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Thursday, July 4, 2019

CBSE CLASS 12 MATHS IMPORTANT QUESTIONS - PROPERTIES OF DETERMINANTS

Properties of Determinants -important Questions

Here we are providing Important Practice Questions based on Properties of Determinants prepared by Experts for 2019-20 Board Examinations.

Properties Of Determinants - Extra Practice  Questions 

  1.Prove that | yz x 2 zx y 2 xy z 2 zx y 2 xy z 2 yz x 2 xy z 2 yz x 2 zx y 2 | is divisible by ( x+y+z ), andhence find the quotient. 2.| ( x+y ) 2 zx zy zx ( z+y ) 2 xy zy xy ( z+x ) 2 |=2xyz ( x+y+z ) 3 3.Show that the ΔABC is an isosceles triangle if the determinant | 1 1 1 1+cosA 1+cosB 1+cosC cos 2 A+cosA cos 2 B+cosB cos 2 C+cosC |=0 4. Write the value of | ab bc ca bc ca ab ca ab bc | 5. Solve forx:| a+x ax ax ax a+x ax ax ax a+x |=0 6.| ( b+c ) 2 a 2 bc ( c+a ) 2 b 2 ac ( a+b ) 2 c 2 ab |=( ab )( bc )( a+b+c )( a 2 + b 2 + c 2 ) 7. If a, b and c all non -zero and | 1+a 1 1 1 1+b 1 1 1 1+c |=0, then prove that 1 a + 1 b + 1 c +1=0 8. if f( x )=| a 1 0 ax a 1 a x 2 ax a |, using properties of determinant, find the value f( 2x )f( x ). 9.| a 3 2 a b 3 2 b c 3 2 c |=2( ab )( bc )( ca )( a+b+c ). 10. Write the value of | x+y y+z z+x z x y 3 3 3 |. 11.| a 2 bc ac+ c 2 a 2 +ab b 2 ac ab b 2 +bc c 2 |=4 a 2 b 2 c 2 . 12.If a+b+c0and| a b c b c a c a b |, then prove that with properties of determinant a = b = c. 13.| 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 14.| x+2 x+6 x1 x+6 x1 x+2 x1 x+2 x+6 |=0, find the value of x. 15.| 1 x x+1 2x x( x1 ) x(x+1) 3x( 1x ) x( x1 )( x2 ) x( x+1 )( x1 ) |=6 x 2 ( 1 x 2 ). 16.| 1 a a 2 a 2 1 a a a 2 1 |= ( 1 a 3 ) 2 17.| 2y yzx 2y 2z 2z xyz xyz 2x 2x |= ( x+y+z ) 3 18.| a+b+2c a b c b+c+2a b c a c+a+2b |=2 ( a+b+c ) 2 19.| x 2 +1 xy xz xy y 2 +1 yz xz yz z 2 +1 |=1+ x 2 + y 2 + z 2 20.| x+y x x 5x+4y 4x 2x 10x+8y 8x 3x |= x 3 21.| 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 |. 22.| 1+a 1 1 1 1+b 1 1 1 1+c |=abc+bc+ca+ab 23.| a+x y z x a+y z x y a+z |= a 2 ( a+x+y+z ). 24.| x+λ 2x 2x 2x x+λ 2x 2x 2x x+λ |=( 5x+λ ) ( λx ) 2 . 25.| a 2 bc ac+ c 2 a 2 +ab b 2 ac ab b 2 +bc c 2 |=4 a 2 b 2 c 2 26.| 1 x x 2 x 2 1 x x x 2 1 |= ( 1 x 3 ) 2 . 27.| x x+y x+2y x+2y x x+y x+y x+2y x |=9 y 2 ( x+y ). 28.| 3x x+y x+z xy 3y zy xz yz 3z |=3( x+y+z )( xy+yz+zx ). 29.| a 2 +1 ab ac ab b 2 +1 bc ca cb c 2 +1 |=1+ a 2 + b 2 + c 2 . 30.| 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 31.| 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 | 32.| 1 1 1 a b c a 3 b 3 c 3 |=( ab )( bc )( ca )( a+b+c ). 33. If a, b, c are the p th , q th and r th terms respectively of a G.P., prove that | loga p 1 logb q 1 logc r 1 | =0

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