Advertisements
Advertisements
Question
The value of determinant `|("a" - "b", "b" + "c", "a"),("b" - "a", "c" + "a", "b"),("c" - "a", "a" + "b", "c")|` is ______.
Options
a3 + b3 + c3
3bc
a3 + b3 + c3 – 3abc
None of these
Advertisements
Solution
The value of determinant `|("a" - "b", "b" + "c", "a"),("b" - "a", "c" + "a", "b"),("c" - "a", "a" + "b", "c")|` is none of these.
Explanation:
Here, we have `|("a" - "b", "b" + "c", "a"),("b" - "a", "c" + "a", "b"),("c" - "a", "a" + "b", "c")|`
C2 → C2 + C3
⇒ `|("a" - "b", "a" + "b" + "c", "a"),("b" - "a", "a" + "b" + "c", "b"),("c" - "a", "a" + "b" + "c", "c")|`
⇒ `("a" + "b" + "c") |("a" - "b", 1, "a"),("b" - "a", 1, "b"),("c" - "a", 1, "c")|` .....(Taking a + b + c common from C2)
R1 → R1 – R2, R2 → R2 – R3
⇒ `("a" + "b" + "c") |(2("a" - "b"), 0, "a" - "b"),("b" - "c", 0, "b" - "c"),("c" - "a", 1, "c")|`
Taking (a – b) and (b – c) common from R1 and R2 respectively
⇒ `("a" + "b" + "c")("a" - "b")("b" - "c") |(2, 0, 1),(1, 0, 1),("c" - "a", 1, "c")|`
Expanding along C2
⇒ `("a" + "b" + "c")("a" - "b")("b" - "c") [-1|(2, 1),(1, 1)|]`
⇒ (a + b + c)(a – b)(b – c)(– 1)
⇒ (a + b + c)(a – b)(c – b)
APPEARS IN
RELATED QUESTIONS
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 properties of determinants, show that ΔABC is isosceles if:`|[1,1,1],[1+cosA,1+cosB,1+cosC],[cos^2A+cosA,cos^B+cosB,cos^2C+cosC]|=0`
Using the properties of determinants, prove the following:
`|[1,x,x+1],[2x,x(x-1),x(x+1)],[3x(1-x),x(x-1)(x-2),x(x+1)(x-1)]|=6x^2(1-x^2)`
Using properties of determinants, prove that
`|[x+y,x,x],[5x+4y,4x,2x],[10x+8y,8x,3x]|=x^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`
Using the property of determinants and without expanding, prove that:
`|(b+c, q+r, y+z),(c+a, r+p, z +x),(a+b, p+q, x + y )| = 2|(a,p,x),(b,q,y),(c, r,z)|`
By using properties of determinants, show that:
`|(-a^2, ab, ac),(ba, -b^2, bc),(ca,cb, -c^2)| = 4a^2b^2c^2`
By using properties of determinants, show that:
`|(x+y+2z, x, y),(z, y+z+2z,y),(z,x,z+x+2y)| = 2(x+y+z)^3`
Evaluate `|(1,x,y),(1,x+y,y),(1,x,x+y)|`
Using properties of determinants, prove that
`|(a^2 + 2a,2a + 1,1),(2a+1,a+2, 1),(3, 3, 1)| = (a - 1)^3`
Using properties of determinants, prove that `|(x,x+y,x+2y),(x+2y, x,x+y),(x+y, x+2y, x)| = 9y^2(x + y)`
Using properties of determinants, prove that `|(1,1,1+3x),(1+3y, 1,1),(1,1+3z,1)| = 9(3xyz + xy + yz+ zx)`
Using properties of determinants, prove the following :
Using properties of determinants, prove the following:
Using properties of determinants, prove that \[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix} = a^2 \left( a + x + y + z \right)\] .
Using properties of determinant prove that
`|(b+c , a , a), (b , c+a, b), (c, c, a+b)|` = 4abc
Without expanding the determinant, find the value of `|(10, 57, 107),(12, 64, 124),(15, 78, 153)|`.
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 the determinants, show that `|(l, "m", "n"),("e", "d", "f"),("u", "v", "w")| = |("n", "f", "w"),(l, "e", "u"),("m", "d", "v")|`
If `|("x"^"k", "x"^("k" + 2), "x"^("k" + 3)),("y"^"k", "y"^("k" + 2), "y"^("k" + 3)),("z"^"k", "z"^("k" + 2), "z"^("k" + 3))|` = (x - y) (y - z) (z - x)`(1/"x"+ 1/"y" + 1/"z") ` then
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 `|(101, 102, 103),(106, 107, 108),(1, 2, 3)|` by using properties
Answer the following question:
By using properties of determinant prove that `|(x + y, y + z, z + x),(z, x, y),(1, 1, 1)|` = 0
Answer the following question:
Without expanding determinant show that
`|("b" + "c", "bc", "b"^2"c"^2),("c" + "a", "ca", "c"^2"a"^2),("a" + "b", "ab", "a"^2"b"^2)|` = 0
Evaluate: `|("a" + x, y, z),(x, "a" + y, z),(x, y, "a" + z)|`
If A + B + C = 0, then prove that `|(1, cos"c", cos"B"),(cos"C", 1, cos"A"),(cos"B", cos"A", 1)|` = 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 ______.
If A, B and C are angles of a triangle, then the determinant `|(-1, cos"C", cos"B"),(cos"C", -1, cos"A"),(cos"B", cos"A", -1)|` is equal to ______.
If cos2θ = 0, then `|(0, costheta, sin theta),(cos theta, sin theta,0),(sin theta, 0, cos theta)|^2` = ______.
`|(x + 1, x + 2, x + "a"),(x + 2, x + 3, x + "b"),(x + 3, x + 4, x + "c")|` = 0, where a, b, c are in A.P.
The determinant `|(sin"A", cos"A", sin"A" + cos"B"),(sin"B", cos"A", sin"B" + cos"B"),(sin"C", cos"A", sin"C" + cos"B")|` is equal to zero.
Let Δ = `|("a", "p", x),("b", "q", y),("c", "r", z)|` = 16, then Δ1 = `|("p" + x, "a" + x, "a" + "p"),("q" + y, "b" + y, "b" + "q"),("r" + z, "c" + z, "c" + "r")|` = 32.
If a, b, c are the roots of the equation x3 - 3x2 + 3x + 7 = 0, then the value of `abs((2 "bc - a"^2, "c"^2, "b"^2),("c"^2, 2 "ac - b"^2, "a"^2),("b"^2, "a"^2, 2 "ab - c"^2))` is ____________.
The value of the determinant `abs ((alpha, beta, gamma),(alpha^2, beta^2, gamma^2),(beta + gamma, gamma + alpha, alpha + beta)) =` ____________.
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)|`
Without expanding evaluate the following determinant.
`|(1, a, b+c), (1, b, c+a), (1, c, a+b)|`
Without expanding determinants, find the value of `|(10, 57, 107),(12, 64, 124),(15, 78, 153)|`
Without expanding evaluate the following determinant.
`|(1, a, b + c),(1, b, c + a),(1, c, a + b)|`
