Consider the following linear equations...

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Advanced 2007Moderate

Consider the following linear equations $a x + b y + cz = 0; b x + cy + a z = 0; c x + a y + b z = 0$ Match the conditions/expressions in Column I with statements in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

List-I

(P)

a + b + c \neq = 0

and

a^{2} + b^{2} + c^{2} = ab + b c + c a

(Q)

a + b + c = 0

and

a^{2} + b^{2} + c^{2} \neq = ab + b c + c a

(R)

a + b + c \neq = 0

and

a^{2} + b^{2} + c^{2} \neq = ab + b c + c a

(S)

a + b + c = 0

and

a^{2} + b^{2} + c^{2} = ab + b c + c a

List-II

(1)

the equations represent planes meeting only at a single point

(2)

the equations represent the line

x = y = z

(3)

the equations represent identical planes.

(4)

the equations represent the whole of the three dimensional space.

Answer:P → 3, Q → 2, R → 1, S → 4

Visualized Solution (English)

System of Homogeneous Equations

Given system of equations:
$a x + b y + cz = 0$
$b x + cy + a z = 0$
$c x + a y + b z = 0$
These are homogeneous linear equations, representing planes passing through the origin $(0, 0, 0)$ .

The Determinant .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Δ

The determinant of the system is:
$Δ = a b c b c a c a b$
This is a cyclic determinant.

Factoring the Determinant

Expanding the determinant:
$Δ = 3 ab c - a^{3} - b^{3} - c^{3}$
Factoring the expression:
$Δ = - (a + b + c) (a^{2} + b^{2} + c^{2} - ab - b c - c a)$

The Sum of Squares Form

Rewriting the second factor:
$a^{2} + b^{2} + c^{2} - ab - b c - c a = \frac{1}{2} [(a - b)^{2} + (b - c)^{2} + (c - a)^{2}]$
This term is zero if and only if $a = b = c$ .

Condition (A): .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a = b = c

Condition (A): $a + b + c \neq = 0$ and $a^{2} + b^{2} + c^{2} = ab + b c + c a$
This implies $a = b = c$ .
Substituting into the equations:
$a x + a y + a z = 0 ⟹ x + y + z = 0$ (for all three)

Case (A) Result: Identical Planes

Since all three equations are the same, they represent identical planes.
Match: (A) $\to$ (r)

Condition (B): .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a + b + c = 0

Condition (B): $a + b + c = 0$ and $a^{2} + b^{2} + c^{2} \neq = ab + b c + c a$
Then $Δ = 0$ , implying infinite solutions.
Summing equations: $(a + b + c) (x + y + z) = 0 ⟹ 0 = 0$ .

Case (B) Result: Line .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x = y = z

Substituting $x = y = z = k$ :
$ak + bk + c k = k (a + b + c) = 0$ .
The equations represent the line $x = y = z$ .
Match: (B) $\to$ (q)

Condition (C): .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Δ \neq = 0

Condition (C): $a + b + c \neq = 0$ and $a^{2} + b^{2} + c^{2} \neq = ab + b c + c a$
This implies $Δ \neq = 0$ .
For a homogeneous system, $Δ \neq = 0$ means only the trivial solution $(0, 0, 0)$ exists.
Match: (C) $\to$ (p)

Condition (D): .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a = b = c = 0

Condition (D): $a + b + c = 0$ and $a^{2} + b^{2} + c^{2} = ab + b c + c a$
This implies $a = b = c$ and $3 a = 0 ⟹ a = b = c = 0$ .
The equations become $0 = 0$ , representing the whole 3D space.
Match: (D) $\to$ (s)

Final Summary

Final Matching Summary:
(A) $\to$ (r): Identical planes
(B) $\to$ (q): Line $x = y = z$
(C) $\to$ (p): Single point (origin)
(D) $\to$ (s): Whole 3D space

00:00 / 00:00

Conceptually Similar Problems

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

(A)

Only one solution

(B)

Only two solutions

(C)

Infinite number of solutions

(D)

None of these

Answer:D

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Advanced 1995Easy

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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

(A)

(a) no solution

(B)

(b) unique solution

(C)

(D)

(d) finitely many solutions

Answer:D

MathematicsEquation of a PlaneJEE Advanced 2015Moderate

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In $R^{3}$ , consider the planes $P_{1} : y = 0$ and $P_{2} : x + z = 1$ . Let $P_{3}$ be the plane, different from $P_{1}$ and $P_{2}$ , which passes through the intersection of $P_{1}$ and $P_{2}$ . If the distance of the point $(0, 1, 0)$ from $P_{3}$ is 1 and the distance of a point $(α, β, γ)$ from $P_{3}$ is 2, then which of the following relations is (are) true?

* Multiple Correct Options

(A)

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(B)

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(C)

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(D)

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MathematicsEquation of a PlaneJEE Advanced 2008Moderate

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Consider three planes $P_{1} : x - y + z = 1$ , $P_{2} : x + y - z = 1$ , $P_{3} : x - 3 y + 3 z = 2$ . Let $L_{1}, L_{2}, L_{3}$ be the lines of intersection of the planes $P_{2}$ and $P_{3}$ , $P_{3}$ and $P_{1}$ , $P_{1}$ and $P_{2}$ , respectively. STATEMENT-1 : At least two of the lines $L_{1}, L_{2}$ and $L_{3}$ are non-parallel and STATEMENT-2 : The three planes do not have a common point.

(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

Answer:D

MathematicsEquation of a PlaneJEE Advanced 2013Moderate

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Consider the lines $L_{1} : \frac{x - 1}{2} = \frac{y}{- 1} = \frac{z + 3}{1}, L_{2} : \frac{x - 4}{1} = \frac{y + 3}{1} = \frac{z + 3}{2}$ and the planes $P_{1} : 7 x + y + 2 z = 3, P_{2} : 3 x + 5 y - 6 z = 4$ . Let $a x + b y + cz = d$ be the equation of the plane passing through the point of intersection of lines $L_{1}$ and $L_{2}$ , and perpendicular to planes $P_{1}$ and $P_{2}$ . Match List I with List II:

List-I

(P)

(Q)

(R)

(S)

List-II

(1)

(2)

-3

(3)

(4)

-2

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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:

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exactly two values of

λ

(B)

exactly three values of

λ

(C)

infinitely many values of

λ

(D)

exactly one value of

λ

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Comprehension Passage

Let

a, b

and

c

be three real numbers satisfying

[a b c] 187923777 = [000] \dots (E)

Question 1:

If the point $P (a, b, c)$ , with reference to (E), lies on the plane $2 x + y + z = 1$ , then the value of $7 a + b + c$ is

(A)

(B)

(C)

(D)

Answer:D

Question 2:

Let $ω$ be a solution of $x^{3} - 1 = 0$ with $Im (ω) > 0$ , if $a = 2$ with $b$ and $c$ satisfying (E), then the value of $\frac{3}{ω ^{a}} + \frac{1}{ω ^{b}} + \frac{3}{ω ^{c}}$ is equal to

(A)

-2

(B)

(C)

(D)

-3

Answer:A

Question 3:

Let $b = 6$ , with $a$ and $c$ satisfying (E). If $α$ and $β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0$ , then $\sum_{n = 0}^{\infty} (\frac{1}{α} + \frac{1}{β})^{n}$ is

(A)

(B)

(C)

6/7

(D)

\infty

Answer:B

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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

(B)

are in A.P

(C)

are in G.P

(D)

are in H.P.

Answer:D

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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 .........

Answer:0

MathematicsVarious Forms of Equations of a LineJEE Advanced 2008Moderate

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Consider the lines given by $L_{1} : x + 3 y - 5 = 0$ ; $L_{2} : 3 x - k y - 1 = 0$ ; $L_{3} : 5 x + 2 y - 12 = 0$ . Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

List-I

(P)

L_{1}, L_{2}, L_{3}

are concurrent, if

(Q)

One of

L_{1}, L_{2}, L_{3}

is parallel to at least one of the other two, if

(R)

L_{1}, L_{2}, L_{3}

form a triangle, if

(S)

L_{1}, L_{2}, L_{3}

do not form a triangle, if

List-II

(1)

k = - 9

(2)

k = - \frac{6}{5}

(3)

k = \frac{5}{6}

(4)

k = 5

Answer:P → 4, P → 2, R → 3, S → NaN

List-I

List-II

Visualized Solution (English)

System of Homogeneous Equations

The Determinant .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Δ

Factoring the Determinant

The Sum of Squares Form

Condition (A): .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a=b=c

Case (A) Result: Identical Planes

Condition (B): .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a+b+c=0

Case (B) Result: Line .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=y=z

Condition (C): .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Δ=0

Condition (D): .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a=b=c=0

Final Summary

Conceptually Similar Problems

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List-I

List-II

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Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the point P(a,b,c), with reference to (E), lies on the plane 2x+y+z=1, then the value of 7a+b+c is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ω be a solution of x3−1=0 with Im(ω)>0, if a=2 with b and c satisfying (E), then the value of ωa3​+ωb1​+ωc3​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let b=6, with a and c satisfying (E). If α and β are the roots of the quadratic equation ax2+bx+c=0, then ∑n=0∞​(α1​+β1​)n 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; } 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 .........

List-I

List-II

List-I

List-II

.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; } 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

List-I

List-II

.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:

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the point P(a,b,c), with reference to (E), lies on the plane 2x+y+z=1, then the value of 7a+b+c is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ω be a solution of x3−1=0 with Im(ω)>0, if a=2 with b and c satisfying (E), then the value of ωa3​+ωb1​+ωc3​ is equal to

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List-I

List-II

The Determinant $Δ$

Condition (A): $a = b = c$

Condition (B): $a + b + c = 0$

Case (B) Result: Line $x = y = z$

Condition (C): $Δ \neq = 0$

Condition (D): $a = b = c = 0$

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

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:

If the point $P (a, b, c)$ , with reference to (E), lies on the plane $2 x + y + z = 1$ , then the value of $7 a + b + c$ is

Let $ω$ be a solution of $x^{3} - 1 = 0$ with $Im (ω) > 0$ , if $a = 2$ with $b$ and $c$ satisfying (E), then the value of $\frac{3}{ω ^{a}} + \frac{1}{ω ^{b}} + \frac{3}{ω ^{c}}$ is equal to

Let $b = 6$ , with $a$ and $c$ satisfying (E). If $α$ and $β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0$ , then $\sum_{n = 0}^{\infty} (\frac{1}{α} + \frac{1}{β})^{n}$ 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 $a, b, c$

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 .........

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

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:

If the point $P (a, b, c)$ , with reference to (E), lies on the plane $2 x + y + z = 1$ , then the value of $7 a + b + c$ is

Let $ω$ be a solution of $x^{3} - 1 = 0$ with $Im (ω) > 0$ , if $a = 2$ with $b$ and $c$ satisfying (E), then the value of $\frac{3}{ω ^{a}} + \frac{1}{ω ^{b}} + \frac{3}{ω ^{c}}$ is equal to

Let $b = 6$ , with $a$ and $c$ satisfying (E). If $α$ and $β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0$ , then $\sum_{n = 0}^{\infty} (\frac{1}{α} + \frac{1}{β})^{n}$ 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 $a, b, c$

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 .........