150 with positive x-axis and passing through

Transcription

1 ` KUKATPALLY CENTRE IPE MAT IB Important Questions FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa:

2 A Find the equation of line which makes an angle of (-, -) 5 with positive -ais and passing through B Find the equation of straight line passing through origin and making equal angles with coordinate aes C Find the equation of the straight line passing through (-, 4) and making non zero intercepts whose sum is zero D Find the equation of the straight line making an angle of -ais and has y-intercept 3 Tan - æ ç ö è 3ø with the positive E Find the equation of the straight line passing through (3, -4) and making X and Y-intercepts which are in the ratio : 3 F Find the equations of the straight line passing through the following points : a) (, 5), (, 8) b) (3, -3), (7, -3) c) (, -), (-, 3) d) ( at )(, at at, at ) G Find the equations of the straight lines passing through the point (4, -3) and (i) parallel (ii) perpendicular to the line passing through the points (, ) and (, 3) H Prove that the points (-5, ), (5, 5), (, 7) are collinear and find the equation containing the lines I If the portion of a straight line intercepted between the aes of coordinates is bisected at (p, q) Find the equation of the straight line J A straight line passing through A (-, ) makes an angle of 3 with OX suur in the positive direction Find the points on the straight line whose distance from A is 4 units K Find the points on the line 3-4y- = which are at a distance of 5 units from the point (3, ) L If the area of the triangle formed by the straight lines =, y= and 3 4y a Find the value of a M Transform the equation + y+ = into normal form + = ( ) a> is 6 N Find the ratio in which the straight line + 3y- = divides the join of the points (, 3) and (, ) O State whether (3, ) and (-4, -3) are on the same side or on opposite side of the straight line - 3y+ 4= A Find the equation of the straight line passing through the point of intersection of the lines + y+ = and - y+ 5= and containing the point (5, -) B If a, b, c are in AP, show that a+ by+ c= represents a family of concurrent lines and find the concurrency C Find the point of concurrency of the lines represented + 5k k y+ - k = by ( ) ( ) ( ) FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa:

3 D Find the area of the triangle formed by the coordinate aes and the line 3-4y+ = E Find the set of values of a if the points (, ) and (3, 4) lie on the same side of the straight line 3-5y+ a= F Find the value of k, if the angle between the straight lines 4- y+ 7= and k- 5y- 9= is G If 3y 5 a+ b - - = is the perpendicular bisector of the line segment joining (3, -4) and (, ) H Find the length of the perpendicular drawn from (3, 4) to the line 3-4y+ = I Find the distance between the straight lines 5-3y- 4= and - 6y- 9= J Find the circumcentre of the triangle formed by the lines =, y= and + y= FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa: a b Find K Find the value of k if the straight lines y- 3k+ 4= & ( k- ) -( 8k- ) y- 6= are perpendicular L Find the equation of the line perpendicular to the line 3+ 4y+ 6= and making an intercept -4 on the -ais M Find the orthocenter of the triangle whose sides are given by + y+ =, -y- = and + y- 7= N If (-, 6) is the image of the point (4, ) wrt the line L, then find the equation of L O Find the angle which the straight line y= 3-4 makes with y-ais A Find the equation of the plane passing through (,, ) and ( 3, 3,4) - y+ z= 6 B Find the equation of the plane through ( ) and 3+ 3y+ z- 8= - and perpendicular to 4, 4, and perpendicular to the planes + y+ z+ 3= C Find the equation of the plane through the points (,, - ),( 3,4, ),( 7,,6) D A plane meets the coordinate aes in A,B,C If centroid of the ABC y z equation to the plane is + + = 3 a b c D is (,, ) a b c Show that the E Find the equation of the plane if the foot of the perpendicular from origin to the plane is (,3, - 5) F Find the equation to the plane parallel to the ZX-plane and passing through (, 4,4 ) G Find the equation of the plane if the foot of the perpendicular from origin to the plane is (,3, - 5) H Reduce the equation + y- 3z- 6= of the plane to the normal form I Find the equation of the plane passing through the point (-,,3) and having ( 3, 5,4) its normal J Find the angle between the planes + y+ z- 5= and 3+ 3y+ z- 8= - as dr s of

4 K Find the equation of the plane passing through the point (,, ) and parallel to the plane 3 + y+ 3z- 7= æ ö A Find + - ç çè ø æ e ö - B Compute ç è + - ø a - b - C Compute ( a>, b>, b¹ ) æ ö D + ç çè ø G H I J ( - ) sin sin ( -) tan a -a ( a+ b) -sin( a-b) ( -a) ( a¹ ) æ 3 ö - ç è + - ø E F 4 cos p æ ö p ç - çè ø sina cos - cosm sin n K ( m, nî Z) A B C D E æ 4 ö - çè - - 4ø F ( + - ) G H I J K æ ö ç + - çè ø + sin cos cos cos+ sin + 5 A If f is defined by f( ) ìï sin ï if ¹ =í ïï ïî if = continuous at? FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa:

5 B Check the continuity of the following function at = for f( ) C Check the continuity of f given by ( ) ( ) ( ) f ï the point 3 ( ) ìï - 4 if < < = ï í if = -3-8 if > ï ïî ìï -9 / i f < < 5 and ¹ 3 =í ïï ïî 5 if = 3 at D If, f given by f( ) ìï ïk - k if ³ =í ï ïî if <, is a continuous function on R, then find the value of k E Show that f( ) 6 ì cosa- cosb if ¹ =í ï ïï ï ( b - a ) if = ïî where a and b are real constants, is continuous at ' f A If f( ) = , find ( ) B If f( ) = then prove that f ( ) f ( ) ' f C If f( ) = log( sec+ tan ), find ( ) D If y= tan ( < ) - æ ö ç çè - ø E If y= tan ç ( < ) 7, find dy d then find dy d Find the derivatives of the following functions ' ' = A B C e log + - log ( > ) D 7 ( > ) E e sin 4 e log 3+ 4, æ ç >- ö çè 3ø F ( ) G sin - e H log( sin( log )) - I sin( tan ( e ) ) m n J sin cos 8 Find D y and dy for the following functions: FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa:

6 A Find the value of the dy and D y of the function y= when = and D = B If y= +, =, D =, find D y and dy C D E y= , when =, D = y= e when =, D = y= when =, D = F If an error of 3% occurs in measuring the side of a cube, find the percentage error in its volume G Show that the relative error in the n th power of a number is n times the relative error in that number H If the increase in the side of a square is %, find the percentage of change in the area of the square I Area of the DABC is measured by the measures of sides a, b and angle C If measuring C, then what is the percentage error in the area? D C is the error in J The diameter of a sphere is measured to be cm If the error of cm occurs in this, find the errors in volume and surface area of the sphere 9 A State the point at which the following functions are increasing and the points at which they are decreasing i) ii) 3 ( - ) iii) B Find the intervals in which ( ) f( ) is decreasing A If z= log( tan+ tan y), show that ( ) z ( ) z z = +, show that - y = y B If z f( y ) 3 f = is increasing and the intervals in which sin + sin y zy = + y = + + show that C If z e f( ) g( y) y zy = e + z z D Find, y for i) y z tan - æ ö = ç çè ø ii) z= + + y E If v=p r h, show that rvr + hvh= 4 v F If z sin( y) log( y) = show that z= zyy A Find the equation of the locus of P, if the ratio of the distances from P to A (5, -4) and B (7, 6) is : 3 FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa:

7 B A (, 3), B (-3, 4) be two given points Find the equation of locus of P so that the area of the triangle PAB is 85 C Find the equation of locus of a point P such that the distance of P from origin is twice the distance of P from A (, ) D Find the equation of locus of P, if the line segment joining (, 3) and (-, 5) subtends a right angle at P E Find the equation of locus of a point, the difference of whose distances from (-5, ) and (5, ) is 8 F Find the equation of locus of P, if A(, 3), B (, -3) and PA + PB = 8 G Find the equation of locus of the point which is collinear with the points (3, 4) and (-4, 3) H An iron rod of length l is sliding on two mutually perpendicular lines Find the equation of locus of the midpoint of the rod I A straight rod of length 9 slides with its ends A, B always on the X and Y aes respectively Then find the locus of the centroid of D AOB J A (5, 3), B (3, -) and C (, -) are three points If P is a point such that the area of the quadrilateral PABC is then find the locus of P K O (,), A (6, ) and B (, 4) are three points If P is a point such that the area of D POB is twice the area of D POA, then find the locus of P L A (, 3), B (, 5) and C (-, ) are three points If P is a point such that the locus of P PA + PB = PC, then find A Find the point to which the origin is to be shifted by the translation of aes so as to remove the first degree terms from the eq a + hy+ by + g+ fy+ c=, where h ¹ ab -æ h ö B Show that the aes are to be rotated through an angle of Tan ç so as to remove the XY è a-bø term from the equation a p + hy+ by = if a¹ b, and through an angle if a = b 4 C When the origin is shifted to the point (, 3), the transferred equation of a curve is + 3y- y + 7-7y- = Find the original equation of the curve D When the aes are rotated through an angle 45, the transformed equation of a curve is 7-6y+ 7y = 5 Find the original equation of the curve E When the aes are rotated through an angle + 3y- y = a p, find the transformed equation of 6 F When the aes are rotated through an angle 3 y 3y = p, find the transformed equation of 4 FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa:

8 G Find the transformed equation of æ ö ç, by translation of aes è ø 5 4y 8y y = when the origin is shifted to H Find the angle through which the aes may be turned so that the equation A+ By+ C= 3 ( A> ) reduces to the form = constant and determine the value of this constant A A straight line parallel to the line y= 3 passes through Q (, 3) and cuts the line + 4y- 7= at P Find the length PQ B Transform the equation 3+ 4y+ = into (i) slope-intercept form (ii) intercept form (iii) normal form C A straight line Q (, 3) makes an angle 3 p with the negative direction of the X-ais If the straight 4 line intersects the line + y- 7= at P, find the distance PQ D E A straight line L with negative slope passes through the point (8, ) and cuts positive coordinate aes at the points P & Q Find the minimum value of OP + OQ as L varies, where O is the origin A straight line L is drawn through the point A (, ) such that its point of intersection with the straight line + y= 9 is at a distance of 3 from A Find the angle which the line L makes with the positive direction of the X-ais F Find the value of k, if the lines - 3y+ k=, 3-4y- 3= and 8- y- 33= are concurrent y G A variable straight line drawn through the point of intersection of the straight lines + = and a b y + = meets the coordinate aes at A and B Show that the locus of the midpoint of AB is b a a+ b y= ab + y ( ) ( ) H Show that the origin is within the triangle whose angular points are (, ) (3, -) and (-4, -) I Find the equations of the straight lines passing through the point (-3, ) and making an angle of 45 with the straight line 3- y+ 4= J Each side of a square is of length 4 units The centre of the square is (3, 7) and one of its diagonals is parallel to y= Find the coordinates of its vertices K If P and Q are the lengths of the perpendiculars from the origin to the straight lines seca+ y coseca= a and cosa- y sina = a cos a Prove that 4P + Q = a L Find the area of the rhombus enclosed by the four straight lines a± by± c= M Find the equations of the straight lines passing through (, ) and which are at a distance of 3 units from (-, 3) FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa:

9 y N Transform the equation + = into the normal form when a >, b > If the perpendicular a b distance of the straight line from the origin is P, deduce that = + p a b O If the straight lines a+ by+ c=, b+ cy+ a= and c+ ay+ b= are concurrent, then prove that a b c 3abc + + = P Find the point on the straight line 3+ y+ 4=, which is equidistant from the point (-5, 6) and (3, ) Q Find the area of the triangle formed by the straight lines -y- 5=, - 5y+ = and + y- = R If the four straight lines a+ by+ p=, a+ by+ q=, c+ dy+ r= and c+ dy+ s= form a ( )( ) parallelogram, show that the area of the parallelogram so formed is p - q r - s bc-ad S The base of an equilateral triangle is + y- = and the opposite verte is (, -) Find the equations of the remaining sides T If the opposite vertices of a square are (-, 3) and (8, 5) Find the equations of the sides of the square 4 A P is a variable point which moves such that 3PA PB that P satisfies the equation + y + z + 8- y+ z- 47= = If A= (-,,3) and ( 3, 3, 3) B Find the ratio in which YZ -plane divides the line joining A (,4,5) and B( 3,5, 4) point of intersection C Show that the points A( 3, -,4 ), B(,,) and (,4, ) C - - are collinear B - prove - Also find the D Find the fourth verte of the parallelogram whose consecutive vertices are (, 4, -),( 3,6, - ) and ( 4,5, ) E A( 5,4,6 ), B(,,3 ), C( 4,3,) bisector of - are three points Find the coordinates of the point in which the Ð BAC meets the side BC F Show that the points ( 5, 4, ), ( 6,, - ) and ( 8,, 7) - - are collinear G Show that the points A( 3,, -4), B( 5, 4, - 6) and C( 9, 8, ) which B divides AC - are collinear and find the ratio in 5 A Find the derivatives of the following functions f( ) from the first principles i) sin ii) cos a iiii) tan iv) sin FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa:

10 B If y y = e -, then show that dy = d log ( + log) C If y ( a y) sin = sin +, then show that dy sin ( a+ y) = (a is not a multiple of p ) d sina D Find the derivatives of the following function i) é ù 3-3a tan - êa( a - 3 ) ú ë û ii) æ - ö tan + - ç çè ø E Find dy d for the following function i) a( cost t sin t), y a( sint t cost) = + = - ii) æ t ö - bt = a, y = ç è+ t ø + t '' f F If f( ) = sin sin sin 3, find ( ) G Show that y= + tan satisfies d y cos y d + = H If a( t sin t), y a( cost) = - = + find d y d I If y= sin( sin ), show that ( ) '' ' y + tan y + y cos = J Find the second order derivatives of the following functions ( ) K Prove the following: f log( 4 9) - i) If n+ -n y a b n y n n y = + then = ( + ) ii) If ( ) y= acos+ b+ sin then = then iii) If y 6( ) ( a b) e 3 '' y + y= 4 cos ' y'' - 6y + 9y= '' ' = + then ( ) iv) If ay 4 ( b) 5 5yy v) If y= acos( sin) + bsin( sin) then ( ) L If a + hy+ by = the prove that M If y ae cos( c d) = y '' ' y + tan y + y cos = d y h -ab = 3 d ( h+ by) -b '' ' = + then prove that y + by + ( b + c ) y= k - = + prove that then N If y e ( a cos n b sin n) '' ' æ k ö y + ky + n + y= ç 4 çè ø FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa:

11 6 A A point is moving on the curve y= 4- The coordinate of the point is decreasing at the rate of 5 units per second Find the rate at which y coordinate of the point is changing when the point is at (, ) B A balloon is in the shape of an inverted cone surmounted by hemisphere Diameter of the sphere is equal to the height of the cone If h is the total height of the balloon, then how does the volume of the balloon changes with h? What is the rate of change in volume when h= 9 units? C A particle moving along a straight line has the relation s= t + t+ 3 connecting the distance s described by the particle in time t Find the velocity and acceleration of the particle at time t= 3 seconds D Sand is poured from a pipe at the rate of cc/sec The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sith of the radius of the base How fast is the height of the sand-cone increasing when the height is 4 cm 3 E Water is dripping out from a conical funnel, at a uniform rate of cm /sec through a tiny hole at the verte at the bottom When the slant height of the water is 4 cm Find the rate of decrease of the slant height of the water given that the vertical angle of the funnel is F A man 6 feet high walks at a uniform rate of 4 miles per hour away from a lamp feet high Find the rate at which the length of his shadow increases ( mile = 58 feet) G A man 8 cm high walks at a uniform rate of km per hour away from a lamp post of 45 cm high Find the rate at which the length of his shadow increases H A light is at the top of a pole 64 m high A ball is dropped from the same height ( 64m ) from a point m from the light Assuming that the ball falls according to the law shadow of the ball moving along the ground seconds later? s= 5t, how fast is the I A ship starts moving from a point at am and heading due east at km/hr Another ship starts moving from the same point at am and heading due south at km/hr How fast is the distance between the ships increasing at noon? 7 A If ( ) -æyö - f, y = tan y tan æ ö - ; ¹, y¹ ç è ø çè yø show that -y fy= + y B If r ( a) ( y b) = - + -, find the value of r + r yy C If z= f( y) + yg( ), show that z y zy z yzy + = + D For the following functions f, show that f+ fyy= i) y + y ii) y tan - æ ç ö çè ø + ay E If ( z+ a) e = b, then show that z ( z z ) + =- zy F If u =, then show that + y + z å u = FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa:

12 G Let f(, y ) be a homogenous function of degree a whose first order partial derivatives eist Then f f + y =af, y for all, y in the domain f H Verify Euler s theorem for the function f(, y) + y = + y I Using Euler s theorem show that u+ yuy = tan u for the function - æ + y ö u= sin ç y è + ø J If æ + y ö u= tan ç + y, show that u+ yuy = sin u çè ø K If u logv = and (, ) v y is a homogenous function of degree n, prove that u+ yuy = n L If æ - y ö u= tan ç 3 3 è + y ø then show that u+ yuy = 8 A If Q(h, k) is the foot of the perpendicular from (, ) h- k-y -( a + by + c) show that = = a b a + b p y on the straight line a+ by+ c=, then And hence find the foot of the perpendicular from (-, 3) on the straight line 5-y- 8= B If Q(h, k) is the image of the point (, ) h- k-y - ( a + by + c) = = a b a + b - 3y+ 5= p y wrt the straight line a+ by+ c= then show that And hence find the image of (, -) wrt the straight line C Find the orthocenter of the triangle with the vertices (-, -) (6, -) and (, 5) D Find the circumcenter of the triangle whose vertices are (, 3), (, -) and (-3, ) E If the equations of the sides of a triangle are 7+ y- =, - y+ 5= and + y+ =, find the orthocenter of the triangle F Find the circumcenter of the triangle whose sides are 3-y- 5=, + y- 4= and 5+ 3y+ = G Find the incentre of the triangle whose sides are + y- 7=, - y+ = and - 3y+ 5= H Two adjacent sides of a parallelogram are given by 4+ 5y= and 7+ y= and one diagonal is + 7y= 9 Find the equations of the remaining sides and the other diagonal I Two vertices of a triangle are (5, -) and (-, 3) If the orthocentre of the triangle is the origin, find the third verte FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa:

13 J A triangle is formed by the lines a+ by+ c=, l+ my+ n= and p+ qy+ r= Given that the a+ by+ c l+ my+ n triangle is not right angled, Show that the straight line = passes through the ap+ bq lp+ mq orthocenter of the triangle 9 A If the equation a + hy+ by = represents a pair of distinct (ie, intersecting) lines, then the h - y = a- b y combined equation of the pair of bisectors of the angles between the lines is ( ) ( ) B Prove that the lines represented by the equations equilateral triangle - 4y+ y = and + y= 3 form an C Show that the product of the perpendicular distances from a point ( a, b ) to the pair of straight lines a + hy+ by = is aa + hab+ bb - + 4h ( a b) D Show that the area of the triangle formed by the lines n h -ab am - hlm+ bl a + hy+ by = and l+ my+ n= is E Find the centroid and the area of the triangle formed by the following lines i) y -y- 6 =, + y+ 4= ii) 3-4y+ y =, - y= 6 F Show that the straight lines represented by y+ 3y = and 3- y+ 3= form an equilateral triangle of area 3 3 squnits G If ( a, b ) is the centroid of the triangle formed by the lines prove that a b = = bl-hm am-hl 3 bl - hlm+ am ( ) a + hy+ by = and l+ my=, H The straight line l+ my+ n= bisects the angle between the pair of lines of which one is p+ qy+ r= Show that the other line is ( p+ qy+ r)( l + m )- ( lp+ mq)( l+ my+ n) = I If the equation then ( i ) h Sº a + hy+ by + g+ fy+ c= represents a pair of parallel straight lines, = ab ( ii ) g -ac f -bc = a a+ b + ( ) b( a b) af bg = and ( ) iii the distance between the parallel lines = A If the pair of lines represented by a + hy+ by = and form a rhombus, prove that ( a b) fg h( f g ) = a + hy+ by + g+ fy+ c= FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa:

14 B If two of the sides of the parallelogram are represented by a + hy+ by = and p+ qy= is one of its diagonals, prove that the other diagonal is y( bp hq) ( aq hp) - = - C Find the value of k, if the equation + ky- 6y + 3+ y+ = represent a pair of straight lines Find the point of intersection of the lines and the angle between the straight lines for this value of k D Show that the straight lines y - 4y+ 3= and + 4y+ 4y + 5+ y+ 4= form a parallelogram and find the lengths of its sides E If the equation a + hy+ by + g+ fy+ c= represents a pair of intersecting lines, then show that the square of the distance of their point of intersection from the origin is c( a+ b) - f -g Also show that the square of this distance is ab-h perpendicular f + g h + b if the given lines are F Show that the lines joining the origin to the points of intersection of the curve - y+ y y- = and the straight line -y- = are mutually perpendicular G Find the value of k, if the lines joining the origin to the points of intersection of the curve - y+ 3y + -y- = and the line + y= k are mutually perpendicular H Find the angle between the lines joining the origin to the points of intersection of the curve + y+ y + + y- 5= and the line 3- y+ = I Write down the equation of the pair of straight lines joining the origin to the points of intersection of the line 6- y+ 8= with the pair of straight lines 3 + 4y- 4y - + y+ 6= Show that the lines so obtained make equal angles with the coordinate aes A Find the direction cosines of two lines which are connected by the relations l+ m+ n= and mn- nl- lm= B Show that the lines whose dc s are given by l+ m+ n=, mn+ 3nl- 5lm= are perpendicular to each other C Find the angle between the lines whose direction cosines satisfy the equations l+ m+ n=, l + m - n = D If a ray makes angles a, b, g, d with the four diagonals of a cube find cos a+ cos b+ cos g+ cos d E Find the angle between the lines whose direction cosines are given by the equations 3l+ m+ 5n= and 6mn- nl+ 5lm= FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa:

15 F If a variable line in two adjacent positions has direction cosines ( l, m, n ) and ( l l, m m, n n) +d +d +d, dq = d + d + d show that the small angle dq between the two positions is given by ( ) ( l) ( m) ( n) A If é ù y= tan ê ú ë û for < <, find dy d B If tan ( sin ) cos y= +, find dy d = - then C If y a( y) dy -y = d - D if æ ö y= a + + a log ç + a + çè ø then dy a d = + E If logy = log then dy y é -loglogy ù = d êë log úû F Find the derivative dy /3-3/4 d of each of the following function ( - ) ( + 3) y= ( - 6) + ( + 7) G Find the derivatives of the following function ( sin) H Establish the following 3 i) If y b + y = a then ii) If f( ) - -b = sin iii) If a> b> and ; y dy é - y + y logy ù =- d y - êë log+ y úû a-b and ( ) g = tan - -b < <p f( ) ( a b ) A If the tangent at any point on the curve B, the show that the length AB is a constant log sin + ' 5/6-6/7 a- then f ( ) = g ( ) ( b< <a ) -/ - acos b cos æ + = - ö ç çè a+ b cosø ' then f ' ( ) ( a b cos) - = + /3 /3 /3 + y = a intersects the coordinate aes in A and B If the tangent at any point P on the cure then show that AP : BP is a constant m n m n y a + = ( ) mn¹ meets the coordinate aes in A,B, m+ n m-n n = > + ¹, mth power of the length of C Show that at any point on the curve a y ( a, m n ) the subtangent varies as the n th power of the length of the subnormal D At any point t on the curve a( t sin t), y a( cost) subtangent and subnormal = + = -, find the lengths of the tangent, normal, FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa:

16 E Find the lengths of subtangent, subnormal at a point t on the curve = a cost+ t sin t, y= a sint- t cost ( ) ( ) F Find the angle between the curves y= and + 4y= G Show that the condition for the orthogonality of the curves - = - a b a b a + by = and a + by = is H Show that the cures y = 4( + ) and y 36( 9 ) = - intersect orthogonally 4 A The sum of the height and diameter of the base of a right circular cylinder is given as 3 units Find the radius of the base and height of the cylinder so that the volume is maimum B A jet of an enemy is flying along the curve y= + A soldier is placed at the point ( 3, ) What is the nearest distance between the soldier and the jet C From a rectangular sheet of dimensions 3 cm 8 cm Four equal squares of side cm are removed at the corners, and the sides are then turned up so as to form an open rectangular bo Find the value of, so that the volume of the bo is the greatest D A window is in the shape of rectangle surrounded by a semicircle If the perimeter of the window be feet Find the maimum area E Show that when the curved surface of a right circular cylinder inscribed in a sphere of radius R is maimum, then the height of the cylinder is R F A wire of length l is cut into two parts which are bent respectively in the form of a square and a circle What are the lengths of the pieces of the wire so that the sum of the areas is the least G Show that the semi-vertical angle of the right circular cone of a maimum volume and of given slant height is - tan H Assume that the petrol burnt (per hour) is driving a motor boat varies as the cube of its velocity Show that the most economical speed when going against a current of 6 km per hour is 9 km per hour Xwish you all the bestx FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6 Ph: , , Fa: