SUMMER VACATION HOMEWORKCLASS-XII

PREPARED BY- V. D. DUBEY PGT (MATHS)

1. If Aα = [ cosα sinα−sinα cosα ] ,then prove that (i) Aα Aβ = Aα+β

(ii) (Aα)n=[ cosnα sin nα−sin nα cosnα ] for every positive integer n.

2. If A= [1 2 22 1 22 2 1] ,then prove that A2-4A-5I=0 . (CBSE2008)

3. Express the following matrices as the sum of symmetric and skew symmetric matrix-[ 3 2 71 4 3

−2 5 8]4. Using properties of determinants prove

(i) | x y zx2 y2 z2

x3 y3 z3|=xyz (x− y ) ( y−z ) ( z−x ) . (CBSE2010,11)(ii) | x y z

x2 y2 z2

y+z z+x x+ y|=( x− y ) ( y−z ) ( z−x ) ( x+ y+ z ) . (CBSE2007,08,10)(iii) If x≠ y ≠ z∧|x x2 1+x3

y y2 1+ y3

z z2 1+z3|=0 , then prove that xyz=−1. (CBSE2011)(iv) For any scalar p prove that

|x x2 1+ p x3

y y2 1+p y3

z z2 1+ pz3|=(1+ pxyz ) ( x− y ) ( y−z ) ( z−x ) .

(v) Without expanding prove that|1 a a2−bc1 b b2−ca1 c c2−ab|=0. (CBSE 2002)

(vi) Without expanding prove that|1 1+ p 1+ p+q2 3+2 p 1+3 p+2q3 6+3 p 1+6 p+3q|=1. (CBSE2009)

(vii) Show that |b+c c+a a+bq+r r+ p p+qy+z z+x x+ y|=2|

a b cp q rx y z|. (CBSE2004,06,10,12)

(viii) |1+a 1 11 1+b 11 1 1+c|=abc(1+ 1a +

1b+1c )=abc+bc+ca+ab . (CBSE2004,09,12)

(ix) |(b+c)2 a2 a2

b2 (c+a)2 b2

c2 c2 (a+b)2|=2abc (a+b+c )3. (CBSE2006,10)(x) |(b+c)

2 ba caab (c+a)2 cbac bc (a+b)2|=2abc (a+b+c )3. (CBSE2006,10)

5. Usingelementry operations find the inverseof the followingmatrices(i) [2 5

1 3] (ii) [2 −1 44 0 23 −2 7 ]6. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}. (CBSE 2009)7. Show that the relation R defined by R = {(a, b) : a- b is divisible by 3; a,b∈ Z} is an equivalence relation. (CBSE 2008)8. Show that the function f :N→N definedby

f ( x )={n+12 ,∧if n is odd

n2,∧if n is even

Is many –one onto function. (CBSE 2009)9. Considerf :R+¿→¿ givenby f ( x )= x2+4.¿ Show that f is invertible with inverse f−1of f given by f −1 ( x )=√ x−4 ,where R+¿ ¿ is the set of all non-negative real numbers. (CBSE 2013)

10. Considerf :R+¿→[−5 ,∞ ) givenby f ( x )=9 x2+6x−5.¿ Show that f is invertible with f−1 (x )=√ x+6−1

3.11. A binary operation on set {0,1,2,3,4,5} is defined sas a∗b=

{ a+b , if a+b≤6a+b−6 , if a+b≥6 . Show that zero is the identity for this operation and each element ʻaʼ of the set is invertible with 6−a, being the inverse of ʻaʼ. (CBSE 2011)12. Let f :R→R bea function given by f ( x )=10x+7. Find the function g:R→R suchthat gof=fog=IR . (CBSE 2011)