Number of ways of selecting three integers from `{1,2,3, , n}` if their sum is divisible by 3 is a. `b . c.3d .(^(e . f. n//3g .)h . i. C_j .3k . l.)+m . n.(n//3)^(o.3p .)q .r .` s. if `t . u. n=3k ,k in N v.` w. x. `y . z.2a a .(^(b b . cc. n-1//3d d .)e e . ff. C_g g .3h h . ii.)+j j .(^(k k . ll.(m m . nn. n+2o o .)//3p p .)q q . rr. C_s s .3t t . uu.)+v v . ww.((x x . yy. n-1z z .)//3)^(a a a .2b b b .)c c c .(d d d . eee. n+2f f f .),g g g .` hhh. if `i i i . jjj. n=3k+1,k in N kkk.` lll. mmm. `n n n . ooo.2p p p .(^(q q q . rrr. n-1//3s s s .)t t t . uuu. C_v v v .3w w w . xxx.)+y y y .(^(z z z . aaaa.(b b b b . cccc. n+2d d d d .)//3e e e e .)f f f f . gggg. C_h h h h .3i i i i . jjjj.)+k k k k . llll.((m m m m . nnnn. n-1o o o o .)//3)^(p p p p .2q q q q .)r r r r .(s s s s . tttt. n+2u u u u .),v v v v .` wwww. if `x x x x . yyyy. n=3k+2,k in N zzzz.` aaaaa. bbbbb. Independent of `c c c c c . n ddddd.` eeeee.

The total number of ways of selecting two numbers from the set {1,2, 3, 4, ........3n} so that their sum is divisible by 3 is equal to a. (2n^2-n)/2 b. (3n^2-n)/2 c. 2n^2-n d. 3n^2-n

The solution of the differential equation {1+xsqrt((x^2+y^2))}dx+{sqrt((x^2+y^2))-1}ydy=0 is equal to (a) ( b ) (c) (d) x^(( e )2( f ))( g )+( h )(( i ) (j) y^(( k )2( l ))( m ))/( n )2( o ) (p)+( q )1/( r )3( s ) (t) (u) (v)(( w ) (x) (y) x^(( z )2( a a ))( b b )+( c c ) y^(( d d )2( e e ))( f f ) (gg))^(( h h ) (ii) (jj)3/( k k )2( l l ) (mm) (nn))( o o )=c (pp) (qq) (rr) ( s s ) (tt) x+( u u )(( v v ) (ww) y^(( x x )3( y y ))( z z ))/( a a a )3( b b b ) (ccc)+( d d d )1/( e e e )2( f f f ) (ggg) (hhh) (iii)(( j j j ) (kkk) (lll) x^(( m m m )2( n n n ))( o o o )+( p p p ) y^(( q q q )2( r r r ))( s s s ) (ttt))^(( u u u ) (vvv) (www)1/( x x x )2( y y y ) (zzz) (aaaa))( b b b b )=c (cccc) (dddd) (eeee) ( f f f f ) (gggg) x-( h h h h )(( i i i i ) (jjjj) y^(( k k k k )2( l l l l ))( m m m m ))/( n n n n )2( o o o o ) (pppp)+( q q q q )1/( r r r r )3( s s s s ) (tttt) (uuuu) (vvvv)(( w w w w ) (xxxx) (yyyy) x^(( z z z z )2( a a a a a ))( b b b b b )+( c c c c c ) y^(( d d d d d )2( e e e e e ))( f f f f f ) (ggggg))^(( h h h h h ) (iiiii) (jjjjj)3/( k k k k k )2( l l l l l ) (mmmmm) (nnnnn))( o o o o o )=c (ppppp) (qqqqq)

The solution of the differential equation {1+xsqrt((x^2+y^2))}dx+{sqrt((x^2+y^2))-1}ydy=0 is equal to (a) ( b ) (c) (d) x^(( e )2( f ))( g )+( h )(( i ) (j) y^(( k )2( l ))( m ))/( n )2( o ) (p)+( q )1/( r )3( s ) (t) (u) (v)(( w ) (x) (y) x^(( z )2( a a ))( b b )+( c c ) y^(( d d )2( e e ))( f f ) (gg))^(( h h ) (ii) (jj)3/( k k )2( l l ) (mm) (nn))( o o )=c (pp) (qq) (rr) ( s s ) (tt) x+( u u )(( v v ) (ww) y^(( x x )3( y y ))( z z ))/( a a a )3( b b b ) (ccc)+( d d d )1/( e e e )2( f f f ) (ggg) (hhh) (iii)(( j j j ) (kkk) (lll) x^(( m m m )2( n n n ))( o o o )+( p p p ) y^(( q q q )2( r r r ))( s s s ) (ttt))^(( u u u ) (vvv) (www)1/( x x x )2( y y y ) (zzz) (aaaa))( b b b b )=c (cccc) (dddd) (eeee) ( f f f f ) (gggg) x-( h h h h )(( i i i i ) (jjjj) y^(( k k k k )2( l l l l ))( m m m m ))/( n n n n )2( o o o o ) (pppp)+( q q q q )1/( r r r r )3( s s s s ) (tttt) (uuuu) (vvvv)(( w w w w ) (xxxx) (yyyy) x^(( z z z z )2( a a a a a ))( b b b b b )+( c c c c c ) y^(( d d d d d )2( e e e e e ))( f f f f f ) (ggggg))^(( h h h h h ) (iiiii) (jjjjj)3/( k k k k k )2( l l l l l ) (mmmmm) (nnnnn))( o o o o o )=c (ppppp) (qqqqq)

The solution of (y+x+5)dy=(y-x+1)dx is (a) ( b ) (c)log(( d ) (e) (f) (g)(( h ) (i) y+3( j ))^(( k )2( l ))( m )+( n ) (o)(( p ) (q) x+2( r ))^(( s )2( t ))( u ) (v))+( w ) (x)tan^(( y ) (z)-1( a a ))( b b ) (cc)(( d d ) y+3)/( e e )(( f f ) y+2)( g g ) (hh)+C (ii) (jj) (kk) ( l l ) (mm)log(( n n ) (oo) (pp) (qq)(( r r ) (ss) y+3( t t ))^(( u u )2( v v ))( w w )+( x x ) (yy)(( z z ) (aaa) x-2( b b b ))^(( c c c )2( d d d ))( e e e ) (fff))+( g g g ) (hhh)tan^(( i i i ) (jjj)-1( k k k ))( l l l ) (mmm)(( n n n ) y-3)/( o o o )(( p p p ) y-2)( q q q ) (rrr)+C (sss) (ttt) (uuu) ( v v v ) (www)log(( x x x ) (yyy) (zzz) (aaaa)(( b b b b ) (cccc) y+3( d d d d ))^(( e e e e )2( f f f f ))( g g g g )+( h h h h ) (iiii)(( j j j j ) (kkkk) x+2( l l l l ))^(( m m m m )2( n n n n ))( o o o o ) (pppp))+2( q q q q ) (rrrr)tan^(( s s s s ) (tttt)-1( u u u u ))( v v v v ) (wwww)(( x x x x ) y+3)/( y y y y )(( z z z z ) x+2)( a a a a a ) (bbbbb)+C (ccccc) (ddddd) (eeeee) ( f f f f f ) (ggggg)log(( h h h h h ) (iiiii) (jjjjj) (kkkkk)(( l l l l l ) (mmmmm) y+3( n n n n n ))^(( o o o o o )2( p p p p p ))( q q q q q )+( r r r r r ) (sssss)(( t t t t t ) (uuuuu) x+2( v v v v v ))^(( w w w w w )2( x x x x x ))( y y y y y ) (zzzzz))-2( a a a a a a ) (bbbbbb)tan^(( c c c c c c ) (dddddd)-1( e e e e e e ))( f f f f f f ) (gggggg)(( h h h h h h ) y+3)/( i i i i i i )(( j j j j j j ) x+2)( k k k k k k ) (llllll)+C (mmmmmm) (nnnnnn)

The differential equation whose general solution is given by y=(c_1cos(x+c_2)-(c_3e^((-x+c4))+(c_5sinx), where c_1,c_2,c_3,c_4,c_5 are arbitrary constants, is (a) ( b ) (c) (d)(( e ) (f) d^(( g )4( h ))( i ) y)/( j )(( k ) d (l) x^(( m )4( n ))( o ))( p ) (q)-( r )(( s ) (t) d^(( u )2( v ))( w ) y)/( x )(( y ) d (z) x^(( a a )2( b b ))( c c ))( d d ) (ee)+y=0( f f ) (gg) (hh) ( i i ) (jj) (kk)(( l l ) (mm) d^(( n n )3( o o ))( p p ) y)/( q q )(( r r ) d (ss) x^(( t t )3( u u ))( v v ))( w w ) (xx)+( y y )(( z z ) (aaa) d^(( b b b )2( c c c ))( d d d ) y)/( e e e )(( f f f ) d (ggg) x^(( h h h )2( i i i ))( j j j ))( k k k ) (lll)+( m m m )(( n n n ) dy)/( o o o )(( p p p ) dx)( q q q ) (rrr)+y=0( s s s ) (ttt) (uuu) ( v v v ) (www) (xxx)(( y y y ) (zzz) d^(( a a a a )5( b b b b ))( c c c c ))/( d d d d )(( e e e e ) d (ffff) x^(( g g g g )5( h h h h ))( i i i i ))( j j j j ) (kkkk)+y=0( l l l l ) (mmmm) (nnnn) ( o o o o ) (pppp) (qqqq)(( r r r r ) (ssss) d^(( t t t t )3( u u u u ))( v v v v ) y)/( w w w w )(( x x x x ) d (yyyy) x^(( z z z z )3( a a a a a ))( b b b b b ))( c c c c c ) (ddddd)-( e e e e e )(( f f f f f ) (ggggg) d^(( h h h h h )2( i i i i i ))( j j j j j ) y)/( k k k k k )(( l l l l l ) d (mmmmm) x^(( n n n n n )2( o o o o o ))( p p p p p ))( q q q q q ) (rrrrr)+( s s s s s )(( t t t t t ) dy)/( u u u u u )(( v v v v v ) dx)( w w w w w ) (xxxxx)-y=0( y y y y y ) (zzzzz)

In which of the following differential equation degree is not defined? (a) ( b ) (c) (d)(( e ) (f) d^(( g )2( h ))( i ) y)/( j )(( k ) d (l) x^(( m )2( n ))( o ))( p ) (q)+3( r ) (s)(( t ) (u) (v)(( w ) dy)/( x )(( y ) dx)( z ) (aa) (bb))^(( c c )2( d d ))( e e )=xlog( f f )(( g g ) (hh) d^(( i i )2( j j ))( k k ) y)/( l l )(( m m ) d (nn) x^(( o o )2( p p ))( q q ))( r r ) (ss) (tt) (uu) (vv) ( w w ) (xx) (yy)(( z z ) (aaa) d^(( b b b )2( c c c ))( d d d ) y)/( e e e )(( f f f ) d (ggg) x^(( h h h )2( i i i ))( j j j ))( k k k ) (lll)+( m m m ) (nnn)(( o o o ) (ppp) (qqq)(( r r r ) dy)/( s s s )(( t t t ) dx)( u u u ) (vvv) (www))^(( x x x )2( y y y ))( z z z )=xsin( a a a a )(( b b b b ) (cccc) d^(( d d d d )2( e e e e ))( f f f f ) y)/( g g g g )(( h h h h ) d (iiii) x^(( j j j j )2( k k k k ))( l l l l ))( m m m m ) (nnnn) (oooo) (pppp) (qqqq) ( r r r r ) (ssss) x=sin(( t t t t ) (uuuu) (vvvv)(( w w w w ) dy)/( x x x x )(( y y y y ) dx)( z z z z ) (aaaaa)-2y (bbbbb)),|x|<1( c c c c c ) (ddddd) (eeeee) ( f f f f f ) (ggggg) x-2y=log(( h h h h h ) (iiiii) (jjjjj)(( k k k k k ) dy)/( l l l l l )(( m m m m m ) dx)( n n n n n ) (ooooo) (ppppp))( q q q q q ) (rrrrr)

The solution of the differential equation (dy)/(dx)=(3x^2y^4+2x y)/(x^2-2x^3y^3) is (a) ( b ) (c) (d)(( e ) (f) y^(( g )2( h ))( i ))/( j ) x (k) (l)-( m ) x^(( n )3( o ))( p ) (q) y^(( r )2( s ))( t )=c (u) (v) (w) ( x ) (y) (z)(( a a ) (bb) x^(( c c )2( d d ))( e e ))/( f f )(( g g ) (hh) y^(( i i )2( j j ))( k k ))( l l ) (mm)+( n n ) x^(( o o )3( p p ))( q q ) (rr) y^(( s s )3( t t ))( u u )=c (vv) (ww) (xx) ( y y ) (zz) (aaa)(( b b b ) (ccc) x^(( d d d )2( e e e ))( f f f ))/( g g g ) y (hhh) (iii)+( j j j ) x^(( k k k )3( l l l ))( m m m ) (nnn) y^(( o o o )2( p p p ))( q q q )=c (rrr) (sss) (d) ( t t t ) (uuu) (vvv)(( w w w ) (xxx) x^(( y y y )2( z z z ))( a a a a ))/( b b b b )(( c c c c )3y)( d d d d ) (eeee)-2( f f f f ) x^(( g g g g )3( h h h h ))( i i i i ) (jjjj) y^(( k k k k )2( l l l l ))( m m m m )=c (nnnn) (oooo)

The differential equation of the family of curves y=e^x(Acosx+Bsinx), where A and B are arbitrary constants is (a) ( b ) (c) (d)(( e ) (f) d^(( g )2( h ))( i ) y)/( j )(( k ) d (l) x^(( m )2( n ))( o ))( p ) (q)-2( r )(( s ) dy)/( t )(( u ) dx)( v ) (w)+2y=0( x ) (y) (z) ( a a ) (bb) (cc)(( d d ) (ee) d^(( f f )2( g g ))( h h ) y)/( i i )(( j j ) d (kk) x^(( l l )2( m m ))( n n ))( o o ) (pp)+2( q q )(( r r ) dy)/( s s )(( t t ) dx)( u u ) (vv)+2y=0( w w ) (xx) (yy) ( z z ) (aaa) (bbb)(( c c c ) (ddd) d^(( e e e )2( f f f ))( g g g ) y)/( h h h )(( i i i ) d (jjj) x^(( k k k )2( l l l ))( m m m ))( n n n ) (ooo)+( p p p ) (qqq)(( r r r ) (sss) (ttt)(( u u u ) dy)/( v v v )(( w w w ) dx)( x x x ) (yyy) (zzz))^(( a a a a )2( b b b b ))( c c c c )+y=0( d d d d ) (eeee) (ffff) ( g g g g ) (hhhh) (iiii)(( j j j j ) (kkkk) d^(( l l l l )2( m m m m ))( n n n n ) y)/( o o o o )(( p p p p ) d (qqqq) x^(( r r r r )2( s s s s ))( t t t t ))( u u u u ) (vvvv)-7( w w w w )(( x x x x ) dy)/( y y y y )(( z z z z ) dx)( a a a a a ) (bbbbb)+2y=0( c c c c c ) (ddddd)

The solution of the differential equation (d^2y)/(dx^2)=sin3x+e^x+x^2 when y_1(0)=1 and y(0) is (a) ( b ) (c) (d)(( e )-sin3x)/( f )9( g ) (h)+( i ) e^(( j ) x (k))( l )+( m )(( n ) (o) x^(( p )4( q ))( r ))/( s )(( t ) 12)( u ) (v)+( w )1/( x )3( y ) (z) x-1( a a ) (bb) (cc) ( d d ) (ee) (ff)(( g g )-sin3x)/( h h )9( i i ) (jj)+( k k ) e^(( l l ) x (mm))( n n )+( o o )(( p p ) (qq) x^(( r r )4( s s ))( t t ))/( u u )(( v v ) 12)( w w ) (xx)+( y y )1/( z z )3( a a a ) (bbb) x (ccc) (ddd) (eee) ( f f f ) (ggg) (hhh)(( i i i )-cos3x)/( j j j )3( k k k ) (lll)+( m m m ) e^(( n n n ) x (ooo))( p p p )+( q q q )(( r r r ) (sss) x^(( t t t )4( u u u ))( v v v ))/( w w w )(( x x x ) 12)( y y y ) (zzz)+( a a a a )1/( b b b b )3( c c c c ) (dddd) x+1( e e e e ) (ffff) (d) None of these

The general solution of the differential equation (dy)/(dx)+sin(x+y)/2=sin(x-y)/2 is (a) ( b ) (c)logtan(( d ) (e) (f) y/( g )2( h ) (i) (j))=c-2sinx (k) (l) (m) ( n ) (o)logtan(( p ) (q) (r) y/( s )4( t ) (u) (v))=c-2sin(( w ) (x) (y) x/( z )2( a a ) (bb) (cc))( d d ) (ee) (ff) ( g g ) (hh)logtan(( i i ) (jj) (kk) y/( l l )2( m m ) (nn)+( o o )pi/( p p )4( q q ) (rr) (ss))=c-2sinx (tt) (uu) (vv) ( w w ) (xx)logtan(( y y ) (zz) (aaa) y/( b b b )4( c c c ) (ddd)+( e e e )pi/( f f f )4( g g g ) (hhh) (iii))=c-2sin(( j j j ) (kkk) (lll) x/( m m m )2( n n n ) (ooo) (ppp))( q q q ) (rrr)