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Matrices and Determinants question

2013 · Shift 0 · Q42
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Matrices and Determinants question

2013 · Shift 0 · Q42

JEE MainMathematicsMatrices and DeterminantsMCQ+4 / −1
The number of values of kkk, for which the system of equations : (k+1)x+8y=4kkx+(k+3)y=3k−1\begin{matrix} {\left( {k + 1} \right)x + 8y = 4k} \\ {kx + \left( {k + 3} \right)y = 3k - 1} \\ \end{matrix}(k+1)x+8y=4kkx+(k+3)y=3k−1​ has no solution, is
  1. A
    infinite
  2. B
    1
  3. C
    2
  4. D
    3
View written solutionFree

Correct answer: B

  1. Write the system in matrix form

The system is

(k+1)x+8y=4k,(k+1)x+8y=4k,(k+1)x+8y=4k, kx+(k+3)y=3k−1.kx+(k+3)y=3k-1.kx+(k+3)y=3k−1.

For a pair of linear equations in two variables, the system has no solution when the two lines are parallel and distinct, i.e.

a1a2=b1b2≠c1c2.\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq \frac{c_1}{c_2}.a2​a1​​=b2​b1​​=c2​c1​​.

Here,

a1=k+1,b1=8,c1=4k,a_1=k+1,\quad b_1=8,\quad c_1=4k,a1​=k+1,b1​=8,c1​=4k, a2=k,b2=k+3,c2=3k−1.a_2=k,\quad b_2=k+3,\quad c_2=3k-1.a2​=k,b2​=k+3,c2​=3k−1.
  1. First make the coefficient ratios equal

We need

k+1k=8k+3.\frac{k+1}{k}=\frac{8}{k+3}.kk+1​=k+38​.

Cross-multiplying,

(k+1)(k+3)=8k.(k+1)(k+3)=8k.(k+1)(k+3)=8k.

Expand:

k2+4k+3=8kk^2+4k+3=8kk2+4k+3=8k k2−4k+3=0k^2-4k+3=0k2−4k+3=0 (k−1)(k−3)=0.(k-1)(k-3)=0.(k−1)(k−3)=0.

So possible values are

k=1ork=3.k=1 \quad \text{or} \quad k=3.k=1ork=3.
  1. Check whether these give no solution or infinitely many solutions

For no solution, we must also have

k+1k=8k+3≠4k3k−1.\frac{k+1}{k}=\frac{8}{k+3}\neq \frac{4k}{3k-1}.kk+1​=k+38​=3k−14k​.

Case 1: k=1k=1k=1

Then the equations become

2x+8y=4,2x+8y=4,2x+8y=4, x+4y=2.x+4y=2.x+4y=2.

The first equation is exactly twice the second, so both represent the same line. Hence there are infinitely many solutions, not no solution.

Case 2: k=3k=3k=3

Then the equations become

4x+8y=12,4x+8y=12,4x+8y=12, 3x+6y=8.3x+6y=8.3x+6y=8.

Now,

43=86=43,\frac{4}{3}=\frac{8}{6}=\frac{4}{3},34​=68​=34​,

but

128=32≠43.\frac{12}{8}=\frac{3}{2}\neq \frac{4}{3}.812​=23​=34​.

So the lines are parallel and distinct. Hence the system has no solution.

  1. Count the number of such values of kkk

Only one value works:

k=3.k=3.k=3.

Therefore, the number of values of kkk is

1.\boxed{1}.1​.
  1. Compare with stored answer

Stored correct answer: B

Our derived answer: B

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