The slope of the tangent at `(x , y)` to a curve passing through `(1,pi/4)` is given by `y/x-cos^2(y/x),` then the equation of the curve is (a) `( b ) (c) y=( d ) (e)tan^(( f ) (g)-1( h ))( i )(( j ) (k)log(( l ) (m) (n) e/( o ) x (p) (q) (r))( s ))( t )` (u) (v) `( w ) (x) y=x (y) (z)tan^(( a a ) (bb)-1( c c ))( d d )(( e e ) (ff)log(( g g ) (hh) (ii) x/( j j ) e (kk) (ll) (mm))( n n ))( o o )` (pp) (qq) `( r r ) (ss) y=x (tt) (uu)tan^(( v v ) (ww)-1( x x ))( y y )(( z z ) (aaa)log(( b b b ) (ccc) (ddd) e/( e e e ) x (fff) (ggg) (hhh))( i i i ))( j j j )` (kkk) (d) none of these

The slope of the tangent at (x , y) to a curve passing through a point (2,1) is (x^2+y^2)/(2x y) , then the equation of the curve is

The slope of the tangent at (x , y) to a curve passing through a point (2,1) is (x^2+y^2)/(2x y) , then the equation of the curve is

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 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)

The curve with the property that the projection of the ordinate on the normal is constant and has a length equal to a is (a) ( b ) (c) a1n(( d ) (e)sqrt(( f ) (g) (h) y^(( i )2( j ))( k )-( l ) a^(( m )2( n ))( o ) (p))( q )+y (r))=x+c (s) (t) (u) ( v ) (w) x+sqrt(( x ) (y) (z) a^(( a a )2( b b ))( c c )-( d d ) y^(( e e )2( f f ))( g g ) (hh))( i i )=c (jj) (kk) (ll) ( m m ) (nn) (oo) (pp)(( q q ) (rr) y-a (ss))^(( t t )2( u u ))( v v )=c x (ww) (xx) (yy) ( z z ) (aaa) a y=( b b b ) (ccc)tan^(( d d d ) (eee)-1( f f f ))( g g g )(( h h h ) (iii) x+c (jjj))( k k k ) (lll)

Solution of (dy)/(dx)+2x y=y is (a) ( b ) (c) y=c (d) e^( e ) (f) x-( g ) x^((( h )2( i ))( j ) (k))( l ) (m) (n) (b) ( o ) (p) y=c( q ) e^( r ) (s) (t) x^((( u )2( v ))( w )-x (x))( y ) (z) (aa) (c) ( d ) (e) y=c( f ) e^(( g ) x (h))( i ) (j) (k) (d) ( l ) (m) y=c (n) e^( o ) (p)-( q ) x^((( r )2( s ))( t ) (u))( v ) (w) (x)

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 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)