If the determinant \(\begin{vmatrix} a & b & 2a\,\alpha+3b \\[0.3em]

1.	If the determinant \(\begin{vmatrix} a & b & 2a\,\alpha+3b \\[0.3em] b& c &2b\,\alpha+3c \\[0.3em] 2a\,\alpha+3b & 2b\,\alpha+3c & 0 \end{vmatrix}\) , thenA. a, b, c are in H.P. B. α is a root of 4ax2 + 12bx + 9c = 0 or, a, b, c are in G.P. C. a, b, c are in G.P. only D. a, b, c are in A.P.
Answer» B. α is a root of 4ax² + 12bx + 9c = 0 or, a, b, c are in G.P. Expend the determinats : a[-(2bα+3c)² ]-b[-(2bα+3c)(2aα+3b)]+ (2aα+3b)[b(2bα+3c)-c(2aα+3b)] = 0 -a(2bα+3c)² + b(2bα+3c)(2aα+3b)+(2aα+3b)[2b²α+3bc-3bc-2acα] = 0 (2bα+3c) [-2abα-3ac+2abα+3b²]+ (2aα+3b)(2α)( b² -ac) = 0 (2bα+3c) [-3ac +3b² ]+ (2aα+3b)(2α)( b²- ac) = 0 (b² – ac)[4aα² + 12bα + ac] = 0 CASE 1→ (b² - ac) = 0 b² = ac {abc are in Gp} CASE 2→ (4aα² +12bα + ac)=0 {Whose one root is α}

If the determinant \(\begin{vmatrix} a & b & 2a\,\alpha+3b \\[0.3em] b& c &2b\,\alpha+3c \\[0.3em] 2a\,\alpha+3b & 2b\,\alpha+3c & 0 \end{vmatrix}\) , thenA. a, b, c are in H.P. B. α is a root of 4ax2 + 12bx + 9c = 0 or, a, b, c are in G.P. C. a, b, c are in G.P. only D. a, b, c are in A.P.

Answer»

B. α is a root of 4ax² + 12bx + 9c = 0 or, a, b, c are in G.P.

Expend the determinats :

a[-(2bα+3c)² ]-b[-(2bα+3c)(2aα+3b)]+ (2aα+3b)[b(2bα+3c)-c(2aα+3b)] = 0

-a(2bα+3c)² + b(2bα+3c)(2aα+3b)+(2aα+3b)[2b²α+3bc-3bc-2acα] = 0

(2bα+3c) [-2abα-3ac+2abα+3b²]+ (2aα+3b)(2α)( b² -ac) = 0

(2bα+3c) [-3ac +3b² ]+ (2aα+3b)(2α)( b²- ac) = 0

(b² – ac)[4aα² + 12bα + ac] = 0

CASE 1→

(b² - ac) = 0

b² = ac

{abc are in Gp}

CASE 2→

(4aα² +12bα + ac)=0

{Whose one root is α}