question_id stringlengths 8 35 | subject stringclasses 3
values | chapter stringclasses 90
values | topic stringclasses 459
values | question stringlengths 17 24.5k | options stringlengths 2 4.26k | correct_option stringclasses 6
values | answer stringclasses 460
values | explanation stringlengths 1 10.6k | question_type stringclasses 3
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|---|---|---|---|---|---|---|---|---|---|---|---|
qp2BEp4dd9iOFnFq | maths | matrices-and-determinants | expansion-of-determinant | If $$D = \left| {\matrix{
1 & 1 & 1 \cr
1 & {1 + x} & 1 \cr
1 & 1 & {1 + y} \cr
} } \right|$$ for $$x \ne 0,y \ne 0,$$ then $$D$$ is : | [{"identifier": "A", "content": "divisible by $$x$$ but not $$y$$"}, {"identifier": "B", "content": "divisible by $$y$$ but not $$x$$"}, {"identifier": "C", "content": "divisible by neither $$x$$ nor $$y$$"}, {"identifier": "D", "content": "divisible by both $$x$$ and $$y$$"}] | ["D"] | null | Given, $$D = \left| {\matrix{
1 & 1 & 1 \cr
1 & {1 + x} & 1 \cr
1 & 1 & {1 + y} \cr
} } \right|$$
<br><br>Apply $$\,\,\,{R^2} \to {R_2} - {R_1}$$ $$\,\,\,\,$$
<br><br>and $$\,\,\,\,$$ $$R \to {R_3} - {R_1}$$
<br><br>$$\therefore$$ $$\,\,\,\,\,D = \left| {\matrix{
1 & 1 &... | mcq | aieee-2007 | 6,950 |
r4Fv71k1mBq9dYh2 | maths | matrices-and-determinants | expansion-of-determinant | If $$\alpha ,\beta \ne 0,$$ and $$f\left( n \right) = {\alpha ^n} + {\beta ^n}$$ and
$$$\left| {\matrix{
3 & {1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} \cr
{1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} & {1 + f\left( 3 \right)} \cr
{1 + f\left( 2 \right)} & {1 + f\left( 3 \... | [{"identifier": "A", "content": "$$1$$ "}, {"identifier": "B", "content": "$$-1$$"}, {"identifier": "C", "content": "$$\\alpha \\beta $$ "}, {"identifier": "D", "content": "$${1 \\over {\\alpha \\beta }}$$ "}] | ["A"] | null | Consider
<br><br>$$\left| {\matrix{
3 & {1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} \cr
{1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} & {1 + f\left( 3 \right)} \cr
{1 + f\left( 2 \right)} & {1 + f\left( 3 \right)} & {1 + f\left( 4 \right)} \cr
} } \right|$$
<br><br>$$\... | mcq | jee-main-2014-offline | 6,952 |
3HijCKA6R6fQjXo8tvuQN | maths | matrices-and-determinants | expansion-of-determinant | If A = $$\left[ {\matrix{
{ - 4} & { - 1} \cr
3 & 1 \cr
} } \right]$$,
<br/><br/>then the determinant of the matrix (A<sup>2016</sup> − 2A<sup>2015</sup> − A<sup>2014</sup>) is :
| [{"identifier": "A", "content": "2014"}, {"identifier": "B", "content": "$$-$$ 175"}, {"identifier": "C", "content": "2016"}, {"identifier": "D", "content": "$$-$$ 25"}] | ["D"] | null | Given,
<br><br>$$A = \left[ {\matrix{
{ - 4} & { - 1} \cr
3 & 1 \cr
} } \right]$$
<br><br>$${A^2} = \left[ {\matrix{
{ - 4} & { - 1} \cr
3 & 1 \cr
} } \right]\left[ {\matrix{
{ - 4} & { - 1} \cr
3 & 1 \cr
} } \right]$$
<br><br>$$ = \left[ {\matrix{
{13} &am... | mcq | jee-main-2016-online-10th-april-morning-slot | 6,953 |
RS4yHNVUSJJseZvzGSGiK | maths | matrices-and-determinants | expansion-of-determinant | The number of distinct real roots of the equation,
<br/><br/>$$\left| {\matrix{
{\cos x} & {\sin x} & {\sin x} \cr
{\sin x} & {\cos x} & {\sin x} \cr
{\sin x} & {\sin x} & {\cos x} \cr
} } \right| = 0$$ in the interval $$\left[ { - {\pi \over 4},{\pi \over 4}} \right]$$ is :... | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "1"}] | ["C"] | null | Given,
<br><br>$$\left| {\matrix{
{\cos x} & {\sin x} & {\sin x} \cr
{\sin x} & {\cos x} & {\sin x} \cr
{\sin x} & {\sin x} & {\cos x} \cr
} } \right| = 0$$
<br><br>R<sub>1</sub> $$ \to $$ R<sub>1</sub> $$-$$ R<sub>3</sub>
<br><br>R<s... | mcq | jee-main-2016-online-9th-april-morning-slot | 6,954 |
eun9UQWukCJ25rjmhngyg | maths | matrices-and-determinants | expansion-of-determinant | If
<br/><br/>$$S = \left\{ {x \in \left[ {0,2\pi } \right]:\left| {\matrix{
0 & {\cos x} & { - \sin x} \cr
{\sin x} & 0 & {\cos x} \cr
{\cos x} & {\sin x} & 0 \cr
} } \right| = 0} \right\},$$
<br/><br/>then $$\sum\limits_{x \in S} {\tan \left( {{\pi \over 3} + x} \right)} $$ i... | [{"identifier": "A", "content": "$$4 + 2\\sqrt 3 $$"}, {"identifier": "B", "content": "$$ - 2 + \\sqrt 3 $$"}, {"identifier": "C", "content": "$$ - 2 - \\sqrt 3 $$"}, {"identifier": "D", "content": "$$-\\,\\,4 - 2\\sqrt 3 $$"}] | ["C"] | null | Given,
<br><br> $$\left| {\matrix{
0 & {\cos x} & { - \sin x} \cr
{\sin x} & 0 & {\cos x} \cr
{\cos x} & {\sin x} & 0 \cr
} } \right|$$ = 0
<br><br>$$ \Rightarrow $$$$\,\,\,$$ 0 (0 $$-$$ cosx sinx) $$-$$ cosx (0 $$-$$ cos<sup>2</su... | mcq | jee-main-2017-online-8th-april-morning-slot | 6,955 |
eHnuQRvur8S3cbRr4ps0G | maths | matrices-and-determinants | expansion-of-determinant | Let $$A$$ be a matrix such that $$A.\left[ {\matrix{
1 & 2 \cr
0 & 3 \cr
} } \right]$$ is a scalar matrix and |3A| = 108.
<br/>Then A<sup>2</sup> equals : | [{"identifier": "A", "content": "$$\\left[ {\\matrix{\n 4 & { - 32} \\cr \n 0 & {36} \\cr \n\n } } \\right]$$"}, {"identifier": "B", "content": "$$\\left[ {\\matrix{\n {36} & 0 \\cr \n { - 32} & 4 \\cr \n\n } } \\right]$$"}, {"identifier": "C", "content": "$$\\left[ {\\matrix{\n 4 & 0 ... | ["D"] | null | According to questions, <br/><br>
A. $$\left[ {\matrix{
1 & 2 \cr
0 & 3 \cr
} } \right]$$ = $$\left[ {\matrix{
\lambda & 0 \cr
0 & \lambda \cr
} } \right]$$<br><br>
$$ \Rightarrow $$ A = $$\left[ {\matrix{
\lambda & 0 \cr
0 & \lambda \cr
} } \right]$$ $$\left... | mcq | jee-main-2018-online-15th-april-morning-slot | 6,956 |
8OQG5ZHmm0j8UPp8rP3rsa0w2w9jxb4ij7c | maths | matrices-and-determinants | expansion-of-determinant | A value of $$\theta \in \left( {0,{\pi \over 3}} \right)$$, for which
<br/>$$\left| {\matrix{
{1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } & {4\cos 6\theta } \cr
{{{\cos }^2}\theta } & {1 + {{\sin }^2}\theta } & {4\cos 6\theta } \cr
{{{\cos }^2}\theta } & {{{\sin }^2}\theta } &... | [{"identifier": "A", "content": "$${\\pi \\over {18}}$$"}, {"identifier": "B", "content": "$${\\pi \\over {9}}$$"}, {"identifier": "C", "content": "$${{7\\pi } \\over {24}}$$"}, {"identifier": "D", "content": "$${{7\\pi } \\over {36}}$$"}] | ["B"] | null | $$\left| {\matrix{
{1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } & {4\cos 6\theta } \cr
{{{\cos }^2}\theta } & {1 + {{\sin }^2}\theta } & {4\cos 6\theta } \cr
{{{\cos }^2}\theta } & {{{\sin }^2}\theta } & {1 + 4\cos 6\theta } \cr
} } \right| = 0$$<br><br>
R<sub>1</sub> $$ \to ... | mcq | jee-main-2019-online-12th-april-evening-slot | 6,959 |
NvW5EmQOQ4vTpvOt9L3rsa0w2w9jx65olji | maths | matrices-and-determinants | expansion-of-determinant | If $$B = \left[ {\matrix{
5 & {2\alpha } & 1 \cr
0 & 2 & 1 \cr
\alpha & 3 & { - 1} \cr
} } \right]$$ is the inverse of a 3 × 3 matrix A, then the sum of all values of $$\alpha $$ for which
det(A) + 1 = 0, is :
| [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "- 1"}, {"identifier": "C", "content": "0"}, {"identifier": "D", "content": "1"}] | ["D"] | null | Given |A| + 1 = 0
<br><br>$$ \Rightarrow $$ |A| = -1
<br><br>$$\left| B \right| = \left| {{A^{ - 1}}} \right| = {1 \over {\left| A \right|}} = - 1$$<br><br>
$$\left| {\matrix{
5 & {2\alpha } & 1 \cr
0 & 2 & 1 \cr
\alpha & 3 & { - 1} \cr
} } \right| $$ = -1
<br><br>$$ \Rightarr... | mcq | jee-main-2019-online-12th-april-morning-slot | 6,960 |
72ZcOZKf5Yr1mCFPHU3rsa0w2w9jx23i3gx | maths | matrices-and-determinants | expansion-of-determinant | The sum of the real roots of the equation
<br/>$$\left| {\matrix{
x & { - 6} & { - 1} \cr
2 & { - 3x} & {x - 3} \cr
{ - 3} & {2x} & {x + 2} \cr
} } \right| = 0$$, is equal to : | [{"identifier": "A", "content": "- 4"}, {"identifier": "B", "content": "0"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "6"}] | ["B"] | null | x(-3x $$ \times $$ (x + 2) - 2x(x - 3)) + (– 6) (2(x + 2) + 3 (x – 3)) + (–1) (4x + 3 (–3x))<br><br>
$$ \Rightarrow $$ – 5x<sup>3</sup> + 30x –30 + 5x = 0<br><br>
$$ \Rightarrow $$ x<sup>3</sup> – 7x + 6 = 0<br><br>
$$ \therefore $$ sum of roots = 0 | mcq | jee-main-2019-online-10th-april-evening-slot | 6,961 |
2QtHVXivwg1Ceho7jC3rsa0w2w9jwxkqlfg | maths | matrices-and-determinants | expansion-of-determinant | If $${\Delta _1} = \left| {\matrix{
x & {\sin \theta } & {\cos \theta } \cr
{ - \sin \theta } & { - x} & 1 \cr
{\cos \theta } & 1 & x \cr
} } \right|$$ and
<br/>$${\Delta _2} = \left| {\matrix{
x & {\sin 2\theta } & {\cos 2\theta } \cr
{ - \sin 2\theta } & {... | [{"identifier": "A", "content": "$${\\Delta _1} - {\\Delta _2}$$ = x (cos 2$$\\theta $$ \u2013 cos 4$$\\theta $$)"}, {"identifier": "B", "content": "$${\\Delta _1} + {\\Delta _2}$$ = - 2x<sup>3</sup>"}, {"identifier": "C", "content": "$${\\Delta _1} + {\\Delta _2}$$ = \u2013 2(x<sup>3</sup> + x \u20131)"}, {"identifi... | ["B"] | null | $${\Delta _1} = \left| {\matrix{
x & {\sin \theta } & {\cos \theta } \cr
{ - \sin \theta } & { - x} & 1 \cr
{\cos \theta } & 1 & x \cr
} } \right|$$<br><br>
= x(–x<sup>2</sup> –1) – sin$$\theta $$(–xsin$$\theta $$ – cos$$\theta $$) + cos$$\theta $$(–sin$$\theta $$+ xcos$$\theta ... | mcq | jee-main-2019-online-10th-april-morning-slot | 6,962 |
MGCkKbAWU9TkkCJ1F318hoxe66ijvwp7qik | maths | matrices-and-determinants | expansion-of-determinant | Let $$\alpha $$ and $$\beta $$ be the roots of the equation
x<sup>2</sup> + x + 1 = 0. Then for y $$ \ne $$ 0 in R,<br/>
$$$\left| {\matrix{
{y + 1} & \alpha & \beta \cr
\alpha & {y + \beta } & 1 \cr
\beta & 1 & {y + \alpha } \cr
} } \right|$$$
is equal to | [{"identifier": "A", "content": "y(y<sup>2</sup> \u2013 1)"}, {"identifier": "B", "content": "y(y<sup>2</sup> \u2013 3)"}, {"identifier": "C", "content": "y<sup>3</sup>"}, {"identifier": "D", "content": "y<sup>3</sup> \u2013 1"}] | ["C"] | null | $$\alpha $$ and $$\beta $$ are the roots of the equation
x<sup>2</sup> + x + 1 = 0.
<br><br>$$ \therefore $$ $$\alpha $$ = $$\omega $$ and $$\beta $$ = $${\omega ^2}$$
<br><br>$$\left| {\matrix{
{y + 1} & \alpha & \beta \cr
\alpha & {y + \beta } & 1 \cr
\beta & 1 & {y + \alpha }... | mcq | jee-main-2019-online-9th-april-morning-slot | 6,963 |
JRTEaKMvpR1dXBveR2RUC | maths | matrices-and-determinants | expansion-of-determinant | Let the number 2,b,c be in an A.P. and<br/>
A = $$\left[ {\matrix{
1 & 1 & 1 \cr
2 & b & c \cr
4 & {{b^2}} & {{c^2}} \cr
} } \right]$$. If det(A) $$ \in $$ [2, 16], then c
lies in the interval : | [{"identifier": "A", "content": "[2, 3)"}, {"identifier": "B", "content": "[4, 6]"}, {"identifier": "C", "content": "(2 + 2<sup>3/4</sup>, 4)"}, {"identifier": "D", "content": "[3, 2 + 2<sup>3/4</sup>]"}] | ["B"] | null | 2, b, c are in AP.
<br><br>Let common difference = d
<br><br>$$ \therefore $$ b = 2 + d and c = 2 + 2d
<br><br>|A| = $$\left[ {\matrix{
1 & 1 & 1 \cr
2 & b & c \cr
4 & {{b^2}} & {{c^2}} \cr
} } \right]$$
<br><br>C<sub>2</sub> = C<sub>2</sub> - C<sub>1</sub>
<br><br>C<sub>3</sub>... | mcq | jee-main-2019-online-8th-april-evening-slot | 6,964 |
FIXxORXt1C1lR1l1ZhxXD | maths | matrices-and-determinants | expansion-of-determinant | If A = $$\left[ {\matrix{
1 & {\sin \theta } & 1 \cr
{ - \sin \theta } & 1 & {\sin \theta } \cr
{ - 1} & { - \sin \theta } & 1 \cr
} } \right]$$;
<br/><br/>then for all $$\theta $$ $$ \in $$ $$\left( {{{3\pi } \over 4},{{5\pi } \over 4}} \right)$$, det (A) lies in the interva... | [{"identifier": "A", "content": "$$\\left( {{3 \\over 2},3} \\right]$$"}, {"identifier": "B", "content": "$$\\left( {0,{3 \\over 2}} \\right]$$"}, {"identifier": "C", "content": "$$\\left[ {{5 \\over 2},4} \\right)$$"}, {"identifier": "D", "content": "$$\\left( {1,{5 \\over 2}} \\right]$$"}] | ["A"] | null | $$\left| A \right| = \left| {\matrix{
1 & {\sin \theta } & 1 \cr
{ - \sin \theta } & 1 & {\sin \theta } \cr
{ - 1} & { - \sin \theta } & 1 \cr
} } \right|$$
<br><br>= 2(1 + sin<sup>2</sup>$$\theta $$)
<br><br>$$\theta $$ $$ \in $$ $$\left( {{{3\pi } \over 4},{{5\pi } \over 4}} \... | mcq | jee-main-2019-online-12th-january-evening-slot | 6,965 |
o5MdSV4OLnSSysFVE1ztM | maths | matrices-and-determinants | expansion-of-determinant | If $$\left| {\matrix{
{a - b - c} & {2a} & {2a} \cr
{2b} & {b - c - a} & {2b} \cr
{2c} & {2c} & {c - a - b} \cr
} } \right|$$
<br/><br/> = (a + b + c) (x + a + b + c)<sup>2</sup>, x $$ \ne $$ 0,
<br/><br/>then x is equal to : | [{"identifier": "A", "content": "\u20132(a + b + c)"}, {"identifier": "B", "content": "2(a + b + c)"}, {"identifier": "C", "content": "abc"}, {"identifier": "D", "content": "\u2013(a + b + c)"}] | ["A"] | null | $$\left| {\matrix{
{a - b - c} & {2a} & {2a} \cr
{2b} & {b - c - a} & {2b} \cr
{2c} & {2c} & {c - a - b} \cr
} } \right|$$
<br><br>R<sub>1</sub> $$ \to $$ R<sub>1</sub> + R<sub>2</sub> + R<sub>3</sub>
<br><br>$$ = \left| {\matrix{
{a + b + c} & {a + b + c} & {a + b +... | mcq | jee-main-2019-online-11th-january-evening-slot | 6,966 |
jeywfxj9iOKHSYmfc5TKS | maths | matrices-and-determinants | expansion-of-determinant | Let A = $$\left[ {\matrix{
2 & b & 1 \cr
b & {{b^2} + 1} & b \cr
1 & b & 2 \cr
} } \right]$$ where b > 0.
<br/><br/>Then the minimum value of $${{\det \left( A \right)} \over b}$$ is - | [{"identifier": "A", "content": "$$\\sqrt 3 $$"}, {"identifier": "B", "content": "$$-$$ $$2\\sqrt 3 $$"}, {"identifier": "C", "content": "$$ - \\sqrt 3 $$"}, {"identifier": "D", "content": "$$2\\sqrt 3 $$"}] | ["D"] | null | A = $$\left[ {\matrix{
2 & b & 1 \cr
b & {{b^2} + 1} & b \cr
1 & b & 2 \cr
} } \right]$$ (b > 0)
<br><br>$$\left| A \right|$$ = 2(2b<sup>2</sup> + 2 $$-$$ b<sup>2</sup>) $$-$$ b(2b $$-$$ b) + 1(b<sub>2</sub> $$-$$ b<sub>2</sub> $$-$$ 1)
<br><br>$$\left| A \right|$$ = 2(b... | mcq | jee-main-2019-online-10th-january-evening-slot | 6,967 |
gZOLcTTnTzFV5f4sTVG6J | maths | matrices-and-determinants | expansion-of-determinant | Let d $$ \in $$ R, and
<br/><br/>$$A = \left[ {\matrix{
{ - 2} & {4 + d} & {\left( {\sin \theta } \right) - 2} \cr
1 & {\left( {\sin \theta } \right) + 2} & d \cr
5 & {\left( {2\sin \theta } \right) - d} & {\left( { - \sin \theta } \right) + 2 + 2d} \cr
} } \right],$$
<br/><br... | [{"identifier": "A", "content": "$$-$$ 7"}, {"identifier": "B", "content": "$$2\\left( {\\sqrt 2 + 2} \\right)$$ "}, {"identifier": "C", "content": "$$-$$ 5"}, {"identifier": "D", "content": "$$2\\left( {\\sqrt 2 + 1} \\right)$$"}] | ["C"] | null | $$\det A = \left| {\matrix{
{ - 2} & {4 + d} & {\sin \theta - 2} \cr
1 & {\sin \theta + 2} & d \cr
5 & {2\sin \theta - d} & { - \sin \theta + 2 + 2d} \cr
} } \right|$$
<br><br>(R<sub>1</sub> $$ \to $$ R<sub>1</sub> + R<sub>3</sub> $$-$$ 2R<sub>2</sub>)
<br><br>$$ = \left| {\... | mcq | jee-main-2019-online-10th-january-morning-slot | 6,968 |
Mq79PZTU58QxMIUvbvjgy2xukg0cuvx2 | maths | matrices-and-determinants | expansion-of-determinant | Let $$\theta = {\pi \over 5}$$ and $$A = \left[ {\matrix{
{\cos \theta } & {\sin \theta } \cr
{ - \sin \theta } & {\cos \theta } \cr
} } \right]$$. <br/><br/> If B = A + A<sup>4</sup>
, then det (B) : | [{"identifier": "A", "content": "lies in (1, 2)"}, {"identifier": "B", "content": "lies in (2, 3)."}, {"identifier": "C", "content": "is zero.\n"}, {"identifier": "D", "content": "is one."}] | ["A"] | null | $$A = \left[ {\matrix{
{\cos \theta } & {\sin \theta } \cr
{ - \sin \theta } & {\cos \theta } \cr
} } \right]$$
<br><br>A<sup>2</sup> = $$\left[ {\matrix{
{\cos \theta } & {\sin \theta } \cr
{ - \sin \theta } & {\cos \theta } \cr
} } \right]$$$$\left[ {\matrix{
{\cos \theta } ... | mcq | jee-main-2020-online-6th-september-evening-slot | 6,970 |
hqjDMffUsigfBYiX0jjgy2xukfuvg3xv | maths | matrices-and-determinants | expansion-of-determinant | Let m and M be respectively the minimum and maximum values of
<br/><br/>$$\left| {\matrix{
{{{\cos }^2}x} & {1 + {{\sin }^2}x} & {\sin 2x} \cr
{1 + {{\cos }^2}x} & {{{\sin }^2}x} & {\sin 2x} \cr
{{{\cos }^2}x} & {{{\sin }^2}x} & {1 + \sin 2x} \cr
} } \right|$$
<br/><br/>Then the... | [{"identifier": "A", "content": "(\u20133, \u20131)"}, {"identifier": "B", "content": "(\u20134, \u20131)"}, {"identifier": "C", "content": "(1, 3)"}, {"identifier": "D", "content": "(\u20133, 3)"}] | ["A"] | null | $$\left| {\matrix{
{{{\cos }^2}x} & {1 + {{\sin }^2}x} & {\sin 2x} \cr
{1 + {{\cos }^2}x} & {{{\sin }^2}x} & {\sin 2x} \cr
{{{\cos }^2}x} & {{{\sin }^2}x} & {1 + \sin 2x} \cr
} } \right|$$
<br><br>R<sub>1</sub> $$ \to $$ R<sub>1</sub> – R<sub>2</sub>, R<sub>2</sub> $$ \to $$ R<s... | mcq | jee-main-2020-online-6th-september-morning-slot | 6,971 |
t8IVACMgbcUbnsFcWz1klt7w2kq | maths | matrices-and-determinants | expansion-of-determinant | Let A be a 3 $$\times$$ 3 matrix with det(A) = 4. Let R<sub>i</sub> denote the i<sup>th</sup> row of A. If a matrix B is obtained by performing the operation R<sub>2</sub> $$ \to $$ 2R<sub>2</sub> + 5R<sub>3</sub> on 2A, then det(B) is equal to : | [{"identifier": "A", "content": "64"}, {"identifier": "B", "content": "16"}, {"identifier": "C", "content": "128"}, {"identifier": "D", "content": "80"}] | ["A"] | null | $$A = \left[ {\matrix{
{{R_{11}}} & {{R_{12}}} & {{R_{13}}} \cr
{{R_{21}}} & {{R_{22}}} & {{R_{23}}} \cr
{{R_{31}}} & {{R_{32}}} & {{R_{33}}} \cr
} } \right]$$<br><br>$$2A = \left[ {\matrix{
{2{R_{11}}} & {2{R_{12}}} & {2{R_{13}}} \cr
{2{R_{21}}} & {2{R_{22}}... | mcq | jee-main-2021-online-25th-february-evening-slot | 6,974 |
J4wCNBxubchEpe31cN1kmko0m1s | maths | matrices-and-determinants | expansion-of-determinant | If 1, log<sub>10</sub>(4<sup>x</sup> $$-$$ 2) and log<sub>10</sub>$$\left( {{4^x} + {{18} \over 5}} \right)$$ are in arithmetic progression for a real number x, then the value of the determinant $$\left| {\matrix{
{2\left( {x - {1 \over 2}} \right)} & {x - 1} & {{x^2}} \cr
1 & 0 & x \cr
x &... | [] | null | 2 | 1, $$lo{g_{10}}({4^x} - 2),\,lo{g_{10}}\left( {{4^x} + {{18} \over 5}} \right)$$ in AP.<br><br>$$ \therefore $$ 2$$ \times $$$$lo{g_{10}}({4^x} - 2) = 1 + \,lo{g_{10}}\left( {{4^x} + {{18} \over 5}} \right)$$ <br><br>$$lo{g_{10}}{({4^x} - 2)^2} = \,lo{g_{10}}\left( {10.\left( {{4^x} + {{18} \over 5}} \right)} \right)$$... | integer | jee-main-2021-online-17th-march-evening-shift | 6,977 |
0v6oV602XmCfWdoQV31kmli3lt0 | maths | matrices-and-determinants | expansion-of-determinant | The solutions of the equation $$\left| {\matrix{
{1 + {{\sin }^2}x} & {{{\sin }^2}x} & {{{\sin }^2}x} \cr
{{{\cos }^2}x} & {1 + {{\cos }^2}x} & {{{\cos }^2}x} \cr
{4\sin 2x} & {4\sin 2x} & {1 + 4\sin 2x} \cr
} } \right| = 0,(0 < x < \pi )$$, are | [{"identifier": "A", "content": "$${\\pi \\over {12}},{\\pi \\over 6}$$"}, {"identifier": "B", "content": "$${\\pi \\over 6},{{5\\pi } \\over 6}$$"}, {"identifier": "C", "content": "$${{5\\pi } \\over {12}},{{7\\pi } \\over {12}}$$"}, {"identifier": "D", "content": "$${{7\\pi } \\over {12}},{{11\\pi } \\over {12}}$$... | ["D"] | null | By using C<sub>1</sub> $$ \to $$ C<sub>1</sub> $$-$$ C<sub>2</sub> and C<sub>3</sub> $$ \to $$ C<sub>3</sub> $$-$$ C<sub>2</sub> we get<br><br>$$\left| {\matrix{
1 & {{{\sin }^2}x} & 0 \cr
{ - 1} & {1 + {{\cos }^2}x} & { - 1} \cr
0 & {4\sin 2x} & 1 \cr
} } \right| = 0$$<br><br>E... | mcq | jee-main-2021-online-18th-march-morning-shift | 6,978 |
b29ZvX12aHiapPJUir1kmm46ruq | maths | matrices-and-determinants | expansion-of-determinant | Let I be an identity matrix of order 2 $$\times$$ 2 and P = $$\left[ {\matrix{
2 & { - 1} \cr
5 & { - 3} \cr
} } \right]$$. Then the value of n$$\in$$N for which P<sup>n</sup> = 5I $$-$$ 8P is equal to ____________. | [] | null | 6 | $$P = \left[ {\matrix{
2 & { - 1} \cr
5 & { - 3} \cr
} } \right]$$<br><br>$$\left| {\matrix{
{2 - \lambda } & { - 1} \cr
5 & { - 3 - \lambda } \cr
} } \right| = 0$$<br><br>$$ \Rightarrow $$ $$\lambda$$<sup>2</sup> + $$\lambda$$ $$-$$ 1 = 0<br><br>$$ \Rightarrow $$ P<sup>2</sup> +... | integer | jee-main-2021-online-18th-march-evening-shift | 6,979 |
1krq1a1sm | maths | matrices-and-determinants | expansion-of-determinant | Let a, b, c, d in arithmetic progression with common difference $$\lambda$$. If $$\left| {\matrix{
{x + a - c} & {x + b} & {x + a} \cr
{x - 1} & {x + c} & {x + b} \cr
{x - b + d} & {x + d} & {x + c} \cr
} } \right| = 2$$, then value of $$\lambda$$<sup>2</sup> is equal to _______... | [] | null | 1 | $$\left| {\matrix{
{x + a - c} & {x + b} & {x + a} \cr
{x - 1} & {x + c} & {x + b} \cr
{x - b + d} & {x + d} & {x + c} \cr
} } \right| = 2$$<br><br>$${C_2} \to {C_2} - {C_3}$$<br><br>$$ \Rightarrow \left| {\matrix{
{x - 2\lambda } & \lambda & {x + a} \cr
{x - 1}... | integer | jee-main-2021-online-20th-july-morning-shift | 6,980 |
1krzmszlz | maths | matrices-and-determinants | expansion-of-determinant | The number of distinct real roots <br/><br/>of $$\left| {\matrix{
{\sin x} & {\cos x} & {\cos x} \cr
{\cos x} & {\sin x} & {\cos x} \cr
{\cos x} & {\cos x} & {\sin x} \cr
} } \right| = 0$$ in the interval $$ - {\pi \over 4} \le x \le {\pi \over 4}$$ is : | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "3"}] | ["B"] | null | $$\left| {\matrix{
{\sin x} & {\cos x} & {\cos x} \cr
{\cos x} & {\sin x} & {\cos x} \cr
{\cos x} & {\cos x} & {\sin x} \cr
} } \right| = 0, - {\pi \over 4} \le x \le {\pi \over 4}$$<br><br>Apply : $${R_1} \to {R_1} - {R_2}$$ & $${R_2} \to {R_2} - {R_3}$$<br><br>$$ \Righta... | mcq | jee-main-2021-online-25th-july-evening-shift | 6,981 |
1ks0ch3dj | maths | matrices-and-determinants | expansion-of-determinant | Let $$f(x) = \left| {\matrix{
{{{\sin }^2}x} & { - 2 + {{\cos }^2}x} & {\cos 2x} \cr
{2 + {{\sin }^2}x} & {{{\cos }^2}x} & {\cos 2x} \cr
{{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \cos 2x} \cr
} } \right|,x \in [0,\pi ]$$. Then the maximum value of f(x) is equal to ______________. | [] | null | 6 | $$\left| {\matrix{
{ - 2} & { - 2} & 0 \cr
2 & 0 & { - 1} \cr
{{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \cos 2x} \cr
} } \right|\left( \matrix{
{R_1} \to {R_1} - {R_2} \hfill \cr
\& \,{R_2} \to {R_2} - {R_3} \hfill \cr} \right)$$<br><br>= $$ - 2({\cos ^2}x) + 2(2 + 2\cos ... | integer | jee-main-2021-online-27th-july-morning-shift | 6,982 |
1ktfw40qs | maths | matrices-and-determinants | expansion-of-determinant | Let $$A = \left( {\matrix{
{[x + 1]} & {[x + 2]} & {[x + 3]} \cr
{[x]} & {[x + 3]} & {[x + 3]} \cr
{[x]} & {[x + 2]} & {[x + 4]} \cr
} } \right)$$, where [t] denotes the greatest integer less than or equal to t. If det(A) = 192, then the set of values of x is the interval : | [{"identifier": "A", "content": "[68, 69)"}, {"identifier": "B", "content": "[62, 63)"}, {"identifier": "C", "content": "[65, 66)"}, {"identifier": "D", "content": "[60, 61)"}] | ["B"] | null | $$\left| {\matrix{
{[x + 1]} & {[x + 2]} & {[x + 3]} \cr
{[x]} & {[x + 3]} & {[x + 3]} \cr
{[x]} & {[x + 2]} & {[x + 4]} \cr
} } \right| = 192$$<br><br>R<sub>1</sub> $$\to$$ R<sub>1</sub> $$-$$ R<sub>3</sub> & R<sub>2</sub> $$\to$$ R<sub>2</sub> $$-$$ R<sub>3</sub><br><br>$$... | mcq | jee-main-2021-online-27th-august-evening-shift | 6,983 |
1ktirfw8s | maths | matrices-and-determinants | expansion-of-determinant | If $${a_r} = \cos {{2r\pi } \over 9} + i\sin {{2r\pi } \over 9}$$, r = 1, 2, 3, ....., i = $$\sqrt { - 1} $$, then<br/> the determinant $$\left| {\matrix{
{{a_1}} & {{a_2}} & {{a_3}} \cr
{{a_4}} & {{a_5}} & {{a_6}} \cr
{{a_7}} & {{a_8}} & {{a_9}} \cr
} } \right|$$ is equal to : | [{"identifier": "A", "content": "a<sub>2</sub>a<sub>6</sub> $$-$$ a<sub>4</sub>a<sub>8</sub>"}, {"identifier": "B", "content": "a<sub>9</sub>"}, {"identifier": "C", "content": "a<sub>1</sub>a<sub>9</sub> $$-$$ a<sub>3</sub>a<sub>7</sub>"}, {"identifier": "D", "content": "a<sub>5</sub>"}] | ["C"] | null | $${a_r} = {e^{{{i2\pi r} \over 9}}}$$, r = 1, 2, 3, ......, a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, ..... are in G.P.<br><br>$$\left| {\matrix{
{{a_1}} & {{a_2}} & {{a_3}} \cr
{{a_n}} & {{a_5}} & {{a_6}} \cr
{{a_7}} & {{a_8}} & {{a_9}} \cr
} } \right| = \left| {\matrix{
... | mcq | jee-main-2021-online-31st-august-morning-shift | 6,984 |
1l5vzbklz | maths | matrices-and-determinants | expansion-of-determinant | <p>Let $$A = \left[ {\matrix{
1 & { - 2} & \alpha \cr
\alpha & 2 & { - 1} \cr
} } \right]$$ and $$B = \left[ {\matrix{
2 & \alpha \cr
{ - 1} & 2 \cr
4 & { - 5} \cr
} } \right],\,\alpha \in C$$. Then the absolute value of the sum of all values of $$\alpha$$ fo... | [{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "5"}] | ["A"] | null | <p>Given,</p>
<p>$$A = \left[ {\matrix{
1 & { - 2} & \alpha \cr
\alpha & 2 & { - 1} \cr
} } \right]$$</p>
<p>and $$B = \left[ {\matrix{
2 & \alpha \cr
{ - 1} & 2 \cr
4 & { - 5} \cr
} } \right]$$</p>
<p>$$AB = \left[ {\matrix{
1 & { - 2} & \alpha \cr
\alpha & 2 & { - 1} \cr
... | mcq | jee-main-2022-online-30th-june-morning-shift | 6,985 |
1ldv34755 | maths | matrices-and-determinants | expansion-of-determinant | <p>Let $$\mathrm{A_1,A_2,A_3}$$ be the three A.P. with the same common difference d and having their first terms as $$\mathrm{A,A+1,A+2}$$, respectively. Let a, b, c be the $$\mathrm{7^{th},9^{th},17^{th}}$$ terms of $$\mathrm{A_1,A_2,A_3}$$, respective such that $$\left| {\matrix{
a & 7 & 1 \cr
{2b} &a... | [] | null | 495 | $a=A+6 d$
<br/><br/>
$$
\begin{aligned}
& b=A+8 d+1 \\\\
& c=A+16 d+2 \\\\
& \left|\begin{array}{ccc}
a & 7 & 1 \\
26 & 17 & 1 \\
c & 17 & 1
\end{array}\right|=-70 \\\\
& \Rightarrow\left|\begin{array}{ccc}
A+6 d & 7 & 1 \\
2 A+16 d+2 & 17 & 1 \\
A+16 d+2 & 17 & 1
\end{array}\right|=-70 \\\\
& R_{3} \rightarrow R_{3}-R... | integer | jee-main-2023-online-25th-january-morning-shift | 6,987 |
1lgrghmb3 | maths | matrices-and-determinants | expansion-of-determinant | <p>Let $$\mathrm{D}_{\mathrm{k}}=\left|\begin{array}{ccc}1 & 2 k & 2 k-1 \\
n & n^{2}+n+2 & n^{2} \\
n & n^{2}+n & n^{2}+n+2\end{array}\right|$$. If $$\sum_\limits{k=1}^{n} \mathrm{D}_{\mathrm{k}}=96$$, then $$n$$ is equal to _____________.</p> | [] | null | 6 | $$
\begin{aligned}
& \sum_{k=1}^n D_k=\left|\begin{array}{ccc}
\sum 1 & 2 \sum k & 2 \sum k-\sum 1 \\
n & n^2+n+2 & n^2 \\
n & n^2+n & n^2+n+2
\end{array}\right| \\\\
& =\left|\begin{array}{ccc}
n & n(n+1) & n^2 \\
n & n^2+n+2 & n^2 \\
n & n^2+n & n^2+n+2
\end{array}\right| \\\\
& =\left|\begin{array}{ccc}
0 & -2 & 0 \... | integer | jee-main-2023-online-12th-april-morning-shift | 6,988 |
1lgsverwb | maths | matrices-and-determinants | expansion-of-determinant | <p>$$\left|\begin{array}{ccc}x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^{2}\end{array}\right|=\frac{9}{8}(103 x+81)$$, then $$\lambda, \frac{\lambda}{3}$$ are the roots of the equation :</p> | [{"identifier": "A", "content": "$$4 x^{2}+24 x-27=0$$"}, {"identifier": "B", "content": "$$4 x^{2}-24 x+27=0$$"}, {"identifier": "C", "content": "$$4 x^{2}-24 x-27=0$$"}, {"identifier": "D", "content": "$$4 x^{2}+24 x+27=0$$"}] | ["B"] | null | $$\left|\begin{array}{ccc}x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^{2}\end{array}\right|=\frac{9}{8}(103 x+81)$$
<br/><br/>Put $x=0$
<br/><br/>$$
\begin{aligned}
& \left|\begin{array}{ccc}
1 & 0 & 0 \\
0 & \lambda & 0 \\
0 & 0 & \lambda^2
\end{array}\right|=\frac{9}{8} \times 81 \\\\
& \lambda^3=\frac{3^6}... | mcq | jee-main-2023-online-11th-april-evening-shift | 6,989 |
jaoe38c1lscntluu | maths | matrices-and-determinants | expansion-of-determinant | <p>The values of $$\alpha$$, for which $$\left|\begin{array}{ccc}1 & \frac{3}{2} & \alpha+\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha+\frac{1}{3} \\ 2 \alpha+3 & 3 \alpha+1 & 0\end{array}\right|=0$$, lie in the interval</p> | [{"identifier": "A", "content": "$$(-2,1)$$\n"}, {"identifier": "B", "content": "$$\\left(-\\frac{3}{2}, \\frac{3}{2}\\right)$$\n"}, {"identifier": "C", "content": "$$(-3,0)$$\n"}, {"identifier": "D", "content": "$$(0,3)$$"}] | ["C"] | null | <p>$$\left|\begin{array}{ccc}
1 & \frac{3}{2} & \alpha+\frac{3}{2} \\
1 & \frac{1}{3} & \alpha+\frac{1}{3} \\
2 \alpha+3 & 3 \alpha+1 & 0
\end{array}\right|=0$$</p>
<p>$$\begin{aligned}
& \Rightarrow(2 \alpha+3)\left\{\frac{7 \alpha}{6}\right\}-(3 \alpha+1)\left\{\frac{-7}{6}\right\}=0 \\
& \Rightarrow(2 \alpha+3) \cdo... | mcq | jee-main-2024-online-27th-january-evening-shift | 6,990 |
jaoe38c1lseyky4c | maths | matrices-and-determinants | expansion-of-determinant | <p>$$\text { Let } A=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & \alpha & \beta \\
0 & \beta & \alpha
\end{array}\right] \text { and }|2 \mathrm{~A}|^3=2^{21} \text { where } \alpha, \beta \in Z \text {, Then a value of } \alpha \text { is }$$</p> | [{"identifier": "A", "content": "9"}, {"identifier": "B", "content": "17"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "5"}] | ["D"] | null | <p>$$\begin{aligned}
& |\mathrm{A}|=\alpha^2-\beta^2 \\
& |2 \mathrm{~A}|^3=2^{21} \Rightarrow|\mathrm{A}|=2^4 \\
& \alpha^2-\beta^2=16 \\
& (\alpha+\beta)(\alpha-\beta)=16 \Rightarrow \alpha=4 \text { or } 5
\end{aligned}$$</p> | mcq | jee-main-2024-online-29th-january-morning-shift | 6,991 |
lvc57b13 | maths | matrices-and-determinants | expansion-of-determinant | <p>For $$\alpha, \beta \in \mathbb{R}$$ and a natural number $$n$$, let $$A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta \\ 3 r-2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|$$. Then $$2 A_{10}-A_8$$ is</p> | [{"identifier": "A", "content": "$$4 \\alpha+2 \\beta$$\n"}, {"identifier": "B", "content": "0"}, {"identifier": "C", "content": "$$2 n$$\n"}, {"identifier": "D", "content": "$$2 \\alpha+4 \\beta$$"}] | ["A"] | null | <p>$$A_r=\left|\begin{array}{ccc}
r & 1 & \frac{n^2}{2}+\alpha \\
2 r & 2 & n^2-\beta \\
3 r-2 & 3 & \frac{n(3 n-1)}{2}
\end{array}\right|$$</p>
<p>$${A_r} = 2\left| {\matrix{
r & 1 & {{{{n^2}} \over 2} + \alpha } \cr
{2r} & 2 & {{{{n^2}} \over 2} - \beta } \cr
{3r - 2} & 3 & {{{n(3n - 1)} \over 2}} \cr
... | mcq | jee-main-2024-online-6th-april-morning-shift | 6,992 |
L5OkUKHe1Wo61y8B | maths | matrices-and-determinants | inverse-of-a-matrix | Let $$A = \left( {\matrix{
1 & { - 1} & 1 \cr
2 & 1 & { - 3} \cr
1 & 1 & 1 \cr
} } \right).$$ and $$10$$ $$B = \left( {\matrix{
4 & 2 & 2 \cr
{ - 5} & 0 & \alpha \cr
1 & { - 2} & 3 \cr
} } \right)$$. if $$B$$ is
<p>the inverse of matri... | [{"identifier": "A", "content": "$$5$$"}, {"identifier": "B", "content": "$$-1$$ "}, {"identifier": "C", "content": "$$2$$ "}, {"identifier": "D", "content": "$$-2$$"}] | ["A"] | null | Given that $$10B$$ $$\,\,\, = \left[ {\matrix{
4 & 2 & 2 \cr
{ - 5} & 0 & \alpha \cr
1 & { - 2} & 3 \cr
} } \right]$$
<br><br>$$ \Rightarrow B = {1 \over {10}}\left[ {\matrix{
4 & 2 & 2 \cr
{ - 5} & 0 & \alpha \cr
1 & { - 2} & 3 \cr
} }... | mcq | aieee-2004 | 6,993 |
BrIRgABjKyBYb3DI | maths | matrices-and-determinants | inverse-of-a-matrix | Let $$A = \left( {\matrix{
0 & 0 & { - 1} \cr
0 & { - 1} & 0 \cr
{ - 1} & 0 & 0 \cr
} } \right)$$. The only correct
<p>statement about the matrix $$A$$ is</p> | [{"identifier": "A", "content": "$${A^2} = 1$$ "}, {"identifier": "B", "content": "$$A=(-1)I,$$ where $$I$$ is a unit matrix "}, {"identifier": "C", "content": "$${A^{ - 1}}$$ does not exist "}, {"identifier": "D", "content": "$$A$$ is a zero matrix"}] | ["A"] | null | $$A = \left[ {\matrix{
0 & 0 & { - 1} \cr
0 & { - 1} & 0 \cr
{ - 1} & 0 & 0 \cr
} } \right]$$
<br><br>clearly $$\,\,\,A \ne 0.\,$$ Also $$\,\,\left| A \right| = - 1 \ne 0$$
<br><br>$$\therefore$$ $${A^{ - 1}}\,\,$$ exists, further
<br><br>$$\left( { - 1} \right)I = \left[ {\ma... | mcq | aieee-2004 | 6,994 |
gVRbAj7S0qPH4tAt | maths | matrices-and-determinants | inverse-of-a-matrix | If $${A^2} - A + 1 = 0$$, then the inverse of $$A$$ is : | [{"identifier": "A", "content": "$$A+I$$ "}, {"identifier": "B", "content": "$$A$$ "}, {"identifier": "C", "content": "$$A-I$$ "}, {"identifier": "D", "content": "$$I-A$$"}] | ["D"] | null | Given $${A^2} - A + I = 0$$
<br><br>$${A^{ - 1}}{A^2} - {A^{ - 1}}A + {A^{ - 1}}.I = {A^{ - 1}}.0$$
<br><br>(Multiplying $$\,\,\,{A^{ - 1}}$$ on both sides)
<br><br>$$ \Rightarrow A - 1 + {A^{ - 1}} = 0$$
<br><br>or $${A^{ - 1}} = 1 - A$$ | mcq | aieee-2005 | 6,995 |
NZE3QvqIskeDsv0i | maths | matrices-and-determinants | inverse-of-a-matrix | If $$A$$ is a $$3 \times 3$$ non-singular matrix such that $$AA'=A'A$$ and
<br/>$$B = {A^{ - 1}}A',$$ then $$BB'$$ equals: | [{"identifier": "A", "content": "$${B^{ - 1}}$$ "}, {"identifier": "B", "content": "$$\\left( {{B^{ - 1}}} \\right)'$$"}, {"identifier": "C", "content": "$$I+B$$ "}, {"identifier": "D", "content": "$$I$$ "}] | ["D"] | null | $$BB' = B\left( {{A^{ - 1}}A'} \right)' = B\left( {A'} \right)'\left( {{A^{ - 1}}} \right)' = BA\left( {{A^{ - 1}}} \right)'$$
<br><br>$$ = \left( {{A^{ - 1}}A'} \right)\left( {A\left( {{A^{ - 1}}} \right)'} \right)$$
<br><br>$$ = {A^{ - 1}}A.A'.\left( {{A^{ - 1}}} \right)'\,\,\,\,\,\,$$ $$\left\{ {} \right.$$ as $$\,\... | mcq | jee-main-2014-offline | 6,996 |
WdAPCKhiSFPJAR2URgL8E | maths | matrices-and-determinants | inverse-of-a-matrix | Let A be a 3 $$ \times $$ 3 matrix such that A<sup>2</sup> $$-$$ 5A + 7I = 0
<br/><br/><b>Statement - I :</b>
<br/><br/>A<sup>$$-$$1</sup> = $${1 \over 7}$$ (5I $$-$$ A).
<br/><br/><b>Statement - II :</b>
<br/><br/>The polynomial A<sup>3</sup> $$-$$ 2A<sup>2</sup> $$-$$ 3A + I can be reduced to 5(A $$-$$ 4I).
<br/><... | [{"identifier": "A", "content": "Statement-I is true, but Statement-II is false."}, {"identifier": "B", "content": "Statement-I is false, but Statement-II is true."}, {"identifier": "C", "content": "Both the statements are true."}, {"identifier": "D", "content": "Both the statements are false"}] | ["C"] | null | Given,
<br><br>A<sup>2</sup> $$-$$ 5A + 7I = 0
<br><br>$$ \Rightarrow $$ A<sup>2</sup> $$-$$ 5A = $$-$$ 7I
<br><br>$$ \Rightarrow $$ AAA<sup>$$-$$1</sup> $$-$$ 5AA<sup>$$-$$1</sup> = $$-$$ 7IA<sup>$$-$$1</sup>
<br><br>$$ \Rightarrow $$ AI $$-$$ 5I = $$-$$ 7A<sup>$$-$$1... | mcq | jee-main-2016-online-10th-april-morning-slot | 6,997 |
G13VIQ4UxGankd46IX9Mu | maths | matrices-and-determinants | inverse-of-a-matrix | Suppose A is any 3$$ \times $$ 3 non-singular matrix and ( A $$-$$ 3I) (A $$-$$ 5I) = O where I = I<sub>3</sub> and O = O<sub>3</sub>. If $$\alpha $$A + $$\beta $$A<sup>-1</sup> = 4I, then $$\alpha $$ + $$\beta $$ is equal to : | [{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "7"}, {"identifier": "C", "content": "13"}, {"identifier": "D", "content": "12"}] | ["A"] | null | Given,
<br><br>( A $$-$$ 3I) (A $$-$$ 5I) = O
<br><br>$$ \Rightarrow $$ A<sup>2</sup> - 8A + 15I = O
<br><br>Multiplying both sides by A<sup>- 1</sup>, we get,
<br><br>A<sup>- 1</sup>A.A - 8A<sup>- 1</sup>A + 15A<sup>- 1</sup>I = A<sup>- 1</sup>O
<br><br>$$ \Rightarrow $$ A - 8I + 15A<sup>- 1</sup> = O
<br><br>$$ \Rig... | mcq | jee-main-2018-online-15th-april-evening-slot | 6,998 |
1dHMda3zohOtFwklZhLEe | maths | matrices-and-determinants | inverse-of-a-matrix | If $$A = \left[ {\matrix{
{\cos \theta } & { - \sin \theta } \cr
{\sin \theta } & {\cos \theta } \cr
} } \right]$$, then the matrix A<sup>–50</sup> when $$\theta $$ = $$\pi \over 12$$, is equal to :
| [{"identifier": "A", "content": "$$\\left[ {\\matrix{\n { {{\\sqrt 3 } \\over 2}} & { - {1 \\over 2}} \\cr \n {{{ 1} \\over 2}} & {{{\\sqrt 3 } \\over 2}} \\cr \n\n } } \\right]$$"}, {"identifier": "B", "content": "$$\\left[ {\\matrix{\n {{1 \\over 2}} & -{{{\\sqrt 3 } \\over 2}} \\cr \n {{{\\s... | ["C"] | null | (A<sup>$$-$$50</sup>) = (A<sup>$$-$$1</sup>)<sup>50</sup>
<br><br>We know,
<br><br>A<sup>$$-$$1</sup> = $${{adjA} \over {\left| A \right|}}$$
<br><br>$$\left| A \right|$$ = cos<sup>2</sup>$$\theta $$ + sin<sup>2</sup>$$\theta $$ = 1
<br><br>cofactor of A = $$\left[ {\matrix{
{\cos \theta } & { - \sin \theta } ... | mcq | jee-main-2019-online-9th-january-morning-slot | 6,999 |
FDEz8mcROZKQr1vuUT18hoxe66ijvwpuwxp | maths | matrices-and-determinants | inverse-of-a-matrix | If $$\left[ {\matrix{
1 & 1 \cr
0 & 1 \cr
} } \right]\left[ {\matrix{
1 & 2 \cr
0 & 1 \cr
} } \right]$$$$\left[ {\matrix{
1 & 3 \cr
0 & 1 \cr
} } \right]$$....$$\left[ {\matrix{
1 & {n - 1} \cr
0 & 1 \cr
} } \right] = \left[ {\matrix{
1 ... | [{"identifier": "A", "content": "$$\\left[ {\\matrix{\n 1 & { 0} \\cr \n {12} & 1 \\cr \n\n } } \\right]$$"}, {"identifier": "B", "content": "$$\\left[ {\\matrix{\n 1 & { 0} \\cr \n {13} & 1 \\cr \n\n } } \\right]$$"}, {"identifier": "C", "content": "$$\\left[ {\\matrix{\n 1 & { - 13} ... | ["C"] | null | Given<br><br>
$$\left[ {\matrix{
1 & 1 \cr
0 & 1 \cr
} } \right]\left[ {\matrix{
1 & 2 \cr
0 & 1 \cr
} } \right]$$$$\left[ {\matrix{
1 & 3 \cr
0 & 1 \cr
} } \right]$$....$$\left[ {\matrix{
1 & {n - 1} \cr
0 & 1 \cr
} } \right] = \left[ {\ma... | mcq | jee-main-2019-online-9th-april-morning-slot | 7,000 |
rvBAr0Vg54rfbOxUni1klrhw2ua | maths | matrices-and-determinants | inverse-of-a-matrix | Let P = $$\left[ {\matrix{
3 & { - 1} & { - 2} \cr
2 & 0 & \alpha \cr
3 & { - 5} & 0 \cr
} } \right]$$, where $$\alpha $$ $$ \in $$ R. Suppose Q = [ q<sub>ij</sub>] is a matrix satisfying PQ = kl<sub>3</sub> for some non-zero k $$ \in $$ R. <br/>If q<sub>23</sub> = $$ - {k \ove... | [] | null | 17 | As $$PQ = kI \Rightarrow Q = k{P^{ - 1}}I$$<br><br>now $$Q = {k \over {|P|}}(adjP)I $$
<br><br>$$\Rightarrow Q = {k \over {(20 + 12\alpha )}}\left[ {\matrix{
- & - & - \cr
- & - & {( - 3\alpha - 4)} \cr
- & - & - \cr
} } \right]\left[ {\matrix{
1 & 0 &... | integer | jee-main-2021-online-24th-february-morning-slot | 7,001 |
vTciYbdTPPnzWP11Sg1kls5jz0i | maths | matrices-and-determinants | inverse-of-a-matrix | If $$A = \left[ {\matrix{
0 & { - \tan \left( {{\theta \over 2}} \right)} \cr
{\tan \left( {{\theta \over 2}} \right)} & 0 \cr
} } \right]$$ and <br/>$$({I_2} + A){({I_2} - A)^{ - 1}} = \left[ {\matrix{
a & { - b} \cr
b & a \cr
} } \right]$$, then $$13({a^2} + {b^2})$$ is equal... | [] | null | 13 | $$A = \left[ {\matrix{
0 & { - \tan {\theta \over 2}} \cr
{\tan {\theta \over 2}} & 0 \cr
} } \right]$$<br><br>$$ \Rightarrow I + A = \left[ {\matrix{
1 & { - \tan {\theta \over 2}} \cr
{\tan {\theta \over 2}} & 1 \cr
} } \right]$$<br><br>$$ \Rightarrow I - A = \left[ {\matri... | integer | jee-main-2021-online-25th-february-morning-slot | 7,002 |
1ks088t6b | maths | matrices-and-determinants | inverse-of-a-matrix | Let $$A = \left[ {\matrix{
1 & 2 \cr
{ - 1} & 4 \cr
} } \right]$$. If A<sup>$$-$$1</sup> = $$\alpha$$I + $$\beta$$A, $$\alpha$$, $$\beta$$ $$\in$$ R, I is a 2 $$\times$$ 2 identity matrix then 4($$\alpha$$ $$-$$ $$\beta$$) is equal to : | [{"identifier": "A", "content": "5"}, {"identifier": "B", "content": "$${8 \\over 3}$$"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "4"}] | ["D"] | null | $$A = \left[ {\matrix{
1 & 2 \cr
{ - 1} & 4 \cr
} } \right],|A| = 6$$<br><br>$${A^{ - 1}} = {{adjA} \over {|A|}} = {1 \over 6}\left[ {\matrix{
4 & { - 2} \cr
1 & 1 \cr
} } \right] = \left[ {\matrix{
{{2 \over 3}} & { - {1 \over 3}} \cr
{{1 \over 6}} & {{1 \over 6}... | mcq | jee-main-2021-online-27th-july-morning-shift | 7,003 |
1l58gzk5u | maths | matrices-and-determinants | inverse-of-a-matrix | <p>Let $$X = \left[ {\matrix{
0 & 1 & 0 \cr
0 & 0 & 1 \cr
0 & 0 & 0 \cr
} } \right],\,Y = \alpha I + \beta X + \gamma {X^2}$$ and $$Z = {\alpha ^2}I - \alpha \beta X + ({\beta ^2} - \alpha \gamma ){X^2}$$, $$\alpha$$, $$\beta$$, $$\gamma$$ $$\in$$ R. If $${Y^{ - 1}} = \left[ {\m... | [] | null | 100 | <p>$$\because$$ $$X = \left[ {\matrix{
0 & 1 & 0 \cr
0 & 0 & 1 \cr
0 & 0 & 0 \cr
} } \right]$$</p>
<p>$$\therefore$$ $${X^2} = \left[ {\matrix{
0 & 0 & 1 \cr
0 & 0 & 0 \cr
0 & 0 & 0 \cr
} } \right]$$</p>
<p>$$\therefore$$ $$Y = \alpha I + \beta X + \gamma {X^2}\left[ {\matrix{
\alph... | integer | jee-main-2022-online-26th-june-evening-shift | 7,004 |
1l6i06330 | maths | matrices-and-determinants | inverse-of-a-matrix | <p>The number of matrices $$A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$$, where $$a, b, c, d \in\{-1,0,1,2,3, \ldots \ldots, 10\}$$, such that $$A=A^{-1}$$, is ___________.</p> | [] | null | 50 | <p>$$\because$$ $$A = \left[ {\matrix{
a & b \cr
c & d \cr
} } \right]$$ then $${A^2} = \left[ {\matrix{
{{a^2} + bc} & {b(a + d)} \cr
{c(a + d)} & {bc + {d^2}} \cr
} } \right]$$</p>
<p>For A<sup>$$-$$1</sup> must exist $$ad - bc \ne 0$$ ...... (i)</p>
<p>and $$A = {A^{ - 1}} \Rightarrow {A^2} =... | integer | jee-main-2022-online-26th-july-evening-shift | 7,005 |
1ldu5socp | maths | matrices-and-determinants | inverse-of-a-matrix | <p>Let $$A = \left[ {\matrix{
{{1 \over {\sqrt {10} }}} & {{3 \over {\sqrt {10} }}} \cr
{{{ - 3} \over {\sqrt {10} }}} & {{1 \over {\sqrt {10} }}} \cr
} } \right]$$ and $$B = \left[ {\matrix{
1 & { - i} \cr
0 & 1 \cr
} } \right]$$, where $$i = \sqrt { - 1} $$. If $$\mathrm{M=A^T ... | [{"identifier": "A", "content": "$$\\left[ {\\matrix{\n 1 & { - 2023i} \\cr \n 0 & 1 \\cr \n\n } } \\right]$$"}, {"identifier": "B", "content": "$$\\left[ {\\matrix{\n 1 & 0 \\cr \n {2023i} & 1 \\cr \n\n } } \\right]$$"}, {"identifier": "C", "content": "$$\\left[ {\\matrix{\n 1 & {2023i} \\cr \n 0 & 1 ... | ["C"] | null | $$
\begin{aligned}
& \mathrm{AA}^{\mathrm{T}}=\left[\begin{array}{cc}
\frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\
\frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}}
\end{array}\right]\left[\begin{array}{cc}
\frac{1}{\sqrt{10}} & \frac{-3}{\sqrt{10}} \\
\frac{3}{\sqrt{10}} & \frac{1}{\sqrt{10}}
\end{array}\right]=\left[\begin{... | mcq | jee-main-2023-online-25th-january-evening-shift | 7,006 |
1lguucigm | maths | matrices-and-determinants | inverse-of-a-matrix | <p>Let $$\mathrm{A}$$ be a $$2 \times 2$$ matrix with real entries such that $$\mathrm{A}'=\alpha \mathrm{A}+\mathrm{I}$$, where $$\alpha \in \mathbb{R}-\{-1,1\}$$. If $$\operatorname{det}\left(A^{2}-A\right)=4$$, then the sum of all possible values of $$\alpha$$ is equal to :</p> | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "$$\\frac{3}{2}$$"}, {"identifier": "C", "content": "0"}, {"identifier": "D", "content": "$$\\frac{5}{2}$$"}] | ["D"] | null | We have, $A^T=\alpha A+I$, where $A$ is $2 \times 2$ matrix and $\alpha \in R-\{-1,1\}$
<br/><br/>$$
\begin{aligned}
\left(A^T\right)^T & =\alpha A^T+I \\\\
A & =\alpha A^T+I \\\\
A & =\alpha(\alpha A+I)+I \left[\because A^T=\alpha A+I\right]\\\\
A & =\alpha^2 A+(\alpha+1) I \\\\
A & \left(1-\alpha^2\right)=(\alpha+1)... | mcq | jee-main-2023-online-11th-april-morning-shift | 7,007 |
lsbku1aw | maths | matrices-and-determinants | inverse-of-a-matrix | Consider the matrix $f(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$.
<br/><br/>Given below are two statements :
<br/><br/>Statement I : $ f(-x)$ is the inverse of the matrix $f(x)$.
<br/><br/>Statement II : $f(x) f(y)=f(x+y)$.
<br/><br/>I... | [{"identifier": "A", "content": "Statement I is false but Statement II is true"}, {"identifier": "B", "content": "Both Statement I and Statement II are false"}, {"identifier": "C", "content": "Both Statement I and Statement II are true"}, {"identifier": "D", "content": "Statement I is true but Statement II is false"}] | ["C"] | null | <p>$$\begin{aligned}
& f(-x)=\left[\begin{array}{ccc}
\cos x & \sin x & 0 \\
-\sin x & \cos x & 0 \\
0 & 0 & 1
\end{array}\right] \\
& f(x) \cdot f(-x)=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]=I
\end{aligned}$$</p>
<p>Hence statement- I is correct</p>
<p>Now, checking statement II... | mcq | jee-main-2024-online-27th-january-morning-shift | 7,009 |
luxwcxug | maths | matrices-and-determinants | inverse-of-a-matrix | <p>Let $$B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]$$ and $$A$$ be a $$2 \times 2$$ matrix such that $$A B^{-1}=A^{-1}$$. If $$B C B^{-1}=A$$ and $$C^4+\alpha C^2+\beta I=O$$, then $$2 \beta-\alpha$$ is equal to</p> | [{"identifier": "A", "content": "16"}, {"identifier": "B", "content": "10"}, {"identifier": "C", "content": "8"}, {"identifier": "D", "content": "2"}] | ["B"] | null | <p>$$\begin{aligned}
& B=\left[\begin{array}{ll}
1 & 3 \\
1 & 5
\end{array}\right] \\
& A B^{-1}=A^{-1} \\
& \Rightarrow A^2=B
\end{aligned}$$</p>
<p>Also, $$B C B^{-1}=A$$</p>
<p>$$\begin{aligned}
\Rightarrow C & =B^{-1} A B \\
\Rightarrow C^4 & =\left(B^{-1} A B\right)\left(B^{-1} A B\right)\left(B^{-1} A B\right)\le... | mcq | jee-main-2024-online-9th-april-evening-shift | 7,010 |
lv2eqxv2 | maths | matrices-and-determinants | inverse-of-a-matrix | <p>Let $$A$$ be a $$2 \times 2$$ symmetric matrix such that $$A\left[\begin{array}{l}1 \\ 1\end{array}\right]=\left[\begin{array}{l}3 \\ 7\end{array}\right]$$ and the determinant of $$A$$ be 1 . If $$A^{-1}=\alpha A+\beta I$$, where $$I$$ is an identity matrix of order $$2 \times 2$$, then $$\alpha+\beta$$ equals _____... | [] | null | 5 | <p>Let $$A=\left[\begin{array}{ll}a & b \\ b & c\end{array}\right]$$</p>
<p>$$|A|=1 \Rightarrow a c-b^2=0 \quad \text{... (i)}$$</p>
<p>$$\text { Given }\left[\begin{array}{ll}
a & b \\
b & c
\end{array}\right]\left[\begin{array}{l}
1 \\
1
\end{array}\right]=\left[\begin{array}{l}
3 \\
7
\end{array}\right]$$</p>
<p>$$\... | integer | jee-main-2024-online-4th-april-evening-shift | 7,011 |
Dxj4NLInsOwXDlXB | maths | matrices-and-determinants | multiplication-of-matrices | If $$A = \left[ {\matrix{
a & b \cr
b & a \cr
} } \right]$$ and $${A^2} = \left[ {\matrix{
\alpha & \beta \cr
\beta & \alpha \cr
} } \right]$$, then | [{"identifier": "A", "content": "$$\\alpha = 2ab,\\,\\beta = {a^2} + {b^2}$$ "}, {"identifier": "B", "content": "$$\\alpha = {a^2} + {b^2},\\,\\beta = ab$$ "}, {"identifier": "C", "content": "$$\\alpha = {a^2} + {b^2},\\,\\beta = 2ab$$ "}, {"identifier": "D", "content": "$$\\alpha = {a^2} + {b^2},\\,\\beta = {a... | ["C"] | null | $${A^2} = \left[ {\matrix{
\alpha & \beta \cr
\beta & \alpha \cr
} } \right] = \left[ {\matrix{
a & b \cr
b & a \cr
} } \right]\left[ {\matrix{
a & b \cr
b & a \cr
} } \right]$$
<br><br>$$ = \left[ {\matrix{
{{a^2} + {b^2}} & {2ab} \cr
{2ab} &a... | mcq | aieee-2003 | 7,012 |
KAstYuenUMEEwuNAkDNlk | maths | matrices-and-determinants | multiplication-of-matrices | Let A = $$\left[ {\matrix{
1 & 0 & 0 \cr
1 & 1 & 0 \cr
1 & 1 & 1 \cr
} } \right]$$ and B = A<sup>20</sup>. Then the sum of the elements of the first column of B is : | [{"identifier": "A", "content": "210"}, {"identifier": "B", "content": "211"}, {"identifier": "C", "content": "231"}, {"identifier": "D", "content": "251"}] | ["C"] | null | A = $$\left[ {\matrix{
1 & 0 & 0 \cr
1 & 1 & 0 \cr
1 & 1 & 1 \cr
} } \right]$$
<br><br>A<sup>2</sup> = A.A = $$\left[ {\matrix{
1 & 0 & 0 \cr
1 & 1 & 0 \cr
1 & 1 & 1 \cr
} } \right] \times \left[ {\matrix{
1 & 0 & 0 \cr
1 ... | mcq | jee-main-2018-online-16th-april-morning-slot | 7,013 |
qXMCpQCnIQe4sYD3Q2jgy2xukf0ypwyy | maths | matrices-and-determinants | multiplication-of-matrices | Let A = $$\left[ {\matrix{
x & 1 \cr
1 & 0 \cr
} } \right]$$, x $$ \in $$ R and A<sup>4</sup> = [a<sub>ij</sub>].
<br/>If
a<sub>11</sub> = 109, then a<sub>22</sub> is equal to _______ . | [] | null | 10 | $${A^2} = \left[ {\matrix{
x & 1 \cr
1 & 0 \cr
} } \right]\left[ {\matrix{
x & 1 \cr
1 & 0 \cr
} } \right] = \left[ {\matrix{
{{x^2} + 1} & x \cr
x & 1 \cr
} } \right]$$<br><br>$${A^4} = \left[ {\matrix{
{{x^2} + 1} & x \cr
x & 1 \cr
} } \r... | integer | jee-main-2020-online-3rd-september-morning-slot | 7,014 |
W3T58kXuIKp6eHtE5Ejgy2xukf8zff62 | maths | matrices-and-determinants | multiplication-of-matrices | If $$A = \left[ {\matrix{
{\cos \theta } & {i\sin \theta } \cr
{i\sin \theta } & {\cos \theta } \cr
} } \right]$$, $$\left( {\theta = {\pi \over {24}}} \right)$$<br/><br/>
and $${A^5} = \left[ {\matrix{
a & b \cr
c & d \cr
} } \right]$$, where $$i = \sqrt { - 1} $$ then which ... | [{"identifier": "A", "content": "$$a$$<sup>2</sup> - $$c$$<sup>2</sup> = 1"}, {"identifier": "B", "content": "$$0 \\le {a^2} + {b^2} \\le 1$$"}, {"identifier": "C", "content": "$$ a$$<sup>2</sup> - $$d$$<sup>2</sup> = 0"}, {"identifier": "D", "content": "$${a^2} - {b^2} = {1 \\over 2}$$"}] | ["D"] | null | $$ \because $$ $$A = \left[ {\matrix{
{\cos \theta } & {i\sin \theta } \cr
{i\sin \theta } & {\cos \theta } \cr
} } \right]$$<br><br>$$ \therefore $$ $${A^n} = \left[ {\matrix{
{\cos \,n\theta } & {i\sin \,n\theta } \cr
{i\sin \,n\theta } & {\cos \,n\theta } \cr
} } \right],n \in... | mcq | jee-main-2020-online-4th-september-morning-slot | 7,015 |
PxfQPOh5QrObe8wYIi1kluy5ax1 | maths | matrices-and-determinants | multiplication-of-matrices | If the matrix $$A = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 2 & 0 \cr
3 & 0 & { - 1} \cr
} } \right]$$ satisfies the equation<br/><br/> $${A^{20}} + \alpha {A^{19}} + \beta A = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 4 & 0 \cr
0 & 0 & 1 \cr
} } \ri... | [] | null | 4 | $${A^2} = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 2 & 0 \cr
3 & 0 & { - 1} \cr
} } \right]\left[ {\matrix{
1 & 0 & 0 \cr
0 & 2 & 0 \cr
3 & 0 & { - 1} \cr
} } \right] = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 4 & 0 \cr
0... | integer | jee-main-2021-online-26th-february-evening-slot | 7,016 |
KxeIvsh9BKSv7d6i4Q1kmknlkg7 | maths | matrices-and-determinants | multiplication-of-matrices | Let $$A = \left[ {\matrix{
a & b \cr
c & d \cr
} } \right]$$ and $$B = \left[ {\matrix{
\alpha \cr
\beta \cr
} } \right] \ne \left[ {\matrix{
0 \cr
0 \cr
} } \right]$$ such that AB = B and a + d = 2021, then the value of ad $$-$$ bc is equal to ___________. | [] | null | 2020 | $$A = \left[ {\matrix{
a & b \cr
c & d \cr
} } \right],\,B = \left[ {\matrix{
\alpha \cr
\beta \cr
} } \right]$$<br><br>$$AB = B$$<br><br>$$\left[ {\matrix{
a & b \cr
c & d \cr
} } \right]\left[ {\matrix{
\alpha \cr
\beta \cr
} } \right] = \left[ {\mat... | integer | jee-main-2021-online-17th-march-evening-shift | 7,017 |
1kruapkyd | maths | matrices-and-determinants | multiplication-of-matrices | Let $$A = \left[ {\matrix{
0 & 1 & 0 \cr
1 & 0 & 0 \cr
0 & 0 & 1 \cr
} } \right]$$. Then the number of 3 $$\times$$ 3 matrices B with entries from the set {1, 2, 3, 4, 5} and satisfying AB = BA is ____________. | [] | null | 3125 | Let matrix $$B = \left[ {\matrix{
a & b & c \cr
d & e & f \cr
g & h & i \cr
} } \right]$$<br><br>$$\because$$ $$AB = BA$$<br><br>$$\left[ {\matrix{
0 & 1 & 0 \cr
1 & 0 & 0 \cr
0 & 0 & 1 \cr
} } \right]\left[ {\matrix{
a & b & c... | integer | jee-main-2021-online-22th-july-evening-shift | 7,018 |
1l55j80fk | maths | matrices-and-determinants | multiplication-of-matrices | <p>Let $$A = \left( {\matrix{
{1 + i} & 1 \cr
{ - i} & 0 \cr
} } \right)$$ where $$i = \sqrt { - 1} $$. Then, the number of elements in the set { n $$\in$$ {1, 2, ......, 100} : A<sup>n</sup> = A } is ____________.</p> | [] | null | 25 | <p>$$\therefore$$ $${A^2} = \left[ {\matrix{
{1 + i} & 1 \cr
{ - i} & 0 \cr
} } \right]\left[ {\matrix{
{1 + i} & 1 \cr
{ - 1} & 0 \cr
} } \right] = \left[ {\matrix{
i & {1 + i} \cr
{1 - i} & { - i} \cr
} } \right]$$</p>
<p>$${A^4} = \left[ {\matrix{
i & {1 + i} \cr
{1 - i} &... | integer | jee-main-2022-online-28th-june-evening-shift | 7,020 |
1l6hxkaiy | maths | matrices-and-determinants | multiplication-of-matrices | <p>$$
\text { Let } A=\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right] \text { and } B=\left[\begin{array}{ccc}
9^{2} & -10^{2} & 11^{2} \\
12^{2} & 13^{2} & -14^{2} \\
-15^{2} & 16^{2} & 17^{2}
\end{array}\right] \text {, then the value of } A^{\prime} B A \text { is: }
$$</p> | [{"identifier": "A", "content": "1224"}, {"identifier": "B", "content": "1042"}, {"identifier": "C", "content": "540"}, {"identifier": "D", "content": "539"}] | ["D"] | null | <p>$$A'BA = \left[ {\matrix{
1 & 1 & 1 \cr
} } \right]\left[ {\matrix{
{{9^2}} & { - {{10}^2}} & {{{11}^2}} \cr
{{{12}^2}} & {{{13}^2}} & { - {{14}^2}} \cr
{ - {{15}^2}} & {{{16}^2}} & {{{17}^2}} \cr
} } \right]A$$</p>
<p>$$ = \left[ {\matrix{
{{9^2} + {{12}^2} - {{15}^2}} & { - {{10}^2} + {{... | mcq | jee-main-2022-online-26th-july-evening-shift | 7,021 |
1l6jb5z9r | maths | matrices-and-determinants | multiplication-of-matrices | <p>Let $$A=\left(\begin{array}{cc}1 & 2 \\ -2 & -5\end{array}\right)$$. Let $$\alpha, \beta \in \mathbb{R}$$ be such that $$\alpha A^{2}+\beta A=2 I$$. Then $$\alpha+\beta$$ is equal to</p> | [{"identifier": "A", "content": "$$-$$10"}, {"identifier": "B", "content": "$$-$$6"}, {"identifier": "C", "content": "6"}, {"identifier": "D", "content": "10"}] | ["D"] | null | <p>$${A^2} = \left[ {\matrix{
1 & 2 \cr
{ - 2} & { - 5} \cr
} } \right]\left[ {\matrix{
1 & 2 \cr
{ - 2} & { - 5} \cr
} } \right] = \left[ {\matrix{
{ - 3} & { - 8} \cr
8 & {21} \cr
} } \right]$$</p>
<p>$$\alpha {A^2} + \beta A = \left[ {\matrix{
{ - 3\alpha } & { - 8\alpha } \cr... | mcq | jee-main-2022-online-27th-july-morning-shift | 7,022 |
1l6m6njhu | maths | matrices-and-determinants | multiplication-of-matrices | <p>Let $$A=\left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right]$$ and $$B=\left[\begin{array}{cc}\beta & 1 \\ 1 & 0\end{array}\right], \alpha, \beta \in \mathbf{R}$$. Let $$\alpha_{1}$$ be the value of $$\alpha$$ which satisfies $$(\mathrm{A}+\mathrm{B})^{2}=\mathrm{A}^{2}+\left[\begin{array}{l... | [] | null | 2 | <p>$${(A + B)^2} = {A^2} + {B^2} + AB + BA$$</p>
<p>$$ = {A^2} + \left[ {\matrix{
2 & 2 \cr
2 & 2 \cr
} } \right]$$</p>
<p>$$\therefore$$ $${B^2} + AB + BA = \left[ {\matrix{
2 & 2 \cr
2 & 2 \cr
} } \right]$$ ..... (1)</p>
<p>$$AB = \left[ {\matrix{
1 & { - 1} \cr
2 & \alpha \cr
} }... | integer | jee-main-2022-online-28th-july-morning-shift | 7,023 |
1l6rfk48l | maths | matrices-and-determinants | multiplication-of-matrices | <p>Let $$X=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$$ and $$A=\left[\begin{array}{ccc}-1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1\end{array}\right]$$. For $$\mathrm{k} \in N$$, if $$X^{\prime} A^{k} X=33$$, then $$\mathrm{k}$$ is equal to _______.</p> | [] | null | 10 | Given $A=\left[\begin{array}{ccc}-1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1\end{array}\right]$
<br/><br/>
$A^{2}=\left[\begin{array}{lll}1 & 0 & 6 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right], \quad A^{4}=\left[\begin{array}{ccc}1 & 0 & 12 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
<br/><br/>
$\Rightarrow A^{k}=\left[\begi... | integer | jee-main-2022-online-29th-july-evening-shift | 7,024 |
1lgre7f3c | maths | matrices-and-determinants | multiplication-of-matrices | <p>Let $$A=\left[\begin{array}{cc}1 & \frac{1}{51} \\ 0 & 1\end{array}\right]$$. If $$\mathrm{B}=\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right] A\left[\begin{array}{cc}-1 & -2 \\ 1 & 1\end{array}\right]$$, then the sum of all the elements of the matrix $$\sum_\limits{n=1}^{50} B^{n}$$... | [{"identifier": "A", "content": "50"}, {"identifier": "B", "content": "75"}, {"identifier": "C", "content": "100"}, {"identifier": "D", "content": "125"}] | ["C"] | null | $$
\begin{aligned}
& \text { Let } C=\left[\begin{array}{cc}
1 & 2 \\
-1 & -1
\end{array}\right], \mathrm{D}=\left[\begin{array}{cc}
-1 & -2 \\
1 & 1
\end{array}\right] \\\\
& \mathrm{DC}=\left[\begin{array}{cc}
1 & 2 \\
-1 & -1
\end{array}\right]\left[\begin{array}{cc}
-1 & -2 \\
1 & 1
\end{array}\right]=\left[\begin{... | mcq | jee-main-2023-online-12th-april-morning-shift | 7,026 |
lsan7qev | maths | matrices-and-determinants | multiplication-of-matrices | Let $A=I_2-2 M M^T$, where $M$ is a real matrix of order $2 \times 1$ such that the relation $M^T M=I_1$ holds. If $\lambda$ is a real number such that the relation $A X=\lambda X$ holds for some non-zero real matrix $X$ of order $2 \times 1$, then the sum of squares of all possible values of $\lambda$ is equal to ____... | [] | null | 2 | $\begin{aligned} & A=I_2-2 M^T \\\\ & A^2=\left(I_2-2 M M^T\right)\left(I_2-2 M^T\right) \\\\ & =I_2-2 M^T-2 M M^T+4 M^T M^T \\\\ & =I_2-4 M M^T+4 M M^T \\\\ & =I_2\end{aligned}$
<br/><br/>$\begin{aligned} & \mathrm{AX}=\lambda \mathrm{X} \\\\ & \mathrm{A}^2 \mathrm{X}=\lambda \mathrm{AX} \\\\ & \mathrm{X}=\lambda(\lam... | integer | jee-main-2024-online-1st-february-evening-shift | 7,027 |
lsblig15 | maths | matrices-and-determinants | multiplication-of-matrices | Let $A=\left[\begin{array}{lll}2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{array}\right], B=\left[B_1, B_2, B_3\right]$, where $B_1, B_2, B_3$ are column matrics, and
<br/><br/>$$
\mathrm{AB}_1=\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right], \mathrm{AB}_2=\left[\begin{array}{l}
2 \\
3 \\
0
\e... | [] | null | 28 | <p>$$\mathrm{A}=\left[\begin{array}{lll}
2 & 0 & 1 \\
1 & 1 & 0 \\
1 & 0 & 1
\end{array}\right] \quad \mathrm{B}=\left[\mathrm{B}_1, \mathrm{~B}_2, \mathrm{~B}_3\right]$$</p>
<p>$$\mathrm{B}_1=\left[\begin{array}{l}
\mathrm{x}_1 \\
\mathrm{y}_1 \\
\mathrm{z}_1
\end{array}\right], \quad \mathrm{B}_2=\left[\begin{array}{... | integer | jee-main-2024-online-27th-january-morning-shift | 7,028 |
PhR7ljF2bx2QxzpZ | maths | matrices-and-determinants | operations-on-matrices | Let $$A = \left( {\matrix{
1 & 2 \cr
3 & 4 \cr
} } \right)$$ and $$B = \left( {\matrix{
a & 0 \cr
0 & b \cr
} } \right),a,b \in N.$$ Then | [{"identifier": "A", "content": "there cannot exist any $$B$$ such that $$AB=BA$$ "}, {"identifier": "B", "content": "there exist more then one but finite number of $$B'$$s such that $$AB=BA$$"}, {"identifier": "C", "content": "there exists exactly one $$B$$ such that $$AB=BA$$ "}, {"identifier": "D", "content": "there... | ["D"] | null | $$A = \left[ {\matrix{
1 & 2 \cr
3 & 4 \cr
} } \right]\,\,\,\,B = \left[ {\matrix{
a & 0 \cr
0 & b \cr
} } \right]$$
<br><br>$$AB = \left[ {\matrix{
a & {2b} \cr
{3a} & {4b} \cr
} } \right]$$
<br><br>$$BA = \left[ {\matrix{
a & 0 \cr
0 & b \cr... | mcq | aieee-2006 | 7,029 |
BVviBGTP0IGky423R1Ghg | maths | matrices-and-determinants | operations-on-matrices | Let P = $$\left[ {\matrix{
1 & 0 & 0 \cr
3 & 1 & 0 \cr
9 & 3 & 1 \cr
} } \right]$$ and Q = [q<sub>ij</sub>] be two 3 $$ \times $$ 3 matrices such that Q – P<sup>5</sup> = I<sub>3</sub>.
<br/><br/>Then $${{{q_{21}} + {q_{31}}} \over {{q_{32}}}}$$ is equal to : | [{"identifier": "A", "content": "15"}, {"identifier": "B", "content": "9"}, {"identifier": "C", "content": "135 "}, {"identifier": "D", "content": "10"}] | ["D"] | null | $$P = \left[ {\matrix{
1 & 0 & 0 \cr
3 & 1 & 0 \cr
9 & 3 & 1 \cr
} } \right]$$
<br><br>$${P^2} = \left[ {\matrix{
1 & 0 & 0 \cr
{3 + 3} & 1 & 0 \cr
{9 + 9 + 9} & {3 + 3} & 1 \cr
} } \right]$$
<br><br>$${P^3} = \left[ {\matrix{
1 & ... | mcq | jee-main-2019-online-12th-january-morning-slot | 7,031 |
Twer9dbwsJBwRwagoq2cN | maths | matrices-and-determinants | operations-on-matrices | Let $$A = \left( {\matrix{
{\cos \alpha } & { - \sin \alpha } \cr
{\sin \alpha } & {\cos \alpha } \cr
} } \right)$$, ($$\alpha $$ $$ \in $$ R)<br/> such that $${A^{32}} = \left( {\matrix{
0 & { - 1} \cr
1 & 0 \cr
} } \right)$$ then a value of $$\alpha $$ is | [{"identifier": "A", "content": "0"}, {"identifier": "B", "content": "$${\\pi \\over {16}}$$"}, {"identifier": "C", "content": "$${\\pi \\over {32}}$$"}, {"identifier": "D", "content": "$${\\pi \\over {64}}$$"}] | ["D"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265941/exam_images/vlkqlnh7isvrysclmlsm.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263666/exam_images/rf4qtzmkrumvqo5bw5x9.webp"><source media="(max-wid... | mcq | jee-main-2019-online-8th-april-morning-slot | 7,032 |
VLRXTfvhfMWdTaSlS37k9k2k5e2emb4 | maths | matrices-and-determinants | operations-on-matrices | Let $$\alpha $$ be a root of the equation x<sup>2</sup> + x + 1 = 0 and the <br/>matrix A = $${1 \over {\sqrt 3 }}\left[ {\matrix{
1 & 1 & 1 \cr
1 & \alpha & {{\alpha ^2}} \cr
1 & {{\alpha ^2}} & {{\alpha ^4}} \cr
} } \right]$$<br/><br/> then the matrix
A<sup>31</sup> is equal... | [{"identifier": "A", "content": "A<sup>2</sup>"}, {"identifier": "B", "content": "A"}, {"identifier": "C", "content": "I<sub>3</sub>"}, {"identifier": "D", "content": "A<sup>3</sup>"}] | ["D"] | null | x<sup>2</sup> + x + 1 = 0
<br><br>$$ \Rightarrow $$ x = $${{ - 1 + i\sqrt 3 } \over 2}$$ = $$\omega $$ or $${{ - 1 - i\sqrt 3 } \over 2}$$ = $${\omega ^2}$$
<br><br>Let $$\alpha $$ = $$\omega $$
<br><br>$$ \therefore $$ A = $${1 \over {\sqrt 3 }}\left[ {\matrix{
1 & 1 & 1 \cr
1 & \omega & {{\om... | mcq | jee-main-2020-online-7th-january-morning-slot | 7,033 |
WRLXvWOxnX5Sxd5nGV7k9k2k5hjw6xo | maths | matrices-and-determinants | operations-on-matrices | If $$A = \left( {\matrix{
2 & 2 \cr
9 & 4 \cr
} } \right)$$ and $$I = \left( {\matrix{
1 & 0 \cr
0 & 1 \cr
} } \right)$$ then 10A<sup>–1</sup> is
equal to : | [{"identifier": "A", "content": "6I \u2013 A"}, {"identifier": "B", "content": "4I \u2013 A"}, {"identifier": "C", "content": "A \u2013 6I"}, {"identifier": "D", "content": "A \u2013 4I"}] | ["C"] | null | According to Cayley Hamilton equation
<br>|A – $$\lambda $$I| = 0
<br><br>$$ \Rightarrow $$ $$\left| {\matrix{
{2 - \lambda } & 2 \cr
9 & {4 - \lambda } \cr
} } \right|$$ = 0
<br><br>$$ \Rightarrow $$ (2 – $$\lambda $$)(4 – $$\lambda $$) – 18 = 0
<br><br>$$ \Rightarrow $$ 8 – 2$$\lambda $$ – 4$$\lam... | mcq | jee-main-2020-online-8th-january-evening-slot | 7,034 |
VQhiXPYmhIls9RqlD81kmizzhfm | maths | matrices-and-determinants | operations-on-matrices | Let $$A = \left[ {\matrix{
{{a_1}} \cr
{{a_2}} \cr
} } \right]$$ and $$B = \left[ {\matrix{
{{b_1}} \cr
{{b_2}} \cr
} } \right]$$ be two 2 $$\times$$ 1 matrices with real entries such that A = XB, where <br/><br/>$$X = {1 \over {\sqrt 3 }}\left[ {\matrix{
1 & { - 1} \cr
1 & k \c... | [] | null | 1 | $$XB = A$$
<br><br>$$ \Rightarrow $$ $${1 \over {\sqrt 3 }}\left[ {\matrix{
1 & { - 1} \cr
1 & k \cr
} } \right]\left[ {\matrix{
{{b_1}} \cr
{{b_2}} \cr
} } \right] = \left[ {\matrix{
{{a_1}} \cr
{{a_2}} \cr
} } \right]$$
<br><br>$$ \Rightarrow $$ $${1 \over {\sqrt 3 }}\left... | integer | jee-main-2021-online-16th-march-evening-shift | 7,035 |
1krq0aujn | maths | matrices-and-determinants | operations-on-matrices | Let $$A = \left( {\matrix{
1 & { - 1} & 0 \cr
0 & 1 & { - 1} \cr
0 & 0 & 1 \cr
} } \right)$$ and B = 7A<sup>20</sup> $$-$$ 20A<sup>7</sup> + 2I, where I is an identity matrix of order 3 $$\times$$ 3. If B = [b<sub>ij</sub>], then b<sub>13</sub>is equal to _____________. | [] | null | 910 | Let $$A = \left( {\matrix{
1 & { - 1} & 0 \cr
0 & 1 & { - 1} \cr
0 & 0 & 1 \cr
} } \right) = I + C$$<br><br>where, $$I = \left( {\matrix{
1 & 0 & 0 \cr
0 & 1 & 0 \cr
0 & 0 & 1 \cr
} } \right),C = \left( {\matrix{
0 & { - 1} & 0... | integer | jee-main-2021-online-20th-july-morning-shift | 7,036 |
1kru3wirg | maths | matrices-and-determinants | operations-on-matrices | Let A = [a<sub>ij</sub>] be a real matrix of order 3 $$\times$$ 3, such that a<sub>i1</sub> + a<sub>i2</sub> + a<sub>i3</sub> = 1, for i = 1, 2, 3. Then, the sum of all the entries of the matrix A<sup>3</sup> is equal to : | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "9"}] | ["C"] | null | $$A = \left[ {\matrix{
{{a_{11}}} & {{a_{12}}} & {{a_{13}}} \cr
{{a_{21}}} & {{a_{22}}} & {{a_{23}}} \cr
{{a_{31}}} & {{a_{32}}} & {{a_{33}}} \cr
} } \right]$$<br><br>Let $$x = \left[ {\matrix{
1 \cr
1 \cr
1 \cr
} } \right]$$<br><br>$$AX = \left[ {\matrix{
{{... | mcq | jee-main-2021-online-22th-july-evening-shift | 7,037 |
1krygt7fm | maths | matrices-and-determinants | operations-on-matrices | If $$A = \left[ {\matrix{
1 & 1 & 1 \cr
0 & 1 & 1 \cr
0 & 0 & 1 \cr
} } \right]$$ and M = A + A<sup>2</sup> + A<sup>3</sup> + ....... + A<sup>20</sup>, then the sum of all the elements of the matrix M is equal to _____________. | [] | null | 2020 | $${A^n} = \left[ {\matrix{
1 & n & {{{{n^2} + n} \over 2}} \cr
0 & 1 & n \cr
0 & 0 & 1 \cr
} } \right]$$<br><br>So, required sum<br><br>$$ = 20 \times 3 + 2 \times \left( {{{20 \times 21} \over 2}} \right) + \sum\limits_{r = 1}^{20} {\left( {{{{r^2} + r} \over 2}} \right)} $$<br... | integer | jee-main-2021-online-27th-july-evening-shift | 7,038 |
1krzn7q3c | maths | matrices-and-determinants | operations-on-matrices | If $$P = \left[ {\matrix{
1 & 0 \cr
{{1 \over 2}} & 1 \cr
} } \right]$$, then P<sup>50</sup> is : | [{"identifier": "A", "content": "$$\\left[ {\\matrix{\n 1 & 0 \\cr \n {25} & 1 \\cr \n\n } } \\right]$$"}, {"identifier": "B", "content": "$$\\left[ {\\matrix{\n 1 & {50} \\cr \n 0 & 1 \\cr \n\n } } \\right]$$"}, {"identifier": "C", "content": "$$\\left[ {\\matrix{\n 1 & {25} \\cr \n ... | ["A"] | null | $$P = \left[ {\matrix{
1 & 0 \cr
{{1 \over 2}} & 1 \cr
} } \right]$$<br><br>$${P^2} = \left[ {\matrix{
1 & 0 \cr
{{1 \over 2}} & 1 \cr
} } \right]\left[ {\matrix{
1 & 0 \cr
{{1 \over 2}} & 1 \cr
} } \right] = \left[ {\matrix{
1 & 0 \cr
1 & 1 \... | mcq | jee-main-2021-online-25th-july-evening-shift | 7,039 |
1ktbfkr1u | maths | matrices-and-determinants | operations-on-matrices | If $$A = \left( {\matrix{
{{1 \over {\sqrt 5 }}} & {{2 \over {\sqrt 5 }}} \cr
{{{ - 2} \over {\sqrt 5 }}} & {{1 \over {\sqrt 5 }}} \cr
} } \right)$$, $$B = \left( {\matrix{
1 & 0 \cr
i & 1 \cr
} } \right)$$, $$i = \sqrt { - 1} $$, and Q = A<sup>T</sup>BA, then the inverse of the ... | [{"identifier": "A", "content": "$$\\left( {\\matrix{\n {{1 \\over {\\sqrt 5 }}} & { - 2021} \\cr \n {2021} & {{1 \\over {\\sqrt 5 }}} \\cr \n\n } } \\right)$$"}, {"identifier": "B", "content": "$$\\left( {\\matrix{\n 1 & 0 \\cr \n { - 2021i} & 1 \\cr \n\n } } \\right)$$"}, {"identifier": "C... | ["B"] | null | $$A{A^T} = \left( {\matrix{
{{1 \over {\sqrt 5 }}} & {{2 \over {\sqrt 5 }}} \cr
{{{ - 2} \over {\sqrt 5 }}} & {{1 \over {\sqrt 5 }}} \cr
} } \right)\left( {\matrix{
{{1 \over {\sqrt 5 }}} & {{{ - 2} \over {\sqrt 5 }}} \cr
{{2 \over {\sqrt 5 }}} & {{1 \over {\sqrt 5 }}} \cr
} } \r... | mcq | jee-main-2021-online-26th-august-morning-shift | 7,040 |
1kteid0ux | maths | matrices-and-determinants | operations-on-matrices | If the matrix $$A = \left( {\matrix{
0 & 2 \cr
K & { - 1} \cr
} } \right)$$ satisfies $$A({A^3} + 3I) = 2I$$, then the value of K is : | [{"identifier": "A", "content": "$${1 \\over 2}$$"}, {"identifier": "B", "content": "$$-$$$${1 \\over 2}$$"}, {"identifier": "C", "content": "$$-$$1"}, {"identifier": "D", "content": "1"}] | ["A"] | null | Given matrix $$A = \left[ {\matrix{
0 & 2 \cr
k & { - 1} \cr
} } \right]$$<br><br>$${A^4} + 3IA = 2I$$<br><br>$$ \Rightarrow {A^4} = 2I - 3A$$<br><br>Also characteristic equation of A is $$|A - \lambda I|\, = 0$$<br><br>$$ \Rightarrow \left| {\matrix{
{0 - \lambda } & 2 \cr
k & { - 1... | mcq | jee-main-2021-online-27th-august-morning-shift | 7,041 |
1ktkekk3h | maths | matrices-and-determinants | operations-on-matrices | The number of elements in the set $$\left\{ {A = \left( {\matrix{
a & b \cr
0 & d \cr
} } \right):a,b,d \in \{ - 1,0,1\} \,and\,{{(I - A)}^3} = I - {A^3}} \right\}$$, where I is 2 $$\times$$ 2 identity matrix, is : | [] | null | 8 | $${(I - A)^3} = {I^3} - {A^3} - 3A(I - A) = I - {A^3}$$<br><br>$$ \Rightarrow 3A(I - A) = 0$$ or $${A^2} = A$$<br><br>$$ \Rightarrow \left[ {\matrix{
{{a^2}} & {ab + bd} \cr
0 & {{d^2}} \cr
} } \right] = \left[ {\matrix{
a & b \cr
0 & d \cr
} } \right]$$<br><br>$$ \Rightarrow {a^... | integer | jee-main-2021-online-31st-august-evening-shift | 7,042 |
1l5bb0wm7 | maths | matrices-and-determinants | operations-on-matrices | <p>Let $$S = \left\{ {\left( {\matrix{
{ - 1} & a \cr
0 & b \cr
} } \right);a,b \in \{ 1,2,3,....100\} } \right\}$$ and let $${T_n} = \{ A \in S:{A^{n(n + 1)}} = I\} $$. Then the number of elements in $$\bigcap\limits_{n = 1}^{100} {{T_n}} $$ is ___________.</p> | [] | null | 100 | $$
\begin{aligned}
&\mathrm{A}=\left[\begin{array}{cc}
-1 & \mathrm{a} \\\\
0 & \mathrm{~b}
\end{array}\right] \\\\
&\mathrm{A}^2=\left[\begin{array}{cc}
-1 & \mathrm{a} \\\\
0 & \mathrm{~b}
\end{array}\right]\left[\begin{array}{cc}
-1 & \mathrm{a} \\\\
0 & \mathrm{~b}
\end{array}\right] \\\\
&=\left[\begin{array}{cc}
... | integer | jee-main-2022-online-24th-june-evening-shift | 7,046 |
1l6dwytee | maths | matrices-and-determinants | operations-on-matrices | <p>Let $$A=\left(\begin{array}{rrr}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right)$$ and $$B=A-I$$. If $$\omega=\frac{\sqrt{3} i-1}{2}$$, then the number of elements in the $$\operatorname{set}\left\{n \in\{1,2, \ldots, 100\}: A^{n}+(\omega B)^{n}=A+B\right\}$$ is equal to ____________... | [] | null | 17 | Here $A=\left(\begin{array}{ccc}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right)$
<br/><br/>
We get $A^{2}=A$ and similarly for
<br/><br/>
$$
B=A-I=\left[\begin{array}{lll}
1 & -1 & -1 \\
1 & -1 & -1 \\
1 & -1 & -1
\end{array}\right]
$$
<br/><br/>
We get $B^{2}=-B \Rightarrow B^{3}=B$
<br/><br/>
$$
\therefore ... | integer | jee-main-2022-online-25th-july-morning-shift | 7,047 |
1l6rdsrjt | maths | matrices-and-determinants | operations-on-matrices | <p>Which of the following matrices can NOT be obtained from the matrix $$\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right]$$ by a single elementary row operation ?</p> | [{"identifier": "A", "content": "$$\\left[\\begin{array}{cc}0 & 1 \\\\ 1 & -1\\end{array}\\right]$$"}, {"identifier": "B", "content": "$$\\left[\\begin{array}{cc}1 & -1 \\\\ -1 & 2\\end{array}\\right]$$"}, {"identifier": "C", "content": "$$\\left[\\begin{array}{rr}-1 & 2 \\\\ -2 & 7\\end{array}\\right]$$"}, {"identifie... | ["C"] | null | <p>Given matrix $$A = \left[ {\matrix{
{ - 1} & 2 \cr
1 & { - 1} \cr
} } \right]$$</p>
<p>For option A :</p>
<p>$${R_1} \to {R_1} + {R_2}$$</p>
<p>$$A = \left[ {\matrix{
0 & 1 \cr
1 & { - 1} \cr
} } \right]$$</p>
<p>$$\therefore$$ Option A can be obtained.</p>
<p>For option B :</p>
<p>$${R_1} \l... | mcq | jee-main-2022-online-29th-july-evening-shift | 7,048 |
1ldsuoc5l | maths | matrices-and-determinants | operations-on-matrices | <p>Let $$\alpha$$ and $$\beta$$ be real numbers. Consider a 3 $$\times$$ 3 matrix A such that $$A^2=3A+\alpha I$$. If $$A^4=21A+\beta I$$, then</p> | [{"identifier": "A", "content": "$$\\alpha=1$$"}, {"identifier": "B", "content": "$$\\alpha=4$$"}, {"identifier": "C", "content": "$$\\beta=8$$"}, {"identifier": "D", "content": "$$\\beta=-8$$"}] | ["D"] | null | $\mathrm{A}^{2}=3 \mathrm{~A}+\alpha \mathrm{I}$
<br/><br/>
$A^{3}=3 A^{2}+\alpha A$
<br/><br/>
$\mathrm{A}^{3}=3(3 \mathrm{~A}+\alpha \mathrm{I})+\alpha \mathrm{A}$
<br/><br/>
$\mathrm{A}^{3}=9 \mathrm{~A}+\alpha \mathrm{A}+3 \alpha \mathrm{I}$
<br/><br/>
$\mathrm{A}^{4}=(9+\alpha) \mathrm{A}^{2}+3 \alpha \mathrm{A}$
... | mcq | jee-main-2023-online-29th-january-morning-shift | 7,049 |
1ldybe3tm | maths | matrices-and-determinants | operations-on-matrices | <p>If A and B are two non-zero n $$\times$$ n matrices such that $$\mathrm{A^2+B=A^2B}$$, then :</p> | [{"identifier": "A", "content": "$$\\mathrm{A^2B=I}$$"}, {"identifier": "B", "content": "$$\\mathrm{A^2=I}$$ or $$\\mathrm{B=I}$$"}, {"identifier": "C", "content": "$$\\mathrm{A^2B=BA^2}$$"}, {"identifier": "D", "content": "$$\\mathrm{AB=I}$$"}] | ["C"] | null | Given : $A^{2}+B=A^{2} B\quad...(i)$
<br/><br/>
$\Rightarrow A^{2}+B-I=A^{2} B-I$
<br/><br/>
$\Rightarrow A^{2} B-A^{2}-B+I=I$
<br/><br/>
$\Rightarrow A^{2}(B-I)-I(B-I)=I$
<br/><br/>
$\Rightarrow\left(A^{2}-I\right)(B-I)=I$
<br/><br/>
$\therefore A^{2}-I$ is the inverse matrix of $B-I$ and vice versa.
<br/><br/>
So, $(... | mcq | jee-main-2023-online-24th-january-morning-shift | 7,050 |
1lguwckg8 | maths | matrices-and-determinants | operations-on-matrices | <p>Let $$A=\left[\begin{array}{lll}0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0\end{array}\right]$$, where $$a, c \in \mathbb{R}$$. If $$A^{3}=A$$ and the positive value of $$a$$ belongs to the interval $$(n-1, n]$$, where $$n \in \mathbb{N}$$, then $$n$$ is equal to ___________.</p> | [] | null | 2 | $$
\text { We have, } A=\left[\begin{array}{lll}
0 & 1 & 2 \\
a & 0 & 3 \\
1 & c & 0
\end{array}\right] \text {, where } a, c \in R
$$
<br/><br/>$$
\begin{aligned}
A^2 & =\left[\begin{array}{lll}
0 & 1 & 2 \\
a & 0 & 3 \\
1 & c & 0
\end{array}\right]\left[\begin{array}{lll}
0 & 1 & 2 \\
a & 0 & 3 \\
1 & c & 0
\end{arra... | integer | jee-main-2023-online-11th-april-morning-shift | 7,051 |
1lh21bstu | maths | matrices-and-determinants | operations-on-matrices | <p>Let $$\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{2 \times 2}$$, where $$\mathrm{a}_{\mathrm{ij}} \neq 0$$ for all $$\mathrm{i}, \mathrm{j}$$ and $$\mathrm{A}^{2}=\mathrm{I}$$. Let a be the sum of all diagonal elements of $$\mathrm{A}$$ and $$\mathrm{b}=|\mathrm{A}|$$. Then $$3 a^{2}+4 b^{2}$$ is equal to :</p... | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "14"}, {"identifier": "D", "content": "7"}] | ["A"] | null | Given, $A^2=I$
<br/><br/>and $b=|A|$
<br/><br/>Let
$$
A=\left[\begin{array}{ll}
a_1 & b_1 \\
a_2 & b_2
\end{array}\right]
$$
<br/><br/>$$
\begin{aligned}
\therefore \quad A^2 & =\left[\begin{array}{ll}
a_1 & b_1 \\
a_2 & b_2
\end{array}\right]\left[\begin{array}{ll}
a_1 & b_1 \\
a_2 & b_2
\end{array}\right] \\\\
& =\le... | mcq | jee-main-2023-online-6th-april-morning-shift | 7,052 |
lv5grw76 | maths | matrices-and-determinants | operations-on-matrices | <p>Let $$A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]$$. If $$A^3=4 A^2-A-21 I$$, where $$I$$ is the identity matrix of order $$3 \times 3$$, then $$2 a+3 b$$ is equal to</p> | [{"identifier": "A", "content": "$$-10$$\n"}, {"identifier": "B", "content": "$$-12$$\n"}, {"identifier": "C", "content": "$$-13$$\n"}, {"identifier": "D", "content": "$$-9$$"}] | ["C"] | null | <p>$$\begin{aligned}
& |A-\lambda I|=0 \\
& \left|\begin{array}{ccc}
2-\lambda & a & 0 \\
1 & 3-\lambda & 1 \\
0 & 5 & b-\lambda
\end{array}\right|=0 \\
& (2-\lambda)[(3-\lambda)(b-\lambda)-5]-a[b-\lambda-0]+0=0 \\
& (2-\lambda)\left[3 b-3 \lambda-b \lambda+\lambda^2-5\right]-a b+a \lambda=0 \\
& \lambda^3-(b+5) \lambd... | mcq | jee-main-2024-online-8th-april-morning-shift | 7,055 |
YSvNld4KWlRuXhMh | maths | matrices-and-determinants | properties-of-determinants | Let $$A = \left| {\matrix{
5 & {5\alpha } & \alpha \cr
0 & \alpha & {5\alpha } \cr
0 & 0 & 5 \cr
} } \right|.$$ If $$\,\,\left| {{A^2}} \right| = 25,$$ then $$\,\left| \alpha \right|$$ equals | [{"identifier": "A", "content": "$$1/5$$ "}, {"identifier": "B", "content": "$$5$$"}, {"identifier": "C", "content": "$${5^2}$$ "}, {"identifier": "D", "content": "$$1$$"}] | ["A"] | null | $$\left| {{A^2}} \right| = 25 \Rightarrow {\left| A \right|^2} = 25$$
<br><br>$$ \Rightarrow {\left( {25\alpha } \right)^2} = 25 \Rightarrow \left| \alpha \right| = {1 \over 5}$$ | mcq | aieee-2007 | 7,056 |
XrzcXWoZKBi1HBgV | maths | matrices-and-determinants | properties-of-determinants | Let $$A$$ be a square matrix all of whose entries are integers.
<br/>Then which one of the following is true? | [{"identifier": "A", "content": "If det $$A = \\pm 1,$$ then $${A^{ - 1}}$$ exists but all its entries are not necessarily integers"}, {"identifier": "B", "content": "If det $$A \\ne \\pm 1,$$ then $${A^{ - 1}}$$ exists and all its entries are non integers"}, {"identifier": "C", "content": "If det $$A = \\pm 1,$$ t... | ["C"] | null | As all entries of square matrix $$A$$ are integers, therefore all co-factors should also be integers.
<br><br>If det $$A = \pm 1\,\,$$ then $${A^{ - 1}}\,\,$$ exists. Also all entries of $${A^{ - 1}}$$ are integers. | mcq | aieee-2008 | 7,057 |
5Mqcuw2868tUWChI | maths | matrices-and-determinants | properties-of-determinants | Let $$P$$ and $$Q$$ be $$3 \times 3$$ matrices $$P \ne Q.$$ If $${P^3} = {Q^3}$$ and
<br/> $${P^2}Q = {Q^2}P$$ then determinant of $$\left( {{P^2} + {Q^2}} \right)$$ is equal to : | [{"identifier": "A", "content": "$$-2$$ "}, {"identifier": "B", "content": "$$1$$ "}, {"identifier": "C", "content": "$$0$$ "}, {"identifier": "D", "content": "$$-1$$"}] | ["C"] | null | Given
<br><br>$${P^3} = {q^3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$
<br><br>$${P^2}Q = {Q^2}p\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 2 \right)$$
<br><br>Subtracting $$(1)$$ and $$(2)$$, we get
<br><br>$${P^3} - {P^2}Q = {Q^3} - {Q^2}P$$
<br><br>$$ \Rightarrow {P... | mcq | aieee-2012 | 7,059 |
tQwnUOEU3VBd7zli | maths | matrices-and-determinants | properties-of-determinants | If $$P = \left[ {\matrix{
1 & \alpha & 3 \cr
1 & 3 & 3 \cr
2 & 4 & 4 \cr
} } \right]$$ is the adjoint of a $$3 \times 3$$ matrix $$A$$ and
<br/>$$\left| A \right| = 4,$$ then $$\alpha $$ is equal to : | [{"identifier": "A", "content": "$$4$$ "}, {"identifier": "B", "content": "$$11$$ "}, {"identifier": "C", "content": "$$5$$ "}, {"identifier": "D", "content": "$$0$$"}] | ["B"] | null | $$\left| P \right| = 1\left( {12 - 12} \right) - \alpha \left( {4 - 6} \right) + $$
<br><br>$$\,\,\,\,\,\,\,\,\,\,\,3\left( {4 - 6} \right) = 2\alpha - 6$$
<br><br>Now, $$adj\,\,A = P\,$$ $$\,\,\,\,\,\,\,\, \Rightarrow \left| {adj\,A} \right| = \left| P \right|$$
<br><br>$$ \Rightarrow {\left| A \right|^2} = \left| ... | mcq | jee-main-2013-offline | 7,061 |
9cnuwNKo0TcaRlwfihNII | maths | matrices-and-determinants | properties-of-determinants | Let A and B be two invertible matrices of order 3 $$ \times $$ 3. If det(ABA<sup>T</sup>) = 8 and det(AB<sup>–1</sup>) = 8,
<br/>then det (BA<sup>–1</sup> B<sup>T</sup>) is equal to :
| [{"identifier": "A", "content": "$${1 \\over 4}$$"}, {"identifier": "B", "content": "16"}, {"identifier": "C", "content": "$${1 \\over {16}}$$"}, {"identifier": "D", "content": "1"}] | ["C"] | null | $${\left| A \right|^2}.\left| B \right| = 8$$
<br><br>and $${{\left| A \right|} \over {\left| B \right|}} = 8 \Rightarrow \left| A \right| = 4$$
<br><br>and $$\left| B \right| = {1 \over 2}$$
<br><br>$$ \therefore $$ det(BA<sup>$$-$$1</sup>. B<sup>T</sup>) $$ = {1 \over 4} \times {1 \over 4} = {1 \over ... | mcq | jee-main-2019-online-11th-january-evening-slot | 7,062 |
YYS2DGHZEoLw2DMefI7k9k2k5iqzn1t | maths | matrices-and-determinants | properties-of-determinants | If the matrices A = $$\left[ {\matrix{
1 & 1 & 2 \cr
1 & 3 & 4 \cr
1 & { - 1} & 3 \cr
} } \right]$$,
<br/><br/>B = adjA and
C = 3A, then $${{\left| {adjB} \right|} \over {\left| C \right|}}$$ is equal to : | [{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "72"}, {"identifier": "D", "content": "16"}] | ["A"] | null | A = $$\left[ {\matrix{
1 & 1 & 2 \cr
1 & 3 & 4 \cr
1 & { - 1} & 3 \cr
} } \right]$$
<br><br>$$ \Rightarrow $$ |A| = 6
<br><br>$${{\left| {adjB} \right|} \over {\left| C \right|}}$$
<br><br>= $${{\left| {adj\left( {adjA} \right)} \right|} \over {\left| {3A} \right|}}$$
<br><br>= ... | mcq | jee-main-2020-online-9th-january-morning-slot | 7,064 |
8DDQ9pLxEGNcY3sEdujgy2xukf4552qe | maths | matrices-and-determinants | properties-of-determinants | Let A be a 3 $$ \times $$ 3 matrix such that
<br/>adj A = $$\left[ {\matrix{
2 & { - 1} & 1 \cr
{ - 1} & 0 & 2 \cr
1 & { - 2} & { - 1} \cr
} } \right]$$ and B = adj(adj A).
<br/><br/>If |A| = $$\lambda $$ and |(B<sup>-1</sup>)<sup>T</sup>| = $$\mu $$ , then the ordered pair,
<br... | [{"identifier": "A", "content": "(3, 81)"}, {"identifier": "B", "content": "$$\\left( {9,{1 \\over 9}} \\right)$$"}, {"identifier": "C", "content": "$$\\left( {3,{1 \\over {81}}} \\right)$$"}, {"identifier": "D", "content": "$$\\left( {9,{1 \\over {81}}} \\right)$$"}] | ["C"] | null | $$adj\,A = \left[ {\matrix{
2 & { - 1} & 1 \cr
{ - 1} & 0 & 2 \cr
1 & { - 2} & { - 1} \cr
} } \right]$$<br><br>$$B = adj\,(adj\,A)$$<br><br>$$ = |A{|^{n - 2}}A$$<br><br>$$ = |A{|^{3 - 2}}.A$$ [As here n = 3]<br><br>$$ = |A|.A$$ .....(1)<br><br>Now, $$|adj\,A| = \left[ {\matrix{
... | mcq | jee-main-2020-online-3rd-september-evening-slot | 7,065 |
BLrpVgxd8BhsNuoTtj1kls5lrip | maths | matrices-and-determinants | properties-of-determinants | Let $$A = \left[ {\matrix{
x & y & z \cr
y & z & x \cr
z & x & y \cr
} } \right]$$, where x, y and z are real numbers such that x + y + z > 0 and xyz = 2. If $${A^2} = {I_3}$$, then the value of $${x^3} + {y^3} + {z^3}$$ is ____________. | [] | null | 7 | $$A = \left[ {\matrix{
x & y & z \cr
y & z & x \cr
z & x & y \cr
} } \right]$$
<br><br>$$ \therefore $$ $$|A| = \left( {{x^3} + {y^3} + {z^3} - 3xyz} \right)$$<br><br>Given $${A^2} = {I_3}$$<br><br>$$|{A^2}| = 1$$<br><br>$$ \therefore $$ $${({x^3} + {y^3} + {z^3} - 3xyz)^2} = ... | integer | jee-main-2021-online-25th-february-morning-slot | 7,066 |
6D7g0xG8HmUmFaTFHj1kmjbrw8g | maths | matrices-and-determinants | properties-of-determinants | If $$A = \left[ {\matrix{
2 & 3 \cr
0 & { - 1} \cr
} } \right]$$, then the value of det(A<sup>4</sup>) + det(A<sup>10</sup> $$-$$ (Adj(2A))<sup>10</sup>) is equal to _____________. | [] | null | 16 | $$A = \left[ {\matrix{
2 & 3 \cr
0 & { - 1} \cr
} } \right]$$
<br><br>$$|A|\, = - 2 \Rightarrow |A{|^4} = 16$$
<br><br>$${A^2} = \left[ {\matrix{
4 & 3 \cr
0 & 1 \cr
} } \right]$$
<br><br>$${A^3} = \left[ {\matrix{
8 & 9 \cr
0 & { - 1} \cr
} } \right]$$
<br><... | integer | jee-main-2021-online-17th-march-morning-shift | 7,069 |
1krrv7h3p | maths | matrices-and-determinants | properties-of-determinants | Let $$A = \{ {a_{ij}}\} $$ be a 3 $$\times$$ 3 matrix, <br/><br/>where $${a_{ij}} = \left\{ {\matrix{
{{{( - 1)}^{j - i}}} & {if} & {i < j,} \cr
2 & {if} & {i = j,} \cr
{{{( - 1)}^{i + j}}} & {if} & {i > j} \cr
} } \right.$$ <br/><br/>then $$\det (3Adj(2{A^{ - 1}}))$$ is e... | [] | null | 108 | $$A = \left[ {\matrix{
2 & { - 1} & 1 \cr
{ - 1} & 2 & { - 1} \cr
1 & { - 1} & 2 \cr
} } \right]$$<br><br>$$|A| = 4$$<br><br>$$\det (3adj(2{A^{ - 1}}))$$<br><br>$$ = {3^3}\left| {adj(2{a^{ - 1}})} \right|$$<br><br>$$ = {3^2}{\left| {2{A^{ - 1}}} \right|^2}$$<br><br>$$ = {3^3}{.2... | integer | jee-main-2021-online-20th-july-evening-shift | 7,070 |
1krw2sssh | maths | matrices-and-determinants | properties-of-determinants | Let $$M = \left\{ {A = \left( {\matrix{
a & b \cr
c & d \cr
} } \right):a,b,c,d \in \{ \pm 3, \pm 2, \pm 1,0\} } \right\}$$. Define f : M $$\to$$ Z, as f(A) = det(A), for all A$$\in$$M, where z is set of all integers. Then the number of A$$\in$$M such that f(A) = 15 is equal to _____________. | [] | null | 16 | | A | = ad $$-$$ bc = 15<br><br>where $${a,b,c,d \in \{ \pm 3, \pm 2, \pm 1,0\} }$$<br><br>Case I ad = 9 & bc = $$-$$6<br><br>For ad possible pairs are (3, 3), ($$-$$3, $$-$$3)<br><br>For bc possible pairs are (3, $$-$$2), ($$-$$3, 2), ($$-$$2, 3), (2, $$-$$3)<br><br>So total matrix = 2 $$\times$$ 4 = 8<br><br>Cas... | integer | jee-main-2021-online-25th-july-morning-shift | 7,071 |
1kryf4lkx | maths | matrices-and-determinants | properties-of-determinants | Let A and B be two 3 $$\times$$ 3 real matrices such that (A<sup>2</sup> $$-$$ B<sup>2</sup>) is invertible matrix. If A<sup>5</sup> = B<sup>5</sup> and A<sup>3</sup>B<sup>2</sup> = A<sup>2</sup>B<sup>3</sup>, then the value of the determinant of the matrix A<sup>3</sup> + B<sup>3</sup> is equal to : | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "0"}] | ["D"] | null | C = A<sup>2</sup> $$-$$ B<sup>2</sup>; | C | $$\ne$$ 0<br><br>A<sup>2</sup> = B<sup>5</sup> and A<sup>3</sup>B<sup>2</sup> = A<sup>2</sup>B<sup>2</sup><br><br>Now, A<sup>5</sup> $$-$$ A<sup>3</sup>B<sup>2</sup> = B<sup>5</sup> $$-$$ A<sup>2</sup>B<sup>3</sup><br><br>$$\Rightarrow$$ A<sup>3</sup> (A<sup>2</sup> $$-$$ B<... | mcq | jee-main-2021-online-27th-july-evening-shift | 7,072 |
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