The function `f(theta)=d/(dtheta)int_0^theta(dx)/(1-costhetacosx)` satisfies the differential equation (a) `( b ) (c) (d)(( e ) df(( f )theta( g )))/( h )(( i ) dtheta)( j ) (k)+2f(( l )theta( m ))=0( n )` (o) (p) `( q ) (r) (s)(( t ) df)/( u )(( v ) dtheta)( w ) (x)-2f(( y )theta( z ))cottheta=0( a a )` (bb) (cc) `( d d ) (ee) (ff)(( g g ) df)/( h h )(( i i ) dtheta)( j j ) (kk)+2f(( l l )theta( m m ))=0( n n )` (oo) (d) `( p p ) (qq) (rr)(( s s ) df)/( t t )(( u u ) dtheta)( v v ) (ww)-2(( x x )theta( y y ))=0( z z )` (aaa)

The function f(theta)=d/(d(theta))int_0^theta(dx)/(1-costhetacosx) satisfies the differential equation

The differential equation of all non-horizontal lines in a plane is (a) ( b ) (c) (d)(( e ) (f) d^(( g )2( h ))( i ) y)/( j )(( k ) d (l) x^(( m )2( n ))( o ))( p ) (q) (r) (s) (b) ( t ) (u) (v)(( w ) (x) d^(( y )2( z ))( a a ) x)/( b b )(( c c ) d (dd) y^(( e e )2( f f ))( g g ))( h h ) (ii)=0( j j ) (kk) (c) ( d ) (e) (f)(( g ) dy)/( h )(( i ) dx)( j ) (k)=0( l ) (m) (d) ( n ) (o) (p)(( q ) dx)/( r )(( s ) dy)( t ) (u)=0( v ) (w)

An angle is double of its supplement. The measure of the angle is (a) ( b ) (c) (d) (e) 60^(( f )0( g ))( h )\ ( i ) (j) (b) ( k ) (l) (m) (n) 120^(( o )0( p ))( q ) (r) (s) (t) ( u ) (v) (w) (x) 40^(( y )0( z ))( a a ) (bb) (cc) (d) ( d d ) (ee) (ff) (gg) 80^(( h h )0( i i ))( j j ) (kk) (ll)

Tangent to a curve intercepts the y-axis at a point Pdot A line perpendicular to this tangent through P passes through another point (1,0). The differential equation of the curve is (a) ( b ) (c) y (d)(( e ) dy)/( f )(( g ) dx)( h ) (i)-x (j) (k)(( l ) (m) (n)(( o ) dy)/( p )(( q ) dx)( r ) (s) (t))^(( u )2( v ))( w )=1( x ) (y) (b) ( z ) (aa) (bb)(( c c ) x (dd) d^(( e e )2( f f ))( g g ))/( h h )(( i i ) d (jj) x^(( k k )2( l l ))( m m ))( n n ) (oo)+( p p ) (qq)(( r r ) (ss) (tt)(( u u ) dy)/( v v )(( w w ) dx)( x x ) (yy) (zz))^(( a a a )2( b b b ))( c c c )=0( d d d ) (eee) (c) ( d ) (e) y (f)(( g ) dx)/( h )(( i ) dy)( j ) (k)+x=1( l ) (m) (d) None of these

Tangent to a curve intercepts the y-axis at a point Pdot A line perpendicular to this tangent through P passes through another point (1,0). The differential equation of the curve is (a) ( b ) (c) y (d)(( e ) dy)/( f )(( g ) dx)( h ) (i)-x (j) (k)(( l ) (m) (n)(( o ) dy)/( p )(( q ) dx)( r ) (s) (t))^(( u )2( v ))( w )=1( x ) (y) (b) ( z ) (aa) (bb)(( c c ) x (dd) d^(( e e )2( f f ))( g g ))/( h h )(( i i ) d (jj) x^(( k k )2( l l ))( m m ))( n n ) (oo)+( p p ) (qq)(( r r ) (ss) (tt)(( u u ) dy)/( v v )(( w w ) dx)( x x ) (yy) (zz))^(( a a a )2( b b b ))( c c c )=0( d d d ) (eee) (c) ( d ) (e) y (f)(( g ) dx)/( h )(( i ) dy)( j ) (k)+x=1( l ) (m) (d) None of these

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 solution of the differential equation (x+(x^3)/(3!)+(x^5)/(5!)+)/(1+(x^2)/(2!)+(x^4)/(4!)+)=(dx-dy)/(dx+dy)i s (a) ( b ) (c)2y( d ) e^(( e ) (f)2x (g))( h )=C( i ) e^(( j ) (k)2x (l))( m )+1( n ) (o) (p) ( q ) (r)2y( s ) e^(( t ) (u)2x (v))( w )=C( x ) e^(( y ) (z)2x (aa))( b b )-1( c c ) (dd) (ee) ( f f ) (gg) y( h h ) e^(( i i ) (jj)2x (kk))( l l )=C( m m ) e^(( n n ) (oo)2x (pp))( q q )+2( r r ) (ss) (d) None of these

The solution of differential equation (2y+x y^3)dx+(x+x^2y^2)dy=0 is (a) ( b ) (c) (d) x^(( e )2( f ))( g ) y+( h )(( i ) (j) x^(( k )3( l ))( m ) (n) y^(( o )3( p ))( q ))/( r )3( s ) (t)=c (u) (v) (b) ( w ) (x) x (y) y^(( z )2( a a ))( b b )+( c c )(( d d ) (ee) x^(( f f )3( g g ))( h h ) (ii) y^(( j j )3( k k ))( l l ))/( m m )3( n n ) (oo)=c (pp) (qq) (c) ( d ) (e) (f) x^(( g )2( h ))( i ) y+( j )(( k ) (l) x^(( m )4( n ))( o ) (p) y^(( q )4( r ))( s ))/( t )4( u ) (v)=c (w) (x) (d) None of these

The solution of the differential equation (x^2y^2-1)dy+2xy^3dx=0 is (a) ( b ) (c)1+( d ) x^(( e )2( f ))( g ) (h) y^(( i )2( j ))( k )=c x (l) (m) (b) ( n ) (o)1+( p ) x^(( q )2( r ))( s ) (t) y^(( u )2( v ))( w )=c y (x) (y) (c) ( d ) (e) y=0( f ) (g) (d) ( h ) (i) y=-( j )1/( k )(( l ) (m) x^(( n )2( o ))( p ))( q ) (r) (s) (t)

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)