Why can't we find unique values for $a$, $b$, and $c$?

Because the determinant of the matrix is zero, the system is dependent. This means there are infinitely many solutions, not just one. We find a family of solutions, which is why we use a parameter $a$ to define the others.

How do I handle $\omega$ powers when they are large or negative?

Always use the fundamental property $\omega^3 = 1$. For any power $\omega^n$, simply divide $n$ by 3 and use the remainder. If the power is negative, multiply by $\omega^3$ (or $\omega^6$, etc.) until the exponent becomes positive.

Why does the infinite GP sum work here?

The sum of an infinite geometric progression $S = \frac{A}{1-R}$ is only valid if the common ratio $|R| < 1$. In this problem, our ratio was $6/7$, which is less than 1, so the series converges perfectly.

JEE Advanced 2011

LEVELJEE Advanced

Animated Solution for Mathematics - Matrices and Determinants: 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

Select Answer:

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

Select Answer:

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

Select Answer:

Progressive Hints

Hint 1

The Spark: Do not let the matrix notation intimidate you. This is simply a system of three linear equations in disguise. Write them out explicitly by multiplying the row vector with each column. What do you notice about the third equation compared to the others?

Hint 2

The Arsenal: You have a homogeneous system of equations. Since the right-hand side is zero, you are looking for a relationship between the variables

a

b

, and

c

. Use the third equation to simplify the system immediately—it is the key to unlocking the ratios between the variables.

Hint 3

The Execution: Once you have the relation

a + b + c = 0

, substitute

c = - a - b

into the first equation. This will give you a direct relationship between

b

and

a

. Express everything in terms of

a

to find the parametric solution

(a, 6 a, - 7 a)

Visualized Solution

Decoding the Matrix Equation $(E)$

Given matrix equation: $[a b c] 187923777 = [000]$
This represents a homogeneous system of linear equations.

Generating the System of Equations

Multiplying the row vector by each column of the matrix:
$a (1) + b (8) + c (7) = 0 \Rightarrow a + 8 b + 7 c = 0$
$a (9) + b (2) + c (3) = 0 \Rightarrow 9 a + 2 b + 3 c = 0$
$a (7) + b (7) + c (7) = 0 \Rightarrow 7 a + 7 b + 7 c = 0$

Simplifying the Third Equation

Look at equation (3): $7 a + 7 b + 7 c = 0$
Divide the entire equation by $7$ :
$a + b + c = 0$
Express $c$ in terms of $a$ and $b$ : $c = - a - b$

Solving for $b$ and $c$

Substitute $c = - a - b$ into equation (1):
$a + 8 b + 7 (- a - b) = 0$
$a + 8 b - 7 a - 7 b = 0 \Rightarrow b - 6 a = 0 \Rightarrow b = 6 a$
Substitute $b = 6 a$ back into $c$ :
$c = - a - 6 a = - 7 a$
Parametric solution: $(a, b, c) = (a, 6 a, - 7 a)$

Sub-question 1: Plane Intersection

Point $P (a, b, c)$ lies on the plane $2 x + y + z = 1$ .
Substitute $P (a, 6 a, - 7 a)$ into the plane equation:
$2 (a) + (6 a) + (- 7 a) = 1$

Finding the Value of $a$

Solve for $a$ :
$8 a - 7 a = 1 \Rightarrow a = 1$
Find $b$ and $c$ :
$b = 6 (1) = 6$
$c = - 7 (1) = - 7$

Final Answer for Sub-question 1

We need to find the value of $7 a + b + c$ .
Substitute the values: $a = 1, b = 6, c = - 7$
$7 (1) + 6 + (- 7) = 7 - 1 = 6$
Correct Option: (d)

Sub-question 2: Complex Numbers Setup

Given $a = 2$ .
Using our parametric relations: $b = 6 (2) = 12$ and $c = - 7 (2) = - 14$ .
We need to evaluate: $\frac{3}{ω ^{a}} + \frac{1}{ω ^{b}} + \frac{3}{ω ^{c}}$
Where $ω$ is a complex cube root of unity ( $ω^{3} = 1$ ).

Substituting Values into the Expression

Substitute $a = 2, b = 12, c = - 14$ into the expression:
$\frac{3}{ω ^{2}} + \frac{1}{ω ^{12}} + \frac{3}{ω ^{- 14}}$

Simplifying Powers of $ω$

Using $ω^{3} = 1$ :
$\frac{3}{ω ^{2}} = \frac{3 ω}{ω ^{3}} = 3 ω$
$\frac{1}{ω ^{12}} = \frac{1}{( ω ^{3} ) ^{4}} = \frac{1}{1} = 1$
$\frac{3}{ω ^{- 14}} = 3 ω^{14} = 3 (ω^{3})^{4} \cdot ω^{2} = 3 ω^{2}$

Final Answer for Sub-question 2

The expression becomes: $3 ω + 1 + 3 ω^{2}$
Group the terms: $3 (ω + ω^{2}) + 1$
Recall the property: $1 + ω + ω^{2} = 0 \Rightarrow ω + ω^{2} = - 1$
Result: $3 (- 1) + 1 = - 3 + 1 = - 2$
Correct Option: (a)

Sub-question 3: Quadratic Roots Setup

Given $b = 6$ .
Using $b = 6 a \Rightarrow 6 = 6 a \Rightarrow a = 1$ .
Using $c = - 7 a \Rightarrow c = - 7 (1) = - 7$ .
The quadratic equation is $a x^{2} + b x + c = 0 \Rightarrow x^{2} + 6 x - 7 = 0$ .

Finding the Roots $α$ and $β$

Factorize the quadratic equation:
$x^{2} + 6 x - 7 = 0$
$(x + 7) (x - 1) = 0$
The roots are $α = 1$ and $β = - 7$ .
Calculate the sum of reciprocals:
$\frac{1}{α} + \frac{1}{β} = \frac{1}{1} + \frac{1}{- 7} = 1 - \frac{1}{7} = \frac{6}{7}$

Infinite Geometric Progression

We need to evaluate the infinite sum: $\sum_{n = 0}^{\infty} (\frac{1}{α} + \frac{1}{β})^{n}$
Substitute the value: $\sum_{n = 0}^{\infty} (\frac{6}{7})^{n}$
This is an infinite GP: $1 + \frac{6}{7} + (\frac{6}{7})^{2} + \dots$
Sum formula: $S_{\infty} = \frac{A}{1 - R}$ where $A = 1, R = \frac{6}{7}$
$S_{\infty} = \frac{1}{1 - \frac{6}{7}} = \frac{1}{\frac{1}{7}} = 7$
Correct Option: (b)

00:00 / 00:00

The Sigma Insight: Solution of System of Linear Equations (Matrix Method and Cramer's Rule)

Solution Diagram

Analyzing the Matrix System

We begin with the matrix equation:

[a b c] 187923777 = [000]

This represents a system of linear equations. By performing the matrix multiplication, we extract three distinct constraints.

The third column provides the equation

7 a + 7 b + 7 c = 0

. This simplifies elegantly to:

a + b + c = 0

Solving the Parametric Relationship

Using the constraint

c = - a - b

, we substitute this into the first equation derived from the matrix multiplication, which is

a + 8 b + 7 c = 0

. Substituting

c

yields

a + 8 b + 7 (- a - b) = 0

, which simplifies to

- 6 a + b = 0

, or

b = 6 a

Consequently, we find

c = - a - 6 a = - 7 a

. The entire system is defined by the parametric vector:

(a, 6 a, - 7 a)

This transformation reduces a complex 3D problem into a simple 1D parametric journey.

Geometry of the Plane

For the first sub-question, we consider the plane defined by

2 x + y + z = 1

. Substituting our parametric coordinates

(a, 6 a, - 7 a)

into the plane equation, we get:

2 (a) + (6 a) + (- 7 a) = 1

Solving this linear equation, we find $a = 1$ .

Complex Numbers and Roots of Unity

For the second sub-question, we set

a = 2

, which implies

b = 12

and

c = - 14

. We are tasked with evaluating the expression:

\frac{3}{ω ^{a}} + \frac{1}{ω ^{b}} + \frac{3}{ω ^{c}}

Given the fundamental property

ω^{3} = 1

, we simplify the powers. The term

\frac{1}{ω ^{12}}

becomes

1

, while

\frac{3}{ω ^{2}}

becomes

3 ω

and

\frac{3}{ω ^{- 14}}

becomes

3 ω^{2}

Using the identity

1 + ω + ω^{2} = 0

, the expression simplifies to

3 (ω + ω^{2}) + 1 = 3 (- 1) + 1

. The final result is $- 2$ .

Infinite Geometric Series

Finally, we consider the case where

b = 6

, which implies

a = 1

and

c = - 7

. We examine the quadratic equation

x^{2} + 6 x - 7 = 0

, which has roots

1

and

- 7

The sum of the reciprocals of the roots is

\frac{6}{7}

. Since the absolute value

∣ \frac{6}{7} ∣ < 1

, the infinite geometric series converges:

S = \frac{1}{1 - \frac{6}{7}} = 7

The final value of the series is $7$ .

Animated Solution for Mathematics - Matrices and Determinants: Comprehension Passage

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

Select Answer:

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

Select Answer:

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

Select Answer:

Visualized Solution

Decoding the Matrix Equation $(E)$

Generating the System of Equations

Simplifying the Third Equation

Solving for $b$ and $c$

Sub-question 1: Plane Intersection

Finding the Value of $a$

Final Answer for Sub-question 1

Sub-question 2: Complex Numbers Setup

Substituting Values into the Expression

Simplifying Powers of $ω$

Final Answer for Sub-question 2

Sub-question 3: Quadratic Roots Setup

Finding the Roots $α$ and $β$

Infinite Geometric Progression

The Sigma Insight: Solution of System of Linear Equations (Matrix Method and Cramer's Rule)

Analyzing the Matrix System

Solving the Parametric Relationship

Geometry of the Plane

Complex Numbers and Roots of Unity

Infinite Geometric Series

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Animated Solution for Mathematics - Matrices and Determinants: 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

Select Answer:

.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

Select Answer:

.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

Select Answer:

Visualized Solution

Decoding the Matrix Equation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (E)

Generating the System of Equations

Simplifying the Third Equation

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } c

Sub-question 1: Plane Intersection

Finding the Value of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a

Final Answer for Sub-question 1

Sub-question 2: Complex Numbers Setup

Substituting Values into the Expression

Simplifying Powers of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ω

Final Answer for Sub-question 2

Sub-question 3: Quadratic Roots Setup

Finding the Roots .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } α and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } β

Infinite Geometric Progression

The Sigma Insight: Solution of System of Linear Equations (Matrix Method and Cramer's Rule)

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Geometry of the Plane

Complex Numbers and Roots of Unity

Infinite Geometric Series

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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

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