KENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS CLASS - XII MATHEMATICS (Relations and Functions & Binary Operations)
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1 KENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS 6-7 CLASS - XII MATHEMATICS (Reltions nd Funtions & Binry Opertions) For Slow Lerners: - A Reltion is sid to e Reflexive if.. every A where A is non empty set. - A Reltion is sid to e Symmetri if.,, A. - A Reltion is sid to e Trnsitive if (,) R,,, A. - Let T e the set of ll tringles in plne with R reltion in T given y R = {(T, T): T is ongruent to T}. Show tht R is n equivlene reltion. - Show tht the reltion R in the set Z of integers given y R = {(, ) : divides -} is n equivlene reltion. 6- Chek whether the reltion R defined in the set {,,,,, 6} s R = {(, ): = +} is reflexive, symmetri or trnsitive. 7- Prove tht the funtion f: R R, given y f(x) = x, is one one. 8- Stte whether the funtion is one one, onto or ijetive f: R R defined y f(x) = + x 9- Find gof f(x) = x, g(x) = x +. - If f : R R e defined s f(x) = x, then find f f(x) - Let * e the inry opertion on H given y * = L. C. M of nd. find () * 6 () Is * ommuttive () Is * ssoitive (d) Find the identity of * in N. - Show tht f: RR defined y f(x)= x + is one-one, onto. Show tht f - (x)=(x - ) /. - Show tht the reltion R on N x N defined y (,)R(,d) +d = + is n equivlene reltion. - Let N denote the set of ll nturl numers nd R e the reltion on N x N defined y (,)R(,d) d( ) ( d). Show tht R is n equivlene reltion on N x N. - Let reltion R on the set R of rel numers defined s (x,y)r x xy y. Show tht R is reflexive ut neither symmetri nor trnsitive. - Show tht the reltion in the set R of rel no. defined R = {(, ) : }, is neither reflexive nor symmetri nor trnsitive. - Let A = N N nd * e the inry opertion on A define y (, ) * (, d) = ( +, + d). Show tht * is ommuttive nd ssoitive. 6- Let f : R [,) given y f(x) = 9x +6x-. Show tht f is invertile with f (y) +y+6
2 7- Let L e the set of ll lines in xy plne nd R e the reltion in L define s R = {(L, L): L L}. Show then R is on equivlene reltion. Find the set of ll lines relted to the line Y=x+. 8- Define inry opertion * on the set {,,,,,} s, if 6 * = 6, 6 Show tht zero in the identity for this opertion & eh element of the set is invertile with 6 - eing the inverse of. CLASS - XII MATHEMATICS : (Inverse Trigonometri Funtions) For Slow Lerners: - Write the prinipl vlue of the following.os..tn.os - Write the prinipl vlue of π π os os. x - Write the following in simplest form : tn x - Prove tht 8 77 tn Prove tht π tn tn tn tn Pr ove tht tn tn tn 7 7, x. Prove tht. Prove tht. Solve ot x x x x, x x x tn os x x x tn x tn x π / x,
3 . Solve tn x tn x tn x x π. Solve tn tn x x 8 6. Prove tht tn x x x x os x, x, 7. Prove tht 8. Prove tht os tn 6 6 x x y tn tn y x y CLASS - XII MATHEMATICS : (Mtries & Determinnt) For Slow Lerners:. If mtrix hs elements, wht re the possile orders it n hve?. Construt mtrix whose elements re given y ij = i j. If A =, B =, then find A B.. If A = nd B =, write the order of AB nd BA.. For the following mtries A nd B, verify (AB) T = B T A T, where A =, B = 6. Give exmple of mtries A & B suh tht AB = O, ut BA O, where O is zero mtrix nd A, B re oth non zero mtries. 7. If B is skew symmetri mtrix, write whether the mtrix (ABA T ) is Symmetri or skew symmetri. 8. If A = nd I =, find nd so tht A + I = A 9. Find the djoint of the mtrix A =. If A =, find A - nd hene solve the following system of equtions: x y + z =, x + y z = -, x + y z = -. Ug mtries, solve the following system of equtions:. x + y - z = - x + y + z = x - y z =
4 . x + y + z = 6 x + y + z = 6x + y + z = 7. Find the produt AB, where A =, B = nd use it to solve the equtions x y =, x + y + z = 7, y + z = 7. Ug mtries, solve the following system of equtions: - + = + - = + + =. Ug elementry trnsformtions, find the inverse of the mtrix.. Ug properties of determinnts, prove tht : q r r p p q y z z x x y p q r x y z. Ug properties of determinnts, prove tht :. Ug properties of determinnts, prove tht :.. Express A = s the sum of symmetri nd skew-symmetri mtrix. n n. Let A =, prove y mthemtil indution tht : A n. n n 6. If A =, find x nd y suh tht A + xi = ya. Hene find A. 7
5 7. Let A=. Prove tht. 8. Solve the following system of equtions : x + y + z = 7, x + z =, x y =. 9. Find the produt AB, where A = nd use it to solve the equtions x y + z =, x y z = 9, x + y + z =.. Find the mtrix P stisfying the mtrix eqution.. Ug properties of determinnts, prove the following : = ( + pxyz)(x - y)(y - z) (z - x) tn tn I nd os os A) (I A I B nd 7 P
+ = () i =, find the values of x & y. 4. Write the function in the simplifies from. ()tan i. x x. 5. Find the derivative of. 6.
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