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1

WB JEE 2022

MCQ (More than One Correct Answer)
English
Bengali

Let $$\Delta = \left| {\matrix{ {\sin \theta \cos \phi } & {\sin \theta \sin \phi } & {\cos \theta } \cr {\cos \theta \cos \phi } & {\cos \theta \sin \phi } & { - \sin \theta } \cr { - \sin \theta \sin \phi } & {\sin \theta \cos \phi } & 0 \cr } } \right|$$. Then

A
$$\Delta$$ is independent of $$\theta$$
B
$$\Delta$$ is independent of $$\varphi $$
C
$$\Delta$$ is a constant
D
$${\left( {{{d\Delta } \over {d\theta }}} \right)_{\theta = {\pi \over 2}}} = 0$$

মনে কর $$\Delta = \left| {\matrix{ {\sin \theta \cos \phi } & {\sin \theta \sin \phi } & {\cos \theta } \cr {\cos \theta \cos \phi } & {\cos \theta \sin \phi } & { - \sin \theta } \cr { - \sin \theta \sin \phi } & {\sin \theta \cos \phi } & 0 \cr } } \right|$$, সেক্ষেত্রে

A
$$\Delta$$, $$\theta$$-এর উপর নির্ভরশীল নয়
B
$$\Delta$$, $$\varphi $$-এর উপর নির্ভরশীল নয়
C
$$\Delta$$ ধ্রুবক
D
$${\left( {{{d\Delta } \over {d\theta }}} \right)_{\theta = {\pi \over 2}}} = 0$$
2

WB JEE 2021

MCQ (More than One Correct Answer)
English
Bengali
$$\left| {\matrix{ x & {3x + 2} & {2x - 1} \cr {2x - 1} & {4x} & {3x + 1} \cr {7x - 2} & {17x + 6} & {12x - 1} \cr } } \right| = 0$$ is true for
A
only one value of x
B
only two value of x
C
only three values of x
D
infinitely many value of x

Explanation

We have,

$$\left| {\matrix{ x & {3x + 2} & {2x - 1} \cr {2x - 1} & {4x} & {3x + 1} \cr {7x - 2} & {17x + 6} & {12x - 1} \cr } } \right| = 0$$

R3 $$\to$$ R3 $$-$$ 3R1 $$-$$ 2R2, we get

$$\left| {\matrix{ x & {3x + 2} & {2x - 1} \cr {2x - 1} & {4x} & {3x + 1} \cr 0 & 0 & 0 \cr } } \right| = 0$$

$$\therefore$$ infinite value of x is possible.
$$\left| {\matrix{ x & {3x + 2} & {2x - 1} \cr {2x - 1} & {4x} & {3x + 1} \cr {7x - 2} & {17x + 6} & {12x - 1} \cr } } \right| = 0$$ হলে
A
x-এর মান অনন্য হবে
B
x-এর শুধুমাত্র দুটি মান হবে
C
x-এর শুধুমাত্র তিনটি মান হবে
D
x-এর অসংখ্য মান হবে

Explanation

আমাদের কাছে,

$$\left| {\matrix{ x & {3x + 2} & {2x - 1} \cr {2x - 1} & {4x} & {3x + 1} \cr {7x - 2} & {17x + 6} & {12x - 1} \cr } } \right| = 0$$

R3 $$\to$$ R3 $$-$$ 3R1 $$-$$ 2R2, আমরা পায়

$$\left| {\matrix{ x & {3x + 2} & {2x - 1} \cr {2x - 1} & {4x} & {3x + 1} \cr 0 & 0 & 0 \cr } } \right| = 0$$

$$\therefore$$ x এর অসংখ্য মান সম্ভব।
3

WB JEE 2019

MCQ (More than One Correct Answer)
English
Bengali
Let $$A = \left[ {\matrix{ 3 & 0 & 3 \cr 0 & 3 & 0 \cr 3 & 0 & 3 \cr } } \right]$$. Then, the roots of the equation det $$(A - \lambda {I_3})$$ = 0 (where I3 is the identity matrix of order 3) are
A
3, 0, 3
B
0, 3, 6
C
1, 0, $$-$$6
D
3, 3, 6

Explanation

We have, $$A = \left[ {\matrix{ 3 & 0 & 3 \cr 0 & 3 & 0 \cr 3 & 0 & 3 \cr } } \right]$$

Again, we have $$|A - \lambda I|\, = \,0$$

$$ \Rightarrow \left| {\matrix{ {3 - \lambda } & 0 & 3 \cr 0 & {3 - \lambda } & 3 \cr 3 & 0 & {3 - \lambda } \cr } } \right|\, = \,0$$

$$ \Rightarrow (3 - \lambda )[{(3 - \lambda )^2} - 0] - 0 + 3[0 - 3(3 - \lambda )] = 0$$

$$ \Rightarrow {(3 - \lambda )^3} - 9(3 - \lambda ) = 0$$

$$ \Rightarrow (3 - \lambda )[{(3 - \lambda )^2} - 9] = 0$$

$$ \Rightarrow (3 - \lambda )(9 + {\lambda ^2} - 6\lambda - 9) = 0$$

$$ \Rightarrow (3 - \lambda )({\lambda ^2} - 6\lambda ) = 0$$

$$ \Rightarrow (3 - \lambda )\lambda (\lambda - 6) = 0$$

$$ \Rightarrow \lambda = 3,0,6$$

মনে করো $$A = \left( {\matrix{ 3 & 0 & 3 \cr 0 & 3 & 0 \cr 3 & 0 & 3 \cr } } \right)$$ তাহলে সমীকরণ $$\det (A - \lambda {I_3}) = 0$$ (I3 হল 3 ক্রমের একসম ম্যাট্রিক্স)-এর বীজগুলি হল -

A
3, 0, 3
B
0, 3, 6
C
1, 0, $$-$$6
D
3, 3, 6

Explanation

$$A = \left( {\matrix{ 3 & 0 & 3 \cr 0 & 3 & 0 \cr 3 & 0 & 3 \cr } } \right)$$

$$\det .(A - \lambda {I_3}) = \left| {\matrix{ {3 - \lambda } & 0 & 3 \cr 0 & {3 - \lambda } & 0 \cr 3 & 0 & {3 - \lambda } \cr } } \right| = 0$$

বা, $$(3 - \lambda )\,.\,{(3 - \lambda )^2} + 3\{ 0 - 3(3 - \lambda )\} = 0$$

বা, $$(3 - \lambda )\{ {(3 - \lambda )^2} - 9\} = 0$$

বা, $$(3 - \lambda )(6 - \lambda )( - \lambda ) = 0$$

$$\therefore$$ $$\lambda = 0,3,6$$

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