SATHEE: Matrices and Determinants 2 Question 17

Matrices and Determinants 2 Question 17

19. The parameter on which the value of the determinant

$| \begin{matrix} 1 & a & a^{2} \\ \cos (p - d) x & \cos p x & \cos (p + d) x \\ \sin (p - d) x & \sin p x & \sin (p + d) x \end{matrix} |$

does not depend upon, is

(1997, 2M)

(a) $a$

(b) $p$

(d) $x$

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Answer:

Correct Answer: 19. (b)

Solution:

Let $Δ =$ $| \begin{matrix} 1 & a & a^{2} \\ \cos (p - d) x & \cos p x & \cos (p + d) x \\ \sin (p - d) x & \sin p x & \sin (p + d) x \end{matrix} |$

Applying $C_{1} \to C_{1} + C_{3}$

$\begin{aligned} \Rightarrow Δ = | \begin{array}{c} 1 + a^{2} & a & a^{2} \\ \cos (p - d) x + \cos (p + d) x & \cos p x & \cos (p + d) x \\ \sin (p - d) x + \sin (p + d) x & \sin p x & \sin (p + d) x \end{array} | \end{aligned}$

$\begin{aligned} \Rightarrow Δ = | \begin{array}{c} 1 + a^{2} & a & a^{2} \\ 2 \cos p x \cos d x & \cos p x & \cos (p + d) x \\ 2 \sin p x \cos d x & \sin p x & \sin (p + d) x \end{array} | \end{aligned}$

Applying $C_{1} \to C_{1} - 2 \cos d x C_{2}$

$\Rightarrow Δ = | \begin{matrix} 1 + a^{2} - 2 a \cos d x & a & a^{2} \\ 0 & \cos p x & \cos (p + d) x \\ 0 & \sin p x & \sin (p + d) x \end{matrix} |$

$\Rightarrow Δ = (1 + a^{2} - 2 a \cos d x) [\sin (p + d) x \cos p x$ $- \sin p x \cos (p + d) x]$

$\Rightarrow Δ = (1 + a^{2} - 2 a \cos d x) \sin d x$

which is independent of $p$ .

Matrices and Determinants 2 Question 17

19. The parameter on which the value of the determinant

Answer:

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Matrices and Determinants 2 Question 17

19. The parameter on which the value of the determinant

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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