English

Using Properties of Determinants, Prove That: |(1, 1+P, 1+P+Q),(2, 3+2p, 4+3p+2q),(3,6+3p,10+6p+3q)| = 1 - Mathematics

Advertisements
Advertisements

Question

Using properties of determinants, prove that:

`|(1, 1+p, 1+p+q),(2, 3+2p, 4+3p+2q),(3,6+3p,10+6p+3q)| =  1`                 

Advertisements

Solution

shaalaa.com
  Is there an error in this question or solution?
Chapter 4: Determinants - Exercise 4.7 [Page 142]

APPEARS IN

NCERT Mathematics Part 1 and 2 [English] Class 12
Chapter 4 Determinants
Exercise 4.7 | Q 14 | Page 142

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 prove the following: `|[1,x,x^2],[x^2,1,x],[x,x^2,1]|=(1-x^3)^2`


Using properties of determinants, prove that

`|((x+y)^2,zx,zy),(zx,(z+y)^2,xy),(zy,xy,(z+x)^2)|=2xyz(x+y+z)^3`

 


 

If ` f(x)=|[a,-1,0],[ax,a,-1],[ax^2,ax,a]| ` , using properties of determinants find the value of f(2x) − f(x).

 

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:

`|(a-b,b-c,c-a),(b-c,c-a,a-b),(a-a,a-b,b-c)| = 0`


By using properties of determinants, show that:

`|(0,a, -b),(-a,0, -c),(b, c,0)| = 0`


By using properties of determinants, show that:

`|(y+k,y, y),(y, y+k, y),(y, y, y+k)| = k^2(3y + k)`


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`


By using properties of determinants, show that:

`|(1,x,x^2),(x^2,1,x),(x,x^2,1)| = (1-x^3)^2`


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 the following :

\[\begin{vmatrix}1 & a & a^2 \\ a^2 & 1 & a \\ a & a^2 & 1\end{vmatrix} = \left( 1 - a^3 \right)^2\].

Using properties of determinants, prove the following:

\[\begin{vmatrix}x^2 + 1 & xy & xz \\ xy & y^2 + 1 & yz \\ xz & yz & z^2 + 1\end{vmatrix} = 1 + x^2 + y^2 + z^2\] .

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 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`


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


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


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 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 value of the determinant `|(x , x + y, x + 2y),(x + 2y, x, x + y),(x + y, x + 2y, x)|` is ______.


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.


The determinant `abs (("a","bc","a"("b + c")),("b","ac","b"("c + a")),("c","ab","c"("a + b"))) =` ____________


Using properties of determinants `abs ((1, "a", "a"^2 - "bc"),(1, "b", "b"^2 - "ca"),(1, "c", "c"^2 - "ab")) =` ____________.


Let P be any non-empty set containing p elements. Then, what is the number of relations on P?


The A.M., H.M. and G.M. between two numbers are `144/15`, 15 and 12, but not necessarily in this order then, H.M., G.M. and A.M. respectively are


If f(α) = `[(cosα, -sinα, 0),(sinα, cosα, 0),(0, 0, 1)]`, prove that f(α) . f(– β) = f(α – β).


The value of the determinant `|(6, 0, -1),(2, 1, 4),(1, 1, 3)|` is ______.


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


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


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 the determinant, find the value of `|(10,57,107),(12,64,124),(15,78,153)|`


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×