हिंदी

By Using Properties of Determinants, Show That: |(Asqrt2+1, Ab, Ac),(Ab, Bsqrt2+1, Bc),(Ca, Cb, Csqrt2+1)| = 1+Asqrt2bsqrt2+Csqrt2 - Mathematics

Advertisements
Advertisements

प्रश्न

By using properties of determinants, show that:

`|(a^2+1, ab, ac),(ab, b^2+1, bc),(ca, cb, c^2+1)| = 1+a^2+b^2+c^2`

Advertisements

उत्तर

shaalaa.com
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 4: Determinants - Exercise 4.2 [पृष्ठ १२१]

APPEARS IN

एनसीईआरटी Mathematics Part 1 and 2 [English] Class 12
अध्याय 4 Determinants
Exercise 4.2 | Q 14 | पृष्ठ १२१

संबंधित प्रश्न

 

Using properties of determinants, prove that 

`|[b+c,c+a,a+b],[q+r,r+p,p+q],[y+z,z+x,x+y]|=2|[a,b,c],[p,q,r],[x,y,z]|`

 

By using properties of determinants, show that:

`|(1,1,1),(a,b,c),(a^3, b^3,c^3)|` = (a-b)(b-c)(c-a)(a+b+c)


By using properties of determinants, show that:

`|(1+a^2-b^2, 2ab, -2b),(2ab, 1-a^+b^2, 2a),(2b, -2a, 1-a^2-b^2)| = (1+a^2+b^2)`


Without expanding the determinant, prove that

`|(a, a^2,bc),(b,b^2, ca),(c, c^2,ab)| = |(1, a^2, a^3),(1, b^2, b^3),(1, c^2, c^3)|`


Using properties of determinants, prove that:

`|(alpha, alpha^2,beta+gamma),(beta, beta^2, gamma+alpha),(gamma, gamma^2, alpha+beta)|` =  (β – γ) (γ – α) (α – β) (α + β + γ)


Using properties of determinants, prove that:

`|(1+a^2-b^2, 2ab, -2b),(2ab, 1-a^2+b^2, 2a),(2b, -2a, 1-a^2-b^2)| = (1 + a^2 + b^2)^3`


Using properties of determinants, prove that

`|[b+c , a ,a  ] ,[ b , a+c, b ] ,[c , c, a+b ]|` = 4abc 


Using properties of determinant prove that 

`|(b+c , a , a), (b , c+a, b), (c, c, a+b)|` = 4abc


Without expanding evaluate the following determinant:

`|(1, "a", "b" + "c"),(1, "b", "c" + "a"),(1, "c", "a" + "b")|`


If `|(4 + x, 4 - x, 4 - x),(4 - x, 4 + x, 4 - x),(4 - x, 4 - x, 4 + x)|` = 0, then find the values of x.


Find the value (s) of x, if `|(1, 4, 20),(1, -2, -5),(1, 2x, 5x^2)|` = 0


Without expanding evaluate the following determinant:

`|(2, 3, 4),(5, 6, 8),(6x, 9x, 12x)|`


If  `|(4 + x, 4 - x, 4 - x),(4 - x,4 + x,4 - x),(4 - x,4 - x, 4 + x)|` = 0, then find the values of x.


Select the correct option from the given alternatives:

The determinant D = `|("a", "b", "a" + "b"),("b", "c", "b" + "c"),("a" + "b", "b" + "c", 0)|` = 0 if


Select the correct option from the given alternatives:

If x = –9 is a root of `|(x, 3, 7),(2, x, 2),(7, 6, x)|` = 0 has other two roots are


Select the correct option from the given alternatives:

If `|(6"i", -3"i", 1),(4, 3"i", -1),(20, 3, "i")|` = x + iy then


The value of `|(1, 1, 1),(""^"n""C"_1, ""^("n" + 2)"C"_1, ""^("n" + 4)"C"_1),(""^"n""C"_2, ""^("n" + 2)"C"_2, ""^("n" + 4)"C"_2)|` is 8.


Evaluate: `|(x^2 - x + 1, x - 1),(x + 1, x + 1)|`


Prove that: `|(y + z, z, y),(z, z + x, x),(y, x, x + y)|` = 4xyz


If `[(4 - x, 4 + x, 4 + x),(4 + x, 4 - x, 4 + x),(4 + x, 4 + x, 4 - x)]` = 0, then find values of x.


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 ______.


If cos2θ = 0, then `|(0, costheta, sin theta),(cos theta, sin theta,0),(sin theta, 0, cos theta)|^2` = ______.


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 ____________.


If the ratio of the H.M. and GM. between two numbers a and bis 4 : 5, then a: b is


A system of linear equations represented in matrix form Ax = 0, A is n × n matrix, has a non-zero solution if the determinant of A (i.e., det(A)) is


Without expanding determinants find the value of `|(10,57,107), (12, 64, 124), (15, 78, 153)|`


Without expanding evaluate the following determinant.

`|(1, a, a + 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)|`


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 evaluate the following determinant.

`|(1, a, b+c), (1, b, c+a), (1, c, a+b)|`


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 evaluate the following determinant.

`|(1, a, b + c),(1, b, c + a),(1, c, a + b)|`


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×