Hardayal Public School Holiday Assignment of ... - Educomp Online

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5.⎡32 0⎤If, A =⎢ ⎥⎢1 4 0⎥⎢⎣0 0 5⎥⎦then verify that A 2 – 7a + 10I = 0 and hence find A -1⎡ 2 3 ⎤ ⎡10 −1⎤6. Verify that (AB)’ = B’A’ where (i) A = ⎢ ⎥ , B =⎣ 4 1⎢ ⎥ ⎦ ⎣21 3 ⎦(ii) B = [ − 2 −1− 4], A’ = [ − 1 2 3]7.⎡5− 2⎤If A = ⎢ ⎥ then verify A. (Adj A) = (Adj A).A = ⏐A ⏐I.⎣3− 2 ⎦8. If points A(3, –2); B(x, 2) and C(8, 8) are collinear, find the value of x usingdeterminants.9. Using elementary row transformation find the inverse of the following :a.⎡22 1 ⎤⎡ 3 1 ⎤⎡11 ⎤⎢ ⎥ b.⎣ 2 5⎢ ⎥ c.⎢ ⎥⎦ ⎣52⎢2 1 2⎥⎦⎢⎣1 2 2⎥⎦10.⎡ 1 tan x⎤If A = ⎢⎥ show that A’A -1 ⎡Cos2x− Sin 2x⎤=⎣− tan x 1⎢⎥ ⎦ ⎣ Sin 2xCos 2x⎦11. Find inverse of matrix diog. {a b c}12.⎡ab ⎤Find the inverse of ⎢ 1 + bc ⎥ and verify that aA⎢c= (a 2 + bc + 1)I – aA⎣ a ⎥⎦13.⎛ 8 − 6 2 ⎞⎜⎟Find the value of λ if ⎜ − 6 7 − 4 ⎟⎜⎟⎝ 2 − 4 λ ⎠is a singular matrix.14.⎡1+ 2n− 4n⎤⎡3− 4⎤Prove by mathematical induction that A = ⎢⎥ where A =⎣ n 1−2n⎢ ⎥ ⎦ ⎣1− 1 ⎦15. Find p and q such that (pI + qA) 2 ⎡01= A where A = ⎢⎣ An : p = q = ±− 1316. Points (a 1 , b 1 ); (a 2 , b 2 ); (a 1 , –a 2 , b 1 , –b 2 ) are collinear prove that a 1 b 2 = a 2 b 117.⎡ 2 − 1 − 1⎤A =⎢⎥⎢− 1 2 − 1⎥verify that A 3 – 6A 2 + 9A – 4I = 0 and hence find A -1⎢⎣1 − 1 2 ⎥⎦18.⎡2− 3⎤A = ⎢ ⎥ verify that (Adj A) -1 = adj (A -1 )⎣46 ⎦19.⎡02 y + z ⎤find x, y, z if matrix A =⎢⎥⎢x 6 − z⎥obeys the law A’A = I.⎢⎣x − y z ⎥⎦20. Solve the following equations by inverse method:
(i)2 x − y = 1x + 3 z= 11x + 2 y + z =Ans : 2 , 1, 37(ii)x − 2 y = 102 x + y + 3 z= 8− 2 y + z = 7Ans : 4 , − 3, 1(iii)x +y +z +y = 5z = 3x = 4Ans : 3, 2 ,121.⎡01 2⎤A =⎢ ⎥⎢1 2 3⎥⎢⎣3 1 1 ⎥⎦find A -1 Hence solve the equations y + 2z = 4, x + 2y + 3z = 6,3x + y + z = 4 by using A -1 (Ans: 1, –2, 3)22.⎡11 2 ⎤A =⎢⎥⎢0 2 − 3⎥⎢⎣3 1 1 ⎥⎦find A -1 and hence solve x + 3z = 10; x + 2y – 2z = –7, 2x – 3y + 4z= 1723.⎡−5 1 3 ⎤ ⎡11 2⎤Find the product of A =⎢⎥⎢7 1 − 5⎥B =⎢ ⎥⎢3 2 1⎥and using AB solve the⎢⎣1 − 1 1 ⎥⎦⎢⎣2 1 3⎥⎦equations x + y + 2z = 1, 3x + 2y + z = 7 and 2x + y + 3z = 2a + x a − x a − x24. Solve for x : (i) a − x a + x a − x = 0 (ii)a − x a − x a + xx + a b ca x + b c = 0a b x + c25. Using properties of determinants evaluate or prove the following:x + y x x2 2b c bc b + c(i)(iii)5 x + 4 y 4 x 2 x = x10 x 8 y 8 x 3 xay + z x yz + x z x3= ( x + y + z )( z − x )22 2(ii) c a ca c + a = 02b2log aab q + bp 1(iv) log b q 1 = 0 wherex + y y zlog c r 1a, b, c ore p th , q th and r th term of a G.P.1 x + y2x +2y(v) 1 y + z2y +2z = ( x − y )( y − z )( z − x )1 z + x z2+ x2
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Hardayal Public School Holiday Assignment of ... - Educomp Online

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