Advertisements
Advertisements
प्रश्न
Prove that `|("a"^2, "bc", "ac" + "c"^2),("a"^2 + "ab", "b"^2, "ac"),("ab", "b"^2 + "bc", "c"^2)| = 4"a"^2"b"^2"c"^2`
उत्तर
Let Δ = `|("a"^2, "bc", "ac" + "c"^2),("a"^2 + "ab", "b"^2, "ac"),("ab", "b"^2 + "bc", "c"^2)|`
Δ = `|(2"a"^2 + 2"ab", 2"b"^2 + 2"bc", 2"c"^2 + 2"ac"),("a"^2 + 2"ab", 2"b"^2 + "bc", "c"^2 + "ac"),("ab", "b"^2 + "bc", "c"^2)| {:("R"_1 -> "R"_1 + "R"_2 + "R"_3),("R"_2 -> "R"_2 + "R"_3):}`
= `2 |("a"^2 + "ab", "b"^2 + 2"bc", "c"^2 + "ac"),("a"^2 + 2"ab", 2"b"^2 + "bc", "c"^2 + "ac"),("ab", "b"^2 + "bc", "c"^2)|`
= `2 |("a"^2 + "ab", "b"^2 + "bc", "c"^2 + "ac"),("a"^2 + "ab", "b"^2, "ac"),("ab", "b"^2 + "bc", "c"^2)| "R"_1 -> "R"_1 - "R"_2`
= `2|(0, "bc", "c"^2),("a"^2, -"bc", "ac" - "c"^2),("ab", "b"^2 + "bc", "c"^2)| {:("R"_1 -> "R"_1 - "R"_2),("R"_2 -> "R"_2 - "R"_3):}`
= `2|(0, "bc", "c"^2),("a"^2, 0, "ac"),("ab", "b"^2, 0)| {:("R"_2 -> "R"_2 + "R"_1),("R"_3 -> "R"_3 - "R"_11):}`
= `2"abc" |(0, "b", "c"),("a", 0, "c"),("a", "b", 0)|`
= 2abc [0 – b(0 – ac) + c(ab – 0)]
= 2abc [abc + abc]
= 2abc × 2abc
Δ = 4a2b2c2
APPEARS IN
संबंधित प्रश्न
Show that `|("b" + "c", "bc", "b"^2"C"^2),("c" + "a", "ca", "c"^2"a"^2),("a" + "b", "ab", "a"^2"b"^2)|` = 0
Prove that `|(sec^2theta, tan^2theta, 1),(tan^2theta, sec^2theta, -1),(38, 36, 2)|` = 0
Show that `|("a"^2 + x^2, "ab", "ac"),("ab", "b"^2 + x^2, "bc"),("ac", "bc", "c"^2 + x^2)|` is divisiible by x4
Find the value of `|(1, log_x y, log_x z),(log_y x, 1, log_y z),(log_z x, log_z y, 1)|` if x, y, z ≠ 1
If A = `[(1/2, alpha),(0, 1/2)]`, prove that `sum_("k" = 1)^"n" det("A"^"k") = 1/3(1 - 1/4)`
If A is a Square, matrix, and |A| = 2, find the value of |A AT|
Using cofactors of elements of second row, evaluate |A|, where A = `[(5, 3, 8),(2, 0, 1),(1, 2, 3)]`
Solve the following problems by using Factor Theorem:
Show that `|(x, "a", "a"),("a", x, "a"),("a", "a", x)|` = (x – a)2 (x + 2a)
Show that `|("b" + "c", "a" - "c", "a" - "b"),("b" - "c", "c" + "a", "b" - "a"),("c" - "b", "c" - "a", "a" + b")|` = 8abc
Solve `|(4 - x, 4 + x, 4 + x),(4 + x, 4 - x, 4 + x),(4 + x, 4 + x, 4 - x)|` = 0
Find the area of the triangle whose vertices are (0, 0), (1, 2) and (4, 3)
Determine the values of a and b so that the following matrices are singular:
A = `[(7, 3),(-2, "a")]`
Choose the correct alternative:
If A = `[(1, -1),(2, -1)]`, B = `[("a", 1),("b", -1)]` and (A + B)2 = A2 + B2, then the values of a and b are
Choose the correct alternative:
If `|(2"a", x_1, y_1),(2"b", x_2, y_2),(2"c", x_3, y_3)| = "abc"/2 ≠ 0`, then the area of the triangle whose vertices are `(x_1/"a", y_1/"a"), (x_2/"b", y_2/"b"), (x_3/"c", y_3/"c")` is
Choose the correct alternative:
If A = `|(-1, 2, 4),(3, 1, 0),(-2, 4, 2)|` and B = `|(-2, 4, 2),(6, 2, 0),(-2, 4, 8)|`, then B is given by
If f(x) = `|(cos^2x, cosx.sinx, -sinx),(cosx sinx, sin^2x, cosx),(sinx, -cosx, 0)|`, then for all x
If `x∈R|(8, 2, x),(2, x, 8),(x, 8, 2)|` = 0, then `|x/2|` is equal to ______.
If a, b, c, are non zero complex numbers satisfying a2 + b2 + c2 = 0 and `|(b^2 + c^2, ab, ac),(ab, c^2 + a^2, bc),(ac, bc, a^2 + b^2)|` = ka2b2c2, then k is equal to ______.
