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
values | paper_id stringclasses 154
values | __index_level_0__ int64 2 13.4k |
|---|---|---|---|---|---|---|---|---|---|---|---|
1ktd4vbhk | maths | matrices-and-determinants | properties-of-determinants | Let A be a 3 $$\times$$ 3 real matrix. If det(2Adj(2 Adj(Adj(2A)))) = 2<sup>41</sup>, then the value of det(A<sup>2</sup>) equal __________. | [] | null | 4 | adj (2A) = 2<sup>2</sup> adjA<br><br>$$\Rightarrow$$ adj(adj (2A)) = adj(4 adjA) = 16 adj (adj A)<br><br>= 16 | A | A<br><br>$$\Rightarrow$$ adj (32 | A | A) = (32 | A |)<sup>2</sup> adj A<br><br>12(32| A |)<sup>2</sup> |adj A | = 2<sup>3</sup> (32 | A |)<sup>6</sup> | adj A |<br><br>2<sup>3</sup> . 2<sup>30</sup> | A ... | integer | jee-main-2021-online-26th-august-evening-shift | 7,073 |
1ktg05qbp | maths | matrices-and-determinants | properties-of-determinants | <sub></sub>Let A(a, 0), B(b, 2b + 1) and C(0, b), b $$\ne$$ 0, |b| $$\ne$$ 1, be points such that the area of triangle ABC is 1 sq. unit, then the sum of all possible values of a is : | [{"identifier": "A", "content": "$${{ - 2b} \\over {b + 1}}$$"}, {"identifier": "B", "content": "$${{2b} \\over {b + 1}}$$"}, {"identifier": "C", "content": "$${{2{b^2}} \\over {b + 1}}$$"}, {"identifier": "D", "content": "$${{ - 2{b^2}} \\over {b + 1}}$$"}] | ["D"] | null | $$\left| {{1 \over 2}\left| {\matrix{
a & 0 & 1 \cr
b & {2b + 1} & 1 \cr
0 & b & 1 \cr
} } \right|} \right| = 1$$<br><br>$$ \Rightarrow \left| {\matrix{
a & 0 & 1 \cr
b & {2b + 1} & 1 \cr
0 & b & 1 \cr
} } \right| = \pm \,2$$<br><br>$$ \... | mcq | jee-main-2021-online-27th-august-evening-shift | 7,074 |
1l5667sum | maths | matrices-and-determinants | properties-of-determinants | <p>Let A be a matrix of order 3 $$\times$$ 3 and det (A) = 2. Then det (det (A) adj (5 adj (A<sup>3</sup>))) is equal to _____________.</p> | [{"identifier": "A", "content": "512 $$\\times$$ 10<sup>6</sup>"}, {"identifier": "B", "content": "256 $$\\times$$ 10<sup>6</sup>"}, {"identifier": "C", "content": "1024 $$\\times$$ 10<sup>6</sup>"}, {"identifier": "D", "content": "256 $$\\times$$ 10<sup>11</sup>"}] | ["A"] | null | <p>$$|A| = 2$$</p>
<p>$$||A| = adj\,(5\,adj\,{A^3})|$$</p>
<p>$$ = |25|A|adj\,(adj\,{A^3})|$$</p>
<p>$$ = {25^3}|A{|^3}\,.\,|adj\,{A^3}{|^2}$$</p>
<p>$$ = {25^3}\,.\,{2^3}\,.\,|{A^3}{|^4}$$</p>
<p>$$ = {25^3}\,.\,{2^3}\,.\,{2^{12}} = {10^6}\,.\,512$$</p> | mcq | jee-main-2022-online-28th-june-morning-shift | 7,076 |
1l56q3mwj | maths | matrices-and-determinants | properties-of-determinants | <p>Let $$f(x) = \left| {\matrix{
a & { - 1} & 0 \cr
{ax} & a & { - 1} \cr
{a{x^2}} & {ax} & a \cr
} } \right|,\,a \in R$$. Then the sum of the squares of all the values of a, for which $$2f'(10) - f'(5) + 100 = 0$$, is</p> | [{"identifier": "A", "content": "117"}, {"identifier": "B", "content": "106"}, {"identifier": "C", "content": "125"}, {"identifier": "D", "content": "136"}] | ["C"] | null | <p>$$f(x) = \left| {\matrix{
a & { - 1} & 0 \cr
{ax} & a & { - 1} \cr
{a{x^2}} & {ax} & a \cr
} } \right|,\,a \in R$$</p>
<p>$$f(x) = a({a^2} + ax) + 1({a^2}x + a{x^2})$$</p>
<p>$$ = a{(x + a)^2}$$</p>
<p>$$f'(x) = 2a(x + a)$$</p>
<p>Now, $$2f'(10) - f'(5) + 100 = 0$$</p>
<p>$$ \Rightarrow 2.\,2a(10 + ... | mcq | jee-main-2022-online-27th-june-evening-shift | 7,077 |
1l56q60a3 | maths | matrices-and-determinants | properties-of-determinants | <p>Let A and B be two 3 $$\times$$ 3 matrices such that $$AB = I$$ and $$|A| = {1 \over 8}$$. Then $$|adj\,(B\,adj(2A))|$$ is equal to</p> | [{"identifier": "A", "content": "16"}, {"identifier": "B", "content": "32"}, {"identifier": "C", "content": "64"}, {"identifier": "D", "content": "128"}] | ["C"] | null | <p>A and B are two matrices of order 3 $$\times$$ 3.</p>
<p>and $$AB = I$$,</p>
<p>$$|A| = {1 \over 8}$$</p>
<p>Now, $$|A||B| = 1$$</p>
<p>$$|B| = 8$$</p>
<p>$$\therefore$$ $$|adj(B(adj(2A))| = |B(adj(2A)){|^2}$$</p>
<p>$$ = |B{|^2}|adj(2A){|^2}$$</p>
<p>$$ = {2^6}|2A{|^{2 \times 2}}$$</p>
<p>$$ = {2^6}.\,{2^{12}}.\,{1... | mcq | jee-main-2022-online-27th-june-evening-shift | 7,078 |
1l57p0n9c | maths | matrices-and-determinants | properties-of-determinants | <p>The positive value of the determinant of the matrix A, whose</p>
<p>Adj(Adj(A)) = $$\left( {\matrix{
{14} & {28} & { - 14} \cr
{ - 14} & {14} & {28} \cr
{28} & { - 14} & {14} \cr
} } \right)$$, is _____________.</p> | [] | null | 14 | <p>$$\left| {adj(adj(A))} \right| = {\left| A \right|^{{2^2}}} = {\left| A \right|^4}$$</p>
<p>$$\therefore$$ $${\left| A \right|^4} = \left| {\matrix{
{14} & {28} & { - 14} \cr
{ - 14} & {14} & {28} \cr
{28} & { - 14} & {14} \cr
} } \right|$$</p>
<p>$$ = {(14)^3}\left| {\matrix{
1 & 2 & { - 1} \cr... | integer | jee-main-2022-online-27th-june-morning-shift | 7,079 |
1l587f1jw | maths | matrices-and-determinants | properties-of-determinants | <p>Let A be a 3 $$\times$$ 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|<sup>2</sup> is equal to :</p> | [{"identifier": "A", "content": "6<sup>6</sup>"}, {"identifier": "B", "content": "2<sup>12</sup>"}, {"identifier": "C", "content": "2<sup>6</sup>"}, {"identifier": "D", "content": "1"}] | ["C"] | null | <p>We know, $$|adj\,A| = |A{|^{n - 1}}$$</p>
<p>Now, $$|adj\,24A| = |adj\,3(adj\,2A)|$$</p>
<p>$$ \Rightarrow |24A{|^{3 - 1}} = |3\,adj\,2A{|^{3 - 1}}$$</p>
<p>$$ \Rightarrow |24A{|^2} = |3\,adj\,2A{|^2}$$</p>
<p>Also, we know, $$|KA| = {K^n}|A|$$</p>
<p>$$ \Rightarrow {\left( {{{(24)}^2}} \right)^2}|A{|^2} = {\left( {... | mcq | jee-main-2022-online-26th-june-morning-shift | 7,080 |
1l5c14prc | maths | matrices-and-determinants | properties-of-determinants | <p>Let S = {$$\sqrt{n}$$ : 1 $$\le$$ n $$\le$$ 50 and n is odd}.</p>
<p>Let a $$\in$$ S and $$A = \left[ {\matrix{
1 & 0 & a \cr
{ - 1} & 1 & 0 \cr
{ - a} & 0 & 1 \cr
} } \right]$$.</p>
<p>If $$\sum\limits_{a\, \in \,S}^{} {\det (adj\,A) = 100\lambda } $$, then $$\lambda$$ is eq... | [{"identifier": "A", "content": "218"}, {"identifier": "B", "content": "221"}, {"identifier": "C", "content": "663"}, {"identifier": "D", "content": "1717"}] | ["B"] | null | <p>Given, $$A = {\left[ {\matrix{
1 & 0 & a \cr
{ - 1} & 1 & 0 \cr
{ - a} & 0 & 1 \cr
} } \right]_{3 \times 3}}$$</p>
<p>S = {$$\sqrt{n}$$ : 1 $$\le$$ n $$\le$$ 50 and n is odd}</p>
<p>$$ \therefore $$ S = $$\left\{ {1,\sqrt 3 ,\sqrt 5 ,\sqrt 7 ,....,\sqrt {49} } \right\}$$</p>
<p>We know,</p>
<p>$$\le... | mcq | jee-main-2022-online-24th-june-morning-shift | 7,081 |
1l6kik0ub | maths | matrices-and-determinants | properties-of-determinants | <p>Let $$A=\left(\begin{array}{rr}4 & -2 \\ \alpha & \beta\end{array}\right)$$.</p>
<p>If $$\mathrm{A}^{2}+\gamma \mathrm{A}+18 \mathrm{I}=\mathrm{O}$$, then $$\operatorname{det}(\mathrm{A})$$ is equal to _____________.</p> | [{"identifier": "A", "content": "$$-$$18"}, {"identifier": "B", "content": "18"}, {"identifier": "C", "content": "$$-$$50"}, {"identifier": "D", "content": "50"}] | ["B"] | null | <p>Characteristic equation of A is given by</p>
<p>$$\left| {A - \lambda I} \right| = 0$$</p>
<p>$$\left| {\matrix{
{4 - \lambda } & { - 2} \cr
\alpha & {\beta - \lambda } \cr
} } \right| = 0$$</p>
<p>$$ \Rightarrow {\lambda ^2} - (4 + \beta )\lambda + (4\beta + 2\alpha ) = 0$$</p>
<p>So, $${A^2} - (4 +... | mcq | jee-main-2022-online-27th-july-evening-shift | 7,084 |
1l6klber7 | maths | matrices-and-determinants | properties-of-determinants | <p>Consider a matrix $$A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta\end{array}\right]$$, where $$\alpha, \beta, \gamma$$ are three distinct natural numbers.</p>
<p>If $$\frac{\operatorname{det}(\operatorna... | [] | null | 42 | <p>$$\det (A) = \left| {\matrix{
\alpha & \beta & \gamma \cr
{{\alpha ^2}} & {{\beta ^2}} & {{\gamma ^2}} \cr
{\beta + \gamma } & {\gamma + \alpha } & {\alpha + \beta } \cr
} } \right|$$</p>
<p>$${R_3} \to {R_3} + {R_1}$$</p>
<p>$$ \Rightarrow (\alpha + \beta + \gamma )\left| {\matrix{
\alph... | integer | jee-main-2022-online-27th-july-evening-shift | 7,085 |
1l6m5y7bn | maths | matrices-and-determinants | properties-of-determinants | <p>Let the matrix $$A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]$$ and the matrix $$B_{0}=A^{49}+2 A^{98}$$. If $$B_{n}=A d j\left(B_{n-1}\right)$$ for all $$n \geq 1$$, then $$\operatorname{det}\left(B_{4}\right)$$ is equal to :</p> | [{"identifier": "A", "content": "$$3^{28}$$"}, {"identifier": "B", "content": "$$3^{30}$$"}, {"identifier": "C", "content": "$$3^{32}$$"}, {"identifier": "D", "content": "$$3^{36}$$"}] | ["C"] | null | <p>$$A = \left[ {\matrix{
0 & 1 & 0 \cr
0 & 0 & 1 \cr
1 & 0 & 0 \cr
} } \right]$$</p>
<p>$$ \Rightarrow {A^2} = \left[ {\matrix{
0 & 1 & 0 \cr
0 & 0 & 1 \cr
1 & 0 & 0 \cr
} } \right] \times \left[ {\matrix{
0 & 1 & 0 \cr
0 & 0 & 1 \cr
1 & 0 & 0 \cr
} } \right] = \left[... | mcq | jee-main-2022-online-28th-july-morning-shift | 7,086 |
1ldr7bd4v | maths | matrices-and-determinants | properties-of-determinants | <p>Let $$A=\left(\begin{array}{cc}\mathrm{m} & \mathrm{n} \\ \mathrm{p} & \mathrm{q}\end{array}\right), \mathrm{d}=|\mathrm{A}| \neq 0$$ and $$\mathrm{|A-d(A d j A)|=0}$$. Then </p> | [{"identifier": "A", "content": "$$1+\\mathrm{d}^{2}=\\mathrm{m}^{2}+\\mathrm{q}^{2}$$"}, {"identifier": "B", "content": "$$1+d^{2}=(m+q)^{2}$$"}, {"identifier": "C", "content": "$$(1+d)^{2}=m^{2}+q^{2}$$"}, {"identifier": "D", "content": "$$(1+d)^{2}=(m+q)^{2}$$"}] | ["D"] | null | <p>$$\left| {A - d\left( {\matrix{
q & { - n} \cr
{ - p} & m \cr
} } \right)} \right| = 0$$</p>
<p>$$\left| {\matrix{
{m - qd} & {n(1 + d)} \cr
{p(1 + d)} & {q - md} \cr
} } \right| = 0$$</p>
<p>$$(m - qd)(q - md) = np{(1 + d)^2}$$</p>
<p>$$mq - ({q^2} + {m^2})d + qm{d^2} = np(1 + {d^2}) + 2npd$... | mcq | jee-main-2023-online-30th-january-morning-shift | 7,089 |
1ldv1q57x | maths | matrices-and-determinants | properties-of-determinants | <p>Let $$x,y,z > 1$$ and $$A = \left[ {\matrix{
1 & {{{\log }_x}y} & {{{\log }_x}z} \cr
{{{\log }_y}x} & 2 & {{{\log }_y}z} \cr
{{{\log }_z}x} & {{{\log }_z}y} & 3 \cr
} } \right]$$. Then $$\mathrm{|adj~(adj~A^2)|}$$ is equal to</p> | [{"identifier": "A", "content": "$$6^4$$"}, {"identifier": "B", "content": "$$2^8$$"}, {"identifier": "C", "content": "$$4^8$$"}, {"identifier": "D", "content": "$$2^4$$"}] | ["B"] | null | $$
\begin{aligned}
& |A|=\frac{1}{\log x \log y \log z}\left|\begin{array}{ccc}
\log x & \log y & \log z \\
\log x & 2 \log y & \log z \\
\log x & \log y & 3 \log z
\end{array}\right|=\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & 2 & 1 \\
1 & 1 & 3
\end{array}\right|=2 \\\\
& \Rightarrow\left|\operatorname{adj}\left(\opera... | mcq | jee-main-2023-online-25th-january-morning-shift | 7,091 |
1ldww8dwi | maths | matrices-and-determinants | properties-of-determinants | <p>Let A be a 3 $$\times$$ 3 matrix such that $$\mathrm{|adj(adj(adj~A))|=12^4}$$. Then $$\mathrm{|A^{-1}~adj~A|}$$ is equal to</p> | [{"identifier": "A", "content": "12"}, {"identifier": "B", "content": "2$$\\sqrt3$$"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "$$\\sqrt6$$"}] | ["B"] | null | $|A|^{(n-1)^{3}}=12^{4}$
<br/><br/>
$$
\begin{aligned}
&|A|^{8}=12^{4} \\\\
&|A|=\sqrt{12} \\\\
&\left|A^{-1} \operatorname{adj} A\right|=\left|A^{-1}\right| \cdot|A|^{2} \\\\
&=|A| = 2\sqrt3
\end{aligned}
$$ | mcq | jee-main-2023-online-24th-january-evening-shift | 7,092 |
lgnxwdvt | maths | matrices-and-determinants | properties-of-determinants | Let the determinant of a square matrix A of order $m$ be $m-n$, where $m$ and $n$<br/><br/> satisfy $4 m+n=22$ and $17 m+4 n=93$.<br/><br/> If $\operatorname{det}(n \operatorname{adj}(\operatorname{adj}(m A)))=3^{a} 5^{b} 6^{c}$ then $a+b+c$ is equal to : | [{"identifier": "A", "content": "96"}, {"identifier": "B", "content": "84"}, {"identifier": "C", "content": "109"}, {"identifier": "D", "content": "101"}] | ["A"] | null | Given that $|A|=m-n$, and let's solve the system of linear equations to find the values of $m$ and $n$ :
<br/><br/>$4m + n = 22$ ...... (1)
<br/><br/>$17m + 4n = 93$ ....... (2)
<br/><br/>We can multiply equation (1) by 4 to make the coefficients of $n$ in both equations equal:
<br/><br/>$16m + 4n = 88$ ......... (3... | mcq | jee-main-2023-online-15th-april-morning-shift | 7,094 |
1lgowmdng | maths | matrices-and-determinants | properties-of-determinants | <p>Let for $$A = \left[ {\matrix{
1 & 2 & 3 \cr
\alpha & 3 & 1 \cr
1 & 1 & 2 \cr
} } \right],|A| = 2$$. If $$\mathrm{|2\,adj\,(2\,adj\,(2A))| = {32^n}}$$, then $$3n + \alpha $$ is equal to</p> | [{"identifier": "A", "content": "11"}, {"identifier": "B", "content": "9"}, {"identifier": "C", "content": "12"}, {"identifier": "D", "content": "10"}] | ["A"] | null | $$
\begin{aligned}
& A=\left[\begin{array}{lll}
1 & 2 & 3 \\
\alpha & 3 & 1 \\
1 & 1 & 2
\end{array}\right] \\\\
& |A|=2
\end{aligned}
$$
<br/><br/>$$
\begin{aligned}
\Rightarrow&1(6-1)-2(2 \alpha-1)+3(\alpha-3)=2\\\\
\Rightarrow&5-4 \alpha+2+3 \alpha-9=2\\\\
\Rightarrow&-\alpha-4=0\\\\
\Rightarrow&\alpha=-4
\end{align... | mcq | jee-main-2023-online-13th-april-evening-shift | 7,095 |
1lgvqep21 | maths | matrices-and-determinants | properties-of-determinants | <p>If $$\mathrm{A}=\frac{1}{5 ! 6 ! 7 !}\left[\begin{array}{ccc}5 ! & 6 ! & 7 ! \\ 6 ! & 7 ! & 8 ! \\ 7 ! & 8 ! & 9 !\end{array}\right]$$, then $$|\operatorname{adj}(\operatorname{adj}(2 \mathrm{~A}))|$$ is equal to :</p> | [{"identifier": "A", "content": "$$2^{12}$$"}, {"identifier": "B", "content": "$$2^{20}$$"}, {"identifier": "C", "content": "$$2^{8}$$"}, {"identifier": "D", "content": "$$2^{16}$$"}] | ["D"] | null | Given that
<br/><br/>$$
\begin{aligned}
& A=\frac{1}{5 ! 6 ! 7 !}\left[\begin{array}{lll}
5 ! & 6 ! & 7 ! \\
6 ! & 7 ! & 8 ! \\
7 ! & 8 ! & 9 !
\end{array}\right] \\\\
& \Rightarrow|A|=\frac{1}{5 ! 6 ! 7 !}\left|\begin{array}{lll}
5 ! & 6 ! & 7 ! \\
6 ! & 7 ! & 8 ! \\
7 ! & 8 ! & 9 !
\end{array}\right| \\\\
& \Rightarr... | mcq | jee-main-2023-online-10th-april-evening-shift | 7,096 |
1lgxt8e8l | maths | matrices-and-determinants | properties-of-determinants | <p>If A is a 3 $$\times$$ 3 matrix and $$|A| = 2$$, then $$|3\,adj\,(|3A|{A^2})|$$ is equal to :</p> | [{"identifier": "A", "content": "$${3^{12}}\\,.\\,{6^{10}}$$"}, {"identifier": "B", "content": "$${3^{11}}\\,.\\,{6^{10}}$$"}, {"identifier": "C", "content": "$${3^{12}}\\,.\\,{6^{11}}$$"}, {"identifier": "D", "content": "$${3^{10}}\\,.\\,{6^{11}}$$"}] | ["B"] | null | Given that $A$ is $3 \times 3$ matrix and $|A|=2$
<br/><br/>$$
\begin{aligned}
& \text { Now, | 3adj }\left(|3 A| A^2\right) \text { | } \\\\
& =3^3\left|\operatorname{adj}\left(|3 A| A^2\right)\right| \\\\
& =3^3\left|\operatorname{adj}\left(54 A^2\right)\right| \\\\
& =3^3\left|54 A^2\right|^2 \\\\
& =3^3 \times\left... | mcq | jee-main-2023-online-10th-april-morning-shift | 7,097 |
lv0vxdmu | maths | matrices-and-determinants | properties-of-determinants | <p>Let $$A$$ be a $$3 \times 3$$ matrix of non-negative real elements such that $$A\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=3\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$$. Then the maximum value of $$\operatorname{det}(\mathrm{A})$$ is _________.</p> | [] | null | 27 | <p>Let $$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]$$</p>
<p>Now</p>
<p>$$A\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right]=3\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right]$$</p>
... | integer | jee-main-2024-online-4th-april-morning-shift | 7,098 |
lv3ve470 | maths | matrices-and-determinants | properties-of-determinants | <p>If $$\alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c}$$ and $$\left|\begin{array}{lll}\alpha & \mathrm{b} & \mathrm{c} \\ \mathrm{a} & \beta & \mathrm{c} \\ \mathrm{a} & \mathrm{b} & \gamma\end{array}\right|=0$$, then $$\frac{\mathrm{a}}{\alpha-\mathrm{a}}+\frac{\mathrm{b... | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "0"}] | ["D"] | null | <p>$$\left|\begin{array}{lll}
\alpha & b & c \\
a & \beta & c \\
a & b & \gamma
\end{array}\right|=0$$</p>
<p>$$\begin{aligned}
& R_1 \rightarrow R_1-R_2, R_2 \rightarrow R_2-R_3 \\
& \Rightarrow\left|\begin{array}{ccc}
\alpha-a & b-\beta & 0 \\
0 & \beta-b & c-\gamma \\
a & b & \gamma
\end{array}\right|=0
\end{aligned... | mcq | jee-main-2024-online-8th-april-evening-shift | 7,099 |
lv7v3k3w | maths | matrices-and-determinants | properties-of-determinants | <p>Let A and B be two square matrices of order 3 such that $$\mathrm{|A|=3}$$ and $$\mathrm{|B|=2}$$. Then $$|\mathrm{A}^{\mathrm{T}} \mathrm{A}(\operatorname{adj}(2 \mathrm{~A}))^{-1}(\operatorname{adj}(4 \mathrm{~B}))(\operatorname{adj}(\mathrm{AB}))^{-1} \mathrm{AA}^{\mathrm{T}}|$$ is equal to :</p> | [{"identifier": "A", "content": "32"}, {"identifier": "B", "content": "81"}, {"identifier": "C", "content": "64"}, {"identifier": "D", "content": "108"}] | ["C"] | null | <p>$$\begin{aligned}
& |A|=3 \\
& |B|=2 \\
& \left.\left|A^T\right||A| \mid(\operatorname{adj}(2 A))^{-1}\|\operatorname{adj}(4 B)\|(\operatorname{adj}(A B))^{-1}\right)|A|\left|A^T\right| \\
& 3 \cdot 3 \frac{1}{64 \cdot 9}(64)^2 \cdot 4 \cdot \frac{1}{9 \cdot 4} 3 \cdot 3 \\
& =64
\end{aligned}$$</p> | mcq | jee-main-2024-online-5th-april-morning-shift | 7,100 |
jlh0quVNcdwOiEkm | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | If the system of linear equations
<br/>$$x + 2ay + az = 0;$$ $$x + 3by + bz = 0;\,\,x + 4cy + cz = 0;$$
<br/>has a non - zero solution, then $$a, b, c$$. | [{"identifier": "A", "content": "satisfy $$a+2b+3c=0$$"}, {"identifier": "B", "content": "are in A.P"}, {"identifier": "C", "content": "are in G.P"}, {"identifier": "D", "content": "are in H.P."}] | ["D"] | null | For homogeneous system of equations to have non zero solution, $$\Delta = 0$$
<br><br>$$\left| {\matrix{
1 & {2a} & a \cr
1 & {3b} & b \cr
1 & {4c} & c \cr
} } \right| = 0\,{C_2} \to {C_2} - 2{C_3}$$
<br><br>$$\left| {\matrix{
1 & 0 & a \cr
1 & b & b \c... | mcq | aieee-2003 | 7,101 |
58rNudkmSRYVl7JA | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | The system of equations
<br/><p>$$\matrix{
{\alpha \,x + y + z = \alpha - 1} \cr
{x + \alpha y + z = \alpha - 1} \cr
{x + y + \alpha \,z = \alpha - 1} \cr
} $$</p>
<p>has no solutions, if $$\alpha $$ is :</p> | [{"identifier": "A", "content": "$$-2$$ "}, {"identifier": "B", "content": "either $$-2$$ or $$1$$ "}, {"identifier": "C", "content": "not $$-2$$ "}, {"identifier": "D", "content": "$$1$$"}] | ["A"] | null | $$ax + y + z = \alpha - 1$$
<br><br>$$x + \alpha \,y + z = \alpha - 1;$$
<br><br>$$x + y + z\alpha = \alpha - 1$$
<br><br>$$\Delta = \left| {\matrix{
\alpha & 1 & 1 \cr
1 & \alpha & 1 \cr
1 & 1 & \alpha \cr
} } \right|$$
<br><br>$$ = \alpha \left( {{\alpha ^2} - 1} \righ... | mcq | aieee-2005 | 7,102 |
xbDxrjEriuFOr9Rq | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | Consider the system of linear equations;
$$$\matrix{
{{x_1} + 2{x_2} + {x_3} = 3} \cr
{2{x_1} + 3{x_2} + {x_3} = 3} \cr
{3{x_1} + 5{x_2} + 2{x_3} = 1} \cr
} $$$
<br/>The system has : | [{"identifier": "A", "content": "exactly $$3$$ solutions "}, {"identifier": "B", "content": "a unique solution "}, {"identifier": "C", "content": "no solution "}, {"identifier": "D", "content": "infinitenumber of solutions "}] | ["C"] | null | $$D = \left| {\matrix{
1 & 2 & 1 \cr
2 & 3 & 1 \cr
3 & 5 & 2 \cr
} } \right| = 0$$
<br><br>$${D_1}\left| {\matrix{
3 & 2 & 1 \cr
3 & 3 & 1 \cr
1 & 5 & 2 \cr
} } \right| \ne 0$$
<br><br>$$ \Rightarrow $$ Given system, does not have any sol... | mcq | aieee-2010 | 7,104 |
XGQveL8rrGXCZ8eT | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | The number of values of $$k$$ for which the linear equations
<br/>$$4x + ky + 2z = 0,kx + 4y + z = 0$$ and $$2x+2y+z=0$$ possess a non-zero solution is : | [{"identifier": "A", "content": "$$2$$ "}, {"identifier": "B", "content": "$$1$$ "}, {"identifier": "C", "content": "zero"}, {"identifier": "D", "content": "$$3$$ "}] | ["A"] | null | $$\Delta = 0 \Rightarrow \left| {\matrix{
4 & k & 2 \cr
k & 4 & 1 \cr
2 & 2 & 1 \cr
} } \right| = 0$$
<br><br>$$ \Rightarrow 4\left( {4 - 2} \right) - k\left( {k - 2} \right) + $$
<br><br>$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\left( {2k - 8} \right) = 0$$
<br><br>$$ \Right... | mcq | aieee-2011 | 7,105 |
DNXywk8s38SOeEaA | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | The number of values of $$k$$, for which the system of equations : $$$\matrix{
{\left( {k + 1} \right)x + 8y = 4k} \cr
{kx + \left( {k + 3} \right)y = 3k - 1} \cr
} $$$
<br/>has no solution, is <br/> | [{"identifier": "A", "content": "infinite "}, {"identifier": "B", "content": "1 "}, {"identifier": "C", "content": "2 "}, {"identifier": "D", "content": "3"}] | ["B"] | null | From the given system, we have
<br><br>$${{k + 1} \over k} = {8 \over {k + 3}} \ne {{4k} \over {3k - 1}}$$
<br><br>( as System has no solution)
<br><br>$$ \Rightarrow {k^2} + 4k + 3 = 8k$$
<br><br>$$ \Rightarrow k = 1,3$$
<br><br>If $$k = 1$$ then $${8 \over {1 + 3}} \ne {{4.1} \over 2}$$ which is false
<br><br>And if... | mcq | jee-main-2013-offline | 7,106 |
brxKL055e9VwbMXT | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | The set of all values of $$\lambda $$ for which the system of linear equations:<br/><br/>
$$\matrix{
{2{x_1} - 2{x_2} + {x_3} = \lambda {x_1}} \cr
{2{x_1} - 3{x_2} + 2{x_3} = \lambda {x_2}} \cr
{ - {x_1} + 2{x_2} = \lambda {x_3}} \cr
} $$<br/><br/>
has a non-trivial solution | [{"identifier": "A", "content": "contains two elements "}, {"identifier": "B", "content": "contains more than two elements "}, {"identifier": "C", "content": "in an empty set "}, {"identifier": "D", "content": "is a singleton"}] | ["A"] | null | $$\left. {\matrix{
{2{x_1} - 2{x_2} + {x^3} = \lambda {x_1}} \cr
{2{x_1} - 3{x_2} + 2{x_3} = \lambda {x_2}} \cr
{\,\,\,\,\,\,\,\,\,\, - {x_1} + 2{x_2} = \lambda {x_3}} \cr
} } \right\}$$
<br><br>$$\eqalign{
& \Rightarrow \,\,\,\,\,\,\,\left( {2 - \lambda } \right){x_1} - 2{x_2} + {x_3} = 0 \cr ... | mcq | jee-main-2015-offline | 7,107 |
cZjTe4dWg3qrDPLQ | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>The system of linear equations </p>
<p>$$\matrix{
{x + \lambda y - z = 0} \cr
{\lambda x - y - z = 0} \cr
{x + y - \lambda z = 0} \cr
} $$ </p>
has a non-trivial solution for : | [{"identifier": "A", "content": "infinitely many values of $$\\lambda .$$ "}, {"identifier": "B", "content": "exactly one value of $$\\lambda .$$ "}, {"identifier": "C", "content": "exactly two values of $$\\lambda .$$ "}, {"identifier": "D", "content": "exactly three values of $$\\lambda .$$ "}] | ["D"] | null | <p>For non-trivial solution, we have</p>
<p>$$\left| {\matrix{
1 & \lambda & { - 1} \cr
\lambda & { - 1} & { - 1} \cr
1 & 1 & { - \lambda } \cr
} } \right| = 0$$</p>
<p>$$ \Rightarrow 1(\lambda + 1) - \lambda ( - {\lambda ^2} + 1) - 1(\lambda + 1) = 0$$</p>
<p>$$ \Rightarrow \lambda ({\lambda ^2} -... | mcq | jee-main-2016-offline | 7,108 |
1JH2i525goy0oScG | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | If S is the set of distinct values of 'b' for which the following system of linear equations
<br/><br/>x + y + z = 1
<br/>x + ay + z = 1
<br/>ax + by + z = 0
<br/><br/>has no solution, then S is : | [{"identifier": "A", "content": "an empty set"}, {"identifier": "B", "content": "an infinite set"}, {"identifier": "C", "content": "a finite set containing two or more elements"}, {"identifier": "D", "content": "a singleton "}] | ["D"] | null | $$\left| {\matrix{
1 & 1 & 1 \cr
1 & a & 1 \cr
a & b & 1 \cr
} } \right| = 0$$
<br><br>$$ \Rightarrow $$ 1 [a – b] – 1 [1 – a] + 1 [b – a<sup>2</sup>] = 0
<br><br>$$ \Rightarrow $$ (a - 1)<sup>2</sup> = 0
<br><br>$$ \Rightarrow $$ a = 1
<br><br>For a = 1, the equations become
<b... | mcq | jee-main-2017-offline | 7,109 |
xcycrqfea8aJoM29XdHeA | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | The number of real values of $$\lambda $$ for which the system of linear equations
<br/><br/>2x + 4y $$-$$ $$\lambda $$z = 0
<br/><br/>4x + $$\lambda $$y + 2z = 0
<br/><br/>$$\lambda $$x + 2y + 2z = 0
<br/><br/>has infinitely many solutions, is : | [{"identifier": "A", "content": "0"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "3"}] | ["B"] | null | <p>The system of equations can be written in the matrix form as</p>
<p>$$\left[ {\matrix{
2 & 4 & { - \lambda } \cr
4 & \lambda & 2 \cr
\lambda & 2 & 2 \cr
} } \right]\left[ {\matrix{
x \cr
y \cr
z \cr
} } \right] = \left[ {\matrix{
0 \cr
0 \cr
0 ... | mcq | jee-main-2017-online-8th-april-morning-slot | 7,110 |
jz7BmhJjLuRJkhaJ | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | If the system of linear equations
<br/><br/>x + ky + 3z = 0
<br/>3x + ky - 2z = 0
<br/>2x + 4y - 3z = 0
<br/><br/>has a non-zero solution (x, y, z), then $${{xz} \over {{y^2}}}$$ is equal to | [{"identifier": "A", "content": "30"}, {"identifier": "B", "content": "-10"}, {"identifier": "C", "content": "10"}, {"identifier": "D", "content": "-30"}] | ["C"] | null | System of equations has non-zero solution when determinant of coefficient = 0.
<br><br>So, in this questions,
<br><br>$$\left| {\matrix{
1 & K & 3 \cr
3 & K & { - 2} \cr
2 & 4 & { - 3} \cr
} } \right| = 0$$
<br><br>$$ \Rightarrow \,\,\,\,$$ ($$-$$ 3K + 8) $$-$$ K ($$-$$9 + 4) +... | mcq | jee-main-2018-offline | 7,111 |
Kd4je4zs0Nem9nzKX70N6 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | Let S be the set of all real values of k for which the systemof linear equations
<br/>x + y + z = 2
<br/>2x + y $$-$$ z = 3
<br/>3x + 2y + kz = 4
<br/>has a unique solution. Then S is : | [{"identifier": "A", "content": "an empty set "}, {"identifier": "B", "content": "equal to {0}"}, {"identifier": "C", "content": "equal to <b>R</b>"}, {"identifier": "D", "content": "equal to <b>R</b> $$-$$ {0}"}] | ["D"] | null | As system of linear equations have unique solutions so, determinant of coefficient $$ \ne $$ 0<br><br>
$$ \therefore $$ $$\left| {\matrix{
1 & 1 & 1 \cr
2 & 1 & { - 1} \cr
3 & 2 & k \cr
} } \right|$$ $$ \ne $$ 0<br><br>
$$ \Rightarrow $$ k + 2 - (2k + 3) + 1 $$ \ne $$ 0<br><br>
... | mcq | jee-main-2018-online-15th-april-morning-slot | 7,112 |
MtVekwWI8fkPYsMPN0LRs | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | The number of values of k for which the system of linear equations,
<br/>(k + 2)x + 10y = k
<br/>kx + (k +3)y = k -1
<br/>has <b>no solution,</b> is : | [{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "infinitely many"}] | ["A"] | null | System of linear equation have no solution,
<br><br>$$\therefore\,\,\,$$ determinant of coefficient = 0
<br><br>$$\left| {\matrix{
{k + 2} & {10} \cr
k & {k + 3} \cr
} } \right| = 0$$
<br><br>$$ \Rightarrow $$ $$\,\,\,\,$$ (k + 2) (k + 3) $$-$$ 10 K = 0
<br><br>$$ \Rightarrow $$ $$\,\,\,\,$$ k<sup... | mcq | jee-main-2018-online-16th-april-morning-slot | 7,114 |
1wm32XZzhcGEpKn931wYK | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | An ordered pair ($$\alpha $$, $$\beta $$) for which the system of linear equations
<br/>(1 + $$\alpha $$) x + $$\beta $$y + z = 2
<br/>$$\alpha $$x + (1 + $$\beta $$)y + z = 3
<br/>$$\alpha $$x + $$\beta $$y + 2z = 2
<br/>has a unique solution, is : | [{"identifier": "A", "content": "(\u20133, 1) "}, {"identifier": "B", "content": "(1, \u20133) "}, {"identifier": "C", "content": "(\u20134, 2) "}, {"identifier": "D", "content": "(2, 4) "}] | ["D"] | null | For unique solution
<br><br>$$\Delta $$ $$ \ne $$ 0 $$ \Rightarrow $$ $$\left| {\matrix{
{1 + \alpha } & \beta & 1 \cr
\alpha & {1 + \beta } & 1 \cr
\alpha & \beta & 2 \cr
} } \right| \ne 0$$
<br><br>$$\left| {\matrix{
1 & { - 1} & 0 \cr
0 & 1 & { -... | mcq | jee-main-2019-online-12th-january-morning-slot | 7,115 |
HMsP8Zu83OwHy7SXri18hoxe66ijvwu7iyt | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | If the system of equations 2x + 3y – z = 0, x + ky
– 2z = 0 and 2x – y + z = 0 has a non-trival solution
(x, y, z), then $${x \over y} + {y \over z} + {z \over x} + k$$
is equal to :- | [{"identifier": "A", "content": "-4"}, {"identifier": "B", "content": "$${3 \\over 4}$$"}, {"identifier": "C", "content": "$${1 \\over 2}$$"}, {"identifier": "D", "content": "$$-{1 \\over 4}$$"}] | ["C"] | null | Given 2x + 3y – z = 0,
<br><br>x + ky – 2z = 0
<br><br>2x – y + z = 0
<br><br>For non trivial solution
<br><br>$$\Delta = 0 \Rightarrow \left| {\matrix{
2 & 3 & { - 1} \cr
1 & k & { - 2} \cr
2 & { - 1} & 1 \cr
} } \right| = 0$$<br><br>
$$ \Rightarrow k = {9 \over 2}$$<br><br>
... | mcq | jee-main-2019-online-9th-april-evening-slot | 7,118 |
V1GwSMBgU8TtzC3pJZMkz | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | The greatest value of c $$ \in $$ R for which the system
of linear equations<br/>
x – cy – cz = 0<br/>
cx – y + cz = 0<br/>
cx + cy – z = 0<br/>
has a non-trivial solution, is : | [{"identifier": "A", "content": "-1"}, {"identifier": "B", "content": "0"}, {"identifier": "C", "content": "1/2"}, {"identifier": "D", "content": "2"}] | ["C"] | null | If the system of equations has non-trivial
solutions, then
<br><br>D = 0
<br><br>$$\left| {\matrix{
1 & { - c} & { - c} \cr
c & { - 1} & c \cr
c & c & { - 1} \cr
} } \right| = 0$$
<br><br>$$ \Rightarrow $$ (1 - c<sup>2</sup>) + c(-c - c<sup>2</sup>) - c(c<sup>2</sup> + c) = 0
<b... | mcq | jee-main-2019-online-8th-april-morning-slot | 7,119 |
z0Ep8YB1pu7YM0a4MG1hU | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | The set of all values of $$\lambda $$ for which the system of linear equations
<br/>x – 2y – 2z = $$\lambda $$x
<br/>x + 2y + z = $$\lambda $$y
<br/>– x – y = $$\lambda $$z
<br/>has a non-trivial solutions : | [{"identifier": "A", "content": "is an empty set"}, {"identifier": "B", "content": "contains more than two elements"}, {"identifier": "C", "content": "is a singleton"}, {"identifier": "D", "content": "contains exactly two elements\n"}] | ["C"] | null | $$\left| {\matrix{
{\lambda - 1} & 2 & 2 \cr
1 & {2 - \lambda } & 1 \cr
1 & 1 & 1 \cr
} } \right| = 0$$
<br><br>$$ \Rightarrow {\left( {\lambda - 1} \right)^3} = 0 \Rightarrow \lambda = 1$$ | mcq | jee-main-2019-online-12th-january-evening-slot | 7,120 |
SQdlZGDAX2v9SsT2dj0Rf | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | If the system of linear equations
<br/>2x + 2y + 3z = a
<br/>3x – y + 5z = b
<br/>x – 3y + 2z = c
<br/>where a, b, c are non zero real numbers, has more one solution, then : | [{"identifier": "A", "content": "b \u2013 c \u2013 a = 0"}, {"identifier": "B", "content": "a + b + c = 0"}, {"identifier": "C", "content": "b \u2013 c + a = 0"}, {"identifier": "D", "content": "b + c \u2013 a = 0"}] | ["A"] | null | P<sub>1</sub> : 2x + 2y + 3z = a
<br><br>P<sub>2</sub> : 3x $$-$$ y + 5z = b
<br><br>P<sub>3</sub> : x $$-$$ 3y + 2z = c
<br><br>We find
<br><br>P<sub>1</sub> + P<sub>3</sub> = P<sub>2</sub> $$ \Rightarrow $$ a + c = b
| mcq | jee-main-2019-online-11th-january-morning-slot | 7,121 |
MZzFH39aOy3rUWdoZ77Jw | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | The number of values of $$\theta $$ $$ \in $$ (0, $$\pi $$) for which the system of linear equations
<br/><br/>x + 3y + 7z = 0
<br/><br/>$$-$$ x + 4y + 7z = 0
<br/><br/>(sin3$$\theta $$)x + (cos2$$\theta $$)y + 2z = 0.
<br/><br/>has a non-trival solution, is - | [{"identifier": "A", "content": "two"}, {"identifier": "B", "content": "one"}, {"identifier": "C", "content": "four"}, {"identifier": "D", "content": "three"}] | ["A"] | null | $$\left| {\matrix{
1 & 3 & 7 \cr
{ - 1} & 4 & 7 \cr
{\sin 3\theta } & {\cos 2\theta } & 2 \cr
} } \right| = 0$$
<br><br>(8 $$-$$ 7 cos 2$$\theta $$) $$-$$ 3($$-$$2 $$-$$ 7 sin 3$$\theta $$)
<br><br> +7 ($$-$$ cos 2$$\theta $$ $$-$$ 4 sin 3$$\th... | mcq | jee-main-2019-online-10th-january-evening-slot | 7,122 |
mpfwNPbYtgolDevb4dZRy | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | If the system of equations
<br/><br/>x + y + z = 5
<br/><br/>x + 2y + 3z = 9
<br/><br/>x + 3y + az = $$\beta $$
<br/><br/>has infinitely many solutions, then $$\beta $$ $$-$$ $$\alpha $$ equals - | [{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "21"}, {"identifier": "C", "content": "18"}, {"identifier": "D", "content": "5"}] | ["A"] | null | $$D = \left| {\matrix{
1 & 1 & 1 \cr
1 & 2 & 3 \cr
1 & 3 & \alpha \cr
} } \right| = \left| {\matrix{
1 & 1 & 1 \cr
0 & 1 & 2 \cr
0 & 2 & {\alpha - 1} \cr
} } \right|$$
<br><br> $$ = \left( {\alpha - 1... | mcq | jee-main-2019-online-10th-january-morning-slot | 7,123 |
HeOmh2hQUK6GpkFIn8YAj | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | If the system of linear equations
<br/>x $$-$$ 4y + 7z = g
<br/> 3y $$-$$ 5z = h
<br/>$$-$$2x + 5y $$-$$ 9z = k
<br/>is consistent, then : | [{"identifier": "A", "content": "g + 2h + k = 0"}, {"identifier": "B", "content": "g + h + 2k = 0"}, {"identifier": "C", "content": "2g + h + k = 0"}, {"identifier": "D", "content": "g + h + k = 0"}] | ["C"] | null | x $$-$$ $$4y + 7z = g$$
<br> $$3y$$ $$-$$ $$5z = h$$
<br>$$-$$$$2x + 5y$$ $$-$$ $$9z = k$$
<br><br>$$D = \left| {\matrix{
1 & { - 4} & 7 \cr
0 & 3 & { - 5} \cr
{ - 2} & 5 & { - 9} \cr
} } \right|$$
<br><br>$$D = 1\left( { - 27 + 25} \right) - 2... | mcq | jee-main-2019-online-9th-january-evening-slot | 7,124 |
MpyEVVbwxRUBTKEHiR8Ao | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | The system of linear equations
<br/> x + y + z = 2
<br/>2x + 3y + 2z = 5
<br/>2x + 3y + (a<sup>2</sup> – 1) z = a + 1 then
| [{"identifier": "A", "content": "has infinitely many solutions for a = 4 "}, {"identifier": "B", "content": "has a unique solution for |a| = $$\\sqrt3$$"}, {"identifier": "C", "content": "is inconsistent when |a| = $$\\sqrt3$$"}, {"identifier": "D", "content": "is inconsistent when a = 4"}] | ["C"] | null | $$D = \left| {\matrix{
1 & 1 & 1 \cr
2 & 3 & 2 \cr
2 & 3 & {{\alpha ^2} - 1} \cr
} } \right|$$
<br><br>D = 3$$a$$<sup>2</sup> $$-$$ 3 $$-$$ 6 $$-$$ 2$$a$$<sup>2</sup> + 2 + 4 + 2$$a$$<sup>2</sup> $$-$$ 2 $$-$$ 4
<br><br>D = ($$a$$<sup>2</sup> $$-$$ 3)
<br><br>When D $$ \ne $$ 0 ... | mcq | jee-main-2019-online-9th-january-morning-slot | 7,125 |
dWw9wkzgDfk71NagXfjgy2xukf8zx58k | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | If the system of equations<br/>
x - 2y + 3z = 9<br/>
2x + y + z = b<br/>
x - 7y + az = 24, <br/>has infinitely many solutions, then a - b is equal to......... | [] | null | 5 | D = 0<br><br>$$\left| {\matrix{
1 & { - 2} & 3 \cr
2 & 1 & 1 \cr
1 & { - 7} & a \cr
} } \right| = 0$$<br><br>$$1(a + 7) + 2(2a - 1) + 3( - 14 - 1) = 0$$<br><br>$$a + 7 + 4a - 2 - 45 = 0$$<br><br>$$5a = 40$$<br><br>$$a = 8$$<br><br>$${D_1} = \left| {\matrix{
9 & { - 2} &am... | integer | jee-main-2020-online-4th-september-morning-slot | 7,133 |
sSAYfOA3tsGReJtLHNjgy2xukezelwq7 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | Let A = {X = (x, y, z)<sup>T</sup>: PX = 0 and
<br/><br/>x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = 1} where
<br/><br/>$$P = \left[ {\matrix{
1 & 2 & 1 \cr
{ - 2} & 3 & { - 4} \cr
1 & 9 & { - 1} \cr
} } \right]$$,
<br/><br/>then the set A : | [{"identifier": "A", "content": "is an empty set.\n"}, {"identifier": "B", "content": "contains more than two elements."}, {"identifier": "C", "content": "contains exactly two elements."}, {"identifier": "D", "content": "is a singleton."}] | ["C"] | null | Let $$X = \left[ {\matrix{
x \cr
y \cr
z \cr
} } \right]$$<br><br>
PX = O<br><br>
$$\left[ {\matrix{
1 & 2 & 1 \cr
{ - 2} & 3 & { - 4} \cr
1 & 9 & { - 1} \cr
} } \right]\left[ {\matrix{
x \cr
y \cr
z \cr
} } \right] = \left[ {\matrix{
0 \cr ... | mcq | jee-main-2020-online-2nd-september-evening-slot | 7,134 |
xYS1FKI11Dse7mNmPOjgy2xukewnbkm1 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | Let S be the set of all $$\lambda $$ $$ \in $$ R for which the system
of linear equations
<br/><br/>2x – y + 2z = 2
<br/>x – 2y +
$$\lambda $$z = –4
<br/>x +
$$\lambda $$y + z = 4
<br/><br/>has no solution. Then the set S : | [{"identifier": "A", "content": "contains more than two elements."}, {"identifier": "B", "content": "contains exactly two elements."}, {"identifier": "C", "content": "is a singleton."}, {"identifier": "D", "content": "is an empty set."}] | ["B"] | null | For no solution :
<br><br>$$\Delta $$ = 0 and $$\Delta $$<sub>1</sub>/$$\Delta $$<sub>2</sub>/$$\Delta $$<sub>3</sub> $$ \ne $$ 0
<br><br>$$\Delta $$ = $$\left| {\matrix{
2 & { - 1} & 2 \cr
1 & { - 2} & \lambda \cr
1 & \lambda & 1 \cr
} } \right|$$ = 0
<br><br>$$ \Rightarrow $... | mcq | jee-main-2020-online-2nd-september-morning-slot | 7,135 |
NGdiQX4MhPGe9ikRQC7k9k2k5itcig0 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | If for some $$\alpha $$ and $$\beta $$ in R, the intersection of the
following three places<br/>
x + 4y – 2z = 1<br/>
x + 7y – 5z = b<br/>
x + 5y + $$\alpha $$z = 5<br/>
is a line in R<sup>3</sup>, then $$\alpha $$ + $$\beta $$ is equal to : | [{"identifier": "A", "content": "-10"}, {"identifier": "B", "content": "0"}, {"identifier": "C", "content": "10"}, {"identifier": "D", "content": "2"}] | ["C"] | null | For planes to intersect on a line there should be infinite solution of the
given system of equations.
<br><br>For infinite solutions
<br><br>$$\Delta $$ = $$\left| {\matrix{
1 & 4 & { - 2} \cr
1 & 7 & { - 5} \cr
1 & 5 & \alpha \cr
} } \right|$$ = 0
<br><br>$$ \Rightarrow $$ 1(7... | mcq | jee-main-2020-online-9th-january-morning-slot | 7,136 |
0qzGq9uNJLFafpg2SS7k9k2k5gz1yx1 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | For which of the following ordered pairs ($$\mu $$, $$\delta $$),
the system of linear equations
<br/>x + 2y + 3z = 1
<br/>3x + 4y + 5z = $$\mu $$
<br/>4x + 4y + 4z = $$\delta $$
<br/>is inconsistent ? | [{"identifier": "A", "content": "(1, 0)"}, {"identifier": "B", "content": "(4, 3)"}, {"identifier": "C", "content": "(4, 6)"}, {"identifier": "D", "content": "(3, 4)"}] | ["B"] | null | For inconsistent system we need
<br><br>$$\Delta $$ = 0 and atleast one of $$\Delta $$x, $$\Delta $$y, $$\Delta $$z $$ \ne $$ 0
<br><br>$$ \therefore $$ $$\Delta $$ = $$\left| {\matrix{
1 & 2 & 3 \cr
3 & 4 & 5 \cr
4 & 4 & 4 \cr
} } \right|$$ = 0
<br><br>$$\Delta $$<sub>x</sub> =... | mcq | jee-main-2020-online-8th-january-morning-slot | 7,138 |
Q4MVJjbJtMr0dpsrfw7k9k2k5fopwui | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | If the system of linear equations,
<br/>x + y + z = 6
<br/>x + 2y + 3z = 10
<br/>3x + 2y + $$\lambda $$z = $$\mu $$
<br/>has more than two solutions, then $$\mu $$ - $$\lambda $$<sup>2</sup>
is equal to ______. | [] | null | 13 | Given system of equation more than
2 solutions.
Hence system of equation has infinite many
solution.
<br><br>$$ \therefore $$ $$\Delta $$ = $$\Delta $$<sub>1</sub> = $$\Delta $$<sub>2</sub> = $$\Delta $$<sub>3</sub> = 0
<br><br>$$\Delta $$ = $$\left| {\matrix{
1 & 1 & 1 \cr
1 & 2 & 3 \cr
3 ... | integer | jee-main-2020-online-7th-january-evening-slot | 7,139 |
ZCZ7Gft603uNCE9sQX7k9k2k5e3cqx9 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | If the system of linear equations<br/>
2x + 2ay + az = 0<br/>
2x + 3by + bz = 0<br/>
2x + 4cy + cz = 0,<br/>
where a, b, c $$ \in $$ R are non-zero distinct; has a non-zero solution, then: | [{"identifier": "A", "content": "$${1 \\over a},{1 \\over b},{1 \\over c}$$ are in A.P. "}, {"identifier": "B", "content": "a + b + c = 0"}, {"identifier": "C", "content": "a, b, c are in G.P."}, {"identifier": "D", "content": "a,b,c are in A.P."}] | ["A"] | null | For non-zero solution
<br><br>$$\left| {\matrix{
2 & {2a} & a \cr
2 & {3b} & b \cr
2 & {4c} & c \cr
} } \right| = 0$$
<br><br>$$ \Rightarrow $$ $$\left| {\matrix{
1 & {2a} & a \cr
0 & {3b - 2a} & {b - a} \cr
0 & {4c - 2a} & {c - a} \cr
} ... | mcq | jee-main-2020-online-7th-january-morning-slot | 7,140 |
lKP9oeIa262XxKtivNjgy2xukf49oomt | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | Let S be the set of all integer solutions, (x, y, z),
of the system of equations
<br/>x – 2y + 5z = 0
<br/>–2x + 4y + z = 0
<br/>–7x + 14y + 9z = 0
<br/>such that 15 $$ \le $$ x<sup>2</sup>
+ y<sup>2</sup>
+ z<sup>2</sup> $$ \le $$ 150. Then, the
number of elements in the set S is equal to
______ .
| [] | null | 8 | $$x - 2y + 5z = 0$$ ....(1)<br><br>$$ - 2x + 4y + z = 0$$ .....(2)<br><br>$$ - 7x + 14y + 9z = 0$$ ....(3)<br><br>2.(1) + (2) we get z = 0, x = 2y<br><br>15 $$ \le $$ 4y<sup>2</sup> + y<sup>2</sup> $$ \le $$ 150<br><br>$$ \Rightarrow $$ 3 $$ \le $$ y<sup>2</sup> $$ \le $$ 30<br><br>$$y \in \left[ { - \sqrt {30} , - \sq... | integer | jee-main-2020-online-3rd-september-evening-slot | 7,141 |
C0QoIvmtVpCzopvrR11klrhinn0 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | The system of linear equations
<br/>3x - 2y - kz = 10
<br/>2x - 4y - 2z = 6
<br/>x+2y - z = 5m
<br/>is inconsistent if : | [{"identifier": "A", "content": "k $$ \\ne $$ 3, m $$ \\in $$ <b>R</b>"}, {"identifier": "B", "content": "k = 3, m $$ \\ne $$ $${4 \\over 5}$$"}, {"identifier": "C", "content": "k = 3, m $$ = $$ $${4 \\over 5}$$"}, {"identifier": "D", "content": "k $$ \\ne $$ 3, m $$ \\ne $$ $${4 \\over 5}$$"}] | ["B"] | null | $$\Delta = \left| {\matrix{
3 & { - 2} & { - k} \cr
1 & { - 4} & { - 2} \cr
1 & 2 & { - 1} \cr
} } \right| = 0$$<br><br>$$3(4 + 4) + 2( - 2 + 2) - k(4 + 4) = 0$$<br><br>$$ \Rightarrow k = 3$$<br><br>$${\Delta _x} = \left| {\matrix{
{10} & { - 2} & { - 3} \cr
6 &... | mcq | jee-main-2021-online-24th-february-morning-slot | 7,142 |
BaRKThiud28DgJukO31klrkt5yt | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | For the system of linear equations:<br/><br/>$$x - 2y = 1,x - y + kz = - 2,ky + 4z = 6,k \in R$$,<br/><br/>consider the following statements :<br/><br/>(A) The system has unique solution if $$k \ne 2,k \ne - 2$$.<br/><br/>(B) The system has unique solution if k = $$-$$2<br/><br/>(C) The system has unique solution if ... | [{"identifier": "A", "content": "(B) and (E) only"}, {"identifier": "B", "content": "(C) and (D) only"}, {"identifier": "C", "content": "(A) and (E) only"}, {"identifier": "D", "content": "(A) and (D) only"}] | ["D"] | null | $$x - 2y + 0.z = 1$$<br><br>$$x - y + kz = - 2$$<br><br>$$0.x + ky + 4z = 6$$<br><br>$$\Delta = \left| {\matrix{
1 & { - 2} & 0 \cr
1 & { - 1} & k \cr
0 & k & 4 \cr
} } \right| = 4 - {k^2}$$<br><br>For unique solution $$4 - {k^2} \ne 0$$<br><br>$$ \Rightarrow $$ k $$ \ne $$ $$... | mcq | jee-main-2021-online-24th-february-evening-slot | 7,143 |
mwbm36x3PoBzX7mXxC1kls5owix | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | If the system of equations<br/><br/>kx + y + 2z = 1<br/><br/>3x $$-$$ y $$-$$ 2z = 2<br/><br/>$$-$$2x $$-$$2y $$-$$4z = 3<br/><br/>has infinitely many solutions, then k is equal to __________. | [] | null | 21 | D = 0<br><br>$$ \Rightarrow \left| {\matrix{
k & 1 & 2 \cr
3 & { - 1} & { - 2} \cr
{ - 2} & { - 2} & { - 4} \cr
} } \right| = 0$$<br><br>$$ \Rightarrow $$ k (4 $$-$$ 4) $$-$$ 1 ($$-$$ 12 $$-$$ 4) + 2 ($$-$$ 6 $$-$$ 2)<br><br>$$ \Rightarrow $$ 16 $$-$$ 16 = 0<br><br>Also, $${D_1}... | integer | jee-main-2021-online-25th-february-morning-slot | 7,144 |
7FjcmuhVR7Bug874By1klt9gxr2 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | The following system of linear equations<br/><br/>2x + 3y + 2z = 9<br/><br/>3x + 2y + 2z = 9<br/><br/>x $$-$$ y + 4z = 8 | [{"identifier": "A", "content": "does not have any solution"}, {"identifier": "B", "content": "has a solution ($$\\alpha$$, $$\\beta$$, $$\\gamma$$) satisfying $$\\alpha$$ + $$\\beta$$<sup>2</sup> + $$\\gamma$$<sup>3</sup> = 12"}, {"identifier": "C", "content": "has a unique solution"}, {"identifier": "D", "content": "... | ["C"] | null | $$\Delta = \left| {\matrix{
2 & 3 & 2 \cr
3 & 2 & 2 \cr
1 & { - 1} & 4 \cr
} } \right| = - 20 \ne 0$$ $$ \therefore $$ unique solution<br><br>$${\Delta _x} = \left| {\matrix{
9 & 3 & 2 \cr
9 & 2 & 2 \cr
8 & { - 1} & 4 \cr
} } \right| = ... | mcq | jee-main-2021-online-25th-february-evening-slot | 7,145 |
GHIxtxm7Vb75iE0fjX1kluvwlfb | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | Consider the following system of equations :<br/><br/>x + 2y $$-$$ 3z = a<br/><br/>2x + 6y $$-$$ 11z = b<br/><br/>x $$-$$ 2y + 7z = c,<br/><br/>where a, b and c are real constants. Then the system of equations : | [{"identifier": "A", "content": "has no solution for all a, b and c"}, {"identifier": "B", "content": "has a unique solution when 5a = 2b + c"}, {"identifier": "C", "content": "has infinite number of solutions when 5a = 2b + c"}, {"identifier": "D", "content": "has a unique solution for all a, b and c"}] | ["C"] | null | $$D = \left| {\matrix{
1 & 2 & { - 3} \cr
2 & 6 & { - 11} \cr
1 & { - 2} & 7 \cr
} } \right|$$<br><br>= 20 $$-$$ 2(25) $$-$$3($$-$$10)<br><br>= 20 $$-$$ 50 + 30 = 0<br><br>$${D_1} = \left| {\matrix{
a & 2 & { - 3} \cr
b & 6 & { - 11} \cr
c & { - ... | mcq | jee-main-2021-online-26th-february-evening-slot | 7,146 |
C4InkmGv7kT3bDLA8n1kmhx159q | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | Let $$A = \left[ {\matrix{
i & { - i} \cr
{ - i} & i \cr
} } \right],i = \sqrt { - 1} $$. Then, the system of linear equations $${A^8}\left[ {\matrix{
x \cr
y \cr
} } \right] = \left[ {\matrix{
8 \cr
{64} \cr
} } \right]$$ has : | [{"identifier": "A", "content": "Exactly two solutions"}, {"identifier": "B", "content": "Infinitely many solutions"}, {"identifier": "C", "content": "A unique solution"}, {"identifier": "D", "content": "No solution"}] | ["D"] | null | $$A = \left[ {\matrix{
i & { - i} \cr
{ - i} & i \cr
} } \right]$$<br><br>$${A^2} = \left[ {\matrix{
i & { - i} \cr
{ - i} & i \cr
} } \right]\left[ {\matrix{
i & { - i} \cr
{ - i} & i \cr
} } \right] = \left[ {\matrix{
{ - 2} & 2 \cr
2 & { - 2... | mcq | jee-main-2021-online-16th-march-morning-shift | 7,147 |
852vLD8DVyX1VNm4W01kmli9o9n | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | Let $$\alpha$$, $$\beta$$, $$\gamma$$ be the real roots of the equation, x<sup>3</sup> + ax<sup>2</sup> + bx + c = 0, (a, b, c $$\in$$ R and a, b $$\ne$$ 0). If the system of equations (in u, v, w) given by $$\alpha$$u + $$\beta$$v + $$\gamma$$w = 0, $$\beta$$u + $$\gamma$$v + $$\alpha$$w = 0; $$\gamma$$u + $$\alpha$$v... | [{"identifier": "A", "content": "5"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "0"}] | ["B"] | null | x<sup>3</sup> + ax<sup>2</sup> + bx + c = 0 <br><br>Roots are $$\alpha$$, $$\beta$$, $$\gamma$$.<br><br>For non-trivial solutions,<br><br>$$\left| {\matrix{
\alpha & \beta & \gamma \cr
\beta & \gamma & \alpha \cr
\gamma & \alpha & \beta \cr
} } \right| = 0$$<br><br>$$ \... | mcq | jee-main-2021-online-18th-march-morning-shift | 7,149 |
ejJWcC5YgNq1NbYw5C1kmm2kpfg | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | Let the system of linear equations <br/><br/>4x + $$\lambda$$y + 2z = 0<br/><br/>2x $$-$$ y + z = 0<br/><br/>$$\mu$$x + 2y + 3z = 0, $$\lambda$$, $$\mu$$$$\in$$R.<br/><br/>has a non-trivial solution. Then which of the following is true? | [{"identifier": "A", "content": "$$\\mu$$ = 6, $$\\lambda$$$$\\in$$R"}, {"identifier": "B", "content": "$$\\lambda$$ = 3, $$\\mu$$$$\\in$$R"}, {"identifier": "C", "content": "$$\\mu$$ = $$-$$6, $$\\lambda$$$$\\in$$R"}, {"identifier": "D", "content": "$$\\lambda$$ = 2, $$\\mu$$$$\\in$$R"}] | ["A"] | null | <p>Given, system of linear equations</p>
<p>4x + $$\lambda$$y + 2z = 0</p>
<p>2x $$-$$ y + z = 0</p>
<p>$$\mu$$x + 2y + 3z = 0</p>
<p>For non-trivial solution, $$\Delta$$ = 0</p>
<p>$$\left| {\matrix{
4 & \lambda & 2 \cr
2 & { - 1} & 1 \cr
\mu & 2 & 3 \cr
} } \right| = 0$$</p>
<p>$$ \Rightarrow 4( - ... | mcq | jee-main-2021-online-18th-march-evening-shift | 7,150 |
1krruoai8 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | The value of k $$\in$$R, for which the following system of linear equations<br/><br/>3x $$-$$ y + 4z = 3,<br/><br/>x + 2y $$-$$ 3z = $$-$$2<br/><br/>6x + 5y + kz = $$-$$3,<br/><br/>has infinitely many solutions, is : | [{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "$$-$$5"}, {"identifier": "C", "content": "5"}, {"identifier": "D", "content": "$$-$$3"}] | ["B"] | null | $$\left| {\matrix{
3 & { - 1} & 4 \cr
1 & 2 & { - 3} \cr
6 & 5 & k \cr
} } \right| = 0$$<br><br>$$\Rightarrow$$ 3(2k + 15) + K + 18 $$-$$ 28 = 0<br><br>$$\Rightarrow$$ 7k + 35 = 0 <br><br>$$\Rightarrow$$ k = $$-$$ 5 | mcq | jee-main-2021-online-20th-july-evening-shift | 7,151 |
1krthv2f5 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | The values of $$\lambda$$ and $$\mu$$ such that the system of equations $$x + y + z = 6$$, $$3x + 5y + 5z = 26$$, $$x + 2y + \lambda z = \mu $$ has no solution, are : | [{"identifier": "A", "content": "$$\\lambda$$ = 3, $$\\mu$$ = 5"}, {"identifier": "B", "content": "$$\\lambda$$ = 3, $$\\mu$$ $$\\ne$$ 10"}, {"identifier": "C", "content": "$$\\lambda$$ $$\\ne$$ 2, $$\\mu$$ = 10"}, {"identifier": "D", "content": "$$\\lambda$$ = 2, $$\\mu$$ $$\\ne$$ 10"}] | ["D"] | null | $$x + y + z = 6$$ ..... (i)<br><br>$$3x + 5y + 5z = 26$$ .... (ii)<br><br>$$x + 2y + \lambda z = \mu $$ ..... (iii)<br><br>$$5 \times (i) - (ii) \Rightarrow 2x = 4 \Rightarrow x = 2$$<br><br>$$\therefore$$ from (i) and (iii)<br><br>$$y + z = 4$$ ..... (iv)<br><br>$$2y + \lambda z = \mu - 2$$ .....(v)<br><br>$$(v) - 2 ... | mcq | jee-main-2021-online-22th-july-evening-shift | 7,152 |
1krw0ttza | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | The values of a and b, for which the system of equations <br/><br/>2x + 3y + 6z = 8<br/><br/>x + 2y + az = 5<br/><br/>3x + 5y + 9z = b<br/><br/>has no solution, are : | [{"identifier": "A", "content": "a = 3, b $$\\ne$$ 13"}, {"identifier": "B", "content": "a $$\\ne$$ 3, b $$\\ne$$ 13"}, {"identifier": "C", "content": "a $$\\ne$$ 3, b = 3"}, {"identifier": "D", "content": "a = 3, b = 13"}] | ["A"] | null | $$D = \left| {\matrix{
2 & 3 & 6 \cr
1 & 2 & a \cr
3 & 5 & 9 \cr
} } \right| = 3 - a$$<br><br>$$D = \left| {\matrix{
2 & 3 & 8 \cr
1 & 2 & 5 \cr
3 & 5 & b \cr
} } \right| = b - 13$$<br><br>If a = 3, b $$\ne$$ 13, no solution. | mcq | jee-main-2021-online-25th-july-morning-shift | 7,153 |
1ktbc2w2e | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | Let $$\theta \in \left( {0,{\pi \over 2}} \right)$$. If the system of linear equations<br/><br/>$$(1 + {\cos ^2}\theta )x + {\sin ^2}\theta y + 4\sin 3\,\theta z = 0$$<br/><br/>$${\cos ^2}\theta x + (1 + {\sin ^2}\theta )y + 4\sin 3\,\theta z = 0$$<br/><br/>$${\cos ^2}\theta x + {\sin ^2}\theta y + (1 + 4\sin 3\,\the... | [{"identifier": "A", "content": "$${{4\\pi } \\over 9}$$"}, {"identifier": "B", "content": "$${{7\\pi } \\over {18}}$$"}, {"identifier": "C", "content": "$${\\pi \\over {18}}$$"}, {"identifier": "D", "content": "$${{5\\pi } \\over {18}}$$"}] | ["B"] | null | $$\left| {\matrix{
{1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } & {4\sin 3\,\theta } \cr
{{{\cos }^2}\theta } & {1 + {{\sin }^2}\theta } & {4\sin 3\,\theta } \cr
{{{\cos }^2}\theta } & {{{\sin }^2}\theta } & {1 + 4\sin 3\,\theta } \cr
} } \right| = 0$$<br><br>$${C_1} \to {C_1... | mcq | jee-main-2021-online-26th-august-morning-shift | 7,155 |
1ktg0cvqr | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | Let [$$\lambda$$] be the greatest integer less than or equal to $$\lambda$$. The set of all values of $$\lambda$$ for which the system of linear equations <br/>x + y + z = 4, <br/>3x + 2y + 5z = 3, <br/>9x + 4y + (28 + [$$\lambda$$])z = [$$\lambda$$] has a solution is : | [{"identifier": "A", "content": "R"}, {"identifier": "B", "content": "($$-$$$$\\infty$$, $$-$$9) $$\\cup$$ ($$-$$9, $$\\infty$$)"}, {"identifier": "C", "content": "[$$-$$9, $$-$$8)"}, {"identifier": "D", "content": "($$-$$$$\\infty$$, $$-$$9) $$\\cup$$ [$$-$$8, $$\\infty$$)"}] | ["A"] | null | $$D = \left| {\matrix{
1 & 1 & 1 \cr
3 & 2 & 5 \cr
9 & 4 & {28 + [\lambda ]} \cr
} } \right| = - 24 - [\lambda ] + 15 = - [\lambda ] - 9$$<br><br>if $$[\lambda ] + 9 \ne 0$$ then unique solution<br><br>if $$[\lambda ] + 9 = 0$$ then D<sub>1</sub> = D<sub>2</sub> = D<sub>3</sub... | mcq | jee-main-2021-online-27th-august-evening-shift | 7,157 |
1ktiqe5l2 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | If the following system of linear equations<br/><br/>2x + y + z = 5<br/><br/>x $$-$$ y + z = 3<br/><br/>x + y + az = b<br/><br/>has no solution, then : | [{"identifier": "A", "content": "$$a = - {1 \\over 3},b \\ne {7 \\over 3}$$"}, {"identifier": "B", "content": "$$a \\ne {1 \\over 3},b = {7 \\over 3}$$"}, {"identifier": "C", "content": "$$a \\ne - {1 \\over 3},b = {7 \\over 3}$$"}, {"identifier": "D", "content": "$$a = {1 \\over 3},b \\ne {7 \\over 3}$$"}] | ["D"] | null | Here $$D = \left| {\matrix{
2 & 1 & 1 \cr
1 & { - 1} & 1 \cr
1 & 1 & a \cr
} } \right|\matrix{
{ = 2(a - 1) - 1(a - 1) + 1 + 1} \cr
{ = 1 - 3a} \cr
} $$<br><br>$${D_3} = \left| {\matrix{
2 & 1 & 5 \cr
1 & { - 1} & 3 \cr
1 & 1 & b ... | mcq | jee-main-2021-online-31st-august-morning-shift | 7,158 |
1ktk3cn98 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | If $$\alpha$$ + $$\beta$$ + $$\gamma$$ = 2$$\pi$$, then the system of equations <br/><br/>x + (cos $$\gamma$$)y + (cos $$\beta$$)z = 0<br/><br/>(cos $$\gamma$$)x + y + (cos $$\alpha$$)z = 0<br/><br/>(cos $$\beta$$)x + (cos $$\alpha$$)y + z = 0<br/><br/>has : | [{"identifier": "A", "content": "no solution"}, {"identifier": "B", "content": "infinitely many solution"}, {"identifier": "C", "content": "exactly two solutions"}, {"identifier": "D", "content": "a unique solution"}] | ["B"] | null | <p>Given $$\alpha$$ + $$\beta$$ + $$\gamma$$ = 2$$\pi$$</p>
<p>$$\Delta = \left| {\matrix{
1 & {\cos \gamma } & {\cos \beta } \cr
{\cos \gamma } & 1 & {\cos \alpha } \cr
{\cos \beta } & {\cos \alpha } & 1 \cr
} } \right|$$</p>
<p>$$ = 1 - {\cos ^2}\alpha - \cos \gamma (\cos \gamma - \cos \alpha \cos... | mcq | jee-main-2021-online-31st-august-evening-shift | 7,159 |
1ktnzxwbq | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | Consider the system of linear equations<br/><br/>$$-$$x + y + 2z = 0<br/><br/>3x $$-$$ ay + 5z = 1<br/><br/>2x $$-$$ 2y $$-$$ az = 7<br/><br/>Let S<sub>1</sub> be the set of all a$$\in$$R for which the system is inconsistent and S<sub>2</sub> be the set of all a$$\in$$R for which the system has infinitely many solution... | [{"identifier": "A", "content": "n(S<sub>1</sub>) = 2, n(S<sub>2</sub>) = 2"}, {"identifier": "B", "content": "n(S<sub>1</sub>) = 1, n(S<sub>2</sub>) = 0"}, {"identifier": "C", "content": "n(S<sub>1</sub>) = 2, n(S<sub>2</sub>) = 0"}, {"identifier": "D", "content": "n(S<sub>1</sub>) = 0, n(S<sub>2</sub>) = 2"}] | ["C"] | null | $$\Delta = \left| {\matrix{
{ - 1} & 1 & 2 \cr
3 & { - a} & 5 \cr
2 & { - 2} & { - a} \cr
} } \right|$$<br><br>$$ = - 1({a^2} + 10) - 1( - 3a - 10) + 2( - 6 + 2a)$$<br><br>$$ = - {a^2} - 10 + 3a + 10 - 12 + 4a$$<br><br>$$\Delta = - {a^2} + 7a - 12$$<br><br>$$\Delta = - [{... | mcq | jee-main-2021-online-1st-september-evening-shift | 7,160 |
1l544wdkc | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>If the system of linear equations</p>
<p>2x + y $$-$$ z = 7</p>
<p>x $$-$$ 3y + 2z = 1</p>
<p>x + 4y + $$\delta$$z = k, where $$\delta$$, k $$\in$$ R has infinitely many solutions, then $$\delta$$ + k is equal to:</p> | [{"identifier": "A", "content": "$$-$$3"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "6"}, {"identifier": "D", "content": "9"}] | ["B"] | null | <p>$$2x + y - z = 7$$</p>
<p>$$x - 3y + 2z = 1$$</p>
<p>$$x + 4y + \delta z = k$$</p>
<p>$$\Delta = \left| {\matrix{
2 & 1 & { - 1} \cr
1 & { - 3} & 2 \cr
1 & 4 & \delta \cr
} } \right| = - 7\delta - 21 = 0$$</p>
<p>$$\delta = - 3$$</p>
<p>$${\Delta _1} = \left| {\matrix{
7 & 1 & { - 1} \cr
... | mcq | jee-main-2022-online-29th-june-morning-shift | 7,161 |
1l55j68za | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>If the system of linear equations <br/>$$2x - 3y = \gamma + 5$$, <br/>$$\alpha x + 5y = \beta + 1$$, where $$\alpha$$, $$\beta$$, $$\gamma$$ $$\in$$ R has infinitely many solutions then the value <br/>of | 9$$\alpha$$ + 3$$\beta$$ + 5$$\gamma$$ | is equal to ____________.</p> | [] | null | 58 | <p>If 2x $$-$$ 3y = $$\gamma$$ + 5 and $$\alpha$$x + 5y = $$\beta$$ + 1 have infinitely many solutions then</p>
<p>$${2 \over \alpha } = {{ - 3} \over 5} = {{\gamma + 5} \over {\beta + 1}}$$</p>
<p>$$ \Rightarrow \alpha = - {{10} \over 3}$$ and $$3\beta + 5\gamma = - 28$$</p>
<p>So $$|9\alpha + 3\beta + 5\gamm... | integer | jee-main-2022-online-28th-june-evening-shift | 7,162 |
1l56668zy | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>If the system of linear equations</p>
<p>$$2x + 3y - z = - 2$$</p>
<p>$$x + y + z = 4$$</p>
<p>$$x - y + |\lambda |z = 4\lambda - 4$$</p>
<p>where, $$\lambda$$ $$\in$$ R, has no solution, then</p> | [{"identifier": "A", "content": "$$\\lambda$$ = 7"}, {"identifier": "B", "content": "$$\\lambda$$ = $$-$$7"}, {"identifier": "C", "content": "$$\\lambda$$ = 8"}, {"identifier": "D", "content": "$$\\lambda$$<sup>2</sup> = 1"}] | ["B"] | null | <p>$$\Delta = \left| {\matrix{
2 & 3 & { - 1} \cr
1 & 1 & 1 \cr
1 & { - 1} & {|\lambda |} \cr
} } \right| = 0 \Rightarrow |\lambda | = 7$$</p>
<p>But at $$\lambda = 7,\,{D_x} = {D_y} = {D_z} = 0$$</p>
<p>$${P_1}:2x + 3y - z = - 2$$</p>
<p>$${P_2}:x + y + z = 4$$</p>
<p>$${P_3}:x - y + |\lambda |z = ... | mcq | jee-main-2022-online-28th-june-morning-shift | 7,163 |
1l57nuni9 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>Let the system of linear equations <br/>$$x + 2y + z = 2$$, <br/>$$\alpha x + 3y - z = \alpha $$, <br/>$$ - \alpha x + y + 2z = - \alpha $$ <br/>be inconsistent. Then $$\alpha$$ is equal to :</p> | [{"identifier": "A", "content": "$${5 \\over 2}$$"}, {"identifier": "B", "content": "$$-$$$${5 \\over 2}$$"}, {"identifier": "C", "content": "$${7 \\over 2}$$"}, {"identifier": "D", "content": "$$-$$$${7 \\over 2}$$"}] | ["D"] | null | <p>$$x + 2y + z = 2$$</p>
<p>$$\alpha x + 3y - z = \alpha $$</p>
<p>$$ - \alpha x + y + 2z = - \alpha $$</p>
<p>$$\Delta = \left| {\matrix{
1 & 2 & 1 \cr
\alpha & 3 & { - 1} \cr
{ - \alpha } & 1 & 2 \cr
} } \right| = 1(6 + 1) - 2(2\alpha - \alpha ) + 1(\alpha + 3\alpha )$$</p>
<p>$$ = 7 + 2\alpha ... | mcq | jee-main-2022-online-27th-june-morning-shift | 7,164 |
1l587hfiu | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>The ordered pair (a, b), for which the system of linear equations</p>
<p>3x $$-$$ 2y + z = b</p>
<p>5x $$-$$ 8y + 9z = 3</p>
<p>2x + y + az = $$-$$1</p>
<p>has no solution, is :</p> | [{"identifier": "A", "content": "$$\\left( {3,{1 \\over 3}} \\right)$$"}, {"identifier": "B", "content": "$$\\left( { - 3,{1 \\over 3}} \\right)$$"}, {"identifier": "C", "content": "$$\\left( { - 3, - {1 \\over 3}} \\right)$$"}, {"identifier": "D", "content": "$$\\left( {3, - {1 \\over 3}} \\right)$$"}] | ["C"] | null | <p>$$\left| {\matrix{
3 & { - 2} & 1 \cr
5 & { - 8} & 9 \cr
2 & 1 & a \cr
} } \right| = 0 \Rightarrow - 14a - 42 = 0 \Rightarrow a = - 3$$</p>
<p>Now 3 (equation (1)) $$-$$ (equation (2)) $$-$$ 2 (equation (3)) is</p>
<p>$$3(3x - 2y + z - b) - (5x - 8y + 9z - 3) - 2(2x + y + az + 1) = 0$$</p>
<p>$$ \... | mcq | jee-main-2022-online-26th-june-morning-shift | 7,165 |
1l58ez9zk | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>If the system of equations</p>
<p>$$\alpha$$x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = $$\beta$$</p>
<p>has infinitely many solutions, then the ordered pair ($$\alpha$$, $$\beta$$) is equal to :</p> | [{"identifier": "A", "content": "(1, $$-$$3)"}, {"identifier": "B", "content": "($$-$$1, 3)"}, {"identifier": "C", "content": "(1, 3)"}, {"identifier": "D", "content": "($$-$$1, $$-$$3)"}] | ["C"] | null | <p>Given system of equations</p>
<p>$$\alpha x + y + z = 5$$</p>
<p>$$x + 2y + 3z = 4$$, has infinite solution</p>
<p>$$x + 3y + 5z = \beta $$</p>
<p>$$\therefore$$ $$\Delta = \left| {\matrix{
\alpha & 1 & 1 \cr
1 & 2 & 3 \cr
1 & 3 & 5 \cr
} } \right| = 0 \Rightarrow \alpha (1) - 1(2) + 1(1) = 0$$</p... | mcq | jee-main-2022-online-26th-june-evening-shift | 7,166 |
1l59js96c | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>The system of equations</p>
<p>$$ - kx + 3y - 14z = 25$$</p>
<p>$$ - 15x + 4y - kz = 3$$</p>
<p>$$ - 4x + y + 3z = 4$$</p>
<p>is consistent for all k in the set</p> | [{"identifier": "A", "content": "R"}, {"identifier": "B", "content": "R $$-$$ {$$-$$11, 13}"}, {"identifier": "C", "content": "R $$-$$ {13}"}, {"identifier": "D", "content": "R $$-$$ {$$-$$11, 11}"}] | ["D"] | null | <p>The system may be inconsistent if</p>
<p>$$\left| {\matrix{
{ - k} & 3 & { - 14} \cr
{ - 15} & 4 & { - k} \cr
{ - 4} & 1 & 3 \cr
} } \right| = 0 \Rightarrow k = \, \pm \,11$$</p>
<p>Hence if system is consistent then $$k \in R - \{ 11, - 11\} $$.</p> | mcq | jee-main-2022-online-25th-june-evening-shift | 7,167 |
1l5ahzzd3 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>Let A be a 3 $$\times$$ 3 real matrix such that</p>
<p>$$A\left( {\matrix{
1 \cr
1 \cr
0 \cr
} } \right) = \left( {\matrix{
1 \cr
1 \cr
0 \cr
} } \right);A\left( {\matrix{
1 \cr
0 \cr
1 \cr
} } \right) = \left( {\matrix{
{ - 1} \cr
0 \cr
1 \cr
} } \... | [{"identifier": "A", "content": "no solution"}, {"identifier": "B", "content": "infinitely many solutions"}, {"identifier": "C", "content": "unique solution"}, {"identifier": "D", "content": "exactly two solutions"}] | ["B"] | null | <p>Let $$A = \left[ {\matrix{
a & b & c \cr
d & e & f \cr
g & h & i \cr
} } \right]$$</p>
<p>$$A = \left[ {\matrix{
1 \cr
1 \cr
0 \cr
} } \right] = \left[ {\matrix{
1 \cr
1 \cr
0 \cr
} } \right] \Rightarrow \left[ {\matrix{
a & b & c \cr
d & e & f \cr
g & ... | mcq | jee-main-2022-online-25th-june-morning-shift | 7,168 |
1l5b7v78i | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>Let the system of linear equations</p>
<p>x + y + $$\alpha$$z = 2</p>
<p>3x + y + z = 4</p>
<p>x + 2z = 1</p>
<p>have a unique solution (x$$^ * $$, y$$^ * $$, z$$^ * $$). If ($$\alpha$$, x$$^ * $$), (y$$^ * $$, $$\alpha$$) and (x$$^ * $$, $$-$$y$$^ * $$) are collinear points, then the sum of absolute values of all p... | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "1"}] | ["C"] | null | <p>Given system of equations</p>
<p>$$x + y + az = 2$$ ..... (i)</p>
<p>$$3x + y + z = 4$$ ..... (ii)</p>
<p>$$x + 2z = 1$$ ..... (iii)</p>
<p>Solving (i), (ii) and (iii), we get</p>
<p>x = 1, y = 1, z = 0 (and for unique solution a $$\ne$$ $$-$$3)</p>
<p>Now, ($$\alpha$$, 1), (1, $$\alpha$$) and (1, $$-$$1) are collin... | mcq | jee-main-2022-online-24th-june-evening-shift | 7,169 |
1l5c0yfg6 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>The number of values of $$\alpha$$ for which the system of equations :</p>
<p>x + y + z = $$\alpha$$</p>
<p>$$\alpha$$x + 2$$\alpha$$y + 3z = $$-$$1</p>
<p>x + 3$$\alpha$$y + 5z = 4</p>
<p>is inconsistent, is</p> | [{"identifier": "A", "content": "0"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "3"}] | ["B"] | null | $\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ \alpha & 2 \alpha & 3 \\ 1 & 3 \alpha & 5\end{array}\right|$
<br/><br/>
$$
\begin{aligned}
&=1(10 \alpha-9 \alpha)-1(5 \alpha-3)+1\left(3 \alpha^{2}-2 \alpha\right) \\\\
&=\alpha-5 \alpha+3+3 \alpha^{2}-2 \alpha \\\\
&=3 \alpha^{2}-6 \alpha+3
\end{aligned}
$$
<br/><br/>
For ... | mcq | jee-main-2022-online-24th-june-morning-shift | 7,170 |
1l6dv2rsx | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>The number of $$\theta \in(0,4 \pi)$$ for which the system of linear equations
</p>
<p>$$
\begin{aligned}
&3(\sin 3 \theta) x-y+z=2 \\\\
&3(\cos 2 \theta) x+4 y+3 z=3 \\\\
&6 x+7 y+7 z=9
\end{aligned}
$$</p>
<p>has no solution, is :</p> | [{"identifier": "A", "content": "6"}, {"identifier": "B", "content": "7"}, {"identifier": "C", "content": "8"}, {"identifier": "D", "content": "9"}] | ["B"] | null | <p>Given,</p>
<p>$$3(\sin 3\theta )x - y + z = 2$$</p>
<p>$$3(\cos 2\theta )x + 4y + 3z = 3$$</p>
<p>$$6x + 7y + 7z = 9$$</p>
<p>For no solutions determinant of coefficient will be = 0</p>
<p>$$\therefore$$ $$D = \left| {\matrix{
{3\sin 3\theta } & { - 1} & 1 \cr
{3\cos 2\theta } & 4 & 3 \cr
6 & 7 & 7 \cr... | mcq | jee-main-2022-online-25th-july-morning-shift | 7,171 |
1l6f0ne7j | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>The number of real values of $$\lambda$$, such that the system of linear equations</p>
<p>2x $$-$$ 3y + 5z = 9</p>
<p>x + 3y $$-$$ z = $$-$$18</p>
<p>3x $$-$$ y + ($$\lambda$$<sup>2</sup> $$-$$ | $$\lambda$$ |)z = 16</p>
<p>has no solutions, is</p> | [{"identifier": "A", "content": "0"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "4"}] | ["C"] | null | <p>$$\Delta = \left| {\matrix{
2 & { - 3} & 5 \cr
1 & 3 & { - 1} \cr
3 & { - 1} & {{\lambda ^2} - |\lambda |} \cr
} } \right| = 2\left( {3{\lambda ^2} - 3|\lambda | - 1} \right) + 3\left( {{\lambda ^2} - |\lambda | + 3} \right) + 5( - 1 - 9)$$</p>
<p>$$ = 9{\lambda ^2} - 9|\lambda | - 43$$</p>
<p>$$ =... | mcq | jee-main-2022-online-25th-july-evening-shift | 7,172 |
1l6p0ni83 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>Let A and B be two $$3 \times 3$$ non-zero real matrices such that AB is a zero matrix. Then</p> | [{"identifier": "A", "content": "the system of linear equations $$A X=0$$ has a unique solution"}, {"identifier": "B", "content": "the system of linear equations $$A X=0$$ has infinitely many solutions"}, {"identifier": "C", "content": "B is an invertible matrix"}, {"identifier": "D", "content": "$$\\operatorname{adj}(... | ["B"] | null | <p>AB is zero matrix</p>
<p>$$ \Rightarrow |A| = |B| = 0$$</p>
<p>So neither A nor B is invertible</p>
<p>If $$|A| = 0$$</p>
<p>$$ \Rightarrow |\mathrm{adj}\,A| = 0$$ so $$\mathrm{adj}\,A$$</p>
<p>$$AX = 0$$ is homogeneous system and $$|A| = 0$$</p>
<p>So, it is having infinitely many solutions</p> | mcq | jee-main-2022-online-29th-july-morning-shift | 7,174 |
1l6rdwkbl | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>If the system of equations</p>
<p>$$
\begin{aligned}
&x+y+z=6 \\
&2 x+5 y+\alpha z=\beta \\
&x+2 y+3 z=14
\end{aligned}
$$</p>
<p>has infinitely many solutions, then $$\alpha+\beta$$ is equal to</p> | [{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "36"}, {"identifier": "C", "content": "44"}, {"identifier": "D", "content": "48"}] | ["C"] | null | <p>Given,</p>
<p>$$x + y + z = 6$$ ...... (1)</p>
<p>$$2x + 5y + \alpha z = \beta $$ ..... (2)</p>
<p>$$x + 2y + 3z = 14$$ ...... (3)</p>
<p>System of equation have infinite many solutions.</p>
<p>$$\therefore$$ $${\Delta _x} = {\Delta _y} = {\Delta _z} = 0$$ and $$\Delta = 0$$</p>
<p>Now, $$\Delta = \left| {\matrix{... | mcq | jee-main-2022-online-29th-july-evening-shift | 7,175 |
1ldo5ix8v | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>For the system of linear equations $$\alpha x+y+z=1,x+\alpha y+z=1,x+y+\alpha z=\beta$$, which one of the following statements is <b>NOT</b> correct?</p> | [{"identifier": "A", "content": "It has infinitely many solutions if $$\\alpha=1$$ and $$\\beta=1$$"}, {"identifier": "B", "content": "It has infinitely many solutions if $$\\alpha=2$$ and $$\\beta=-1$$"}, {"identifier": "C", "content": "$$x+y+z=\\frac{3}{4}$$ if $$\\alpha=2$$ and $$\\beta=1$$"}, {"identifier": "D", "c... | ["B"] | null | For infinite solution $\Delta=\Delta_x=\Delta_y=\Delta_z=0$
<br/><br/>$$
\Delta=\left|\begin{array}{lll}
\alpha & 1 & 1 \\
1 & \alpha & 1 \\
1 & 1 & \alpha
\end{array}\right|=0 \Rightarrow\left(\alpha^3-3 \alpha+2\right)=0 \Rightarrow \alpha=1,-2
$$
<br/><br/>If $\beta=1$, then all planes are overlapping
<br/><br/>$\th... | mcq | jee-main-2023-online-1st-february-evening-shift | 7,176 |
1ldomfjmn | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>Let $$S$$ denote the set of all real values of $$\lambda$$ such that the system of equations</p>
<p>$$\lambda x+y+z=1$$</p>
<p>$$x+\lambda y+z=1$$</p>
<p>$$x+y+\lambda z=1$$</p>
<p>is inconsistent, then $$\sum_\limits{\lambda \in S}\left(|\lambda|^{2}+|\lambda|\right)$$ is equal to</p> | [{"identifier": "A", "content": "12"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "4"}, {"identifier": "D", "content": "6"}] | ["D"] | null | $\left|\begin{array}{lll}\lambda & 1 & 1 \\ 1 & \lambda & 1 \\ 1 & 1 & \lambda\end{array}\right|=0$
<br/><br/>$$
\begin{aligned}
& \lambda\left(\lambda^{2}-1\right)-1(\lambda-1)+1(1-\lambda)=0 \\\\
& \Rightarrow \lambda^{3}-\lambda-\lambda+1+1-\lambda=0 \\\\
& \Rightarrow \lambda^{3}-3 \lambda+2=0 \\\\
& \Rightarrow (... | mcq | jee-main-2023-online-1st-february-morning-shift | 7,177 |
1ldprqy0n | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>For the system of linear equations</p>
<p>$$x+y+z=6$$</p>
<p>$$\alpha x+\beta y+7 z=3$$</p>
<p>$$x+2 y+3 z=14$$</p>
<p>which of the following is <b>NOT</b> true ?</p> | [{"identifier": "A", "content": "If $$\\alpha=\\beta=7$$, then the system has no solution"}, {"identifier": "B", "content": "For every point $$(\\alpha, \\beta) \\neq(7,7)$$ on the line $$x-2 y+7=0$$, the system has infinitely many solutions"}, {"identifier": "C", "content": "There is a unique point $$(\\alpha, \\beta)... | ["B"] | null | $\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ \alpha & \beta & 7 \\ 1 & 2 & 3\end{array}\right|$
<br/><br/>$$
\begin{aligned}
& =1(3 \beta-14)-1(3 \alpha-7)+1(2 \alpha-\beta) \\\\
& =3 \beta-14+7-3 \alpha+2 \alpha-\beta \\\\
& =2 \beta-\alpha-7
\end{aligned}
$$
<br/><br/>So, for $\alpha=\beta \neq 7, \Delta \neq 0$ so... | mcq | jee-main-2023-online-31st-january-morning-shift | 7,178 |
ldqv7zxq | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | For $\alpha, \beta \in \mathbb{R}$, suppose the system of linear equations
<br/><br/>$$
\begin{aligned}
& x-y+z=5 \\
& 2 x+2 y+\alpha z=8 \\
& 3 x-y+4 z=\beta
\end{aligned}
$$
<br/><br/>has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of : | [{"identifier": "A", "content": "$x^2+18 x+56=0$"}, {"identifier": "B", "content": "$x^2-10 x+16=0$"}, {"identifier": "C", "content": "$x^2+14 x+24=0$"}, {"identifier": "D", "content": "$x^2-18 x+56=0$"}] | ["D"] | null | <p>$$\Delta = \left| {\matrix{
1 & { - 1} & 1 \cr
2 & 2 & \alpha \cr
3 & { - 1} & 4 \cr
} } \right| = 0$$</p>
<p>$$ \Rightarrow \alpha = 4$$</p>
<p>$${\Delta _3} = 0$$</p>
<p>$$ = \left| {\matrix{
1 & { - 1} & 5 \cr
2 & 2 & 8 \cr
3 & { - 1} & \beta \cr
} } \right| = 0$$</p>
<p>$$ \... | mcq | jee-main-2023-online-30th-january-evening-shift | 7,179 |
1ldswccvm | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>Consider the following system of equations</p>
<p>$$\alpha x+2y+z=1$$</p>
<p>$$2\alpha x+3y+z=1$$</p>
<p>$$3x+\alpha y+2z=\beta$$</p>
<p>for some $$\alpha,\beta\in \mathbb{R}$$. Then which of the following is NOT correct.</p> | [{"identifier": "A", "content": "It has a solution for all $$\\alpha\\ne-1$$ and $$\\beta=2$$"}, {"identifier": "B", "content": "It has no solution if $$\\alpha=-1$$ and $$\\beta\\ne2$$"}, {"identifier": "C", "content": "It has no solution for $$\\alpha=-1$$ and for all $$\\beta \\in \\mathbb{R}$$"}, {"identifier": "D"... | ["C"] | null | $D=\left|\begin{array}{ccc}\alpha & 2 & 1 \\ 2 \alpha & 3 & 1 \\ 3 & \alpha & 2\end{array}\right|=0 \Rightarrow \alpha=-1,3$
<br/><br/>
$D_{x}=\left|\begin{array}{ccc}2 & 1 & 1 \\ 3 & 1 & 1 \\ \alpha & 2 & \beta\end{array}\right|=0 \Rightarrow \beta=2$
<br/><br/>
$D_{y}=\left|\begin{array}{ccc}\alpha & 1 & 1 \\ 2 \alph... | mcq | jee-main-2023-online-29th-january-morning-shift | 7,181 |
1ldwwf3hd | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>If the system of equations</p>
<p>$$x+2y+3z=3$$</p>
<p>$$4x+3y-4z=4$$</p>
<p>$$8x+4y-\lambda z=9+\mu$$</p>
<p>has infinitely many solutions, then the ordered pair ($$\lambda,\mu$$) is equal to :</p> | [{"identifier": "A", "content": "$$\\left( {{{72} \\over 5},{{21} \\over 5}} \\right)$$"}, {"identifier": "B", "content": "$$\\left( { - {{72} \\over 5}, - {{21} \\over 5}} \\right)$$"}, {"identifier": "C", "content": "$$\\left( { - {{72} \\over 5},{{21} \\over 5}} \\right)$$"}, {"identifier": "D", "content": "$$\\left... | ["D"] | null | For infinite many solution, $\Delta=0$ and $\Delta_x=0$
<br/><br/>$$
\begin{aligned}
& \Delta=\left|\begin{array}{ccc}
1 & 2 & 3 \\
4 & 3 & -4 \\
8 & 4 & -\lambda
\end{array}\right|=0 \\\\
& \Rightarrow 1(-3 \lambda+16)-2(-4 \lambda+32)+3(16-24)=0 \\\\
& \Rightarrow 16-3 \lambda+8 \lambda-64-24=0 \Rightarrow 5 \lambd... | mcq | jee-main-2023-online-24th-january-evening-shift | 7,183 |
1lgpxsxhd | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>For the system of linear equations</p>
<p>$$2 x+4 y+2 a z=b$$</p>
<p>$$x+2 y+3 z=4$$</p>
<p>$$2 x-5 y+2 z=8$$</p>
<p>which of the following is NOT correct?</p> | [{"identifier": "A", "content": "It has infinitely many solutions if $$a=3, b=8$$"}, {"identifier": "B", "content": "It has infinitely many solutions if $$a=3, b=6$$"}, {"identifier": "C", "content": "It has unique solution if $$a=b=8$$"}, {"identifier": "D", "content": "It has unique solution if $$a=b=6$$"}] | ["B"] | null | The given system of equations is :
<br/><br/>1. $$2x + 4y + 2az = b$$
<br/><br/>2. $$x + 2y + 3z = 4$$
<br/><br/>3. $$2x - 5y + 2z = 8$$
<br/><br/>We can write this in matrix form :
<br/><br/>$$
\begin{bmatrix}
2 & 4 & 2a \\
1 & 2 & 3 \\
2 & -5 & 2
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
=
\begin{bma... | mcq | jee-main-2023-online-13th-april-morning-shift | 7,185 |
1lgvqob1i | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>Let $$\mathrm{S}$$ be the set of values of $$\lambda$$, for which the system of equations <br/><br/>$$6 \lambda x-3 y+3 z=4 \lambda^{2}$$, <br/><br/>$$2 x+6 \lambda y+4 z=1$$, <br/><br/>$$3 x+2 y+3 \lambda z=\lambda$$ has no solution. Then $$12 \sum_\limits{i \in S}|\lambda|$$ is equal to ___________.</p> | [] | null | 24 | Given that $S$ be the set of values of $\lambda$ for which given system of equations has no solution.
<br/><br/>Therefore for the given set of equations
<br/><br/>$$
\Delta=\left|\begin{array}{ccc}
6 \lambda & -3 & 3 \\
2 & 6 \lambda & 4 \\
3 & 2 & 3 \lambda
\end{array}\right|=0
$$
<br/><br/>$$
\begin{aligned}
&\Righta... | integer | jee-main-2023-online-10th-april-evening-shift | 7,187 |
1lgxvxhlk | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>For the system of linear equations</p>
<p>$$2x - y + 3z = 5$$</p>
<p>$$3x + 2y - z = 7$$</p>
<p>$$4x + 5y + \alpha z = \beta $$,</p>
<p>which of the following is <b>NOT</b> correct?</p> | [{"identifier": "A", "content": "The system has infinitely many solutions for $$\\alpha=-6$$ and $$\\beta=9$$"}, {"identifier": "B", "content": "The system has a unique solution for $$\\alpha$$ $$ \\ne $$ $$-5$$ and $$\\beta=8$$"}, {"identifier": "C", "content": "The system is inconsistent for $$\\alpha=-5$$ and $$\\be... | ["A"] | null | Given system of linear equation is
<br/><br/>$$
\begin{gathered}
2 x-y+3 z=5 \\\\
3 x+2 y-z=7 \\\\
4 x+5 y+\alpha z=\beta \\\\
\text { Now, } \Delta=\left|\begin{array}{ccc}
2 & -1 & 3 \\
3 & 2 & -1 \\
4 & 5 & \alpha
\end{array}\right|=7(\alpha+5)
\end{gathered}
$$
<br/><br/>So, this system of equation has unique solut... | mcq | jee-main-2023-online-10th-april-morning-shift | 7,188 |
1lgym2kp0 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>Let S be the set of all values of $$\theta \in[-\pi, \pi]$$ for which the system of linear equations</p>
<p>$$x+y+\sqrt{3} z=0$$</p>
<p>$$-x+(\tan \theta) y+\sqrt{7} z=0$$</p>
<p>$$x+y+(\tan \theta) z=0$$</p>
<p>has non-trivial solution. Then $$\frac{120}{\pi} \sum_\limits{\theta \in \mathrm{s}} \theta$$ is equal to... | [{"identifier": "A", "content": "40"}, {"identifier": "B", "content": "30"}, {"identifier": "C", "content": "10"}, {"identifier": "D", "content": "20"}] | ["D"] | null | Since, the given system has a non trivial solution,
<br/><br/>$$
\text { So, } \Delta=0
$$
<br/><br/>$$
\Rightarrow \Delta=\left|\begin{array}{ccc}
1 & 1 & \sqrt{3} \\
-1 & \tan \theta & \sqrt{7} \\
1 & 1 & \tan \theta
\end{array}\right|=0
$$
<br/><br/>$$
\begin{aligned}
& \Rightarrow 1\left(\tan ^2 \theta-\sqrt{7}\rig... | mcq | jee-main-2023-online-8th-april-evening-shift | 7,189 |
1lh204tzs | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>If the system of equations</p>
<p>$$x+y+a z=b$$</p>
<p>$$2 x+5 y+2 z=6$$</p>
<p>$$x+2 y+3 z=3$$</p>
<p>has infinitely many solutions, then $$2 a+3 b$$ is equal to :</p> | [{"identifier": "A", "content": "28"}, {"identifier": "B", "content": "25"}, {"identifier": "C", "content": "20"}, {"identifier": "D", "content": "23"}] | ["D"] | null | Given system of equations,
<br/><br/>$$
\text { and } \quad \begin{aligned}
x+y+a z & =b \\
2 x+5 y+2 z & =6 \\
x+2 y+3 z & =3
\end{aligned}
$$
<br/><br/>Since, given system of equation has infinitely many solutions
<br/><br/>$$
\therefore D=0 \text { and } D_1=D_2=D_3=0
$$
<br/><br/>$$
\begin{aligned}
& \text { Here, ... | mcq | jee-main-2023-online-6th-april-morning-shift | 7,190 |
1lh2yj42z | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>For the system of equations</p>
<p>$$x+y+z=6$$</p>
<p>$$x+2 y+\alpha z=10$$</p>
<p>$$x+3 y+5 z=\beta$$, which one of the following is <b>NOT</b> true?</p> | [{"identifier": "A", "content": "System has a unique solution for $$\\alpha=3,\\beta\\ne14$$."}, {"identifier": "B", "content": "System has infinitely many solutions for $$\\alpha=3, \\beta=14$$."}, {"identifier": "C", "content": "System has no solution for $$\\alpha=3, \\beta=24$$."}, {"identifier": "D", "content": "S... | ["A"] | null | Given system of equations,
<br/><br/>$$
\begin{aligned}
x+y+z & =6 ........(i)\\\\
x+2 y+\alpha z & =10 ........(ii)\\\\
x+3 y+5 z & =\beta ........(iii)
\end{aligned}
$$
<br/><br/>Here,
<br/><br/>$$
\begin{aligned}
\Delta & =\left|\begin{array}{lll}
1 & 1 & 1 \\
1 & 2 & \alpha \\
1 & 3 & 5
\end{array}\right| \\\\
& =... | mcq | jee-main-2023-online-6th-april-evening-shift | 7,191 |
lsam7nu4 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | Let the system of equations $x+2 y+3 z=5,2 x+3 y+z=9,4 x+3 y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda+2 \mu$ is equal to : | [{"identifier": "A", "content": "22"}, {"identifier": "B", "content": "17"}, {"identifier": "C", "content": "15"}, {"identifier": "D", "content": "28"}] | ["B"] | null | $$
\begin{aligned}
& x+2 y+3 z=5 \\\\
& 2 x+3 y+z=9 \\\\
& 4 x+3 y+\lambda z=\mu
\end{aligned}
$$
<br/><br/>For infinite following $\Delta=\Delta_1=\Delta_2=\Delta_3=0$
<br/><br/>$\begin{aligned} & \Delta=\left|\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 1 \\ 4 & 3 & \lambda\end{array}\right|=0 \Rightarrow \lambda=-13 \\\\ ... | mcq | jee-main-2024-online-1st-february-evening-shift | 7,192 |
jaoe38c1lsd3de46 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>Let $$A$$ be a $$3 \times 3$$ real matrix such that</p>
<p>$$A\left(\begin{array}{l}
1 \\
0 \\
1
\end{array}\right)=2\left(\begin{array}{l}
1 \\
0 \\
1
\end{array}\right), A\left(\begin{array}{l}
-1 \\
0 \\
1
\end{array}\right)=4\left(\begin{array}{l}
-1 \\
0 \\
1
\end{array}\right), A\left(\begin{array}{l}
0 \\
1 \... | [{"identifier": "A", "content": "exactly two solutions\n"}, {"identifier": "B", "content": "infinitely many solutions\n"}, {"identifier": "C", "content": "unique solution\n"}, {"identifier": "D", "content": "no solution"}] | ["C"] | null | <p>$$\text { Let } A=\left[\begin{array}{lll}
x_1 & y_1 & z_1 \\
x_2 & y_2 & z_2 \\
x_3 & y_3 & z_3
\end{array}\right]$$</p>
<p>$$\text { Given } A\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right]=\left[\begin{array}{l}
2 \\
0 \\
2
\end{array}\right] \quad \text{ ..... (1)}$$</p>
<p>$$\therefore\left[\begin{array}{... | mcq | jee-main-2024-online-31st-january-evening-shift | 7,194 |
jaoe38c1lse5aryy | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>If the system of linear equations</p>
<p>$$\begin{aligned}
& x-2 y+z=-4 \\
& 2 x+\alpha y+3 z=5 \\
& 3 x-y+\beta z=3
\end{aligned}$$</p>
<p>has infinitely many solutions, then $$12 \alpha+13 \beta$$ is equal to</p> | [{"identifier": "A", "content": "60"}, {"identifier": "B", "content": "54"}, {"identifier": "C", "content": "64"}, {"identifier": "D", "content": "58"}] | ["D"] | null | <p>$$\begin{aligned}
& D=\left|\begin{array}{ccc}
1 & -2 & 1 \\
2 & \alpha & 3 \\
3 & -1 & \beta
\end{array}\right| \\
& =1(\alpha \beta+3)+2(2 \beta-9)+1(-2-3 \alpha) \\
& =\alpha \beta+3+4 \beta-18-2-3 \alpha
\end{aligned}$$</p>
<p>For infinite solutions $$\mathrm{D}=0, \mathrm{D}_1=0, \mathrm{D}_2=0$$ and</p>
<p>$$\... | mcq | jee-main-2024-online-31st-january-morning-shift | 7,195 |
jaoe38c1lsfl56cy | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>Let for any three distinct consecutive terms $$a, b, c$$ of an A.P, the lines $$a x+b y+c=0$$ be concurrent at the point $$P$$ and $$Q(\alpha, \beta)$$ be a point such that the system of equations</p>
<p>$$\begin{aligned}
& x+y+z=6, \\
& 2 x+5 y+\alpha z=\beta \text { and }
\end{aligned}$$</p>
<p>$$x+2 y+3 z... | [] | null | 113 | <p>$$\because \mathrm{a}, \mathrm{b}, \mathrm{c}$$ and in A.P</p>
<p>$$\Rightarrow 2 b=a+c \Rightarrow a-2 b+c=0$$</p>
<p>$$\therefore \mathrm{ax}+\mathrm{by}+\mathrm{c}$$ passes through fixed point $$(1,-2)$$</p>
<p>$$\therefore \mathrm{P}=(1,-2)$$</p>
<p>For infinite solution,</p>
<p>$$\begin{aligned}
& D=D_1=D_2=D_3... | integer | jee-main-2024-online-29th-january-evening-shift | 7,196 |
1lsg4izdm | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>Consider the system of linear equations $$x+y+z=5, x+2 y+\lambda^2 z=9, x+3 y+\lambda z=\mu$$, where $$\lambda, \mu \in \mathbb{R}$$. Then, which of the following statement is NOT correct?</p> | [{"identifier": "A", "content": "System is consistent if $$\\lambda \\neq 1$$ and $$\\mu=13$$\n"}, {"identifier": "B", "content": "System is inconsistent if $$\\lambda=1$$ and $$\\mu \\neq 13$$\n"}, {"identifier": "C", "content": "System has unique solution if $$\\lambda \\neq 1$$ and $$\\mu \\neq 13$$\n"}, {"identifie... | ["C"] | null | <p>$$\begin{aligned}
& \left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & 2 & \lambda^2 \\
1 & 3 & \lambda
\end{array}\right|=0 \\
& \Rightarrow 2 \lambda^2-\lambda-1=0 \\
& \lambda=1,-\frac{1}{2} \\
& \left|\begin{array}{ccc}
1 & 1 & 5 \\
2 & \lambda^2 & 9 \\
3 & \lambda & \mu
\end{array}\right|=0 \Rightarrow \mu=13
\end{align... | mcq | jee-main-2024-online-30th-january-evening-shift | 7,197 |
1lsga8e3h | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>Consider the system of linear equations $$x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^2 z=\mu^2+15$$ where $$\lambda, \mu \in \mathbf{R}$$. Which one of the following statements is NOT correct ?</p> | [{"identifier": "A", "content": "The system has unique solution if $$\\lambda \\neq \\frac{1}{2}$$ and $$\\mu \\neq 1,15$$"}, {"identifier": "B", "content": "The system has infinite number of solutions if $$\\lambda=\\frac{1}{2}$$ and $$\\mu=15$$\n"}, {"identifier": "C", "content": "The system is consistent if $$\\lamb... | ["D"] | null | <p>$$x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda{ }^2 z=\mu^2+15$$,</p>
<p>$$\Delta=\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & 2 & 2 \lambda \\
1 & 3 & 4 \lambda^2
\end{array}\right|=(2 \lambda-1)^2$$</p>
<p>For unique solution $$\Delta \neq 0,2 \lambda-1 \neq 0,\left(\lambda \neq \frac{1}{2}\right)$$</p>
<p>... | mcq | jee-main-2024-online-30th-january-morning-shift | 7,198 |
luxwcbqo | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>Consider the matrices : $$A=\left[\begin{array}{cc}2 & -5 \\ 3 & m\end{array}\right], B=\left[\begin{array}{l}20 \\ m\end{array}\right]$$ and $$X=\left[\begin{array}{l}x \\ y\end{array}\right]$$. Let the set of all $$m$$, for which the system of equations $$A X=B$$ has a negative solution (i.e., $$x<0$$ a... | [] | null | 450 | <p>$$\begin{aligned}
& A X=B \\
& 2 x-5 y=20 \\
& 3 x+m y=m \\
& \Rightarrow 3\left(\frac{20+5 y}{2}\right)+m y=m
\end{aligned}$$</p>
<p>$$\begin{aligned}
& \Rightarrow 30+\frac{15}{2} y+m y=m \\
& \Rightarrow y\left(\frac{15}{2}+m\right)=m-30 \\
& \Rightarrow y=\frac{m-30}{\frac{15}{2}+m}<0 \Rightarrow m \in\left(-\fr... | integer | jee-main-2024-online-9th-april-evening-shift | 7,199 |
luy6z5o7 | maths | matrices-and-determinants | solutions-of-system-of-linear-equations-in-two-or-three-variables-using-determinants-and-matrices | <p>Let $$\lambda, \mu \in \mathbf{R}$$. If the system of equations</p>
<p>$$\begin{aligned}
& 3 x+5 y+\lambda z=3 \\
& 7 x+11 y-9 z=2 \\
& 97 x+155 y-189 z=\mu
\end{aligned}$$</p>
<p>has infinitely many solutions, then $$\mu+2 \lambda$$ is equal to :</p> | [{"identifier": "A", "content": "24"}, {"identifier": "B", "content": "25"}, {"identifier": "C", "content": "27"}, {"identifier": "D", "content": "22"}] | ["B"] | null | <p>$$\begin{aligned}
& 3 x+5 y+\lambda z=3 \\
& 7 x+11 y-9 z=2 \\
& 97 x+155 y-189 z=\mu
\end{aligned}$$</p>
<p>$$\begin{aligned}
& {\left[\begin{array}{ccc}
3 & 5 & \lambda \\
7 & 11 & -9 \\
97 & 155 & -189
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
3 \\
2 \\
\mu
\en... | mcq | jee-main-2024-online-9th-april-morning-shift | 7,200 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.