Solution of System of Linear Equations (Matrix Method and Cramer's Rule)

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JEE Main 2016

The system of linear equations $x + λ y - z = 0$ , $λ x - y - z = 0$ , $x + y - λ z = 0$ has a non-trivial solution for:

λ

(B)

exactly three values of

λ

(C)

infinitely many values of

λ

(D)

exactly one value of

λ

JEE Main 2015

LEVELJEE Main

The set of all values of $λ$ for which the system of linear equations: $2 x_{1} - 2 x_{2} + x_{3} = λ x_{1}$ , $2 x_{1} - 3 x_{2} + 2 x_{3} = λ x_{2}$ , $- x_{1} + 2 x_{2} = λ x_{3}$ has a non-trivial solution

(A)

contains two elements

(B)

contains more than two elements

The number of values of $k$ , for which the system of equations : $(k + 1) x + 8 y = 4 k; k x + (k + 3) y = 3 k - 1$ has no solution, is

The number of values of $k$ for which the linear equations $4 x + k y + 2 z = 0$ , $k x + 4 y + z = 0$ and $2 x + 2 y + z = 0$ possess a non-zero solution is

(A)

2

(B)

1

(C)

zero

(D)

3

JEE Main 2010

LEVELJEE Main

Consider the system of linear equations; $x_{1} + 2 x_{2} + x_{3} = 3$ , $2 x_{1} + 3 x_{2} + x_{3} = 3$ , $3 x_{1} + 5 x_{2} + 2 x_{3} = 1$ . The system has

infinite number of solutions

JEE Main 2008

LEVELBoard

Let $a, b, c$ be any real numbers. Suppose that there are real numbers $x, y, z$ not all zero such that $x = cy + b z, y = a z + c x$ , and $z = b x + a y$ . Then $a^{2} + b^{2} + c^{2} + 2 ab c$ is equal to

(A)

2

(B)

- 1

(C)

0

(D)

1

JEE Main 2005

LEVELJEE Main

The system of equations $α x + y + z = α - 1$ , $x + α y + z = α - 1$ , $x + y + α z = α - 1$ has infinite solutions, if $α$ is

(A)

- 2

(B)

either

- 2

1

(C)

not

- 2

(D)

1

JEE Main 2003

LEVELJEE Main

If the system of linear equations $x + 2 a y + a z = 0$ ; $x + 3 b y + b z = 0$ ; $x + 4 cy + cz = 0$ ; has a non-zero solution, then $a, b, c$

(A)

satisfy

a + 2 b + 3 c = 0

Let $a, λ, μ \in R$ . Consider the system of linear equations $a x + 2 y = λ, 3 x - 2 y = μ$ . Which of the following statement(s) is (are) correct?

* Multiple Correct Options

(A)

a = - 3

, then the system has infinitely many solutions for all values of

λ

and

μ

(B)

a \neq = - 3

, then the system has a unique solution for all values of

λ

and

μ

(C)

λ + μ = 0

, then the system has infinitely many solutions for

a = - 3

(D)

λ + μ \neq = 0

, then the system has no solution for

a = - 3

JEE Advanced 2010

LEVELJEE Main

The number of $3 \times 3$ matrices $A$ whose entries are either 0 or 1 and for which the system $A x y z = 100$ has exactly two distinct solutions, is

2^{9} - 1

Consider the system of equations $a x + b y = 0, c x + d y = 0$ , where $a, b, c, d \in {0, 1}$ . STATEMENT - 1 : The probability that the system of equations has a unique solution is $3/8$ . and STATEMENT - 2 : The probability that the system of equations has a solution is 1.

(A)

STATEMENT - 1 is True, STATEMENT - 2 is True, STATEMENT - 2 is a correct explanation for STATEMENT- 1

(B)

STATEMENT - 1 is True, STATEMENT - 2 is True, STATEMENT - 2 is NOT a correct explanation for STATEMENT- 1

(C)

STATEMENT - 1 is True, STATEMENT - 2 is False

(D)

STATEMENT - 1 is False, STATEMENT - 2 is True

JEE Advanced 2008

LEVELJEE Main

Consider the system of equations $x - 2 y + 3 z = - 1$ , $- x + y - 2 z = k$ , $x - 3 y + 4 z = 1$ . STATEMENT - 1 : The system of equations has no solution for $k \neq = 3$ and STATEMENT - 2 : The determinant $1 - 1 1 3 - 2 4 - 1 k 1 \neq = 0$ , for $k \neq = 3$ .

(A)

STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is a correct explanation for STATEMENT - 1

(B)

STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is NOT a correct explanation for STATEMENT - 1

(C)

STATEMENT - 1 is True, STATEMENT - 2 is False

(D)

STATEMENT - 1 is False, STATEMENT - 2 is True

JEE Advanced 2004

LEVELJEE Main

Given $2 x - y + 2 z = 2$ , $x - 2 y + z = - 4$ , $x + y + λ z = 4$ then the value of $λ$ such that the given system of equation has NO solution, is

- 3

JEE Advanced 2003

LEVELJEE Main

If the system of equations $x + a y = 0$ , $a z + y = 0$ and $a x + z = 0$ has infinite solutions, then the value of $a$ is

(A)

(a)

- 1

The number of values of $k$ for which the system of equations $(k + 1) x + 8 y = 4 k$ ; $k x + (k + 3) y = 3 k - 1$ has infinitely many solutions is

If the system of equations $x - k y - z = 0$ , $k x - y - z = 0$ , $x + y - z = 0$ has a non-zero solution, then the possible values of $k$ are

(A)

(a)

- 1, 2

(B)

(b)

1, 2

(C)

(c)

0, 1

(D)

(d)

- 1, 1

JEE Advanced 1995

LEVELJEE Main

Let $a, b, c$ be the real numbers. Then following system of equations in $x, y$ and $z$ $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} - \frac{z ^{2}}{c ^{2}} = 1$ , $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} + \frac{z ^{2}}{c ^{2}} = 1$ , $- \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} + \frac{z ^{2}}{c ^{2}} = 1$ has

(d) finitely many solutions

JEE Advanced 1979

LEVELBoard

The equation $x + 2 y + 2 z = 1$ and $2 x + 4 y + 4 z = 9$ have

Infinite number of solutions

Let $(x, y, z)$ be points with integer coordinates satisfying the system of homogeneous equations : $3 x - y - z = 0; - 3 x + z = 0; - 3 x + 2 y + z = 0$ . Then the number of such points for which $x^{2} + y^{2} + z^{2} \leq 100$ is .........

JEE Advanced 1984

LEVELBoard

The system of equations $λ x + y + z = 0$ , $- x + λ y + z = 0$ , $- x - y + λ z = 0$ will have a non-zero solution if real values of $λ$ are given by .........

JEE Advanced 2004

LEVELJEE Advanced

If $A = a 11 1 b b 0 d c, B = a 0 f 1 d g 1 c h, U = f g h, V = a^{2} 00, X = x y z$ and $A X = U$ has infinitely many solutions, prove that $B X = V$ has no unique solution. Also show that if $a f d \neq = 0$ , then $B X = V$ has no solution.

JEE Advanced 1993

LEVELJEE Main

Let $λ$ and $α$ be real. Find the set of all values of $λ$ for which the system of linear equations $λ x + (sin α) y + (cos α) z = 0, x + (cos α) y + (sin α) z = 0, - x + (sin α) y - (cos α) z = 0$ has a non-trivial solution. For $λ = 1$ , find all values of $α$ .

JEE Advanced 1986

LEVELJEE Main

Consider the system of linear equations in $x, y, z$ : $(sin 3 θ) x - y + z = 0, (cos 2 θ) x + 4 y + 3 z = 0, 2 x + 7 y + 7 z = 0$ . Find the values of $θ$ for which this system has nontrivial solutions.

JEE Advanced 1982

LEVELJEE Main

Find all values of $λ$ , such that $x, y, z \neq = (0, 0, 0)$ and $(\hat{i} + \hat{j} + 3 \hat{k}) x + (3 \hat{i} - 3 \hat{j} + \hat{k}) y + (- 4 \hat{i} + 5 \hat{j}) z = λ (x \hat{i} + y \hat{j} + z \hat{k})$ where $\hat{i}, \hat{j}, \hat{k}$ are unit vectors along the coordinate axes.

JEE Advanced 1980

LEVELJEE Main

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The system of linear equations x+λy−z=0, λx−y−z=0, x+y−λz=0 has a non-trivial solution for:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of values of k, for which the system of equations : (k+1)x+8y=4k;kx+(k+3)y=3k−1 has no solution, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of values of k for which the linear equations 4x+ky+2z=0, kx+4y+z=0 and 2x+2y+z=0 possess a non-zero solution is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider the system of linear equations; x1​+2x2​+x3​=3, 2x1​+3x2​+x3​=3, 3x1​+5x2​+2x3​=1. The system has

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be any real numbers. Suppose that there are real numbers x,y,z not all zero such that x=cy+bz,y=az+cx, and z=bx+ay. Then a2+b2+c2+2abc is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The system of equations αx+y+z=α−1, x+αy+z=α−1, x+y+αz=α−1 has infinite solutions, if α is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the system of linear equations x+2ay+az=0; x+3by+bz=0; x+4cy+cz=0; has a non-zero solution, then a,b,c

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,λ,μ∈R. Consider the system of linear equations ax+2y=λ,3x−2y=μ. Which of the following statement(s) is (are) correct?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of 3×3 matrices A whose entries are either 0 or 1 and for which the system A​xyz​​=​100​​ has exactly two distinct solutions, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given 2x−y+2z=2, x−2y+z=−4, x+y+λz=4 then the value of λ such that the given system of equation has NO solution, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the system of equations x+ay=0, az+y=0 and ax+z=0 has infinite solutions, then the value of a is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of values of k for which the system of equations (k+1)x+8y=4k; kx+(k+3)y=3k−1 has infinitely many solutions is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the system of equations x−ky−z=0, kx−y−z=0, x+y−z=0 has a non-zero solution, then the possible values of k are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be the real numbers. Then following system of equations in x,y and z a2x2​+b2y2​−c2z2​=1, a2x2​−b2y2​+c2z2​=1, −a2x2​+b2y2​+c2z2​=1 has

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation x+2y+2z=1 and 2x+4y+4z=9 have

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The system of equations λx+y+z=0, −x+λy+z=0, −x−y+λz=0 will have a non-zero solution if real values of λ are given by .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider the system of linear equations in x,y,z: (sin3θ)x−y+z=0,(cos2θ)x+4y+3z=0,2x+7y+7z=0. Find the values of θ for which this system has nontrivial solutions.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For what values of m, does the system of equations 3x+my=m;2x−5y=20 has solution satisfying the conditions x>0,y>0.

The system of linear equations $x + λ y - z = 0$ , $λ x - y - z = 0$ , $x + y - λ z = 0$ has a non-trivial solution for:

The number of values of $k$ , for which the system of equations : $(k + 1) x + 8 y = 4 k; k x + (k + 3) y = 3 k - 1$ has no solution, is

The number of values of $k$ for which the linear equations $4 x + k y + 2 z = 0$ , $k x + 4 y + z = 0$ and $2 x + 2 y + z = 0$ possess a non-zero solution is

Consider the system of linear equations; $x_{1} + 2 x_{2} + x_{3} = 3$ , $2 x_{1} + 3 x_{2} + x_{3} = 3$ , $3 x_{1} + 5 x_{2} + 2 x_{3} = 1$ . The system has

Let $a, b, c$ be any real numbers. Suppose that there are real numbers $x, y, z$ not all zero such that $x = cy + b z, y = a z + c x$ , and $z = b x + a y$ . Then $a^{2} + b^{2} + c^{2} + 2 ab c$ is equal to

The system of equations $α x + y + z = α - 1$ , $x + α y + z = α - 1$ , $x + y + α z = α - 1$ has infinite solutions, if $α$ is

If the system of linear equations $x + 2 a y + a z = 0$ ; $x + 3 b y + b z = 0$ ; $x + 4 cy + cz = 0$ ; has a non-zero solution, then $a, b, c$

Let $a, λ, μ \in R$ . Consider the system of linear equations $a x + 2 y = λ, 3 x - 2 y = μ$ . Which of the following statement(s) is (are) correct?

The number of $3 \times 3$ matrices $A$ whose entries are either 0 or 1 and for which the system $A x y z = 100$ has exactly two distinct solutions, is

Given $2 x - y + 2 z = 2$ , $x - 2 y + z = - 4$ , $x + y + λ z = 4$ then the value of $λ$ such that the given system of equation has NO solution, is

If the system of equations $x + a y = 0$ , $a z + y = 0$ and $a x + z = 0$ has infinite solutions, then the value of $a$ is

The number of values of $k$ for which the system of equations $(k + 1) x + 8 y = 4 k$ ; $k x + (k + 3) y = 3 k - 1$ has infinitely many solutions is

If the system of equations $x - k y - z = 0$ , $k x - y - z = 0$ , $x + y - z = 0$ has a non-zero solution, then the possible values of $k$ are

The equation $x + 2 y + 2 z = 1$ and $2 x + 4 y + 4 z = 9$ have

The system of equations $λ x + y + z = 0$ , $- x + λ y + z = 0$ , $- x - y + λ z = 0$ will have a non-zero solution if real values of $λ$ are given by .........

Consider the system of linear equations in $x, y, z$ : $(sin 3 θ) x - y + z = 0, (cos 2 θ) x + 4 y + 3 z = 0, 2 x + 7 y + 7 z = 0$ . Find the values of $θ$ for which this system has nontrivial solutions.

For what values of $m$ , does the system of equations $3 x + m y = m; 2 x - 5 y = 20$ has solution satisfying the conditions $x > 0, y > 0$ .