Without expanding - find the value of a-b 2a-b 3a-b 2a-b 3a-b 4a-b 4a-b 5a-b 6a-b

a+b amp; 2a + b amp; 3a + b 2a + b amp; 3a + b amp; 4a + b 4a + b amp; 5a + b amp; 6a + b R_3 → R_3 R_2 R_2 → R_2 ...

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Byju's Answer

Standard XII

Mathematics

Properties of Determinants

Without expan...

Question

Without expanding , find the value of $∣ ∣ ∣ ∣ \begin{matrix} a + b & 2 a + b & 3 a + b 2 a + b & 3 a + b & 4 a + b 4 a + b & 5 a + b & 6 a + b \end{matrix} ∣ ∣ ∣ ∣$

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Solution

$∣ ∣ ∣ ∣ \begin{matrix} a + b & 2 a + b & 3 a + b 2 a + b & 3 a + b & 4 a + b 4 a + b & 5 a + b & 6 a + b \end{matrix} ∣ ∣ ∣ ∣$
$R_{3} \to R_{3} - R_{2} R_{2} \to R_{2} - R_{1}$
$∣ ∣ ∣ ∣ \begin{matrix} a + b & a & a 2 a + b & a & a 4 a + b & a & a \end{matrix} ∣ ∣ ∣ ∣$
$0$ .

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Evaluate :

(i) (a+ 1)(a-1)(a^2 + 1)

(ii) (a + b)(a-b)(a^2 + b^2)

(iii) (2a-b)(2a +b)(4a^2 + b^2)

(iv) (3 - 2x)(3 + 2x)(9 + 4x^2)

(v) (3x - 4y)(3x + 4y)( 9x^2 + 16y^2)

∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} a & b & c & d a & a + b & a + b + c & a + b + c + d a & 2 a + b & 3 a + 2 b + c & 4 a + 3 b + 2 c + d a & 3 a + b & 6 a + 3 b + c & 10 a + 6 b + 3 c + d \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣

Q. Factorize:

16 (a + b)^{2} - 4 a - 4 b

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Without expanding , find the value of ∣∣ ∣∣a+b2a+b3a+b2a+b3a+b4a+b4a+b5a+b6a+b∣∣ ∣∣

∣∣ ∣∣a+b2a+b3a+b2a+b3a+b4a+b4a+b5a+b6a+b∣∣ ∣∣R3→R3−R2R2→R2−R1∣∣ ∣∣a+baa2a+baa4a+baa∣∣ ∣∣0.

Without expanding , find the value of $∣ ∣ ∣ ∣ \begin{matrix} a + b & 2 a + b & 3 a + b 2 a + b & 3 a + b & 4 a + b 4 a + b & 5 a + b & 6 a + b \end{matrix} ∣ ∣ ∣ ∣$