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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 :
Options:
[{"identifier": "A", "content": "an empty set "}, {"identifier": "B", "content": "equal to {0}"}, {"identifier": "C", "cont... | ["D"]
Explanation:
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 ... |
If the system of linear equations
<br/>x + ay + z = 3
<br/>x + 2y + 2z = 6
<br/>x + 5y + 3z = b
<br/>has no solution, then :
Options:
[{"identifier": "A", "content": "a = $$-$$ 1, b = 9"}, {"identifier": "B", "content": "a = $$-$$ 1, b $$ \\ne $$ 9"}, {"identifier": "C", "content": "a $$ \\n... | ["B"]
Explanation:
As the given system of equations has no solution then
<br><br> $$\Delta $$ = 0 and at least one of $$\Delta $$<sub>1</sub>, $$\Delta $$<sub>2</sub> and $$\Delta $$<sub>2</sub> should not be zero.
<br><br>$$ \therefore $$ $$\Delta $$ = $$\left| {\matrix{
1 & a & 1 \cr
1 & 2 &am... |
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 :
Options:
[{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "infinitely m... | ["A"]
Explanation:
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>$$ \Rightarrow $... |
An ordered pair ($$\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 $$y + 2z = 2
<br/>has a unique solution, is :
Options:
[{"identifier": "A", "content": "(\u20133, 1) "}, {"ide... | ["D"]
Explanation:
For unique solution
<br><br>$$\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
... |
Let $$\lambda $$ be a real number for which the system of linear equations x + y + z = 6, 4x + $$\lambda $$y β $$\lambda $$z = $$\lambda $$ β 2,
3x + 2y β 4z = β 5 has infinitely many solutions. Then $$\lambda $$ is a root of the quadratic equation:
Options:
[{"identifier": "A", "content": "$$\\lambda $$<sup>2</sup> +... | ["B"]
Explanation:
$$\Delta = 0$$<br><br>
$$\left| {\matrix{
1 & 1 & 1 \cr
4 & \lambda & { - \lambda } \cr
3 & 2 & { - 4} \cr
} } \right| = 0$$<br><br>
On solving we get $$\lambda $$ = 3 |
If the system of linear equations
<br/>x + y + z = 5
<br/>x + 2y + 2z = 6
<br/>x + 3y + $$\lambda $$z = $$\mu $$, ($$\lambda $$, $$\mu $$ $$ \in $$ R), has infinitely many solutions, then the value of $$\lambda $$ + $$\mu $$ is :
Options:
[{"identifier": "A", "content": "10"}, {"identifier": "B", "content": "9"}, {"i... | ["A"]
Explanation:
x + y + z = 5<br><br>
x + 2y + 2z = 6<br><br>
x + 3y + $$\lambda $$z = $$\mu $$ have infinite solution<br><br>
$$\Delta $$ = 0, $$\Delta $$x = $$\Delta $$y = $$\Delta $$z = 0<br><br>
$$\Delta = \left| {\matrix{
1 & 1 & 1 \cr
1 & 2 & 2 \cr
1 & 3 & \lambda \cr
... |
If the system of equations 2x + 3y β z = 0, x + ky
β 2z = 0 and 2x β y + z = 0 has a non-trival solution
(x, y, z), then $${x \over y} + {y \over z} + {z \over x} + k$$
is equal to :-
Options:
[{"identifier": "A", "content": "-4"}, {"identifier": "B", "content": "$${3 \\over 4}$$"}, {"identifier": "C", "content": "$${... | ["C"]
Explanation:
Given 2x + 3y β z = 0,
<br><br>x + ky β 2z = 0
<br><br>2x β 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>
$$ \Rightarrow k = {9... |
The greatest value of c $$ \in $$ R for which the system
of linear equations<br/>
x β cy β cz = 0<br/>
cx β y + cz = 0<br/>
cx + cy β z = 0<br/>
has a non-trivial solution, is :
Options:
[{"identifier": "A", "content": "-1"}, {"identifier": "B", "content": "0"}, {"identifier": "C", "content": "1/2"}, {"identifier": "D... | ["C"]
Explanation:
If the system of equations has non-trivial
solutions, then
<br><br>D = 0
<br><br>$$\left| {\matrix{
1 & { - c} & { - c} \cr
c & { - 1} & c \cr
c & c & { - 1} \cr
} } \right| = 0$$
<br><br>$$ \Rightarrow $$ (1 - c<sup>2</sup>) + c(-c - c<sup>2</sup>) - c(c<sup... |
The set of all values of $$\lambda $$ for which the system of linear equations
<br/>x β 2y β 2z = $$\lambda $$x
<br/>x + 2y + z = $$\lambda $$y
<br/>β x β y = $$\lambda $$z
<br/>has a non-trivial solutions :
Options:
[{"identifier": "A", "content": "is an empty set"}, {"identifier": "B", "content": "contains more than... | ["C"]
Explanation:
$$\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$$ |
If the system of linear equations
<br/>2x + 2y + 3z = a
<br/>3x β y + 5z = b
<br/>x β 3y + 2z = c
<br/>where a, b, c are non zero real numbers, has more one solution, then :
Options:
[{"identifier": "A", "content": "b \u2013 c \u2013 a = 0"}, {"identifier": "B", "content": "a + b + c = 0"}, {"identifier": "C", "conte... | ["A"]
Explanation:
P<sub>1</sub> : 2x + 2y + 3z = a
<br><br>P<sub>2</sub> : 3x $$-$$ y + 5z = b
<br><br>P<sub>3</sub> : x $$-$$ 3y + 2z = c
<br><br>We find
<br><br>P<sub>1</sub> + P<sub>3</sub> = P<sub>2</sub> $$ \Rightarrow $$ a + c = b |
The number of values of $$\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 $$)y + 2z = 0.
<br/><br/>has a non-trival solution, is -
Options:
[{"identifier": "A", "content": "two"}, {"identi... | ["A"]
Explanation:
$$\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 $... |
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 -
Options:
[{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "21"}, {"identifier": "C", "content": "18"},... | ["A"]
Explanation:
$$D = \left| {\matrix{
1 & 1 & 1 \cr
1 & 2 & 3 \cr
1 & 3 & \alpha \cr
} } \right| = \left| {\matrix{
1 & 1 & 1 \cr
0 & 1 & 2 \cr
0 & 2 & {\alpha - 1} \cr
} } \right|$$
<br><br> $$ =... |
If the system of linear equations
<br/>x $$-$$ 4y + 7z = g
<br/>Β Β Β Β Β Β Β 3y $$-$$ 5z = h
<br/>$$-$$2x + 5y $$-$$ 9z = k
<br/>is consistent, then :
Options:
[{"identifier": "A", "content": "g + 2h + k = 0"}, {"identifier": "B", "content": "g + h + 2k = 0"}, {"identifier": "C", "content": "2g + h + k = 0"}, {"identifier... | ["C"]
Explanation:
x $$-$$ $$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>$$D = 1\left( { - ... |
The system of linear equations
<br/> x + y + z = 2
<br/>2x + 3y + 2z = 5
<br/>2x + 3y + (a<sup>2</sup> β 1) z = a + 1 then
Options:
[{"identifier": "A", "content": "has infinitely many solutions for a = 4 "}, {"identifier": "B", "content": "has a unique solution for |a| = $$\\sqrt3$$"}, {"identifier": "C", "content"... | ["C"]
Explanation:
$$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$$<sup>2</sup> $$-$$ 3)
<br><br... |
The following system of linear equations<br/>
7x + 6y β 2z = 0<br/>
3x + 4y + 2z = 0<br/>
x β 2y β 6z = 0, has
Options:
[{"identifier": "A", "content": "no solution"}, {"identifier": "B", "content": "infinitely many solutions, (x, y, z) satisfying\ny = 2z"}, {"identifier": "C", "content": "infinitely many solutions, (... | ["C"]
Explanation:
Given
<br>7x + 6y β 2z = 0 .......(1)<br>
3x + 4y + 2z = 0 ......(2)<br>
x β 2y β 6z = 0 .......(3)
<br><br>$$\Delta $$ = $$\left| {\matrix{
7 & 6 & { - 2} \cr
3 & 4 & 2 \cr
1 & { - 2} & { - 6} \cr
} } \right|$$
<br><br>= 7(β24 + 4) β 6(β18 β 2) β 2(β6 β 4) ... |
The sum of distinct values of $$\lambda $$ for which the
system of equations<br/><br/>$$\left( {\lambda - 1} \right)x + \left( {3\lambda + 1} \right)y + 2\lambda z = 0$$<br/>$$\left( {\lambda - 1} \right)x + \left( {4\lambda - 2} \right)y + \left( {\lambda + 3} \right)z = 0$$<br/>$$2x + \left( {3\lambda + 1} \rig... | 3
Explanation:
$$\left| {\matrix{
{\lambda - 1} & {3\lambda + 1} & {2\lambda } \cr
{\lambda - 1} & {4\lambda - 2} & {\lambda + 3} \cr
2 & {3\lambda + 1} & {3\left( {\lambda - 1} \right)} \cr
} } \right|$$ = 0
<br><br>R<sub>2</sub> $$ \to $$ R<sub>2</sub>
β R<sub>1</sub>... |
The values of $$\lambda $$ and $$\mu $$ for which the system of linear equations
<br/>x + y + z = 2
<br/>x + 2y + 3z = 5
<br/>x + 3y + $$\lambda $$z = $$\mu $$
<br/>has infinitely many solutions are, respectively:
Options:
[{"identifier": "A", "content": "6 and 8"}, {"identifier": "B", "content": "5 and 8"}, {"identif... | ["B"]
Explanation:
For infinite many solutions
<br><br>D = D<sub>1</sub> = D<sub>2</sub> = D<sub>3</sub> = 0
<br><br>Now D = $$\left| {\matrix{
1 & 1 & 1 \cr
1 & 2 & 3 \cr
1 & 3 & \lambda \cr
} } \right|$$ = 0
<br><br>$$ \Rightarrow $$ 1. (2$$\lambda $$ β 9) β1.($$\lambda $$ β... |
If the system of linear equations
<br/>x + y + 3z = 0
<br/>x + 3y + k<sup>2</sup>z = 0
<br/>3x + y + 3z = 0
<br/>has a non-zero solution (x, y, z) for some k $$ \in $$ R,
then x + $$\left( {{y \over z}} \right)$$ is equal to :
Options:
[{"identifier": "A", "content": "9"}, {"identifier": "B", "content": "3"}, {"identi... | ["D"]
Explanation:
x + y + 3z = 0 .....(i)
<br>x + 3y + k<sup>2</sup>z = 0 .........(ii)
<br>3x + y + 3z = 0 ......(iii)
<br><br>$$\left| {\matrix{
1 & 1 & 3 \cr
1 & 3 & {{k^2}} \cr
3 & 1 & 3 \cr
} } \right|$$ = 0
<br><br>$$ \Rightarrow $$ 9 + 3 + 3k<sup>2</sup>
β 27 β k<sup>2... |
Let $$\lambda \in $$ R . The system of linear equations<br/>
2x<sub>1</sub>
- 4x<sub>2</sub> + $$\lambda $$x<sub>3</sub> = 1<br/>
x<sub>1</sub> - 6x<sub>2</sub> + x<sub>3</sub> = 2<br/>
$$\lambda $$x<sub>1</sub> - 10x<sub>2</sub> + 4x<sub>3</sub> = 3<br/>
is inconsistent for:
Options:
[{"identifier": "A", "content": ... | ["B"]
Explanation:
D = $$\left| {\matrix{
2 & { - 4} & \lambda \cr
1 & { - 6} & 1 \cr
\lambda & { - 10} & 4 \cr
} } \right|$$ = 0
<br><br>$$ \Rightarrow $$ $$\lambda $$ = 3, $$ - {2 \over 3}$$
<br><br>D<sub>1</sub> = $$\left| {\matrix{
1 & { - 4} & \lambda \cr
... |
If the system of equations<br/>
x+y+z=2<br/>
2x+4yβz=6<br/>
3x+2y+$$\lambda $$z=$$\mu $$<br/>
has infinitely many solutions, then
Options:
[{"identifier": "A", "content": "2$$\\lambda $$ - $$\\mu $$ = 5"}, {"identifier": "B", "content": "$$\\lambda $$ - 2$$\\mu $$ = -5"}, {"identifier": "C", "content": "2$$\\lambda $$... | ["C"]
Explanation:
$$D = 0\,\left| {\matrix{
1 & 1 & 1 \cr
2 & 4 & { - 1} \cr
3 & 2 & \lambda \cr
} } \right| = 0$$<br><br>$$ \Rightarrow $$ $$(4\lambda + 2) - 1(2\lambda + 3) + 1(4 - 12) = 0$$<br><br>$$ \Rightarrow $$ $$4\lambda + 2$$ $$ - 2\lambda - 3$$$$ - $$8$$ = 0$$<b... |
Suppose the vectors x<sub>1</sub>, x<sub>2</sub> and x<sub>3</sub> are the <br/>solutions of the system of linear equations,<br/> Ax = b when the vector b on the right side is equal to b<sub>1</sub>, b<sub>2</sub> and b<sub>3</sub> respectively. if<br/><br/>
$${x_1} = \left[ {\matrix{
1 \cr
1 \cr
1 \cr
... | ["C"]
Explanation:
Let A = $$\left[ {\matrix{
{{a_1}} & {{a_2}} & {{a_3}} \cr
{{a_4}} & {{a_5}} & {{a_6}} \cr
{{a_7}} & {{a_8}} & {{a_9}} \cr
} } \right]$$
<br><br>For Ax<sub>1</sub> = b<sub>1</sub> :
<br><br>$$ \Rightarrow $$ $$\left[ {\matrix{
{{a_1}} & {{a_2}} & {... |
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.........
Options:
[] | 5
Explanation:
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>$${D_1} = \left| {\matrix{
9 ... |
Let A = {X = (x, y, z)<sup>T</sup>: PX = 0 and
<br/><br/>x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = 1} where
<br/><br/>$$P = \left[ {\matrix{
1 & 2 & 1 \cr
{ - 2} & 3 & { - 4} \cr
1 & 9 & { - 1} \cr
} } \right]$$,
<br/><br/>then the set A :
Options:
[{"identifier": "A", "c... | ["C"]
Explanation:
Let $$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[ ... |
Let S be the set of all $$\lambda $$ $$ \in $$ R for which the system
of linear equations
<br/><br/>2x β y + 2z = 2
<br/>x β 2y +
$$\lambda $$z = β4
<br/>x +
$$\lambda $$y + z = 4
<br/><br/>has no solution. Then the set S :
Options:
[{"identifier": "A", "content": "contains more than two elements."}, {"identifier": "B... | ["B"]
Explanation:
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>... |
If for some $$\alpha $$ and $$\beta $$ in R, the intersection of the
following three places<br/>
x + 4y β 2z = 1<br/>
x + 7y β 5z = b<br/>
x + 5y + $$\alpha $$z = 5<br/>
is a line in R<sup>3</sup>, then $$\alpha $$ + $$\beta $$ is equal to :
Options:
[{"identifier": "A", "content": "-10"}, {"identifier": "B", "content... | ["C"]
Explanation:
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>$... |
The system of linear equations<br/>
$$\lambda $$x + 2y + 2z = 5<br/>
2$$\lambda $$x + 3y + 5z = 8<br/>
4x + $$\lambda $$y + 6z = 10 has
Options:
[{"identifier": "A", "content": "a unique solution when $$\\lambda $$ = \u20138"}, {"identifier": "B", "content": "no solution when $$\\lambda $$ = 2"}, {"identifier": "C", "... | ["B"]
Explanation:
$$\Delta $$ = $$\left| {\matrix{
\lambda & 2 & 2 \cr
{2\lambda } & 3 & 5 \cr
4 & \lambda & 6 \cr
} } \right|$$
<br><br>= $$\lambda $$ ( 18 β 5$$\lambda $$) β 2(12$$\lambda $$ β 20) + 2(2$$\lambda $$<sup>2</sup>
β 12)
<br><br>= 18$$\lambda $$ β 5$$\lambda ... |
For which of the following ordered pairs ($$\mu $$, $$\delta $$),
the system of linear equations
<br/>x + 2y + 3z = 1
<br/>3x + 4y + 5z = $$\mu $$
<br/>4x + 4y + 4z = $$\delta $$
<br/>is inconsistent ?
Options:
[{"identifier": "A", "content": "(1, 0)"}, {"identifier": "B", "content": "(4, 3)"}, {"identifier": "C", "co... | ["B"]
Explanation:
For inconsistent system we need
<br><br>$$\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|$$ = 0
<br><br>$$\De... |
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 ______.
Options:
[] | 13
Explanation:
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>$$\Delta $$ = $$\left| {\matrix{
1 & 1 & 1 \cr
1 & 2 &a... |
If the system of linear equations<br/>
2x + 2ay + az = 0<br/>
2x + 3by + bz = 0<br/>
2x + 4cy + cz = 0,<br/>
where a, b, c $$ \in $$ R are non-zero distinct; has a non-zero solution, then:
Options:
[{"identifier": "A", "content": "$${1 \\over a},{1 \\over b},{1 \\over c}$$ are in A.P. "}, {"identifier": "B", "content"... | ["A"]
Explanation:
For non-zero solution
<br><br>$$\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} &... |
Let S be the set of all integer solutions, (x, y, z),
of the system of equations
<br/>x β 2y + 5z = 0
<br/>β2x + 4y + z = 0
<br/>β7x + 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
______ .
Options:
[] | 8
Explanation:
$$x - 2y + 5z = 0$$ ....(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>$$y \in \left[ { - \s... |
The system of linear equations
<br/>3x - 2y - kz = 10
<br/>2x - 4y - 2z = 6
<br/>x+2y - z = 5m
<br/>is inconsistent if :
Options:
[{"identifier": "A", "content": "k $$ \\ne $$ 3, m $$ \\in $$ <b>R</b>"}, {"identifier": "B", "content": "k = 3, m $$ \\ne $$ $${4 \\over 5}$$"}, {"identifier": "C", "content": "k = 3, m $$... | ["B"]
Explanation:
$$\Delta = \left| {\matrix{
3 & { - 2} & { - k} \cr
1 & { - 4} & { - 2} \cr
1 & 2 & { - 1} \cr
} } \right| = 0$$<br><br>$$3(4 + 4) + 2( - 2 + 2) - k(4 + 4) = 0$$<br><br>$$ \Rightarrow k = 3$$<br><br>$${\Delta _x} = \left| {\matrix{
{10} & { - 2} &... |
For the system of linear equations:<br/><br/>$$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$$.<br/><br/>(B) The system has unique solution if k = $$-$$2<br/><br/>(C) The system has unique solution if ... | ["D"]
Explanation:
$$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$$<br><br>$$ \Rightarr... |
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 __________.
Options:
[] | 21
Explanation:
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>... |
The following system of linear equations<br/><br/>2x + 3y + 2z = 9<br/><br/>3x + 2y + 2z = 9<br/><br/>x $$-$$ y + 4z = 8
Options:
[{"identifier": "A", "content": "does not have any solution"}, {"identifier": "B", "content": "has a solution ($$\\alpha$$, $$\\beta$$, $$\\gamma$$) satisfying $$\\alpha$$ + $$\\beta$$<sup>... | ["C"]
Explanation:
$$\Delta = \left| {\matrix{
2 & 3 & 2 \cr
3 & 2 & 2 \cr
1 & { - 1} & 4 \cr
} } \right| = - 20 \ne 0$$ $$ \therefore $$ unique solution<br><br>$${\Delta _x} = \left| {\matrix{
9 & 3 & 2 \cr
9 & 2 & 2 \cr
8 & { - 1} & 4 \... |
Consider the following system of equations :<br/><br/>x + 2y $$-$$ 3z = a<br/><br/>2x + 6y $$-$$ 11z = b<br/><br/>x $$-$$ 2y + 7z = c,<br/><br/>where a, b and c are real constants. Then the system of equations :
Options:
[{"identifier": "A", "content": "has no solution for all a, b and c"}, {"identifier": "B", "conten... | ["C"]
Explanation:
$$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} ... |
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 :
Options:
[{"identifier": "A", "content": "Exac... | ["D"]
Explanation:
$$A = \left[ {\matrix{
i & { - i} \cr
{ - i} & i \cr
} } \right]$$<br><br>$${A^2} = \left[ {\matrix{
i & { - i} \cr
{ - i} & i \cr
} } \right]\left[ {\matrix{
i & { - i} \cr
{ - i} & i \cr
} } \right] = \left[ {\matrix{
{ - 2} & 2 \... |
The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k<sup>2</sup> has no solution if k is equal to :
Options:
[{"identifier": "A", "content": "0"}, {"identifier": "B", "content": "$$-$$1"}, {"identifier": "C", "content": "$$-$$2"}, {"identifier": "D", "content": "1"}] | ["C"]
Explanation:
$$D = \left| {\matrix{
k & 1 & 1 \cr
1 & k & 1 \cr
1 & 1 & k \cr
} } \right| = 0$$<br><br>$$ \Rightarrow k({k^2} - 1) - (k - 1) + (1 - k) = 0$$<br><br>$$ \Rightarrow (k - 1)({k^2} + k - 1 - 1) = 0$$<br><br>$$ \Rightarrow (k - 1)({k^2} + k - 2) = 0$$<br><br>$$... |
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... | ["B"]
Explanation:
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>$$\left| {\matrix{
\alpha & \beta & \gamma \cr
\beta & \gamma & \alpha \cr
\gamma & \alpha & \beta \cr
} } \righ... |
Let the system of linear equations <br/><br/>4x + $$\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?
Options:
[{"identifier": "A", "content": "$$\\mu$$ = 6, $$\\lambda$$$$\\in$$R"}, {... | ["A"]
Explanation:
<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>... |
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 :
Options:
[{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "$$-$$5"}, {"identifier"... | ["B"]
Explanation:
$$\left| {\matrix{
3 & { - 1} & 4 \cr
1 & 2 & { - 3} \cr
6 & 5 & k \cr
} } \right| = 0$$<br><br>$$\Rightarrow$$ 3(2k + 15) + K + 18 $$-$$ 28 = 0<br><br>$$\Rightarrow$$ 7k + 35 = 0 <br><br>$$\Rightarrow$$ k = $$-$$ 5 |
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 :
Options:
[{"identifier": "A", "content": "$$\\lambda$$ = 3, $$\\mu$$ = 5"}, {"identifier": "B", "content": "$$\\lambda$$ = 3, $$\\mu$$ $$\\ne$$ 10"}, {"... | ["D"]
Explanation:
$$x + y + z = 6$$ ..... (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>$$2y + \lambda z = \mu - 2$$ .....(... |
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 :
Options:
[{"identifier": "A", "content": "a = 3, b $$\\ne$$ 13"}, {"identifier": "B", "content": "a $$\\ne$$ 3, b $$\\ne$$ 13"}, {"identifier": "C", "co... | ["A"]
Explanation:
$$D = \left| {\matrix{
2 & 3 & 6 \cr
1 & 2 & a \cr
3 & 5 & 9 \cr
} } \right| = 3 - a$$<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 so... |
For real numbers $$\alpha$$ and $$\beta$$, consider the following system of linear equations :<br/><br/>x + y $$-$$ z = 2, x + 2y + $$\alpha$$z = 1, 2x $$-$$ y + z = $$\beta$$. If the system has infinite solutions, then $$\alpha$$ + $$\beta$$ is equal to ______________.
Options:
[] | 5
Explanation:
For infinite solutions<br><br>$$\Delta$$ = $$\Delta$$<sub>1</sub> = $$\Delta$$<sub>2</sub> = $$\Delta$$<sub>3</sub> = 0<br><br>$$\Delta$$ = $$\left| {\matrix{
1 & 1 & { - 1} \cr
1 & 2 & \alpha \cr
2 & { - 1} & 1 \cr
} } \right| = 0$$<br><br>$$\Delta = \left| {\... |
Let $$\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... | ["B"]
Explanation:
$$\left| {\matrix{
{1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } & {4\sin 3\,\theta } \cr
{{{\cos }^2}\theta } & {1 + {{\sin }^2}\theta } & {4\sin 3\,\theta } \cr
{{{\cos }^2}\theta } & {{{\sin }^2}\theta } & {1 + 4\sin 3\,\theta } \cr
} } \right| = 0$$<br>... |
If the system of linear equations<br/><br/>2x + y $$-$$ z = 3<br/><br/>x $$-$$ y $$-$$ z = $$\alpha$$<br/><br/>3x + 3y + $$\beta$$z = 3<br/><br/>has infinitely many solution, then $$\alpha$$ + $$\beta$$ $$-$$ $$\alpha$$$$\beta$$ is equal to _____________.
Options:
[] | 5
Explanation:
2 $$\times$$ (i) $$-$$ (ii) $$-$$ (iii) gives :<br><br>$$-$$ (1 + $$\beta$$)z = 3 $$-$$ $$\alpha$$<br><br>For infinitely many solution<br><br>$$\beta$$ + 1 = 0 = 3 $$-$$ $$\alpha$$ $$\Rightarrow$$ ($$\alpha$$, $$\beta$$) = (3, $$-$$1)<br><br>Hence, $$\alpha$$ + $$\beta$$ $$-$$ $$\alpha$$$$\beta$$ = 5 |
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 :
Options:
[{"identifier": "A", "content": "R"}, {... | ["A"]
Explanation:
$$D = \left| {\matrix{
1 & 1 & 1 \cr
3 & 2 & 5 \cr
9 & 4 & {28 + [\lambda ]} \cr
} } \right| = - 24 - [\lambda ] + 15 = - [\lambda ] - 9$$<br><br>if $$[\lambda ] + 9 \ne 0$$ then unique solution<br><br>if $$[\lambda ] + 9 = 0$$ then D<sub>1</sub> = D<sub>2<... |
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 :
Options:
[{"identifier": "A", "content": "$$a = - {1 \\over 3},b \\ne {7 \\over 3}$$"}, {"identifier": "B", "content": "$$a \\ne {1 \\over 3},b = {7 \\over 3}$$"}, {"... | ["D"]
Explanation:
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>$${D_3} = \left| {\matrix{
2 & 1 & 5 \cr
1 & { - 1} & 3 \cr
... |
If $$\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$$)y + z = 0<br/><br/>has :
Options:
[{"identifier": "A", "content": "no solution"}, {"... | ["B"]
Explanation:
<p>Given $$\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... |
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... | ["C"]
Explanation:
$$\Delta = \left| {\matrix{
{ - 1} & 1 & 2 \cr
3 & { - a} & 5 \cr
2 & { - 2} & { - a} \cr
} } \right|$$<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><... |
<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>
Options:
[{"identifier": "A", "content": "$$-$$3"}, {"identifier": "B", "content": "3"}, {"identif... | ["B"]
Explanation:
<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>$${\Delta _1} = \left| {\matrix{
7... |
<p>If the system of linear equations <br/>$$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$$ | is equal to ____________.</p>
Options:
[] | 58
Explanation:
<p>If 2x $$-$$ 3y = $$\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 +... |
<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>
Options:
[{"identifier": "A", "content": "$$\\lambda$$ = 7"}, {"identifier": "B", "content": "$$\\lambda$$ = $$-$$7"}... | ["B"]
Explanation:
<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>$${P_3}:x ... |
<p>Let the system of linear equations <br/>$$x + 2y + z = 2$$, <br/>$$\alpha x + 3y - z = \alpha $$, <br/>$$ - \alpha x + y + 2z = - \alpha $$ <br/>be inconsistent. Then $$\alpha$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "$${5 \\over 2}$$"}, {"identifier": "B", "content": "$$-$$$${5 \\over 2}$$"}, ... | ["D"]
Explanation:
<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>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>
Options:
[{"identifier": "A", "content": "$$\\left( {3,{1 \\over 3}} \\right)$$"}, {"identifier": "B", "content": "$$\\left( { - 3,{1 ... | ["C"]
Explanation:
<p>$$\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 + ... |
<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>
Options:
[{"identifier": "A", "content": "(1, $$-$$3)"}, {"identifier": "B", "content": "($$-$$1, 3)"}, {"identifi... | ["C"]
Explanation:
<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>$$\therefore$$ $$\Delta = \left| {\matrix{
\alpha & 1 & 1 \cr
1 & 2 & 3 \cr
1 & 3 & 5 \cr
} } \right| = 0 \Rightarrow \alpha (1) - ... |
<p>The system of equations</p>
<p>$$ - kx + 3y - 14z = 25$$</p>
<p>$$ - 15x + 4y - kz = 3$$</p>
<p>$$ - 4x + y + 3z = 4$$</p>
<p>is consistent for all k in the set</p>
Options:
[{"identifier": "A", "content": "R"}, {"identifier": "B", "content": "R $$-$$ {$$-$$11, 13}"}, {"identifier": "C", "content": "R $$-$$ {13}"},... | ["D"]
Explanation:
<p>The system may be inconsistent if</p>
<p>$$\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\} $$.</p> |
<p>Let A be a 3 $$\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
} } \... | ["B"]
Explanation:
<p>Let $$A = \left[ {\matrix{
a & b & c \cr
d & e & f \cr
g & h & i \cr
} } \right]$$</p>
<p>$$A = \left[ {\matrix{
1 \cr
1 \cr
0 \cr
} } \right] = \left[ {\matrix{
1 \cr
1 \cr
0 \cr
} } \right] \Rightarrow \left[ {\matrix{
a & b & c \cr
d &... |
<p>Let the system of linear equations</p>
<p>x + y + $$\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$$) and (x$$^ * $$, $$-$$y$$^ * $$) are collinear points, then the sum of absolute values of all p... | ["C"]
Explanation:
<p>Given system of equations</p>
<p>$$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$$) and (1... |
<p>The number of values of $$\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$$y + 5z = 4</p>
<p>is inconsistent, is</p>
Options:
[{"identifier": "A", "content": "0"}, {"identifier": "B", "content": "1"}, {"identifier": "C",... | ["B"]
Explanation:
$\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ \alpha & 2 \alpha & 3 \\ 1 & 3 \alpha & 5\end{array}\right|$
<br/><br/>
$$
\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... |
<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>
Options:
[{"identifier": "A", "content": "6"}, {"identifier": "B", ... | ["B"]
Explanation:
<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>$$\therefore$$ $$D = \left| {\matrix{
{3\sin 3\theta } & { - 1} & 1 \cr
{3\cos 2\theta } & 4 & 3 \c... |
<p>The number of real values of $$\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$$ |)z = 16</p>
<p>has no solutions, is</p>
Options:
[{"identifier": "A", "content": "0"}, {"identifier": "B", ... | ["C"]
Explanation:
<p>$$\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 ... |
<p>If the system of linear equations.</p>
<p>$$8x + y + 4z = - 2$$</p>
<p>$$x + y + z = 0$$</p>
<p>$$\lambda x - 3y = \mu $$</p>
<p>has infinitely many solutions, then the distance of the point $$\left( {\lambda ,\mu , - {1 \over 2}} \right)$$ from the plane $$8x + y + 4z + 2 = 0$$ is :</p>
Options:
[{"identifier": "... | ["D"]
Explanation:
<p>$$\Delta = \left| {\matrix{
8 & 1 & 4 \cr
1 & 1 & 1 \cr
\lambda & { - 3} & 0 \cr
} } \right|$$</p>
<p>$$ = 8(3) - 1( - \lambda ) + 4( - 3 - \lambda )$$</p>
<p>$$ = 24 + \lambda - 12 - 4\lambda $$</p>
<p>$$ = 12 - 3\lambda $$</p>
<p>So for $$\lambda = 4$$, it is having infinit... |
<p>Let A and B be two $$3 \times 3$$ non-zero real matrices such that AB is a zero matrix. Then</p>
Options:
[{"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"}, {"ide... | ["B"]
Explanation:
<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> |
<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>
Options:
[{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "36"}, {"identifier": "C", ... | ["C"]
Explanation:
<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, $$\Delta... |
<p>For the system of linear equations $$\alpha x+y+z=1,x+\alpha y+z=1,x+y+\alpha z=\beta$$, which one of the following statements is <b>NOT</b> correct?</p>
Options:
[{"identifier": "A", "content": "It has infinitely many solutions if $$\\alpha=1$$ and $$\\beta=1$$"}, {"identifier": "B", "content": "It has infinitely ... | ["B"]
Explanation:
For infinite solution $\Delta=\Delta_x=\Delta_y=\Delta_z=0$
<br/><br/>$$
\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
$$
<br/><br/>If $\beta=1$, then all planes are overla... |
<p>Let $$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)$$ is equal to</p>
Options:
[{"identifier": "A",... | ["D"]
Explanation:
$\left|\begin{array}{lll}\lambda & 1 & 1 \\ 1 & \lambda & 1 \\ 1 & 1 & \lambda\end{array}\right|=0$
<br/><br/>$$
\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 ... |
<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>
Options:
[{"identifier": "A", "content": "If $$\\alpha=\\beta=7$$, then the system has no solution"}, {"identifier": "B", "content": "For every point... | ["B"]
Explanation:
$\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ \alpha & \beta & 7 \\ 1 & 2 & 3\end{array}\right|$
<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 ... |
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 :
Options:
[{"identifier": "A", "content": "$x^... | ["D"]
Explanation:
<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>$$ = \left| {\matrix{
1 & { - 1} & 5 \cr
2 & 2 & 8 \cr
3 & { - 1} & \beta \cr
} } \righ... |
<p>Let the system of linear equations</p>
<p>$$x+y+kz=2$$</p>
<p>$$2x+3y-z=1$$</p>
<p>$$3x+4y+2z=k$$</p>
<p>have infinitely many solutions. Then the system</p>
<p>$$(k+1)x+(2k-1)y=7$$</p>
<p>$$(2k+1)x+(k+5)y=10$$</p>
<p>has :</p>
Options:
[{"identifier": "A", "content": "unique solution satisfying $$x-y=1$$"}, {"ident... | ["D"]
Explanation:
<p>$$x + y + kz = 2$$ ............(i)</p>
<p>$$2x + 3y - z = 1$$ ..........(ii)</p>
<p>$$3x + 4y + 2z = k$$ ......(iii)</p>
<p>(1) + (2)</p>
<p>$$3x + 4y + z(k - 1) = 3$$</p>
<p>Comparing with (3)</p>
<p>$$k = 3$$</p>
<p>Now, $$4x + 5y = 7$$</p>
<p>$$ \Rightarrow 3x + 3y = 3$$</p>
<p>$$7x + 8y = 10$... |
<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>
Options:
[{"identifier": "A", "content": "It has a solution for all $$\\alpha\\ne-1$$ and ... | ["C"]
Explanation:
$D=\left|\begin{array}{ccc}\alpha & 2 & 1 \\ 2 \alpha & 3 & 1 \\ 3 & \alpha & 2\end{array}\right|=0 \Rightarrow \alpha=-1,3$
<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}\alph... |
<p>Let S$$_1$$ and S$$_2$$ be respectively the sets of all $$a \in \mathbb{R} - \{ 0\} $$ for which the system of linear equations</p>
<p>$$ax + 2ay - 3az = 1$$</p>
<p>$$(2a + 1)x + (2a + 3)y + (a + 1)z = 2$$</p>
<p>$$(3a + 5)x + (a + 5)y + (a + 2)z = 3$$</p>
<p>has unique solution and infinitely many solutions. Then</... | ["C"]
Explanation:
Given system of equations
<br/><br/>
$$
\begin{aligned}
& a x+2 a y-3 a z=1 \\\\
& (2 a+1) x+(2 a+3) y+(a+1) z=2 \\\\
& (3 a+5) x+(a+5) y+(a+2) z=3 \\\\
& \text { Let } A=\left|\begin{array}{ccc}
a & 2 a & -3 a \\\\
2 a+1 & 2 a+3 & a+1 \\\\
3 a+5 & a+5 & a+2
\end{array}\right| \\\\
& =a\left|\begin{... |
<p>If the system of equations</p>
<p>$$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$$) is equal to :</p>
Options:
[{"identifier": "A", "content": "$$\\left( {{{72} \\over 5},{{21} \\over 5}} \\right)$$"}, {"identifier": "B... | ["D"]
Explanation:
For infinite many solution, $\Delta=0$ and $\Delta_x=0$
<br/><br/>$$
\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 ... |
<p>If the system of equations</p>
<p>$$2 x+y-z=5$$</p>
<p>$$2 x-5 y+\lambda z=\mu$$</p>
<p>$$x+2 y-5 z=7$$</p>
<p>has infinitely many solutions, then $$(\lambda+\mu)^{2}+(\lambda-\mu)^{2}$$ is equal to</p>
Options:
[{"identifier": "A", "content": "916"}, {"identifier": "B", "content": "912"}, {"identifier": "C", "cont... | ["A"]
Explanation:
$$
\begin{aligned}
& 2 x+y-z=5 \\
& 2 x-5 y+\lambda z=\mu \\
& x+2 y-5 z=7
\end{aligned}
$$
<br/><br/>For infinite solution $\Delta=0=\Delta_1=\Delta_2=\Delta_3$
<br/><br/>$$
\Delta=\left|\begin{array}{ccc}
2 & 1 & -1 \\
2 & -5 & \lambda \\
1 & 2 & -5
\end{array}\right|=0
$$
<br/><br/>$$
\begin{alig... |
<p>For the system of linear equations</p>
<p>$$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$$</p>
<p>which of the following is NOT correct?</p>
Options:
[{"identifier": "A", "content": "It has infinitely many solutions if $$a=3, b=8$$"}, {"identifier": "B", "content": "It has infinitely many solutio... | ["B"]
Explanation:
The given system of equations is :
<br/><br/>1. $$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{b... |
<p>If the system of linear equations</p>
<p>$$
\begin{aligned}
& 7 x+11 y+\alpha z=13 \\\\
& 5 x+4 y+7 z=\beta \\\\
& 175 x+194 y+57 z=361
\end{aligned}
$$</p>
<p>has infinitely many solutions, then $$\alpha+\beta+2$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "6"}, {"identifier": "B", "con... | ["B"]
Explanation:
Given,
<br/><br/>$$
\begin{aligned}
& 7 x+11 y+\alpha z=13 \\\\
& 5 x+4 y+7 z=\beta \\\\
& 175 x+194 y+57 z=361
\end{aligned}
$$
<br/><br/>$$
\text { For infinite solution, }\left|\begin{array}{ccc}
7 & 11 & \alpha \\
5 & 4 & 7 \\
175 & 194 & 57
\end{array}\right|=0
$$
<br/><br/>$$
\Rightarrow\left... |
<p>Let $$\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|$$ is equal to ___________.</p>
Options:
... | 24
Explanation:
Given that $S$ be the set of values of $\lambda$ for which given system of equations has no solution.
<br/><br/>Therefore for the given set of equations
<br/><br/>$$
\Delta=\left|\begin{array}{ccc}
6 \lambda & -3 & 3 \\
2 & 6 \lambda & 4 \\
3 & 2 & 3 \lambda
\end{array}\right|=0
$$
<br/><br/>$$
\begin{... |
<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>
Options:
[{"identifier": "A", "content": "The system has infinitely many solutions for $$\\alpha=-6$$ and $$\\beta=9$$"}, {"identi... | ["A"]
Explanation:
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 equat... |
<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... | ["D"]
Explanation:
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/>$$
\begin{aligned}
& \Rightarrow 1\left(\tan ^2... |
<p>If the system of equations</p>
<p>$$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$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "28"}, {"identifier": "B", "content": "25"}, {"identifier": "C", "content": "20"}, {"identifier": "D", "co... | ["D"]
Explanation:
Given system of equations,
<br/><br/>$$
\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{align... |
<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>
Options:
[{"identifier": "A", "content": "System has a unique solution for $$\\alpha=3,\\beta\\ne14$$."}, {"identifier": "B", "content": "System has infinitely m... | ["A"]
Explanation:
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/>$$
\begin{aligned}
\Delta & =\left|\begin{array}{lll}
1 & 1 & 1 \\
1 & 2 & \alpha \\
1 & 3 & 5
\end{ar... |
Let the system of equations $x+2 y+3 z=5,2 x+3 y+z=9,4 x+3 y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda+2 \mu$ is equal to :
Options:
[{"identifier": "A", "content": "22"}, {"identifier": "B", "content": "17"}, {"identifier": "C", "content": "15"}, {"identifier": "D", "content": "28"}] | ["B"]
Explanation:
$$
\begin{aligned}
& x+2 y+3 z=5 \\\\
& 2 x+3 y+z=9 \\\\
& 4 x+3 y+\lambda z=\mu
\end{aligned}
$$
<br/><br/>For infinite following $\Delta=\Delta_1=\Delta_2=\Delta_3=0$
<br/><br/>$\begin{aligned} & \Delta=\left|\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 1 \\ 4 & 3 & \lambda\end{array}\right|=0 \Rightarr... |
If the system of equations
<br/><br/>$$
\begin{aligned}
& 2 x+3 y-z=5 \\\\
& x+\alpha y+3 z=-4 \\\\
& 3 x-y+\beta z=7
\end{aligned}
$$
<br/><br/>has infinitely many solutions, then $13 \alpha \beta$ is equal to :
Options:
[{"identifier": "A", "content": "1110"}, {"identifier": "B", "content": "1120"}, {"id... | ["B"]
Explanation:
$\begin{aligned} & \text { Given } 2 x+3 y-z=5 \\\\ & x+\alpha y+3 z=-4 \\\\ & 3 x-y+\beta z=7 \\\\ & \Delta_2=\left|\begin{array}{ccc}2 & -1 & 5 \\ 1 & 3 & -4 \\ 3 & \beta & 7\end{array}\right| \\\\ & \Delta_2=2(21+4 \beta)+1(7+12)+5(\beta-9) \\\\& \Delta_2=42+8 \beta+19+5 \beta-45 \\\\ & \Delta_2... |
<p>Let $$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 \... | ["C"]
Explanation:
<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>$$\therefore... |
<p>If the system of linear equations</p>
<p>$$\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$$ is equal to</p>
Options:
[{"identifier": "A", "content": "60"}, {"identifier": "B", "content": "54"}, {"i... | ["D"]
Explanation:
<p>$$\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}_... |
<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>$$x+2 y+3 z... | 113
Explanation:
<p>$$\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{aligne... |
<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>
Options:
[{"identifier": "A", "content": "System is consistent if $$\\lambda \\neq 1$$ and $$\\mu=13$$\n"}, {"identifier": ... | ["C"]
Explanation:
<p>$$\begin{aligned}
& \left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & 2 & \lambda^2 \\
1 & 3 & \lambda
\end{array}\right|=0 \\
& \Rightarrow 2 \lambda^2-\lambda-1=0 \\
& \lambda=1,-\frac{1}{2} \\
& \left|\begin{array}{ccc}
1 & 1 & 5 \\
2 & \lambda^2 & 9 \\
3 & \lambda & \mu
\end{array}\right|=0 \Rightarr... |
<p>Consider the system of linear equations $$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}$$. Which one of the following statements is NOT correct ?</p>
Options:
[{"identifier": "A", "content": "The system has unique solution if $$\\lambda \\neq \\frac{1}{2}$... | ["D"]
Explanation:
<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 $$\Delta \neq 0,2 \lambda-1 \neq 0,\left(\lambda \neq \frac{1}... |
<p>Consider the matrices : $$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$$ a... | 450
Explanation:
<p>$$\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 \Rightarr... |
<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>
Options:
[{"identifier": "A", "content": "24"}, {"id... | ["B"]
Explanation:
<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>$$\begin{aligned}
& {\left[\begin{array}{ccc}
3 & 5 & \lambda \\
7 & 11 & -9 \\
97 & 155 & -189
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{... |
<p>If the system of equations</p>
<p>$$\begin{aligned}
& x+(\sqrt{2} \sin \alpha) y+(\sqrt{2} \cos \alpha) z=0 \\
& x+(\cos \alpha) y+(\sin \alpha) z=0 \\
& x+(\sin \alpha) y-(\cos \alpha) z=0
\end{aligned}$$</p>
<p>has a non-trivial solution, then $$\alpha \in\left(0, \frac{\pi}{2}\right)$$ is equal to :</... | ["A"]
Explanation:
<p>$$\begin{aligned}
& x+(\sqrt{2} \sin \alpha) y+(\sqrt{2} \cos \alpha) z=0 \\
& x+(\cos \alpha) y+(\sin \alpha) z=0 \\
& x+(\sin \alpha) y-(\cos \alpha) z=0
\end{aligned}$$</p>
<p>$$\because$$ Non-trivial solution</p>
<p>$$\Rightarrow D=0$$</p>
<p>$$\begin{aligned}
& \left|\begin{array}{ccc}
1 & \... |
<p>If the system of equations $$x+4 y-z=\lambda, 7 x+9 y+\mu z=-3,5 x+y+2 z=-1$$ has infinitely many solutions, then $$(2 \mu+3 \lambda)$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "$$-2$$"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "$$-3... | ["D"]
Explanation:
<p>$$\begin{aligned}
& x+4 y-z=\lambda \\
& 7 x+9 y+\mu z=-3 \\
& 5 x+y+2 z=-1 \\
& {\left[\begin{array}{ccc}
1 & 4 & -1 \\
7 & 9 & \mu \\
5 & 1 & 2
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{c}
\lambda \\
-3 \\
-1
\end{array}\right]} \\
& A=\left[\be... |
<p>If the system of equations</p>
<p>$$\begin{array}{r}
11 x+y+\lambda z=-5 \\
2 x+3 y+5 z=3 \\
8 x-19 y-39 z=\mu
\end{array}$$</p>
<p>has infinitely many solutions, then $$\lambda^4-\mu$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "51"}, {"identifier": "B", "content": "45"}, {"identifier": "C", "conte... | ["C"]
Explanation:
<p>$$\begin{aligned}
& 11 x+y+\lambda z=-5 \\
& 2 x+3 y+5 z=3 \\
& 8 x-19 y-39 z=\mu \\
& \Delta=0 \Rightarrow\left|\begin{array}{ccc}
11 & 1 & \lambda \\
2 & 3 & 5 \\
8 & -19 & -39
\end{array}\right|=0 \\
& 11(-39.3+19.5)-1(-39.2-40)+\lambda(-38-24)=0 \\
& =11(-117+95)-1(-118)-62 \lambda=0 \\
& =-2... |
<p>The values of $$m, n$$, for which the system of equations</p>
<p>$$\begin{aligned}
& x+y+z=4, \\
& 2 x+5 y+5 z=17, \\
& x+2 y+\mathrm{m} z=\mathrm{n}
\end{aligned}$$</p>
<p>has infinitely many solutions, satisfy the equation :</p>
Options:
[{"identifier": "A", "content": "$$\\mathrm{m}^2+\\mathrm{n}^2-\... | ["C"]
Explanation:
<p>The given system of linear equations can be represented as,</p>
<p>$$\begin{aligned}
& \left(\begin{array}{ccc|c}
1 & 1 & 1 & 4 \\
2 & 5 & 5 & 17 \\
1 & 2 & m & n
\end{array}\right) \\
& \sim\left(\begin{array}{ccc|c}
1 & 1 & 1 & 4 \\
0 & 3 & 3 & 9 \\
0 & 1 & m-1 & n-4
\end{array}\right) \\
& \si... |
<p>If the system of equations</p>
<p>$$\begin{aligned}
& 2 x+7 y+\lambda z=3 \\
& 3 x+2 y+5 z=4 \\
& x+\mu y+32 z=-1
\end{aligned}$$</p>
<p>has infinitely many solutions, then $$(\lambda-\mu)$$ is equal to ______ :</p>
Options:
[] | 38
Explanation:
<p>To determine if the system of equations:</p>
<p>$$\begin{aligned} 2x + 7y + \lambda z = 3 \\ 3x + 2y + 5z = 4 \\ x + \mu y + 32z = -1 \end{aligned}$$</p>
<p>has infinitely many solutions, we must use Cramer's rule.</p>
<p>The determinants are calculated as follows:</p>
<p>$$\begin{aligned} \Delt... |
<p>Let $$\alpha \beta \gamma=45 ; \alpha, \beta, \gamma \in \mathbb{R}$$. If $$x(\alpha, 1,2)+y(1, \beta, 2)+z(2,3, \gamma)=(0,0,0)$$ for some $$x, y, z \in \mathbb{R}, x y z \neq 0$$, then $$6 \alpha+4 \beta+\gamma$$ is equal to _________.</p>
Options:
[] | 55
Explanation:
<p>Given that $\alpha \beta \gamma = 45$ and $\alpha, \beta, \gamma \in \mathbb{R}$, consider the equation $x(\alpha, 1, 2) + y(1, \beta, 2) + z(2, 3, \gamma) = (0, 0, 0)$ for some $x, y, z \in \mathbb{R}$ where $x y z \neq 0$. To find the value of $6 \alpha + 4 \beta + \gamma$, follow these steps:</p>... |
Let $$A$$ and $$B$$ be two symmetric matrices of order $$3$$.
<br/><br><b>Statement - 1 :</b> $$A(BA)$$ and $$(AB)$$$$A$$ are symmetric matrices.
<br/><br><b>Statement - 2 :</b> $$AB$$ is symmetric matrix if matrix multiplication of $$A$$ with $$B$$ is commutative.</br></br>
Options:
[{"identifier": "A", "content":... | ["A"]
Explanation:
$$\therefore$$ $$A' = A,B' = B$$
<br><br>Now $$\,\,\,\left( {A\left( {BA} \right)} \right)' = \left( {BA} \right)'A'$$
<br><br>$$ = \left( {A'B'} \right)A' = \left( {AB} \right)A = A\left( {BA} \right)$$
<br><br>Similarly $$\left( {\left( {AB} \right)A} \right)' = \left( {AB} \right)A$$
<br><br>S... |
If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = $$\left[ {\matrix{
2 & 3 \cr
5 & { - 1} \cr
} } \right]$$, then AB is equal
to :
Options:
[{"identifier": "A", "content": "$$\\left[ {\\matrix{\n 4 & { - 2} \\cr \n 1 & { - 4} \\cr \n\n } } \\right]$$"}, {"... | ["D"]
Explanation:
$$A + B = \left[ {\matrix{
2 & 3 \cr
5 & { - 1} \cr
} } \right] = P(say)$$<br><br>
Now $$A = {{P + {P^T}} \over 2}\& B = {{P - {P^T}} \over 2}$$<br><br>
So $$A = {1 \over 2}\left( {\left[ {\matrix{
2 & 3 \cr
5 & { - 1} \cr
} } \right] + \left[ {\matrix{
... |
Let A and B be 3 $$\times$$ 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A<sup>2</sup>B<sup>2</sup> $$-$$ B<sup>2</sup>A<sup>2</sup>) X = O, where X is a 3 $$\times$$ 1 column matrix of unknown variables and O is a 3 $$\times$$ 1 null matrix, has :... | ["C"]
Explanation:
A<sup>T</sup> = A, B<sup>T</sup> = $$-$$B<br><br>Let A<sup>2</sup>B<sup>2</sup> $$-$$ B<sup>2</sup>A<sup>2</sup> = P<br><br>P<sup>T</sup> = (A<sup>2</sup>B<sup>2</sup> $$-$$ B<sup>2</sup>A<sup>2</sup>)<sup>T</sup> = (A<sup>2</sup>B<sup>2</sup>)<sup>T</sup> $$-$$ (B<sup>2</sup>A<sup>2</sup>)<sup>T</s... |
Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A<sup>2</sup> is 1, then the possible number of such matrices is :
Options:
[{"identifier": "A", "content": "6"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": ... | ["B"]
Explanation:
Let $$A = \left[ {\matrix{
a & b \cr
b & c \cr
} } \right]$$<br><br>$${A^2} = \left[ {\matrix{
a & b \cr
b & c \cr
} } \right]\left[ {\matrix{
a & b \cr
b & c \cr
} } \right] = \left[ {\matrix{
{{a^2} + {b^2}} & {ab + bc} \cr
{a... |
Let $$A = \left[ {\matrix{
2 & 3 \cr
a & 0 \cr
} } \right]$$, a$$\in$$R be written as P + Q where P is a symmetric matrix and Q is skew symmetric matrix. If det(Q) = 9, then the modulus of the sum of all possible values of determinant of P is equal to :
Options:
[{"identifier": "A", "content": "36"... | ["A"]
Explanation:
$$A = \left[ {\matrix{
2 & 3 \cr
a & 0 \cr
} } \right]$$, $${A^T} = \left[ {\matrix{
2 & a \cr
3 & 0 \cr
} } \right]$$<br/><br/>$$A = {{A + {A^T}} \over 2} + {{A - {A^T}} \over 2}$$<br/><br/>Let $$P = {{A + {A^T}} \over 2}$$ and $$Q = {{A - {A^T}} \over 2}$$<br/><br/>$$Q = \l... |
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