+ = () 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.

Transcription

1 Summer vtions Holid Home Work 7-8 Clss-XII Mths. Give the emple of reltion, whih is trnsitive ut neither refleive nor smmetri.. Find the vlues of unknown quntities if. + + () i =, find the vlues of & 7 () i = 5. If ( ) + = then prove tht. Write the funtion in the simplifies from Cos ()tn i + Sin ( ii) Cot iii)cos + 5. Find the derivtive of Sin Cos Sin + Cos d = d 6. Evlute 5n n ()tn i tn + Cos Cos 6 6 ( ii)tn + Cot + tn Sin 7 7 ( iii)cos Cos + tn t n 6 6 ( iv)sin Sin + Se Prove tht Sin + Sin + Sin = Find d d if Sin = + Cos 9. Let Z e the set of integers, m e positive integer nd R e the reltion on Z defined {(, ):,, } R= Z is divisile m Prove tht R is n equivlene reltion.. Prove tht tn tn tn + = Solve for :Cos tn + = +

2 . If A= R nd funtion : f A A invertile. 7. Show tht tn Sin = is defined + f = 6 Find f if f is. Find the vlue of tn Sin + Cos Let S e n non empt set. (i) Prove tht union nd intersetion re the inr opertions on P(S). (ii) Prove tht oth these opertions re ommuttive nd ssoitive. (iii) Find the identit elements for these opertions. (iv) Find the invertile elements nd their inverses in P(S) for oth the opertions. 6. Find d = log + +. d if m 7. If ( ) = + + then show tht d d m d + d =.. 8. Verif Lgrnge s men vlue theorem for the funtion f = 5, [,] Sin, < Cos 9. Let f =, = find the vlues of nd so tht the given funtion is ( Sin), > ( ) ontinuous t =.. Determine the vlues of,, for whih the funtion Sin + Sin ; < f = ; = ; > M e ontinuous t =. If = + ( Sin ) then find d d. Find d Cos Sin =. If log. Find d if + = tn then show tht d + = d d d if ( Cost tsin t) ; ( Sin t tcos t) = + =.

3 5. For the funtion f = 6 + +, it is given tht f f of &. Verif the Rolle s Theorem on [,]. 6. Find d d if + = Cos 5 Sin t os =, =, Show tht d 7. If t =. d 8. Find d d if = Sin. 9. If = e d log, then show tht = d ( log e). If = tn log, then show tht d d ( + ) + ( ) =. d d. Find d + d if = Sin + 6 t. If = Sin t & = Cos t + log tn,find d d. If = log + +. Differentite tn. show tht d d ( ) d + wrt.. tn 5. Show tht the funtion f defined s: ut not differentile. + + = d, <. f = < 5 > 6. It is given tht the funtion. f 5; [, ] = + find the vlues of &. 7. Find the vlue of k so tht the funtion defined f t =. 8. Find the vlue of k so tht the funtion defined f ontinuous t =. 9. Find the nd order derivtive if = Sin(log ) = =. Find the vlues is ontinuous t = = Rolle s Theorem holds with kcos ; = is ontinuous ; = ksin ( + ); = is tn Sin :

4 . If (i) (ii) (iii) (iv) A = Show tht verif our result. A 6A + 5A+ I = nd hene otin A = 7 5, Find nd suh tht A + l = A. Hene find A.. A. Also A =, find the vlues of & so tht A + l + A: I is the seond order unit mtri. A=, Find n A.. Find the inverse of A = using elementr trnsformtions.. Use properties to prove. + (i) + = (ii) (iii) + = = (iv) + + = ( + + ) + + (v) + + = +. Find the mtri A stisfing the eqution A = 5

5 . For A = + show tht A = + + I A. + p 5. Use properties to prove tht + p = ( + pz)( )( z)( z ) z z + pz nd hene. dedue tht, if, zre, ll different then + + = z z + z onl if z =. 6. (i) (ii) (iii) 5 Find,, nd d if is smmetri mtri. d 7 5 Find,,z if is smmetri mtri. + z 7 Find, if is skew smmetri mti. i 7. Form mtri A= ij ; ij = j find. 8. Show tht B AB is smmetri or skew smmetri ording s A is smmetri or skew smmetri. 9. If A nd B re squre mtries of the sme order suh tht AB=BA then using indution prove tht AB =B A. Further prove tht ( AB) = A B n N. 5. Two Shools A nd B deided to wrd prizes to their students for three vlues honest puntulit nd oediene. Shool A deided to wrd totl of Rs. for three vlues to 5, nd students while shool B deides to wrd Rs. 7 for these vlues to, nd 5 students respetivel. The three prizes together mount to Rs. 7. Represent the ove sitution mtri eqution nd solve. Whih vlue do ou prefer nd wh?

6 Q Q Q Q Q5 Mthemtis Assignments Clss XII 7 8 CHAPTER 5: CONTINUITY AND DIFFERENTIATION k sin ( ) : Find the vlue of k so tht the funtion defined f is ontinuous. tn sin : sin : tn Test the ontinuit of the funtion f : log( ) : e sin : os Let f : ( sin ) : ( ) Find the vlue of nd so tht the given funtion is ontinuous t sin( ) sin : If the funtion f defined f : : Differentite tn m with respet to os Q6 If d d then show tht ( ) m d d Q7 d d If sint; sin pt then prove tht ( ) p d d Q8 Find se sin d Q9 os sin Differentite if (sin ) (os) Q Differentite (os) (sin ) Q t d If ost logtn ; sint find d Q If log( ) tn then show tht Q If tn then prove tht d d d d Q Verif Rolle s Theorem f sin os :, f sin sin :, Q5 Verif Men Vlue Theorem for the funtion

7 Mthemtis Assignments Clss XII 7 8 CHAPTER : INVERSE TRIGONOMETRY Q Epress in simplest form: ) os Q Prove os(tn (sin(ot ))) os ) tn sin Q Evlute tn [ose(tn ) tn(ot )] Q Solve the eqution: tn Q5 Solve: sin Q6 Solve: (tn Q7 Solve: tn Q8 Prove tht ( ) sin ) (ot tn ) sin 5 8 tn os Q9 If ot os tn os Q Prove tht tn os then prove tht sin tn tn sin 7

8 CHAPTER & MATRICES AND DETERMINANTS Q Find nd if 7 Q Find. suh tht A + A =I where A sin os os sin Q Epress 5 s the sum of smmetri nd skew smmetri mtries. Q Find, nd if AA =I A Q5 If A then prove tht A 6A + 7A + I = hene find A. Q6 Find the inverse of A using elementr trnsformtions. Q7 If A then use P.M.I. show tht N n A n n n ) ( Q8 Find the mtri X so tht X Q9 If os sin sin os ) ( f Find f(), f( ). Q If A nd 5 B find the vlues of suh tht A = B Q Find (AB) if A nd B Q Prove tht ) ( ) ( ) ( ) ( Q Prove tht Q Prove tht ) )( ( Q5 Using mtries find A where A hene solve the sstem of liner equtions nd z z z Mthemtis Assignments Clss XII 7 8

9 Summer vtions Holid Home Work 7-8 Clss-XII Mths. Give the emple of reltion, whih is trnsitive ut neither refleive nor smmetri.. Find the vlues of unknown quntities if. + + () i =, find the vlues of & 7 () i = 5. If ( ) + = then prove tht. Write the funtion in the simplifies from Cos ()tn i + Sin ( ii) Cot iii)cos + 5. Find the derivtive of Sin Cos Sin + Cos d = d 6. Evlute 5n n ()tn i tn + Cos Cos 6 6 ( ii)tn + Cot + tn Sin 7 7 ( iii)cos Cos + tn t n 6 6 ( iv)sin Sin + Se Prove tht Sin + Sin + Sin = Find d d if Sin = + Cos 9. Let Z e the set of integers, m e positive integer nd R e the reltion on Z defined {(, ):,, } R= Z is divisile m Prove tht R is n equivlene reltion.. Prove tht tn tn tn + = Solve for :Cos tn + = +

10 . If A= R nd funtion : f A A invertile. 7. Show tht tn Sin = is defined + f = 6 Find f if f is. Find the vlue of tn Sin + Cos Let S e n non empt set. (i) Prove tht union nd intersetion re the inr opertions on P(S). (ii) Prove tht oth these opertions re ommuttive nd ssoitive. (iii) Find the identit elements for these opertions. (iv) Find the invertile elements nd their inverses in P(S) for oth the opertions. 6. Find d = log + +. d if m 7. If ( ) = + + then show tht d d m d + d =.. 8. Verif Lgrnge s men vlue theorem for the funtion f = 5, [,] Sin, < Cos 9. Let f =, = find the vlues of nd so tht the given funtion is ( Sin), > ( ) ontinuous t =.. Determine the vlues of,, for whih the funtion Sin + Sin ; < f = ; = ; > M e ontinuous t =. If = + ( Sin ) then find d d. Find d Cos Sin =. If log. Find d if + = tn then show tht d + = d d d if ( Cost tsin t) ; ( Sin t tcos t) = + =.

11 5. For the funtion f = 6 + +, it is given tht f f of &. Verif the Rolle s Theorem on [,]. 6. Find d d if + = Cos 5 Sin t os =, =, Show tht d 7. If t =. d 8. Find d d if = Sin. 9. If = e d log, then show tht = d ( log e). If = tn log, then show tht d d ( + ) + ( ) =. d d. Find d + d if = Sin + 6 t. If = Sin t & = Cos t + log tn,find d d. If = log + +. Differentite tn. show tht d d ( ) d + wrt.. tn 5. Show tht the funtion f defined s: ut not differentile. + + = d, <. f = < 5 > 6. It is given tht the funtion. f 5; [, ] = + find the vlues of &. 7. Find the vlue of k so tht the funtion defined f t =. 8. Find the vlue of k so tht the funtion defined f ontinuous t =. 9. Find the nd order derivtive if = Sin(log ) = =. Find the vlues is ontinuous t = = Rolle s Theorem holds with kcos ; = is ontinuous ; = ksin ( + ); = is tn Sin :

12 . If (i) (ii) (iii) (iv) A = Show tht verif our result. A 6A + 5A+ I = nd hene otin A = 7 5, Find nd suh tht A + l = A. Hene find A.. A. Also A =, find the vlues of & so tht A + l + A: I is the seond order unit mtri. A=, Find n A.. Find the inverse of A = using elementr trnsformtions.. Use properties to prove. + (i) + = (ii) (iii) + = = (iv) + + = ( + + ) + + (v) + + = +. Find the mtri A stisfing the eqution A = 5

13 . For A = + show tht A = + + I A. + p 5. Use properties to prove tht + p = ( + pz)( )( z)( z ) z z + pz nd hene. dedue tht, if, zre, ll different then + + = z z + z onl if z =. 6. (i) (ii) (iii) 5 Find,, nd d if is smmetri mtri. d 7 5 Find,,z if is smmetri mtri. + z 7 Find, if is skew smmetri mti. i 7. Form mtri A= ij ; ij = j find. 8. Show tht B AB is smmetri or skew smmetri ording s A is smmetri or skew smmetri. 9. If A nd B re squre mtries of the sme order suh tht AB=BA then using indution prove tht AB =B A. Further prove tht ( AB) = A B n N. 5. Two Shools A nd B deided to wrd prizes to their students for three vlues honest puntulit nd oediene. Shool A deided to wrd totl of Rs. for three vlues to 5, nd students while shool B deides to wrd Rs. 7 for these vlues to, nd 5 students respetivel. The three prizes together mount to Rs. 7. Represent the ove sitution mtri eqution nd solve. Whih vlue do ou prefer nd wh?