Advertisements
Advertisements
प्रश्न
Show that `|("a"^2 + x^2, "ab", "ac"),("ab", "b"^2 + x^2, "bc"),("ac", "bc", "c"^2 + x^2)|` is divisiible by x4
उत्तर
Let Δ = `|("a"^2 + x^2, "ab", "ac"),("ab", "b"^2 + x^2, "bc"),("ac", "bc", "c"^2 + x^2)|`
Δ = `"a"/"a"|("a"^2 + x^2, "ab", "ac"),("ab", "b"^2 + x^2, "bc"),("ac", "bc", "c"^2 + x^2)|`
Multiply C1 by a
Δ = `1/"a"|("a"^2 + "a"x^2, "ab", "ac"),("ab", "b"^2 + x^2, "bc"),("ac", "bc", "c"^2 + x^2)|`
Applying `"C"_1 -> "C"_1 + "bC"_2 + "cC"_3`
= `1/"a"|("a"^3 + "a"x^2 +"ab"^2 + "ac"^2, "ab", "ac"),("a"^2"b" + "b"^3 + "b"x^2 + "bc"2, "b"^2 + x^2, "bc"),("a"^2"c" + "b"^2"C" + "c"^3 + "c"x^2, "bc", "c"^2 + x^2)|`
= `1/"a"|("a"("a"^2 + "b"^2 + "c"^2 + x^2), "ab", "ac"),("b"("a"^2 + "b"^2 + "c"^2 + x^2), "b"^2 + x^2, "bc"),("c"("a"^2 + "b"^2 + "c"^2 + x^), "bc", "c"^2 + x^2)|`
= `("a"^2 + "b"^2 + "c"^2 + x^2)/"a" |("a", "bc", "ac"),("b","b"^2 + x^2, "bc"),("c", "bc", "c"^2 + x^2)|`
Applyig `"C"_2 -> "C"_2 - "bC"_1` and `"C"_3 -> "C"_3- "cC"_1`
= `("a"^2 + "b"^2 + "c"^2 + x^2)/"a" |("a", "ab" - "ab", "ac" - "ac"),("b", "b"^2 + x^2 - "b"^2, "bc" - "bc"),("c", "bc" - "bc", "c"^2 + x^2 - "c"^2)|`
= `("a"^2 + "b"^2 + "c"^2 + x^2)/"a" |("a", 0, 0),("b", x^2, 0),("c", 0, x^2)|`
Expanding along the first row
= `("a"^2 + "b"^2 + "c"^2 + x^2)/"a" xx "a"[(x^2) (x^2) - (0) (0)] + 0 + 0`
= `("a"^2 + "b"^2+ "c"^2 + x^2)x^4`
Which is divisible by x4
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
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 `|("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")`
If `|("a", "b", "a"alpha + "b"),("b", "c", "b"alpha + "c"),("a"alpha + "b", "b"alpha + "c", 0)|` = 0, prove that a, b, c are in G. P or α is a root of ax2 + 2bx + c = 0
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
If A = `[(1/2, alpha),(0, 1/2)]`, prove that `sum_("k" = 1)^"n" det("A"^"k") = 1/3(1 - 1/4)`
Solve the following problems by using Factor Theorem:
Show that `|(x, "a", "a"),("a", x, "a"),("a", "a", x)|` = (x – a)2 (x + 2a)
Solve `|(4 - x, 4 + x, 4 + x),(4 + x, 4 - x, 4 + x),(4 + x, 4 + x, 4 - x)|` = 0
If (k, 2), (2, 4) and (3, 2) are vertices of the triangle of area 4 square units then determine the value of k
Determine the values of a and b so that the following matrices are singular:
B = `[("b" - 1, 2, 3),(3, 1, 2),(1, -2, 4)]`
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:
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 ⌊.⌋ denotes the greatest integer less than or equal to the real number under consideration and – 1 ≤ x < 0, 0 ≤ y < 1, 1 ≤ z ≤ 2, then the value of the determinant `[([x] + 1, [y], [z]),([x], [y] + 1, [z]),([x], [y], [z] + 1)]`
The remainder obtained when 1! + 2! + 3! + ......... + 10! is divided by 6 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
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 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 ______.
