x. Lt 'P' b a point on th curv y and tangnt x drawn at P to th curv has gratst slop in magnitud, thn point 'P' is,, (0, 0),. Th quation of common tangnt to th curvs y = 6 x x and xy = x + is : x y = 8 x + y = 0 x + y = x + y = 7. If 0, thn st of valus of for which. If x x 0 has ral roots is : 0,,, [0, ] f ( x ) f ( x) ( x x ) x, x R, thn quation of tangnt to th curv y = f (x) at point (, 8) is : x 8 = 0 y = 0 y 8 = 0 x = 0 5. Any normal to th curv x = a (cos + sin ) ; y = a (sin cos ) at any point ' ' is such that : it passs through (0, 0). it maks constant angl with x-axis. it is at a constant distanc from (0, 0). non of ths. 6. Angl of intrsction btwn th curvs givn by x xy + = 0 and y x y = 0 is : 6 [ 89 ] 7. Equation of normal to curv y x x at x = 0 is : x + y + = 0 x y + = 0 x + y = 0 x + y = 0 y ( ) sin (sin ) 8. Lt at point 'P' on th curv y + x = y, th tangnt is vrtical, thn 'P' may b : (0, 0),,, 9. Acut angl of intrsction btwn th curvs y = x and y = x is givn by : tan 7 cos 9 7 sin 7 cos 9 7 0. If th tangnt and normal to th curv y x at point P(0, ) intrscts th x-axis at 'T' and 'N' rspctivly, thn ara (in sq. units) of quilatral triangl which is circumscribd by th incircl of PTN is : ( ) ( ) ( ) ( ). Lt x + y k = 0 b th tangnt to th curv y = cos(x + y), x, thn possibl valus of 'k' can b : / / / / Mathmatics for JEE-0
. If a function is having horizontal tangnt at origin thn it holds th H-proprty, functions having H-proprty ar : Tangnt and Normal Statmnt is tru but Statmnt is fals. Statmnt is fals but Statmnt is tru. xsin ; x 0 y x 0 ; x 0 x y sin ; x 0 x 0 ; x 0 y = x x y = min{x, x }. Lt a curv in paramtric form b rprsntd by x t, y t for all t R, thn which of th following lins ar tangnt to curv at on point and normal at anothr point of curv? x y 0 x y 0 x y 0 x y 0. Lt f : R R and g : R [0, ) b th functions which ar givn by f (x) = kx and g(x) = log x. If th quation f (x) g (x) = 0 is having thr distinct ral roots, thn possibl valus of 'k' can b : 5. Functions which ar having vrtical tangnt at point x = ar : f ( x) sgn( x ) f ( x) x / f ( x) ( x ) x ; x f ( x) x ; x Following qustions ar assrtion and rasoning typ qustions. Each of ths qustions contains two statmnts, Statmnt (Assrtion) and Statmnt (Rason). Each of ths qustions has four altrnativ answrs, only on of thm is th corrct answr. Slct th corrct answr from th givn options : Both Statmnt and Statmnt ar tru and Statmnt is th corrct xplanation of Statmnt. Both Statmnt and Statmnt ar tru but Statmnt is not th corrct xplanation of Statmnt. 6. Considr th curvs C : y = x and C y. : x Statmnt : Curvs 'C ' and 'C ' form an orthogonal pair of curvs Statmnt : Curvs 'C ' and 'C ' intrsct ach othr at only on point location 7. Lt a (0, ) and b R, whr D ( a b) a b 9 Statmnt : For givn conditions on 'a' and 'b', th minimum valu of 'D' is 8 Statmnt : Th minimum distanc btwn th curvs xy = 9 and x + y = is qual to units. 8. Statmnt : Lt y = f (x) b polynomial function, and tangnt at point A(a, f ) is normal to th curv of y = f (x) at point B(b, f ), thn at last on point (c, f ) xists for which f ' = 0, whr c ( a, b) Statmnt : Product of th slops of tangnts to th curv y = f (x) at 'A' and 'B' is qual to if tangnts ar not paralll to th axs. 9. Considr th curvs C : y = x + x + and C : y = x 5x + 6. Statmnt : Equation of common tangnt to th curvs C and C is givn by 9y + x = 0 Statmnt : Acut angl of intrsction of th curvs 5 C and C is tan. 7 0. Statmnt : Lngth of subtangnt at point P (, ) for th curv x y = is qual to units Statmnt : Lngth of subtangnt at any point (, ) for th curv x y = is qual to. [ 90 ] Mathmatics for JEE-0
Comprhnsion passag () ( Qustions No. - ) 5. Ara (in squar units) of th triangl formd by normal at (, 0), whr, with th co-ordinat axs is qual to : Considr th curv C x x x y. : 5 5 0 6 0 If th normal 'N' to curv C at point P(0, ) mts th curv again at two points Q and R, thn answr th following qustions. 8. Minimum ara (in squar units) of th circl passing through th points Q and R is qual to : 5 8. With rfrnc to lin of normal 'N', which of th following statmnt is corrct? lin 'N' is tangntial to curv C at point Q only. lin 'N' is tangntial to curv C at point R only. lin 'N' is tangntial to curv C at both th points Q and R.. lin 'N' is not tangntial to curv C at ithr of th point Q and R.. Lt th lngth of subtangnts at th points Q and R for th curv C b l and l rspctivly, whr l OQ > OR, 'O' bing th origin, thn is qual to : l 5 Comprhnsion passag () ( Qustions No. -6 ) 6. Lt g( x) f ( x), whr ( g '( x)) g "( x). g( x) 0 is having xactly four distinct ral roots, thn xhaustiv st of valus of ' ' blong to : ( 7, 8) (, ) (, 0) ( 0, ) 7. Lt tangnt at 't ' point to th curv C : y = 8t, x = t + is normal at anothr point 't ' to th curv 6 'C', thn valu of 79( t ) is qual to... 8. Lt any point 'P' lis on th curv y ( x) = (x ), whr th distanc of 'P' from th origin is 'r ' and th distanc of tangnt at 'P' from th origin is 'r '. If point P is (, ), thn valu of qual to... ( r 5) r r is Lt f : R R b dfind as f ( x) ax bx cx 7, whr th curv of y = f (x) touchs th x-axis at point P(, 0) and mts th y-axis at point Q. If f ' (0) = 9, thn answr th following qustions.. If f ( ) = f ( ) = 0 and, thn valu of [ ] [ ] is qual to : ([.] rprsnts th gratst intgr function) 0 5 0 9. Lt l and l b th intrcpts mad on th x-axis and y-axis rspctivly by tangnt at any point of th curv x = a cos ; y bsin, thn th valu l of a l is... b 0. Lt chord PQ of th curv y x 5 x 0 b tangntial to curv y( x) = at th point R(, ), if PR = RQ, thn th last possibl valu of is qual to... [ 9 ] Mathmatics for JEE-0
Tangnt and Normal. Match th following columns (I) and (II) Column (I) Column (II) If th angl btwn th curvs at point (, ) is, thn valu of cos is x yx and y (p) If th acut angl of intrsction of th curvs x = ay and (q) 8a y, a R, is tan ( ), thn ' ' is qual to x a Th lngth of subtangnt at any point on th curv y (r) x / a is qual to (s) 5/ If th slop of tangnt, if xists, varis at vry point of th curv max x x y,, k, thn 'k' can b (t) /. Match th following columns (I) and (II) Column (I) Column (II) If th non-vrtical common tangnt of th curvs xy = (p) and y = 8x is lin 'L', thn ara (in squar units) of th triangl formd by lin 'L' with th co-ordinat axs is If th curvs y = cos x, x and touch ach othr, thn th numbr of possibl valus of ' ' is/ar y x (q) / Th ara (in squar units) of triangl formd by normal at (r) th point (, 0) to th curv x is : sin y with coordinat axs (s) If th inquation x x has at last on ngativ solution, thn th possibl valus of ' ' can b (t) [ 9 ] Mathmatics for JEE-0
.... 5. Ex 6. 7. 8. 9. 0.. (a, d). (a, b, c, d). (a, b). (a, c, d) 5. (a, b) 6. 7. 8. 9. 0..... 5. Ex 6. 7. ( 8 ) 8. ( 9 ) 9. ( ) 0. ( ). r. s p s p q r, s, t p, q, s [ 9 ] Mathmatics for JEE-0