Advertisements
Advertisements
प्रश्न
Solve that `|(x + "a", "b", "c"),("a", x + "b", "c"),("a", "b", x + "c")|` = 0
उत्तर
`|(x + "a", "b", "c"),("a", x + "b", "c"),("a", "b", x + "c")|` = 0
Put x = 0
`|(0 + "a", "b", "c"),("a", 0 + "b", "c"),("a", "b", 0 + "c")|` = 0
`|("a", "b", "c"),("a", "b", "c"),("a", "b", "c")|` = 0
= 0
x = 0 satisfies the given equation. x = 0 is a root of the given equation
Since three rows are identical. x = 0 is a root of multiplicity 2.
Since the degree of the product of the leading diagonal elements (x + a)(x + b)(x + c) is 3.
There is one more root for the given equation.
Put x = – (a + b + c)
`|((-"a" + "b" + "c") + "a", "b", "c"),("a", -("a" + "b" + "c") + "b", "c"),("a", "b", -("a" + "b" + "c") + "c")|` = 0
`|(-"b" - "c", "b", "c"),("a", -"a" - "c", "c"),("a", "b", -"a" - "b")|` = 0
`"C"_1 -> "C"_1 + "C"_2 + "C"_3`
`|(-"b" - "c" + "b" + "c", "b", "c"),("a" - "a" - "c" + "c", -"a" - "c", "c"),("a" + "b" - "a" - "b", "b", -"a" - "b")|` = 0
`|(0, "b", "c"),(0, -"a" - "c", "c"),(0, "b", -"a" - "b")|` = 0
0 = 0
∴ x = – (a + b + c) satisfies the given equation.
Hence, the required roots of the given equation are x = 0, 0, – (a + b + c)
APPEARS IN
संबंधित प्रश्न
Without expanding the determinant, prove that `|("s", "a"^2, "b"^2 + "c"^2),("s", "b"^2, "c"^2 + "a"^2),("s", "c"^2, "a"^2 + "b"^2)|` = 0
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`
Prove that `|(1 + "a", 1, 1),(1, 1 + "b", 1),(1, 1, 1 + "c")| = "abc"(1 + 1/"a" + 1/"b" + 1/"c")`
Prove that `|(1, "a", "a"^2 - "bc"),(1, "b", "b"^2 - "ca"),(1, "c", "c"^2 - "ab")|` = 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
If a, b, c, are all positive, and are pth, qth and rth terms of a G.P., show that `|(log"a", "p", 1),(log"b", "q", 1),(log"c", "r", 1)|` = 0
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 λ = – 2, determine the value of `|(0, lambda, 1),(lambda^2, 0, 3lambda^2 + 1),(-1, 6lambda - 1, 0)|`
Determine the roots of the equation `|(1,4, 20),(1, -2, 5),(1, 2x, 5x^2)|` = 0
Solve `|(4 - x, 4 + x, 4 + x),(4 + x, 4 - x, 4 + x),(4 + x, 4 + x, 4 - x)|` = 0
Determine the values of a and b so that the following matrices are singular:
A = `[(7, 3),(-2, "a")]`
Find the value of the product: `|(log_3 64, log_4 3),(log_3 8, log_4 9)| xx |(log_2 3, log_8 3),(log_3 4, log_3 4)|`
Choose the correct alternative:
The value of x, for which the matrix A = `[("e"^(x - 2), "e"^(7 + x)),("e"^(2 + x), "e"^(2x + 3))]` is singular
Choose the correct alternative:
if Δ = `|("a", "b", "c"),(x, y, z),("p", "q", "r")|` then `|("ka", "kb","kc"),("k"x, "k"y, "k"z),("kp", "kq", "kr")|` is
A pole stands vertically inside a triangular park ΔABC. If the angle of elevation of the top of the pole from each corner of the park is same, then in ΔABC the foot of the pole is at the
`|("b" + "c", "c", "b"),("c", "c" + "a", "a"),("b", "a", "a" + "b")|` = ______.
If a, b, c are positive and are the pth, qth and rth terms respectively of a G.P., then the value of `|(loga, p, 1),(logb, q, 1),(logc, r, 1)|` is ______.
If `x∈R|(8, 2, x),(2, x, 8),(x, 8, 2)|` = 0, then `|x/2|` is equal to ______.
