IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB

Transcription

1 ` KUKATPALLY CENTRE IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

2 Total No of Questions - 4 INTERMEDIATE PUBLIC EXAMINATION, MARCH 017 Total No of Printed Pages - Reg No Part - III MATHEMATICS, Paper-I (B) (English Version) Time : 3 Hours] [Ma Marks : 75 SECTION - A I Very Short Answer Type questions: 1 Find the value of ' y ', if the line joining the points the points 1,4, 0,6 Find the value of concurrent 3 l n m 3, y and 10 = 0 M,7 is parallel to the line joining, if the straight lines p 0, y 0 and 3 y 5 0 are 3 Find the fourth verte of the parallelogram whose consecutive vertices are,4, 1, 3,6, 1 and 4,5,1 4 Find the angle between the planes y z 5 0 and 3 3y z8 0 5 lim 0 1 sin lim If f 7 0, then find ' 8 If tane y 9 Find dy and y of, then show that f y dy e d 1 y at 10 when Verify Rolle s theorem for the function f : 3,8 be defined by f 5 6 II Short Answer Type questions: (i) Attempt any five questions (ii) Each question carries four marks SECTION B 5 4 = 0 M 11 A 5,3 and 3, PAB is 9 sq units B are two fied points Find the equation of locus of P, so that the area of 1 When the aes are rotated through an angle 3 10y 3y 9, find the transformed equation of y 5 0 is the perpendicular bisector of the line segment joining the points A, B If A 1, 3, find the coordinates of ' B ' cos a cosb if 0 1 if 0 b a where a and b are real constants, is continuous at 0 14 Show that f FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

3 4 15 If ay b 5 then 5 yy" y' 16 Find the lengths of subtangent, subnormal at a point ' ' y a sint t cost t on the curve acost t sint, 17 The volume of a cube is increasing at rate of 9 cubic centimeters per second How fast is the surface area increasing when the length of the edge is 10 centimetres? SECTION C 5 7 = 35 M III Long Answer Type questions: (i) Attempt any five questions (ii) Each question carries seven marks 18 Find the orthocentre of the triangle whose vertices are 5,, 1, and 19 Show that the area of the triangle formed by the lines 1,4 a hy by 0 and the line l my n 0 is n h ab am h m bl 0 The condition for the line joining the origin to the point of intersection of the circle and the line l my 1 to coincide y a 1 Find the direction consines of two lines which are connected the relation l m n 0 and mnnl lm 0 If 1 1 y a y 3 At a point, then prove that dy d 1 1 y y on the curve y 3ay, show that the tangent is ay y a y a y A window is in the shape of rectangle surmounted by a semicircle If the perimeter of the window is 0 ft find the maimum area BLUE PRINT (MATHS-IB) SNo Name of the chapter Weightage Marks CO-ORDINATE GEOMETRY 1 Locus 4 (4) Transformation of aes 4 (4) 3 Straight line 15 ( ) 4 Pair of straight lines 14 (7 + 7) 3D GEOMETRY 5 3D-coordinates () 6 Direction Cosines & Direction Rations 7 (7) 7 The plane () CALCULUS 8 Limits & Continuity 8 (4 + + ) 9 Differentiation 15 ( ) 10 Errors Approimations () 11 Tangent & Normal 11 (7 + 4) 1 Rate measure 4 (4) 13 Rolle s & Lagrange s Theorems () 14 Maima & Minima 7 (7) FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

4 1 VERY SHORT ANSWER QUESTIONS A Find the equation of the straight line passing through,4 and making non zero intercepts whose sum is zero B Find the equation of the straight line passing through 3, 4 and making X and Y -intercepts which are in the ratio : 3 C If the area of the triangle formed by the straight lines 0, y 0 and 3 4y a Find the value of a D Transform the equation y 1 0 into normal form 0 a is 6 E Find the ratio in which the line 3y 0 0 divides the join of the points (, 3) and (, 10) F Prove that the points (-5, 1), (5, 5), (10, 7) are collinear and find the equation of the line containing these points G 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 H Find the equations of the straight line passing through the following points : a) (, 5), (, 8) b) (3, -3), (7, -3) c) (1, -), (-, 3) d) at1, at1at,at I 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 (1, 1) and (, 3) J Find the equation of straight line passing through origin and making equal angles with coordinate aes K Find the equation of the straight line making an angle of -ais and has y -intercept 3 Tan 3 L A straight line passing through A,1 makes an angle of 30 with OX direction Find the points on the straight line whose distance from A is 4 units 1 with the positive in the positive M Find the points on the line 3 4y 1 0 which are at a distance of 5 units from the point (3, ) N Find the equation of line which makes an angle of 150 with positive -ais and passing through (-, -1) O State whether (3, ) and (-4, -3) are on the same side or on opposite side of the straight line 3y 4 0 A Find the area of the triangle formed by the coordinate aes and the line 3 4y 1 0 B Find the set of values of ' a ' if the points (1, ) and (3, 4) lie on the same side of the straight line 3 5y a 0 C Find the distance between the straight lines 5 3y 4 0 and 10 6y 9 0 FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

5 D Find the value of ' k ' if the straight lines y 3k 4 0 & k 1 8k 1y 6 0 are perpendicular E Find the orthocenter of the triangle whose sides are given by y 10 0, y 0 and y 7 0 F If a, b, c are in AP, show that a by c 0 represents a family of concurrent lines and find the point of concurrency G Find the equation of the line perpendicular to the line 3 4y 6 0 and making an intercept 4 on the -ais H Find the point of concurrency of the lines represented by k ky k I Find the equation of the straight line passing through the point of intersection of the lines y 1 0 and y 5 0 and containing the point (5, -) J If 3y 5 0 is the perpendicular bisector of the line segment joining (3, 4) and, Find K Find the length of the perpendicular drawn from (3, 4) to the line 3 4y 10 0 L Find the circumcentre of the triangle formed by the lines 1, y 1 and y 1 M Find the value of k, if the angle between the straight lines 4 y 7 0 and k 5y 9 0 is 45 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 3 A Find if the distance between 5, 1,7 and,5,1 B Show that the points 1,,3, 7,0,1 and,3,4 is 9 units are collinear C Show that the points,3,5, 1,5, 1 and 4, 3, form a right angled isosceles triangle D Show that the points 1,,3,,3,1 and 3,1, form an equilateral triangle E P is a variable point which moves such that 3PA = PB If A =,,3 and 13, 3,13 B, prove that P satisfies the equation y z 8 1y 10z 47 0 F Show that ABCD is a square where A, B, C, D are the points0,4,1,,3, 1, 4,5,0 and,6, respectively 4 A Find the equation of the plane if the foot of the perpendicular from origin to the plane is,3, 5 B Reduce the equation y 3z6 0 of the plane to the normal form C Find the angle between the planes y z 5 0 and 3 3y z8 0 FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

6 D Find the equation to the plane parallel to the ZX -plane and passing through 0,4,4 E Find the equation of the plane passing through the point,1,3 and having 3, 5,4 of its normal as dr s F Find the equation of the plane passing through the point 1,1,1 and parallel to the plane y 3z7 0 G Find the equation of the plane through 4,4,0 and perpendicular to the planes y z 3 0 and 3 3y z8 0 H Find the equation of the plane passing through,0,1 and 3, 3,4 y z 6 I Find the equation of the plane through the points,, 1, 3,4,, 7,0,6 J 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 and perpendicular to is,, a b c Show that the 5 A Find B 0 e C D / 0 sin cos / a bsina b E and F a b b G H Evaluate 0 0 a b 1, 0, 0, 1 1 sin a, b 0, a b sin b 0 cos I 0 e sin 1 J K If f, 1 1, 1 then find 1 f and 1 f FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

7 L 0 e 3 1 M Find f 0 N Evaluate O P if 0 where f 0 if 0 1 if 0 log e 1 sin tan a a 0 a Q e e A B C 0 e 1 1 cos cos 007 D Is f defined by f E Show that f F G H 0 sin, if 0 1, if 0 cos a cosb if 0 1 b a if 1cos m 1cosn 3 1 sin 3 n 0 0 continuous at 0 where a and b are real constants, is continuous at 0 I 6 1 J K cos sin 1 FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

8 7 A If f e sin, then find f ' B 100 If f 1 then find f ' 1 C If f logsec tan find f ' 3 D If f 7 0 then find f ' E If f sinlog, 0 find f ' 3 F If 100 find f ' f G If f log7 log, 0 find dy d H If f 3 5 then prove that f f 8 ' 0 3 ' 1 0 A B If If 3 y then find y '' 4 5 n n y a e b e then prove that y'' n y C If y logsinlog D Find the derivative of, find dy d log E Find the second order derivative of f log4 9 F If y sin 1 G If y logcosh, find dy d, find dy d 9 A Find dy and y of y f at 10 when 01 B C If the radius of a sphere is increased from 7 cm to 70 cm then find the approimate increase in the volume of the sphere If the increase in the side of a square is % then find the approimate percentage of increase in its area D Show that the length of the subnormal at any point on the curve y 4a is constant E Show that the length of the subtangent at any point on the curve y a a 0 is a constant F Find dy and y if 1 y, 8 and 00 FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

9 G Find the approimate value of H The side of a square is increased from 3 cm to 301cm Find the approimate increase in the area of the square I The diameter of a sphere is measured to be 40 cm If the error of 00 cm is made in it, then find the approimate errors of 00cm is made in it, then find the approimate errors in volume and surface area of the sphere 1 J Find the slope of the tangent to the curve y at A Find the value of k, so that the length of the subnormal at any point on the curve a constant 1k y a is k B Show that the length of the subnormal at any point on the curve y a varies as the cube of the ordinate of the point C Let f 1 3 Prove that there is more than one ' c ' in 1,3 such that f ' c 0 D Verify Rolle s theorem for the function y f 4 in 3,3 E / Verify Rolle s theorem for the function f 3e in 3,0 F 3 Find the intervals on which the function f is a strictly increasing function G Find the rate of change of area of a circle wrt radius when r 5cm H The distance time formula for the motion of a particle along a straight line is 3 S t 9t 4t 18 Find when and where the velocity is zero I Find the intervals on which f 3 8 is increasing or decreasing J Find the average rate of the change of S f t t 3 between t and t 4 11 A B C SHORT ANSWER QUESTIONS 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 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 (1, ) Find the equation of locus of P, if the line segment joining (, 3) and (-1, 5) subtends a right angle at P D Find the equation of locus of a point, the difference of whose distances from (-5, 0) and (5, 0) is 8 E Find the equation of the locus of a point, the sum of whose distances from (0, ) and (0, ) is 6 FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

10 F G The ends of the hypotenuse of a right angled triangle are (0, 6) and (6, 0) Find the equation of the locus of its third verte A (5, 3) and B (3, ) are two fied points Find the equation of the locus of P, so that the area of triangle PAB is 9 H A (1, ), B (, -3) and C (-, 3) are three points A point P moves such that Show that the equation to the locus of P is 7 7y 4 0 PA PB PC I Find the equation of the locus of P, if A = (4, 0), B = ( 4, 0) and PA PB 4 J 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 K Find the locus of the third verte of a right angled triangle, the ends of whose hypotenuse are (4, 0) and (0, 4) L Find the equation of locus of P, if A (, 3), B (, -3) and PA + PB = 8 1 A Show that the aes are to be rotated through an angle of 1 1 h Tan ab so as to remove the y term from the equation a hy by 0 if a b, and through an angle if a b 4 B C When the origin is shifted to the point (, 3), the transformed equation of a curve is 3y y 17 7y 11 0 Find the original equation of the curve When the aes are rotated through an angle 17 16y 17y 5 Find the original equation of the curve 0 45, the transformed equation of a curve is D When the aes are rotated through an angle 3y y a, find the transformed equation of 6 E When the aes are rotated through an angle 3 10y 3y 9, find the transformed equation of 4 F When the aes are rotated through an angle, find the transformed equation of cos y sin p G 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 0, where h ab H When the origin is shifted to ( 1, ) by the translation of aes, find the transformed equations of the following (i) y 4y 1 0 (ii) y 4 4y 0 FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

11 13 A Transform the equation 3 4y 1 0 into (i) slope-intercept form (ii) intercept form (iii) normal form y B Transform the equation 1 into the normal form when a 0, b 0 If the perpendicular a b distance of the straight line from the origin is p, deduce that p a b C Find the value of p, if the following lines are concurrent (i) 3 4y 5, 3y 4, p 4y 6 (ii) 4 3y 7 0, py 0, 6 5y 1 0 D Find the value of k, if angle between the straight lines 4 y 7 0 and k 5y 9 0 is 45 E A straight line Q (, 3) makes an angle 3 with the negative direction of the X -ais If the 4 straight line intersects the line y 7 0 at P, find the distance PQ F A straight line with slope 1 passes through 3,5 distance PQ Q meets the line y 6 0 at P Find the G Find the equation of straight line parallel to line 3 4y 7 and passing through the point of intersection of lines y 3 0 and 3y 6 0 H I If P and Q are the lengths of the perpendiculars from the origin to the straight lines sec y cos ec a and cos y sin acos Prove that 4P Q a 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 0 J A variable straight line drawn through the point of intersection of the straight lines y 1 a b y and 1 meets the coordinate aes at A and B Show that the locus of the midpoint of AB b a is a by ab y K 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 L 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 M Find the area of the rhombus enclosed by the four straight lines a by c 0 N If the straight lines a by c 0, b cy a 0 and c ay b 0 are concurrent, then prove that a b c 3abc O Find the point on the straight line 3 y 4 0, which is equidistant from (-5, 6) and (3, ) P Find the points on the line 3 4y 1 0 which are at a distance of 5 units from the point 3, FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

12 14 A B C D a 0 sin a asin a cosa cosb Check the continuity of the following function at '' 1 4 if 0 f 0 if 3 8 if Check the continuity of f given by f 9 if 0 5 and if 3 at the point 3 E Show that f cos a cosb 1 b a if if 0 0 where a and b are real constants, is continuous at 0 F 0 1/8 1/8 1 1 G Find real constants a, b so that the function f given by sin if 0 f a if 0 1 b 3 if if 3 H is continuous on I Check the continuity of the function f given below at 1 and 1 if 1 f if 1 1 if J If f, given by f of k k k if, 1, if 1 is a continuous function on, then find the value K Check the continuity of f given by f 4 if 0 5 if if if FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

13 15 A B C Find the derivative of the following functions from first principles (4 marks each) (i) 1 (ii) (v) If a b c (iii) sin (iv) cosa sin (vi) tan (vii) sec3 (viii) y y e Find dy then show that dy d log 1 log d if acost t sint, y asint t cost cos D If /3 /3 /3 dy y a then prove that 3 d y E If y tan F If y tan 1 find dy d find dy d G If sin y sina y H, then show that dy d Find the derivative of tan sin a y a n sin a I Find dy d if 1t a 1 t, bt y 1 t 1 1 J Differentiate f with respect to g for f sec, g 1 1 K Show that the function f 1, is differentiable for all real numbers ecept for 0 & 1 L Show that y tan satisfies d y cos y d n1 n M If y a b then prove that y'' nn 1y 4 N If ay b 5 then prove that 5 yy'' y' 16 A If the slope of the tangent to the curve equations of the tangent and normal at that point y 4y 0 at a point on it is 3 /, then find the B Show that the tangent at any point on the curve c sec, y c tan is given by y sin c cos C Find the equations of tangent and normal to the curve y 10 at,5 FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

14 D Show that at any point, y on the curve the length of the subnormal is y a y b e a, the length of the subtangent is constant and E Show that the equation of the tangent to the curve a 0, b 0 is y a b n n y a b at the point a, b F Find the value of k so that the length of the subnormal at any point on the curve constant / G Find the angle between the curve y e and y ais y k k 1 a is a H Find the equations of the tangent and the normal to the curve y 4 5 at the point 1,5 I Find the lengths of subtangent and subnormal at a point on the curve y b sin a J Find the equations of the tangent and normal to the curve y 4 3 a at a, a K Show that the curves 6 5 y 0 and y 3 touch each other at, 17 A B The volume of a cube is increasing at a rate of 9 cubic centimeters per second How fast is the surface area increasing when the length of the edge is 10 centimeters A point P is moving on the curve y The coordinate of P is increasing at the rate of 4 units per second Find the rate at which the y coordinate is increasing when the point is at,8 C D E A particle is moving in a straight line so that after t seconds its distance is s (in cms) from a fied point on the line is given by 3 8 Find (i) the velocity at time t sec (ii) the s f t t t initial velocity (iii) acceleration at t sec The volume of a cube is increasing at the rate of 8cm 3 /sec How fast is the surface area increasing when the length of an edge is 1 cm? A container in the shape of an inverted cone has height 1 cm and radius 6 cm at the top If it is filled with water at the rate of 1cm 3 /sec, what is the rate of change in the height of water level when the tank is filled 8 cm F A stone is dropped intro a quiet lake and ripples move in circles at the speed of 5cm/sec At the instant when the radius of circular ripple is 8cm, how fast is the enclosed arc increases G A container is in the shape of an inverted cone has height 8m and radius 6m at the top If it is filled with water at the rate of m 3 /minute, how fast is the height of water changing when the level is 4m? FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

15 H The total cost C in rupees associated with production of units of an item is given by Find the marginal cost when 3 units are produced C (marginal cost is the rate of change of total cost) 3 I A particle is moving along a line according to s f t 4t 3t 5t 1 where s is measured in meters and t is measured in seconds Find the velocity and acceleration at time t At what time the acceleration is zero 18 LONG ANSWER QUESTIONS A If Qh, k is the foot of the perpendicular from, show that h k y a b a b a by c P y on the straight line a by c 0, then And hence find the foot of the perpendicular from ( 1, 3) on the straight line 5 y 18 0 B If Qh, k is the image of the point, h k y a b a b 3y 5 0 a by c P y wrt the straight line a by c 0 then show that And hence find the image of (1, ) wrt the straight line C Find the circumcentre of the triangle whose vertices are (1, 3), ( 3, 5) and (5, 1) D Find the circumcentre of the triangle whose sides are 3 y 5 0, y 4 0 and 5 3y 1 0 E Find the orthocenter of the triangle whose vertices are ( 5, 7), (13, ) and ( 5, 6) F If the equations of the sides of a triangle are 7 y 10 0, y 5 0 and y 0, find the orthocenter of the triangle G Two adjacent sides of a parallelogram are given by 4 5y 0 and 7 y 0 and one diagonal is 11 7y 9 Find the equations of the remaining sides and the other diagonal H The base of an equilateral triangle is y 0 equations of the remaining sides and the opposite verte is, 1 Find the I If p and q are the lengths of the perpendiculars from the origin to the straight lines sec ycosec a and cos y sin acos, prove that 4p q a J Find the equations of the straight lines passing through 1,1 and which are at a distance of 3 units from,3 K Find the circumcentre of the triangle formed by the straight lines y 0, y 5 0 and y L Find the orthocenter of the triangle with the vertices (-, -1) (6, -1) and (, 5) FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

16 M Find the orthocenter of the triangle formed by the lines y 0, 4 3y 5 0 and 3 y 0 N Find the incentre of the triangle whose sides are y 7 0, y 1 0 and 3y 5 0 O Find the circumcenter of the triangle whose vertices are (1, 3), (0, -) and (-3, 1) P If the four straight lines a by p 0, a by q 0, c dy r 0 and c dy s 0 form a parallelogram, show that the area of the parallelogram so formed is p qr s bc ad 19 A Show that the product of perpendicular distances from a point, to the pair of straight lines a hy by 0 is a h b ab 4h B If the equation then prove that S a hy by g fy c 0 represents a pair of parallel straight lines (i) h ab (ii) af bg and (iii) the distance between the parallel lines g ac f bc a a b b a b C If the second degree equation and y represents a pair of straight lines, then S a hy by g fy c 0 in the two variables (i) abc fgh af bg ch 0 and (ii) h ab, g ac and f bc D If the equation a hy by 0 represents a pair of intersecting lines, then the combined equation of the pair of bisectors of the angles between these lines is h y a b y E Show that the area of the triangle formed by the lines n h ab am hlm bl a hy by 0 and l my n 0 is F Show that the straight lines represented by 3 48y 3y 0 and 3 y 13 0 form an equilateral triangles of area 13 3 sq units G If the pairs of lines represented by a hy by 0 and form a rhombus, prove that a b fg h f g 0 a hy by g fy c 0 H Show that the product of the perpendicular distances from the origin to the pair of straight lines represented by a hy by g fy c 0 is c ab 4h FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

17 I Show that the lines represented by l my m ly 3 0 and l my n 0 form an equilateral triangle with area 3 l n m J Two equal sides of an isosceles triangle are 7 y 3 0 and y 3 0 and its third side passes through the point 1,0 Find the equation of the third side K If, is the centroid of the triangle formed by the lines a hy by 0 and l my 1 Prove that bl hm am hl 3 bl hlm am L If two of the sides of a parallelogram are represented by one of its diagonals Prove that other diagonal is ybp hq aq hp M If the equation a hy by 0 and p qy 1 is a hy by g fy c 0 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 ab h perpendicular Also show that the square of this distance is f h g b if the given lines are N Let the equation a hy by 0 represents a pair of straight lines Then the angle between the lines is given by cos a b ab 4h 0 A Show that the lines joining the origin to the points of intersection of the curve y y 3 3y 0 and the straight line y 0 are mutually perpendicular B Find the values of k, if the lines joining the origin to the point of intersection of the curve y 3y y 1 0 and the line y k are mutually perpendicular C Find the angle between the lines joining the origin to the points of intersection of the curve y y y 5 0 and the line 3 y 1 0 D Find the condition for the chord l my 1 of the circle the origin y a to subtend a right angle at E Find the lines joining the origin to the points of intersection of the curve 7 4y 8y 4y 8 0 with the straight line 3 y and also the angle between them F Find the condition for the lines joining the origin to the points of intersection of the circle y a and the line l my 1 to coincide FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

18 G Write down the equation of the pair of straight lines joining the origin to the points of intersection of the line 6 y 8 0 with the pair of straight lines 3 4y 4y 11 y 6 0 Show that the lines so obtained make equal angles with coordinate aes H Show that the equation 13y 7y 3y 6 0 represents a pair of straight lines Also find the angles between them and the coordinates of the point of intersection of the lines 1 A Show that the lines whose dc s are given by l m n 0, mn 3nl 5lm 0 are perpendicular to each other B Find the angle between the lines whose direction cosines satisfy the equations l m n 0, l m n 0 C If a ray makes angles,,, with the four diagonals of a cube find cos cos cos cos D Find the direction cosines of two lines which are connected by the relations l 5m 3n 0 and 7l 5m 3n 0 E Find the direction cosines of two lines which are connected by the relations l m n 0 and F mn nl lm 0 Find the angle between the lines whose direction cosines are given by the equations 3l m 5n 0 and 6mn nl 5lm 0 G If a variable line in two adjacent positions has direction cosines l, m, n l l, m m, n n, show that the small angle between the two positions is given by l m n H Find the angle between two diagonals of a cube Find A, B, C I The vertices of triangle are A1, 4,, B,1,, C,3, 4 and A If y tan 1 1 for 0 1, find dy d B If 1 1 y a y then prove that C If y a a log a dy d then prove that y sin, find dy d tan D If cos sin E Find the derivative of the function log sin 1 y 1 FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa: dy a d

19 F If y b y a then prove that y 1 dy y y log y y d 1 log y G If f sin 1 and 1 g tan then Prove f ' g' H If a b 0 and 0 I Find dy 1 acos b cos ab cos, f a b 1/ /3 3/4 d if y /6 6/7 f ' a b cos then 1 J If at sin t, y a1 cost find k K If y e a cos n b sin n d y d then prove that L Differentiate f with respect to g where k y'' ky' n y 0 4 f Tan, 1 g Tan M If y acossin bsinsin then prove y'' tan y' y cos 0 3 A If the tangent at any point on the curve and B, then show that the length AB is constant /3 /3 /3 y a intersects the coordinates aes in A m n mn B If the tangent at any point P on the curve y a mn 0 meets the coordinate aes in A,B then show that AP : BP is constant C Show that the curves y 4 1 and y 369 intersect orthogonally D Find the angle between the curves y and 4y 0 E Find the angle between the curves y 4 and y 5 F Show that the condition for orthogonality of the curves a b a b 1 1 a G Find the lengths of subtangent, subnormal at a point t on the curve cos sin ; y asint t cost a t t t H At any point t on the curve at sin t, y a1 cost subangent and sub normal by 1 and 1 b1y 1 a is Find the lengths of tangent, normal, I Find the equations of the tangents to the curve 3 y 3 where it meets the ais J Show that the tangent at, P y on the curve y a is yy a FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa:

20 K Show that the square of the length of subtangent at any point on the curve by a b 0 varies with the length of the subnormal at that point 3 L Find the angle between the curves y 8 and 4 y 3 M Find the angle between the curves y 9 0 and 3 4y 0 (in the 4 th quadrant) 4 A Find two positive integers and y such that y 60 and 3 y is maimum B From a rectangular sheet of dimensions 30cm80cm, four equal squares of side cm are C D E 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 A window is in the shape of a rectangle surmounted by a semi circle If the perimeter of the window is 0 ft, find the maimum area If the curved surface of a right circular cylinder inscribed in a sphere of radius ' r ' is maimum, show that the height of the cylinder is r 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 respectively so that the sum of the areas is the least F Find two positive numbers whose sum is 15 so that the sum of their squares is minimum G Find the maimum area of the rectangle that can be formed with fied perimeter 0 H Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone 3 I Find the absolute maimum and absolute minimum of f on 8, J The profit function p of a company selling ' ' items per day is given by P Find the number of items that the company should manufacture to get maimum profit Also find the maimum profit wish you all the best FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad Ph: Regd Off: 9A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi Ph: , , Fa: