Solve the following determinant equations-i x-a b cax-b c a b x-c-0ii x-axxxx-axxxx- a -0- a - 0iii 3 x-83333 x-83333 x-8-0iv 1 xx 21 aa 21 bb 2-0- a - bv x-1352 x-2523 x-4-0vi 1 xx 31 bb 31 cc 3-0- b - cvii 15-2 - 11-3 x 7-x 111714101613-0viii 11 xp -1 p -1 p - x 3 x -1 x -2-0ix 3-2 sin 3 -78 cos 2 -11142-0

(i) Let #160; #8710;=x+abcax+bcabx+c=x+a+b+cbcx+a+b+cx+bcx+a+b+cbx+c #160; #160; #160; #160; #160; #160; #160;Applying #160;C1 #8594;C1+C ...

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Byju's Answer

Standard XII

Mathematics

Integration of Irrational Algebraic Fractions - 1

Solve the fol...

Question

Solve the following determinant equations:

(i) $|\begin{matrix} x + a & b & c \\ a & x + b & c \\ a & b & x + c \end{matrix}| = 0$

(ii) $|\begin{matrix} x + a & x & x \\ x & x + a & x \\ x & x & x + a \end{matrix}| = 0, a \neq 0$

(iii) $|\begin{matrix} 3 x - 8 & 3 & 3 \\ 3 & 3 x - 8 & 3 \\ 3 & 3 & 3 x - 8 \end{matrix}| = 0$

(iv) $|\begin{matrix} 1 & x & x^{2} \\ 1 & a & a^{2} \\ 1 & b & b^{2} \end{matrix}| = 0, a \neq b$

(v) $|\begin{matrix} x + 1 & 3 & 5 \\ 2 & x + 2 & 5 \\ 2 & 3 & x + 4 \end{matrix}| = 0$

(vi) $|\begin{matrix} 1 & x & x^{3} \\ 1 & b & b^{3} \\ 1 & c & c^{3} \end{matrix}| = 0, b \neq c$

(vii) $|\begin{matrix} 15 - 2 x & 11 - 3 x & 7 - x \\ 11 & 17 & 14 \\ 10 & 16 & 13 \end{matrix}| = 0$

(viii) $|\begin{matrix} 1 & 1 & x \\ p + 1 & p + 1 & p + x \\ 3 & x + 1 & x + 2 \end{matrix}| = 0$

(ix) $|\begin{matrix} 3 & - 2 & \sin (3 θ) \\ - 7 & 8 & \cos (2 θ) \\ - 11 & 14 & 2 \end{matrix}| = 0$

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Solution

(i)

$Let ∆ = |\begin{matrix} x + a & b & c \\ a & x + b & c \\ a & b & x + c \end{matrix}| = |\begin{matrix} x + a + b + c & b & c \\ x + a + b + c & x + b & c \\ x + a + b + c & b & x + c \end{matrix}| [Applying C_{1} \to C_{1} + C_{2} + C_{3}] = (x + a + b + c) |\begin{matrix} 1 & b & c \\ 1 & x + b & c \\ 1 & b & x + c \end{matrix}| = (x + a + b + c) |\begin{matrix} 1 & b & c \\ 0 & x & 0 \\ 1 & b & x + c \end{matrix}| [Applying R_{2} \to R_{2} - R_{1}] = (x + a + b + c) |\begin{matrix} 1 & b & c \\ 0 & x & 0 \\ 0 & 0 & x \end{matrix}| [Applying R_{3} \to R_{3} - R_{1}] ∆ = (x + a + b + c) (x^{2} - 0) = 0 [Given] \Rightarrow x^{2} = 0 or x + a + b + c = 0 \Rightarrow x = 0 or x = - (a + b + c)$

(ii)
$Let ∆ = |\begin{matrix} x + a & x & x \\ x & x + a & x \\ x & x & x + a \end{matrix}| = |\begin{matrix} 3 x + a & x & x \\ 3 x + a & x + a & x \\ 3 x + a & x & x + a \end{matrix}| [Applying C_{1} \to C_{1} + C_{2} + C_{3}] = (3 x + a) |\begin{matrix} 1 & x & x \\ 1 & x + a & x \\ 1 & x & x + a \end{matrix}| = (3 x + a) |\begin{matrix} 1 & x & x \\ 0 & a & 0 \\ 1 & x & x + a \end{matrix}| [Applying R_{2} \to R_{2} - R_{1}] = (3 x + a) |\begin{matrix} 1 & x & x \\ 0 & a & 0 \\ 0 & 0 & a \end{matrix}| [Applying R_{3} \to R_{3} - R_{1}] ∆ = (3 x + a) (a^{2} - 0) = 0 x = \frac{- a}{3}$

(iii)

$Let ∆ = |\begin{matrix} 3 x - 8 & 3 & 3 \\ 3 & 3 x - 8 & 3 \\ 3 & 3 & 3 x - 8 \end{matrix}| = |\begin{matrix} 3 x - 2 & 3 & 3 \\ 3 x - 2 & 3 x - 8 & 3 \\ 3 x - 2 & 3 & 3 x - 8 \end{matrix}| [Applying C_{1} = C_{1} + C_{2} + C_{3}] = (3 x - 2) |\begin{matrix} 1 & 3 & 3 \\ 1 & 3 x - 8 & 3 \\ 1 & 3 & 3 x - 8 \end{matrix}| = (3 x - 2) |\begin{matrix} 1 & 3 & 3 \\ 0 & 3 x - 11 & 0 \\ 1 & 3 & 3 x - 8 \end{matrix}| [Applying R_{2} \to R_{2} - R_{1}] = (3 x - 2) |\begin{matrix} 1 & 3 & 3 \\ 0 & 3 x - 11 & 0 \\ 0 & 0 & 3 x - 11 \end{matrix}| [Applying R_{3} \to R_{3} - R_{1}] ∆ = (3 x - 2) {(3 x - 11)}^{2} = 0 x = \frac{2}{3}, \frac{11}{3}, \frac{11}{3}$

(iv)
$Let ∆ = |\begin{matrix} 1 & x & x^{2} \\ 1 & a & a^{2} \\ 1 & b & b^{2} \end{matrix}| = |\begin{matrix} 1 & x & x^{2} \\ 0 & x - a & x^{2} - a^{2} \\ 1 & b & b^{2} \end{matrix}| [Applying R_{2} \to R_{1} - R_{2}] = |\begin{matrix} 1 & x & x^{2} \\ 0 & x - a & x^{2} - a^{2} \\ 0 & x - b & x^{2} - b^{2} \end{matrix}| [Applying R_{3} \to R_{1} - R_{3}] = (x - a) (x - b) |\begin{matrix} 1 & x & x^{2} \\ 0 & 1 & x + a \\ 0 & 1 & x + b \end{matrix}| ∆ = (x - a) (x - b) (x + b - x - a) = 0 x = a, b$

(v)

$Let ∆ = |\begin{matrix} x + 1 & 3 & 5 \\ 2 & x + 2 & 5 \\ 2 & 3 & x + 4 \end{matrix}| = |\begin{matrix} x + 9 & 3 & 5 \\ x + 9 & x + 2 & 5 \\ x + 9 & 3 & x + 4 \end{matrix}| [Applying C_{1} = C_{1} + C_{2} + C_{3}] = (x + 9) |\begin{matrix} 1 & 3 & 5 \\ 1 & x + 2 & 5 \\ 1 & 3 & x + 4 \end{matrix}| = (x + 9) |\begin{matrix} 1 & 3 & 5 \\ 0 & x - 1 & 0 \\ 1 & 3 & x + 4 \end{matrix}| [Applying R_{2} \to R_{2} - R_{1}] = (x + 9) |\begin{matrix} 1 & 3 & 5 \\ 0 & x - 1 & 0 \\ 0 & 0 & x - 1 \end{matrix}| [Applying R_{3} \to R_{3} - R_{1}] ∆ = (x + 9) {(x - 1)}^{2} = 0 x = - 9, 1, 1$

(vi)
$Let ∆ = |\begin{matrix} 1 & x & x^{3} \\ 1 & b & b^{3} \\ 1 & c & c^{3} \end{matrix}| = |\begin{matrix} 1 & x & x^{3} \\ 0 & b - x & b^{3} - x^{3} \\ 1 & c & c^{3} \end{matrix}| [Applying R_{2} \to R_{2} - R_{1}] = |\begin{matrix} 1 & x & x^{3} \\ 0 & b - x & b^{3} - x^{3} \\ 0 & c - x & c^{3} - x^{3} \end{matrix}| [Applying R_{3} \to R_{3} - R_{1}] = |\begin{matrix} 1 & x & x^{3} \\ 0 & x - b & x^{3} - b^{3} \\ 0 & x - c & x^{3} - c^{3} \end{matrix}| = (x - b) (x - c) |\begin{matrix} 1 & x & x^{2} \\ 0 & 1 & x^{2} + x b + b^{2} \\ 0 & 1 & x^{2} + x c + c^{2} \end{matrix}| ∆ = (x - b) (x - c) (x (c - b) - b^{2} + c^{2}) = 0 x = b, c, - (b + c)$

(vii)

$Let Δ = |15 - 2 x 11 - 3 x 7 - x 11 17 14 10 16 13| = 0 \Rightarrow |15 - 2 x - 14 + 2 x 11 - 3 x 7 - x 11 - 28 17 14 10 - 26 16 13| = 0 [Applying C_{1} \to C_{1} - 2 C_{3}] \Rightarrow |1 11 - 3 x 7 - x - 17 17 14 - 16 16 13| = 0 \Rightarrow |12 - 3 x 4 - 2 x 7 - x 0 3 14 0 3 13| = 0 [Applying C_{1} \to C_{1} + C_{2} and C_{2} \to C_{2} - C_{3}] \Rightarrow (12 - 3 x) ((3 \times 13) - (3 \times 14)) = 0 \Rightarrow (12 - 3 x) (- 3) = 0 \Rightarrow 12 - 3 x = 0 \Rightarrow 3 x = 12 \Rightarrow x = 4$

(viii)
$Let ∆ = |\begin{matrix} 1 & 1 & x \\ p + 1 & p + 1 & p + x \\ 3 & x + 1 & x + 2 \end{matrix}| = |\begin{matrix} 1 & 1 & x \\ p & p & p \\ 3 & x + 1 & x + 2 \end{matrix}| [Applying R_{2} \to R_{2} - R_{1}] = p |\begin{matrix} 1 & 1 & x \\ 1 & 1 & 1 \\ 3 & x + 1 & x + 2 \end{matrix}| = p |\begin{matrix} 1 & 1 & x \\ 1 & 1 & 1 \\ 2 & x & 2 \end{matrix}| [Applying R_{3} \to R_{3} - R_{1}] = p |\begin{matrix} 0 & 1 & x \\ 0 & 1 & 1 \\ 2 - x & x & 2 \end{matrix}| [Applying C_{1} \to C_{1} - C_{2}] = p \{(2 - x) \times |\begin{matrix} 1 & x \\ 1 & 1 \end{matrix}|\} [Expanding along C_{1}] = p (2 - x) (1 - x) = 0 x = 1, 2$

(ix)

$Let Δ = |\begin{matrix} 3 & - 2 & \sin (3 θ) \\ - 7 & 8 & \cos (2 θ) \\ - 11 & 14 & 2 \end{matrix}| = 0 \Rightarrow |\begin{matrix} 1 & - 2 & \sin (3 θ) \\ 1 & 8 & \cos (2 θ) \\ 3 & 14 & 2 \end{matrix}| = 0 [Applying C_{1} \to C_{1} + C_{2}] \Rightarrow |\begin{matrix} 1 & - 2 & \sin (3 θ) \\ 0 & 10 & \cos (2 θ) - \sin (3 θ) \\ 0 & 20 & 2 - 3 \sin (3 θ) \end{matrix}| = 0 [Applying R_{2} \to R_{2} - R_{1} and R_{3} \to R_{3} - 3 R_{1}] \Rightarrow 10 (2 - 3 \sin (3 θ)) - 20 (\cos (2 θ) - \sin (3 θ)) = 0 \Rightarrow 20 - 10 \sin (3 θ) - 20 \cos (2 θ) = 0 \Rightarrow \sin (3 θ) + 2 \cos (2 θ) - 2 = 0 \Rightarrow 3 \sin θ - 4 \sin^{3} θ + 2 - 4 \sin^{2} θ - 2 = 0 \Rightarrow - \sin θ (4 \sin^{2} θ + 4 \sin θ - 3) = 0 \Rightarrow \sin θ = 0 or 4 \sin^{2} θ + 4 \sin θ - 3 = 0 \Rightarrow θ = n π or (2 \sin θ + 3) (2 \sin θ - 1) = 0 \Rightarrow θ = n π or \sin θ = - \frac{3}{2} or \sin θ = \frac{1}{2} \Rightarrow θ = n π o r θ = n π + {(- 1)}^{n} \frac{π}{6}, n \in ℤ$

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Similar questions

Q. Solve the following determinant equations:

(i)

|\begin{matrix} x + a & b & c \\ a & x + b & c \\ a & b & x + c \end{matrix}| = 0

(ii)

|\begin{matrix} x + a & x & x \\ x & x + a & x \\ x & x & x + a \end{matrix}| = 0, a \neq 0

(iii)

|\begin{matrix} 3 x - 8 & 3 & 3 \\ 3 & 3 x - 8 & 3 \\ 3 & 3 & 3 x - 8 \end{matrix}| = 0

(iv)

|\begin{matrix} 1 & x & x^{2} \\ 1 & a & a^{2} \\ 1 & b & b^{2} \end{matrix}| = 0, a \neq b

(v)

|\begin{matrix} x + 1 & 3 & 5 \\ 2 & x + 2 & 5 \\ 2 & 3 & x + 4 \end{matrix}| = 0

(vi)

|\begin{matrix} 1 & x & x^{3} \\ 1 & b & b^{3} \\ 1 & c & c^{3} \end{matrix}| = 0, b \neq c

(vii)

|\begin{matrix} 15 - 2 x & 11 - 3 x & 7 - x \\ 11 & 17 & 14 \\ 10 & 16 & 13 \end{matrix}| = 0

(viii)

|\begin{matrix} 1 & 1 & x \\ p + 1 & p + 1 & p + x \\ 3 & x + 1 & x + 2 \end{matrix}| = 0

(ix)

|\begin{matrix} 3 & - 2 & \sin (3 θ) \\ - 7 & 8 & \cos (2 θ) \\ - 11 & 14 & 2 \end{matrix}| = 0

(x)

|\begin{matrix} 4 - x & 4 + x & 4 + x \\ 4 + x & 4 - x & 4 + x \\ 4 + x & 4 + x & 4 + x \end{matrix}| = 0

Q. Solve the following determinant equations:

(i)

|\begin{matrix} x + a & b & c \\ a & x + b & c \\ a & b & x + c \end{matrix}| = 0

(ii)

|\begin{matrix} x + a & x & x \\ x & x + a & x \\ x & x & x + a \end{matrix}| = 0, a \neq 0

(iii)

|\begin{matrix} 3 x - 8 & 3 & 3 \\ 3 & 3 x - 8 & 3 \\ 3 & 3 & 3 x - 8 \end{matrix}| = 0

(iv)

|\begin{matrix} 1 & x & x^{2} \\ 1 & a & a^{2} \\ 1 & b & b^{2} \end{matrix}| = 0, a \neq b

(v)

|\begin{matrix} x + 1 & 3 & 5 \\ 2 & x + 2 & 5 \\ 2 & 3 & x + 4 \end{matrix}| = 0

(vi)

|\begin{matrix} 1 & x & x^{3} \\ 1 & b & b^{3} \\ 1 & c & c^{3} \end{matrix}| = 0, b \neq c

(vii)

|\begin{matrix} 15 - 2 x & 11 - 3 x & 7 - x \\ 11 & 17 & 14 \\ 10 & 16 & 13 \end{matrix}| = 0

(viii)

|\begin{matrix} 1 & 1 & x \\ p + 1 & p + 1 & p + x \\ 3 & x + 1 & x + 2 \end{matrix}| = 0

Q. Determine the nature of the roots of the following quadratic equations:

(i) 2x² − 3x + 5 = 0
(ii) 2x² − 6x + 3 = 0
(iii)

\frac{3}{5} x^{2} - \frac{2}{3} x + 1 = 0

(iv)

3 x^{2} - 4 \sqrt{3} x + 4 = 0

(v)

3 x^{2} - 2 \sqrt{6} x + 2 = 0

(vi)

(x - 2 a) (x - 2 b) = 4 a b

(vii)

9 a^{2} b^{2} x^{2} - 24 a b c d x + 16 c^{2} d^{2} = 0, a \neq 0, b \neq 0

(viii)

2 (a^{2} + b^{2}) x^{2} + 2 (a + b) x + 1 = 0

(ix)

(b + c) x^{2} - (a + b + c) x + a = 0

Q. If

p \neq 0

solution set of the equation

Δ = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & x p + 1 & p + 1 & p + x 3 & x + 1 & x + 2 \end{matrix} ∣ ∣ ∣ ∣ = 0

Q. Solve the equation

∣ ∣ ∣ ∣ \begin{matrix} x + a & x & x x & x + a & x x & x & x + a \end{matrix} ∣ ∣ ∣ ∣ = 0, a \neq 0

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​Solve the following determinant equations: (i) x+abcax+bcabx+c=0 (ii) x+axxxx+axxxx+a=0, a≠0 (iii) 3x-83333x-83333x-8=0 (iv) 1xx21aa21bb2=0, a≠b (v) x+1352x+2523x+4=0 (vi) 1xx31bb31cc3=0, b≠c (vii) 15-2x11-3x7-x111714101613=0 (viii) 11xp+1p+1p+x3x+1x+2=0 (ix) 3-2sin3θ-78cos2θ-11142=0