Without expanding- show that the values of each of the following determinants are zero-i 82712351643ii 6-322-12-1052iii 23713175152012iv 1 - aa2bc1-bb2ac1-cc2abv a-b 2 a-b 3 a-b 2 a-b 3 a-b 4 a-b 4 a-b 5 a-b 6 a-bvi 1aa2-bc1bb2-ac1cc2-abvii 491639742623viii 0 xy - x 0 z - y - z 0ix 143673543172x 12223242223242523242526242526272xi abca-2xb-2yc-2zxyzxii 2 x-2-x 22 x-2-x 213 x-3-x 23 x-3-x 214 x-4-x 24 x-4-x 21xiii sin-cos-cos - sin-cos-cos - sin-cos-cos -xiv sin 223-sin 267-cos 180-sin 267-sin 223-cos 2180-cos 180-sin 223-sin 267-

(i) #160; #8710;=82712351643=027035043 #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160;Applying #1 ...

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Byju's Answer

Standard XII

Mathematics

Derivative of One Function w.r.t Another

Without expan...

Question

Without expanding, show that the values of each of the following determinants are zero:
(i) $|\begin{matrix} 8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3 \end{matrix}|$

(ii) $|\begin{matrix} 6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2 \end{matrix}|$

(iii) $|\begin{matrix} 2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12 \end{matrix}|$

(iv) $|\begin{matrix} 1 / a & a^{2} & b c \\ 1 / b & b^{2} & a c \\ 1 / c & c^{2} & a b \end{matrix}|$

(v) $|\begin{matrix} a + b & 2 a + b & 3 a + b \\ 2 a + b & 3 a + b & 4 a + b \\ 4 a + b & 5 a + b & 6 a + b \end{matrix}|$

(vi) $|\begin{matrix} 1 & a & a^{2} - b c \\ 1 & b & b^{2} - a c \\ 1 & c & c^{2} - a b \end{matrix}|$

(vii) $|\begin{matrix} 49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3 \end{matrix}|$

(viii) $|\begin{matrix} 0 & x & y \\ - x & 0 & z \\ - y & - z & 0 \end{matrix}|$

(ix) $|\begin{matrix} 1 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2 \end{matrix}|$

(x) $|\begin{matrix} 1^{2} & 2^{2} & 3^{2} & 4^{2} \\ 2^{2} & 3^{2} & 4^{2} & 5^{2} \\ 3^{2} & 4^{2} & 5^{2} & 6^{2} \\ 4^{2} & 5^{2} & 6^{2} & 7^{2} \end{matrix}|$

(xi) $|\begin{matrix} a & b & c \\ a + 2 x & b + 2 y & c + 2 z \\ x & y & z \end{matrix}|$

(xii) $|\begin{matrix} {(2^{x} + 2^{- x})}^{2} & {(2^{x} - 2^{- x})}^{2} & 1 \\ {(3^{x} + 3^{- x})}^{2} & {(3^{x} - 3^{- x})}^{2} & 1 \\ {(4^{x} + 4^{- x})}^{2} & {(4^{x} - 4^{- x})}^{2} & 1 \end{matrix}|$

(xiii) $|\begin{matrix} \sin α & \cos α & \cos (α + δ) \\ \sin β & \cos β & \cos (β + δ) \\ \sin γ & \cos γ & \cos (γ + δ) \end{matrix}|$

(xiv) $|\begin{matrix} \sin^{2} 23 ° & \sin^{2} 67 ° & \cos 180 ° \\ - \sin^{2} 67 ° & - \sin^{2} 23 ° & \cos^{2} 180 ° \\ \cos 180 ° & \sin^{2} 23 ° & \sin^{2} 67 ° \end{matrix}|$

(xv) $|\begin{matrix} \cos (x + y) & - \sin (x + y) & \cos 2 y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y \end{matrix}|$

(xvi) $|\begin{matrix} \sqrt{23} + \sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} + \sqrt{46} & 5 & \sqrt{10} \\ 3 + \sqrt{115} & \sqrt{15} & 5 \end{matrix}|$

(xvii) $|\begin{matrix} \sin^{2} A & \cot A & 1 \\ \sin^{2} B & \cot B & 1 \\ \sin^{2} C & \cot C & 1 \end{matrix}|, where A, B, C are the angles of ∆ A B C .$

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Solution

$(i) ∆ = |\begin{matrix} 8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3 \end{matrix}| = |\begin{matrix} 0 & 2 & 7 \\ 0 & 3 & 5 \\ 0 & 4 & 3 \end{matrix}| [Applying C_{1} \to C_{1} - 4 C_{2}] \Rightarrow ∆ = 0 (i i) ∆ = |\begin{matrix} 6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2 \end{matrix}| = |\begin{matrix} 0 & - 3 & 2 \\ 0 & - 1 & 2 \\ 0 & 5 & 2 \end{matrix}| [Applying C_{1} \to C_{1} + 2 C_{2}] \Rightarrow ∆ = 0$

$(i i i) ∆ = |\begin{matrix} 2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12 \end{matrix}| = |\begin{matrix} 2 & 3 & 7 \\ 13 & 17 & 5 \\ 13 & 17 & 5 \end{matrix}| [Applying R_{3} \to R_{3} - R_{1}] \Rightarrow ∆ = 0 (i v) ∆ = |\begin{matrix} \frac{1}{a} & a^{2} & b c \\ \frac{1}{b} & b^{2} & a c \\ \frac{1}{c} & c^{2} & a b \end{matrix}| = |\begin{matrix} 1 & a^{3} & a b c \\ 1 & b^{3} & a b c \\ 1 & c^{3} & a b c \end{matrix}| [Applying R_{1} \to a R_{1}, R_{2} \to b R_{2} and R_{3} \to c R_{3}] = a b c |\begin{matrix} 1 & a^{3} & 1 \\ 1 & b^{3} & 1 \\ 1 & c^{3} & 1 \end{matrix}| \Rightarrow ∆ = 0 (v) ∆ = |\begin{matrix} a + b & 2 a + b & 3 a + b \\ 2 a + b & 3 a + b & 4 a + b \\ 4 a + b & 5 a + b & 6 a + b \end{matrix}| = |\begin{matrix} a & a & a \\ 2 a & 2 a & 2 a \\ 4 a + b & 5 a + b & 6 a + b \end{matrix}| [Applying R_{1} \to R_{2} - R_{1} and R_{2} \to R_{3} - R_{2}] = 2 |\begin{matrix} a & a & a \\ a & a & a \\ 4 a + b & 5 a + b & 6 a + b \end{matrix}| = 0 (v i) ∆ = |\begin{matrix} 1 & a & a^{2} - b c \\ 1 & b & b^{2} - a c \\ 1 & c & c^{2} - a b \end{matrix}| = |\begin{matrix} 0 & a - b & a^{2} - b c - b^{2} + a c \\ 0 & b - c & b^{2} - a c - c^{2} + a b \\ 1 & c & c^{2} - a b \end{matrix}| [Applying R_{1} \to R_{1} - R_{2}, R_{2} \to R_{2} - R_{3}] = |\begin{matrix} 0 & a - b & (a - b) (a + b) + c (a - b) \\ 0 & b - c & (b - c) (b + c) + a (b - c) \\ 1 & c & c^{2} - a b \end{matrix}| = (a - b) (b - c) |\begin{matrix} 0 & 1 & (a + b + c) \\ 0 & 1 & (a + b + c) \\ 1 & c & c^{2} - a b \end{matrix}| \Rightarrow ∆ = 0 (v i i) ∆ = |\begin{matrix} 49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3 \end{matrix}| = |\begin{matrix} 1 & 1 & 6 \\ 7 & 7 & 4 \\ 2 & 2 & 3 \end{matrix}| [Applying C_{1} \to C_{1} - 8 C_{3}] \Rightarrow ∆ = 0 (v i i i) ∆ = |\begin{matrix} 0 & x & y \\ - x & 0 & z \\ - y & - z & 0 \end{matrix}| = \frac{x y z}{x y z} |\begin{matrix} 0 & x & y \\ - x & 0 & z \\ - y & - z & 0 \end{matrix}| = \frac{1}{x y z} |\begin{matrix} 0 & x z & y z \\ - x y & 0 & z y \\ - y x & - z x & 0 \end{matrix}| = \frac{1}{x y z} |\begin{matrix} - 2 x y & 0 & 2 y z \\ - x y & 0 & z y \\ - y x & - z x & 0 \end{matrix}| [Applying R_{1} \to R_{1} + R_{2} + R_{3}] = \frac{1}{x y z} |\begin{matrix} 0 & 0 & 0 \\ - x y & 0 & z y \\ - y x & - z x & 0 \end{matrix}| = 0 [Applying R_{1} \to R_{1} - 2 R_{2}] (i x) ∆ = |\begin{matrix} 1 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2 \end{matrix}| = |\begin{matrix} 1 & 1 & 6 \\ 7 & 7 & 4 \\ 3 & 3 & 2 \end{matrix}| = 0 [Applying C_{2} \to C_{2} - 7 C_{3}]$

$x) ∆ = |\begin{matrix} 1^{2} & 2^{2} & 3^{2} & 4^{2} \\ 2^{2} & 3^{2} & 4^{2} & 5^{2} \\ 3^{2} & 4^{2} & 5^{2} & 6^{2} \\ 4^{2} & 5^{2} & 6^{2} & 7^{2} \end{matrix}| = |\begin{matrix} 1 & 4 & 9 & 16 \\ 4 & 9 & 16 & 25 \\ 9 & 16 & 25 & 36 \\ 16 & 25 & 36 & 49 \end{matrix}| = |\begin{matrix} 1 & 4 & 9 & 16 \\ 4 & 9 & 16 & 25 \\ 5 & 7 & 9 & 11 \\ 7 & 9 & 11 & 13 \end{matrix}| [Applying R_{3} \to R_{3} - R_{2} and R_{4} \to R_{4} - R_{3}] = |\begin{matrix} 1 & 4 & 9 & 16 \\ 4 & 9 & 16 & 25 \\ 7 & 9 & 11 & 13 \\ 7 & 9 & 11 & 13 \end{matrix}| = 0 [Applying R_{3} \to 2 + R_{3}] xi) ∆ = |\begin{matrix} a & b & c \\ a + 2 x & b + 2 y & c + 2 z \\ x & y & z \end{matrix}| = |\begin{matrix} a + 2 x & b + 2 y & c + 2 z \\ a + 2 x & b + 2 y & c + 2 z \\ x & y & z \end{matrix}| [Applying R_{1} \to R_{1} + 2 R_{3}] = |\begin{matrix} 0 & 0 & 0 \\ a + 2 x & b + 2 y & c + 2 z \\ x & y & z \end{matrix}| = 0 [Applying R_{1} \to R_{1} - R_{2}]$

(xii)
$|\begin{matrix} {(2^{x} + 2^{- x})}^{2} & {(2^{x} - 2^{- x})}^{2} & 1 \\ {(3^{x} + 3^{- x})}^{2} & {(3^{x} - 3^{- x})}^{2} & 1 \\ {(4^{x} + 4^{- x})}^{2} & {(4^{x} - 4^{- x})}^{2} & 1 \end{matrix}| = |\begin{matrix} (2^{2 x} + 2^{- 2 x} + 2) & (2^{2 x} + 2^{- 2 x} - 2) & 1 \\ (3^{2 x} + 3^{- 2 x} + 2) & (3^{2 x} + 3^{- 2 x} - 2) & 1 \\ (4^{2 x} + 4^{- 2 x} + 2) & (4^{2 x} + 4^{- 2 x} - 2) & 1 \end{matrix}| = |\begin{matrix} 4 & (2^{2 x} + 2^{- 2 x} - 2) & 1 \\ 4 & (3^{2 x} + 3^{- 2 x} - 2) & 1 \\ 4 & (4^{2 x} + 4^{- 2 x} - 2) & 1 \end{matrix}| [Applying C_{1} \to C_{1} - C_{2}] = 4 |\begin{matrix} 1 & (2^{2 x} + 2^{- 2 x} - 2) & 1 \\ 1 & (3^{2 x} + 3^{- 2 x} - 2) & 1 \\ 1 & (4^{2 x} + 4^{- 2 x} - 2) & 1 \end{matrix}| = 0$

(xiii)
$|\begin{matrix} \sin α & \cos α & \cos (α + δ) \\ \sin β & \cos β & \cos (β + δ) \\ \sin γ & \cos γ & \cos (γ + δ) \end{matrix}| = |\begin{matrix} \sin α \sin δ & \cos α \cos δ & \cos (α + δ) \\ \sin β \sin δ & \cos β \cos δ & \cos (β + δ) \\ \sin γ \sin δ & \cos γ \cos δ & \cos (γ + δ) \end{matrix}| [Applying C_{1} \to \sin δ C_{1} and C_{2} \to \cos δ C_{2}] = |\begin{matrix} \sin α \sin δ & \cos (α + δ) & \cos (α + δ) \\ \sin β \sin δ & \cos (β + δ) & \cos (β + δ) \\ \sin γ \sin δ & \cos (γ + δ) & \cos (γ + δ) \end{matrix}| [Applying C_{2} \to C_{2} - C_{1}] = 0$

(xiv)
$|\begin{matrix} \sin^{2} 23 ° & \sin^{2} 67 ° & \cos 180 ° \\ - \sin^{2} 67 ° & - \sin^{2} 23 ° & \cos^{2} 180 ° \\ \cos 180 ° & \sin^{2} 23 ° & \sin^{2} 67 ° \end{matrix}| = |\begin{matrix} \sin^{2} 23 ° & \sin^{2} (90 - 23) ° & - 1 \\ - \sin^{2} (90 - 23) ° & - \sin^{2} 23 ° & 1 \\ - 1 & \sin^{2} 23 ° & \sin^{2} (90 - 23) ° \end{matrix}| = |\begin{matrix} \sin^{2} 23 ° & \cos^{2} 23 ° & - 1 \\ - \cos^{2} 23 ° & - \sin^{2} 23 ° & 1 \\ - 1 & \sin^{2} 23 ° & \cos^{2} 23 ° \end{matrix}| = |\begin{matrix} \sin^{2} 23 ° + \cos^{2} 23 ° & \cos^{2} 23 ° & - 1 \\ - \cos^{2} 23 ° - \sin^{2} 23 ° & - \sin^{2} 23 ° & 1 \\ - 1 + \sin^{2} 23 ° & \sin^{2} 23 ° & \cos^{2} 23 ° \end{matrix}| [A p p l y i n g C_{1} \to C_{1} + C_{2}] = |\begin{matrix} 1 & 1 & - 1 \\ - 1 & - \sin^{2} 23 ° & 1 \\ - \cos^{2} 23 ° & \sin^{2} 23 ° & \cos^{2} 23 ° \end{matrix}| = (- 1) |\begin{matrix} - 1 & 1 & - 1 \\ 1 & - \sin^{2} 23 ° & 1 \\ \cos^{2} 23 ° & \sin^{2} 23 ° & \cos^{2} 23 ° \end{matrix}| = 0$

(xv)
$|\begin{matrix} \cos (x + y) & - \sin (x + y) & \cos 2 y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y \end{matrix}| = \frac{1}{\sin y \cos y} |\begin{matrix} \cos (x + y) & - \sin (x + y) & \cos 2 y \\ \sin x \sin y & \cos x \sin y & \sin^{2} y \\ - \cos x \cos y & \sin x \cos y & - \cos^{2} y \end{matrix}| [Applying R_{2} \to \sin y R_{2} and R_{3} \to \cos y R_{3}] = \frac{1}{\sin y \cos y} |\begin{matrix} \cos (x + y) & - \sin (x + y) & \cos 2 y \\ \sin x \sin y - \cos x \cos y & \cos x \sin y + \sin x \cos y & \sin^{2} y - \cos^{2} y \\ - \cos x \cos y & \sin x \cos y & - \cos^{2} y \end{matrix}| [Applying R_{2} \to R_{2} + R_{3}] = \frac{- 1}{\sin y \cos y} |\begin{matrix} \cos (x + y) & - \sin (x + y) & \cos 2 y \\ \cos (x + y) & - \sin (x + y) & \cos 2 y \\ - \cos x \cos y & \sin x \cos y & - \cos^{2} y \end{matrix}| = 0$

(xvi)
$|\begin{matrix} \sqrt{23} + \sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} + \sqrt{46} & 5 & \sqrt{10} \\ 3 + \sqrt{115} & \sqrt{15} & 5 \end{matrix}| = |\begin{matrix} \sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} & 5 & \sqrt{10} \\ 3 & \sqrt{15} & 5 \end{matrix}| + |\begin{matrix} \sqrt{23} & \sqrt{5} & \sqrt{5} \\ \sqrt{46} & 5 & \sqrt{10} \\ \sqrt{115} & \sqrt{15} & 5 \end{matrix}| = \sqrt{3} |\begin{matrix} 1 & \sqrt{5} & \sqrt{5} \\ \sqrt{5} & 5 & \sqrt{10} \\ \sqrt{3} & \sqrt{15} & 5 \end{matrix}| + \sqrt{23} |\begin{matrix} 1 & \sqrt{5} & \sqrt{5} \\ \sqrt{2} & 5 & \sqrt{10} \\ \sqrt{5} & \sqrt{15} & 5 \end{matrix}| = \sqrt{3} \times \sqrt{5} |\begin{matrix} 1 & 1 & \sqrt{5} \\ \sqrt{5} & \sqrt{5} & \sqrt{10} \\ \sqrt{3} & \sqrt{3} & 5 \end{matrix}| + \sqrt{23} \times \sqrt{5} |\begin{matrix} 1 & \sqrt{5} & 1 \\ \sqrt{2} & 5 & \sqrt{2} \\ \sqrt{5} & \sqrt{15} & \sqrt{5} \end{matrix}| = 0 + 0 = 0$

(xvii)
$|\begin{matrix} \sin^{2} A & \cot A & 1 \\ \sin^{2} B & \cot B & 1 \\ \sin^{2} C & \cot C & 1 \end{matrix}| = |\begin{matrix} \sin^{2} A - \sin^{2} B & \cot A - \cot B & 0 \\ \sin^{2} B & \cot B & 1 \\ \sin^{2} C - \sin^{2} B & \cot C - \cot B & 0 \end{matrix}| [Applying R_{1} \to R_{1} - R_{2} and R_{3} \to R_{3} - R_{2}] = |\begin{matrix} \sin (A + B) \sin (A - B) & \frac{\cos A \sin B - \cos B \sin A}{\sin A \sin B} & 0 \\ \sin^{2} B & \cot B & 1 \\ \sin (C + B) \sin (C - B) & \frac{\cos C \sin B - \cos B \sin C}{\sin B \sin C} & 0 \end{matrix}| = |\begin{matrix} \sin (π - C) \sin (A - B) & \frac{- \sin (A - B)}{\sin A \sin B} & 0 \\ \sin^{2} B & c o t B & 1 \\ \sin (π - A) \sin (C - B) & \frac{- \sin (C - B)}{\sin B \sin C} & 0 \end{matrix}| [∵ A + B + C = π] = |\begin{matrix} \sin C \sin (A - B) & \frac{- \sin (A - B)}{\sin A \sin B} & 0 \\ \sin^{2} B & \frac{\cos B}{\sin B} & 1 \\ \sin A \sin (C - B) & \frac{- \sin (C - B)}{\sin B \sin C} & 0 \end{matrix}| = \frac{\sin (A - B) \sin (C - B)}{\sin B} |\begin{matrix} \sin C & \frac{- 1}{\sin A} & 0 \\ \sin^{2} B & \cos B & 1 \\ \sin A & \frac{- 1}{\sin C} & 0 \end{matrix}| = \frac{\sin (A - B) \sin (C - B)}{\sin B \sin A \sin C} |\begin{matrix} \sin C \sin A & - 1 & 0 \\ \sin^{2} B & \cos B & 1 \\ \sin A \sin C & - 1 & 0 \end{matrix}| [Applying R_{1} \to \sin A R_{1} and R_{3} \to \sin C R_{3}] = \frac{\sin (A - B) \sin (C - B)}{\sin B \sin A \sin C} |\begin{matrix} 0 & 0 & 0 \\ \sin^{2} B & \cos B & 1 \\ \sin A \sin C & - 1 & 0 \end{matrix}| [Applying R_{1} \to R_{1} - R_{3}] = 0$

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