে কমন a, b, c নেওয়া

$$ = {{abc} \over {abc}}\left| {\matrix{ {{a^2} + 10} & {{b^2}} & {{c^2}} \cr {{a^2}} & {{b^2} + 10} & {{c^2}} \cr {{a^2}} & {{b^2}} & {{c^2} + 10} \cr } } \right|$$

C1 $$\to$$ C1 + C2 + C3, প্রয়োগ করলে আমরা পাই

$$ = \left| {\matrix{ {{a^2} + {b^2} + {c^2} + 10} & {{b^2}} & {{c^2}} \cr {{a^2} + {b^2} + {c^2} + 10} & {{b^2} + 10} & {{c^2}} \cr {{a^2} + {b^2} + {c^2} + 0} & {{b^2}} & {{c^2} + 10} \cr } } \right|$$

$$ = ({a^2} + {b^2} + {c^2} + 10)\left| {\matrix{ 1 & {{b^2}} & {{c^2}} \cr 1 & {{b^2} + 10} & {{c^2}} \cr 1 & {{b^2}} & {{c^2} + 10} \cr } } \right|$$

R2 $$\to$$ R2 $$-$$ R1 এবং R3 $$\to$$ R3 $$-$$ R1

$$ = ({a^2} + {b^2} + {c^2} + 10)\left| {\matrix{ 1 & {{b^2}} & {{c^2}} \cr 0 & {10} & 0 \cr 0 & 0 & {10} \cr } } \right|$$

$$ = ({a^2} + {b^2} + {c^2} + 10)\,(100)$$

$$\because$$ যদি 100 দ্বারা বিভাজ্য হয়।

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