Advertisements
Advertisements
प्रश्न
If a, b, c are pth, qth and rth terms of an A.P, find the value of `|("a", "b", "c"),("p", "q", "r"),(1, 1, 1)|`
Advertisements
उत्तर
Given a, b, c are pth, qth and rth terms of an A.P.
tp = a = A + (p – 1)D,
tq = b = A + (q – 1)D,
tr = c = A + (r – 1) D
Where A – first term, D – Common difference of the A.P.
`|("a", "b", "c"),("p", "q", "r"),(1, 1, 1)| = |("A" + ("p" - 1)"D", "A" + ("q" - 1)"D", "A" + ("r" - 1)"D"),("p", "q", "r"),(1, 1, 1)|`
= `|("A", "A","A"),("p","q", "r"),(1, 1, 1)| + |(("p" - 1)"D", ("q" - 1)"D", ("r" - 1)"D"),("p", "q", "r"),(1, 1, 1)|`
= `"A"|(1, 1, 1),("p", "q", "r"),(1, 1, 1)| + "D"|("p" - 1, "q" - 1, "r" - 1),("p", "q", "r"),(1, 1, 1)|`
= `"A" xx 0 + "D" |("p", "q", "r"),("p", "q", "r"),(1, 1, 1)| "R"_1 -> "R"_1+ "R"_3`
= 0 + D × 0 Two rows are same
`|("a", "b", "c"),("p", "q", "r"),(1, 1, 1)|` = 0
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 `|(1 + "a", 1, 1),(1, 1 + "b", 1),(1, 1, 1 + "c")| = "abc"(1 + 1/"a" + 1/"b" + 1/"c")`
Prove that `|(sec^2theta, tan^2theta, 1),(tan^2theta, sec^2theta, -1),(38, 36, 2)|` = 0
Show that `|(x + 2"a", y + 2"b", z + 2"c"),(x, y, z),("a", "b", "c")|` = 0
Write the general form of a 3 × 3 skew-symmetric matrix and prove that its determinant is 0
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 = `[(1/2, alpha),(0, 1/2)]`, prove that `sum_("k" = 1)^"n" det("A"^"k") = 1/3(1 - 1/4)`
Without expanding, evaluate the following determinants:
`|(x + y, y + z, z + x),(z, x, y),(1, 1, 1)|`
Show that `|("b" + "c", "a" - "c", "a" - "b"),("b" - "c", "c" + "a", "b" - "a"),("c" - "b", "c" - "a", "a" + b")|` = 8abc
If (k, 2), (2, 4) and (3, 2) are vertices of the triangle of area 4 square units then determine the value of k
Identify the singular and non-singular matrices:
`[(1, 2, 3),(4, 5, 6),(7, 8, 9)]`
Determine the values of a and b so that the following matrices are singular:
A = `[(7, 3),(-2, "a")]`
If cos 2θ = 0, determine `[(theta, costheta, sintheta),(costheta, sintheta, 0),(sintheta, 0, costheta)]^2`
If P1, P2, P3 are respectively the perpendiculars from the vertices of a triangle to the opposite sides, then `cosA/P_1 + cosB/P_2 + cosC/P_3` is equal to
If f(x) = `|(cos^2x, cosx.sinx, -sinx),(cosx sinx, sin^2x, cosx),(sinx, -cosx, 0)|`, then for all x
Choose the correct option:
Let `|(0, sin theta, 1),(-sintheta, 1, sin theta),(1, -sin theta, 1 - a)|` where 0 ≤ θ ≤ 2n, then
Let a, b, c, d be in arithmetic progression with common difference λ. If `|(x + a - c, x + b, x + a),(x - 1, x + c, x + b),(x - b + d, x + d, x + c)|` = 2, then value of λ2 is equal to ______.
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 ______.
