Let a, b and c be three real numbers...

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Advanced 2011Moderate

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

Visualized Solution (English)

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

Given matrix equation: $[a b c] 187923777 = [000]$

Generating the System of Equations

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

Simplifying the Third Equation

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

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

Substitute $c = - a - b$ into (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 $b = 6 a$ into $c = - a - b$ :
$c = - a - 6 a = - 7 a$
General solution: $(a, b, c) = (a, 6 a, - 7 a)$

Applying Condition for Sub-question 1

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

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

Calculate $7 a + b + c$ :
$7 (1) + 6 + (- 7) = 7 + 6 - 7 = 6$
Correct Option: (d)

Analyzing Sub-question 2: Complex Numbers

Given $a = 2$ , then $b = 6 (2) = 12$ and $c = - 7 (2) = - 14$
Expression: $\frac{3}{ω ^{a}} + \frac{1}{ω ^{b}} + \frac{3}{ω ^{c}}$
Substitute values: $\frac{3}{ω ^{2}} + \frac{1}{ω ^{12}} + \frac{3}{ω ^{- 14}}$

Simplifying with Cube Roots of Unity

Using $ω^{3} = 1$ :
$\frac{3}{ω ^{2}} = 3 ω$
$\frac{1}{ω ^{12}} = \frac{1}{1} = 1$
$\frac{3}{ω ^{- 14}} = 3 ω^{14} = 3 ω^{2}$
Expression: $3 ω + 1 + 3 ω^{2} = 3 (ω + ω^{2}) + 1$
Since $1 + ω + ω^{2} = 0 \Rightarrow ω + ω^{2} = - 1$
Result: $3 (- 1) + 1 = - 2$ . Correct Option: (a)

Sub-question 3: Quadratic Equation

Given $b = 6 \Rightarrow 6 a = 6 \Rightarrow a = 1$ and $c = - 7$
Equation: $x^{2} + 6 x - 7 = 0 \Rightarrow (x + 7) (x - 1) = 0$
Roots: $α = 1, β = - 7$
Calculate $\frac{1}{α} + \frac{1}{β} = \frac{1}{1} + \frac{1}{- 7} = 1 - \frac{1}{7} = \frac{6}{7}$

Infinite Geometric Progression

Sum: $\sum_{n = 0}^{\infty} (\frac{6}{7})^{n} = 1 + \frac{6}{7} + (\frac{6}{7})^{2} + \dots$
Using $S_{\infty} = \frac{a}{1 - r}$ where $a = 1, r = \frac{6}{7}$ :
$S_{\infty} = \frac{1}{1 - \frac{6}{7}} = \frac{1}{\frac{1}{7}} = 7$
Correct Option: (b)

Summary and Key Takeaways

Key Takeaways:
1. Matrix equations represent systems of linear equations.
2. Homogeneous systems often yield parametric solutions (like $b = 6 a, c = - 7 a$ ).
3. Multi-concept problems require careful substitution and property knowledge (like $ω^{3} = 1$ or Infinite GP sum).

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Visualized Solution (English)

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

Applying Condition for Sub-question 1

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

Analyzing Sub-question 2: Complex Numbers

Simplifying with Cube Roots of Unity

Sub-question 3: Quadratic Equation

Infinite Geometric Progression

Summary and Key Takeaways

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