Advertisements
Advertisements
प्रश्न
The determinant `|("b"^2 - "ab", "b" - "c", "bc" - "ac"),("ab" - "a"^2, "a" - "b", "b"^2 - "ab"),("bc" - "ac", "c" - "a", "ab" - "a"^2)|` equals ______.
विकल्प
abc (b–c) (c – a) (a – b)
(b–c) (c – a) (a – b)
(a + b + c) (b – c) (c – a) (a – b)
None of these
Advertisements
उत्तर
The determinant `|("b"^2 - "ab", "b" - "c", "bc" - "ac"),("ab" - "a"^2, "a" - "b", "b"^2 - "ab"),("bc" - "ac", "c" - "a", "ab" - "a"^2)|` equals none of these.
Explanation:
We have, `|("b"^2 - "ab", "b" - "c", "bc" - "ac"),("ab" - "a"^2, "a" - "b", "b"^2 - "ab"),("bc" - "ac", "c" - "a", "ab" - "a"^2)|`
= `|("b"("b" - "a"), "b" - "c", "c"("b" - "a")),("a"("b" - "a"), "a" - "b", "b"("b" - "a")),("c"("b" - "a"), "c" - "a", "a"("b" - "a"))|`
[Taking (b – a) common from C1 and C3 each]
= `("b" - "a")^2 |("b", "b" - "c", "c"),("a", "a" - "b", "b"),("c", "c" - "a", "a")|`
[Applying C2 → C2 + C3]
= `("b" - "a")^2 |("b", "b", "c"),("a", "a", "b"),("c", "c", "a")|`
= 0 .....[As C1 and C2 are identical]
APPEARS IN
संबंधित प्रश्न
Using properties of determinants, prove that `|[2y,y-z-x,2y],[2z,2z,z-x-y],[x-y-z,2x,2x]|=(x+y+z)^3`
Using the property of determinants and without expanding, prove that:
`|(1, bc, a(b+c)),(1, ca, b(c+a)),(1, ab, c(a+b))| = 0`
By using properties of determinants, show that:
`|(y+k,y, y),(y, y+k, y),(y, y, y+k)| = k^2(3y + k)`
Using properties of determinants, prove that
`|(sin alpha, cos alpha, cos(alpha+ delta)),(sin beta, cos beta, cos (beta + delta)),(sin gamma, cos gamma, cos (gamma+ delta))| = 0`
Using properties of determinants, prove the following :
Using properties of determinants, prove that:
`|[a^2 + 1, ab, ac], [ba, b^2 + 1, bc ], [ca, cb, c^2+1]| = a^2 + b^2 + c^2 + 1`
Solve for x : `|("a"+"x","a"-"x","a"-"x"),("a"-"x","a"+"x","a"-"x"),("a"-"x","a"-"x","a"+"x")| = 0`, using properties of determinants.
Without expanding determinants, show that
`|(1, 3, 6),(6, 1, 4),(3, 7, 12)| + |(2, 3, 3),(2, 1, 2),(1, 7, 6)| = 10|(1, 2, 1),(3, 1, 7),(3, 2, 6)|`
By using properties of determinants, prove that `|(x + y, y + z, z + x),(z, x, y),(1, 1, 1)|` = 0.
Without expanding the determinants, show that `|("b" + "c", "bc", "b"^2"c"^2),("c" + "a", "ca", "c"^2"a"^2),("a" + "b", "ab", "a"^2"b"^2)|` = 0
Without expanding evaluate the following determinant:
`|(1, "a", "b" + "c"),(1, "b", "c" + "a"),(1, "c", "a" + "b")|`
Select the correct option from the given alternatives:
The value of a for which system of equation a3x + (a + 1)3 y + (a + 2)3z = 0 ax + (a +1)y + (a + 2)z = 0 and x + y + z = 0 has non zero Soln. is
Select the correct option from the given alternatives:
If `|(6"i", -3"i", 1),(4, 3"i", -1),(20, 3, "i")|` = x + iy then
Answer the following question:
Evaluate `|(2, 3, 5),(400, 600, 1000),(48, 47, 18)|` by using properties
Answer the following question:
Evaluate `|(101, 102, 103),(106, 107, 108),(1, 2, 3)|` by using properties
Answer the following question:
Without expanding determinant show that
`|(l, "m", "n"),("e", "d", "f"),("u", "v", "w")| = |("n", "f", "w"),(l, "e", "u"),("m", "d", "v")|`
Evaluate: `|("a" - "b" - "c", 2"a", 2"a"),(2"b", "b" - "c" - "a", 2"b"),(2"c", 2"c", "c" - "a" - "b")|`
Prove that: `|(y^2z^2, yz, y + z),(z^2x^2, zx, z + x),(x^2y^2, xy, x + y)|` = 0
The number of distinct real roots of `|(sinx, cosx, cosx),(cosx, sinx, cosx),(cosx, cosx, sinx)|` = 0 in the interval `pi/4 x ≤ pi/4` is ______.
The maximum value of Δ = `|(1, 1, 1),(1, 1 + sin theta, 1),(1 + cos theta, 1, 1)|` is ______. (θ is real number)
The determinant `abs (("a","bc","a"("b + c")),("b","ac","b"("c + a")),("c","ab","c"("a + b"))) =` ____________
Let P be any non-empty set containing p elements. Then, what is the number of relations on P?
`f : {1, 2, 3) -> {4, 5}` is not a function, if it is defined by which of the following?
Let 'A' be a square matrix of order 3 × 3, then |KA| is equal to:
Which of the following is correct?
If A, B and C are the angles of a triangle ABC, then `|(sin2"A", sin"C", sin"B"),(sin"C", sin2"B", sin"A"),(sin"B", sin"A", sin2"C")|` = ______.
Let a, b, c be such that b(a + c) ≠ 0 if
`|(a, a + 1, a - 1),(-b, b + 1, b - 1),(c, c - 1, c + 1)| + |(a + 1, b + 1, c - 1),(a - 1, b - 1, c + 1),((-1)^(n + 2)a, (-1)^(n + 1)b, (-1)^n c)|` = 0, then the value of n is ______.
If f(α) = `[(cosα, -sinα, 0),(sinα, cosα, 0),(0, 0, 1)]`, prove that f(α) . f(– β) = f(α – β).
Without expanding determinants find the value of `|(10,57,107),(12,64,124),(15,78,153)|`
By using properties of determinant prove that `|(x+y,y+z,z+x),(z,x,y),(1,1,1)|` = 0.
Without expanding determinant find the value of `|(10,57,107),(12,64,124),(15,78,153)|`
By using properties of determinants, prove that
`|(x+y, y+z, z+x),(z, x, y),(1, 1, 1)|` = 0
The value of the determinant of a matrix A of order 3 is 3. If C is the matrix of cofactors of the matrix A, then what is the value of the determinant of C2?
Without expanding determinants, find the value of `|(10, 57, 107),(12, 64, 124),(15, 78, 153)|`
By using properties of determinant prove that `|(x+y,y+z,z+x),(z,x,y),(1,1,1)|` = 0.
