SATHEE: Matrices and Determinants 4 Question 16

Matrices and Determinants 4 Question 16

17. The number of value of $k$ , for which the system of equation

$(k + 1) x + 8 y = 4 y \Rightarrow k x + (k + 3) y = 3 k - 1$

(2013 Main)

has no solution, is

(a) infinite

(b) 1

(d) 3

Show Answer

Answer:

Correct Answer: 17. (d)

Solution:

Given equations can be written in matrix form $A X = B$

where, $A = \begin{array}{cc} k + 1 & 8 k & k + 3 \end{array}, X = \frac{x}{y}$ and $B = \frac{4 k}{3 k - 1}$

For no solution, $| A | = 0$ and $(adj A) B \neq 0$

Now, $| A | = \begin{array}{cc} k + 1 & 8 k & k + 3 \end{array} = 0$

$\begin{aligned} \Rightarrow & (k^{2} + 1) (k + 3) - 8 k & = 0 \\ k^{2} + 4 k + 3 - 8 k & = 0 \\ k^{2} - 4 k \times 3 & = 0 \\ \Rightarrow & (k - 1) (k - 3) & = 0 \\ k = 1, k & = 3, \end{aligned}$

$Now adj A = \begin{array}{ll} k + 3 & - 8 \\ - k & k + 1 \end{array}$

Now, $(adj A) B = \begin{array}{ccc} k + 3 & - 8 & 4 k - k & k + 1 & 3 k + 1 \end{array}$

$\begin{aligned} = & (k + 3) (4 k) - 8 (3 k - 1) \\ - 4 k^{2} + (k + 1) (3 k - 1) \\ = & 4 k^{2} - 12 k + 8 \\ - k^{2} + 2 k - 1 \end{aligned}$

Put $k = 1$

$(adj A) B = \begin{array}{r} 4 - 12 + 8 \\ - 1 + 2 - 1 \end{array} = \begin{aligned} 0 \\ 0 \end{aligned} not true$

Put $k = 3$

$(adj A) B = \begin{matrix} 36 - 36 + 8 \\ - 9 + 6 - 1 \end{matrix} =_{- 4}^{8} \neq 0 true$

Hence, required value of $k$ is 3 .

Alternate Solution

Condition for the system of equations has no solution is

$\begin{aligned} \frac{a_{1}}{a_{2}} & = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} \\ ∴ \frac{k + 1}{k} & = \frac{8}{k + 3} \neq \frac{4 k}{3 k - 1} \end{aligned}$

Take $\frac{k + 1}{k} = \frac{8}{k + 3}$

$\Rightarrow k^{2} + 4 k + 3 = 8 k$

$\Rightarrow k^{2} - 4 k + 3$

$\Rightarrow (k - 1) (k - 3) = 0$

$k = 1, 3$

If $k - 1$ , then $\frac{8}{1 + 3} \neq \frac{4.1}{2}$ , false

And, if $k = 3$ , then $\frac{8}{6} \neq \frac{4.3}{9 - 1}$ , true

Therefore, $k = 3$

Hence, only one value of $k$ exist.

$\begin{array}{ll} x & 1 \end{array}$

पिछला

अगला

Matrices and Determinants 4 Question 16

17. The number of value of $k$ , for which the system of equation

Answer:

Alternate Solution

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Matrices and Determinants 4 Question 16

17. The number of value of k, for which the system of equation

Answer:

Alternate Solution

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Menu

17. The number of value of $k$ , for which the system of equation