Objective Mathematics

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Aubrey Nicholson
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1 6. In angle etween the pair of tangents drawn from a 1. If straight line y = mx + c is tangential to paraola y 16( x 4), then exhaustive set of values of 'c' is given y (a) R /( 4, 4) () R /(, ) (c) R /( 1, 1) R /[ 4, 4]. Minimum distance etween the paraolic curves y x 4 and (a) 15 4 (c) 15 x y 4 is () Locus of the point of intersection of tangents to paraola y = 4(x + 1) and y = (x + ) which are perpendicular to each other is given y : (a) x = 0 () x + = 0 (c) x + = 0 x = 0 4. If ti, 6ti represents the feet of normals to the paraola y = 1x from (1, ), then to : (a) 6 (c) () 5 i1 1 is equal ti 5. If chords of contact of the pair of tangents drawn from each point on the line y = x + to the curve y x = 0 are concurrent, then the point of concurrency is : (a) (, 0) (c), (),,1 [ 161 ] point 'P' to the paraola y = 4ax is 4, then locus of point 'P' is : (a) paraola. (c) hyperola. () line. ellipse. 7. From a point 'P' if common tangents are drawn to circle x + y = and paraola y = 16x, then the area (in sq. units) of quadrilateral formed y the common tangents, the chords of contact of circle and paraola is given y : (a) 60 () 0 (c) Let P(h, k) lies on the curve f (x) = x x, such that h (0, 1), where 'O' and 'A' are (0, 0) and (1, 0) respectively, then maximum area of POA is: (a) 1 (c) 1 sq. units. () 1 4 sq. units sq. units. sq. units. 9. If curves C 1 : x + y = 5 and C : y 4x = 0 intersect at 'P' and 'Q' and tangents to curve 'C 1 ' and 'C ' at 'P' and 'Q' intersect the x-axis at R and S respectively, then ratio of area of PQR and PQS is : (a) 1 : () 1 : (c) : 1 : If tangent at P(, 4) to paraola y = x meets the curve y = x + 5 at Q and R, then mid-point of QR is : (a) (, 4) () (4, ) (c) (7, 9) (, 5) 11. If two paraola y = 4ax and y = 4c (x ) can-not have common normal other than x-axis, then : a c (a) () a c (c) a c c a Mathematics for JEE-01

2 Paraola 1. If y x 0 cuts the paraola and B, where P,0 ; then PA.PB is : 4 (a) () 4 x y at A 1. Normals PO, PA and PB are drawn to paraola y = 4x from P (h, 0), where 'O' is origin and then area of quadrilateral OAPB is : (a) 1 sq. units (c) 6 sq. units () 4 sq. units 1 sq. units AOB o 90, (c) 4 5 None of these 1. If y = 4a (x ) and x = 4a (y ) always touch one another, and eing oth varying, then locus of point of contact is : (a) xy = 4a () xy = 4a (c) xy = a xy = a/ 14. The locus of the vertex points of the family of a x a x paraolic curve y a, where 'a' is the parameter, is given y : (a) xy (c) xy () xy 01 xy A paraola has its vertex and focus in I st quadrant and axis along the line y = x, if the distances of the vertex and focus from the origin are and respectively, then equation of paraola is : (a) (x + y) = x y + () (x y) = x + y (c) (x y) = (x + y ) (x + y) = (x y + ) 16. If,, then maximum length of latus rect um of paraola whose focus is (a sin, a cos ) and directrix is y a = 0, is : (a) a (c) a () 4a 1 a 19. If normals at the end of a variale chord 'PQ' of the paraola y = 4y + x are perpendicular to each other, then locus of the point of intersection of the tangents at 'P' and 'Q' is given y : (a) 5x + = 0 () x y + = 0 (c) x + 5 = 0 5y = 0 0. The focal chord to y 16x is tangent to the circle ( x 6) y, then the possile values of the slope of this chord, are : (a) 1, 1 (c) (),, 1/, 1/ 1. Let PQ e a chord of the paraola y = 4x and circle on PQ as diameter passes through the vertex 'V' of PVQ is 0 square unit, the paraola. If the area of then the possile co-ordinates for 'P' can e : (a) (, 1) () (1, ) (c) (16, ) ( 16, ). Let a R and the curves x = 4a (y ) and y x = a intersect each other at four distinct points, then the values of '' may lie in the interval : (a) ( a, a) 5a () a, 4 (c) ( a, a) (0, a). Let any point 'P' lies on the paraola y = x. If tangent and normal is drawn to paraola at point 'P' which intersects the x-axis at 'T' and 'N' respectively, then locus of the centroid of triangle PTN is paraolic curve for which : 17. Locus of all points on the curve y = 4a x asin x a at which the tangent is parallel to x-axis is : (a) straight line. () circle. (c) paraola. hyperola. 4 (a) vertex is, 0 () the equation of directrix is x = 0 (c) focus is (, 0) equation of latus rectum is x = 0 [ 16 ] Mathematics for JEE-01

3 4. Let a moving paraola with length of latus rectum units touches a fixed equal paraola, where the axes of moving paraola and fixed paraola eing parallel. If the locus of the vertex of moving paraolic curve is conic 'S', then : (a) eccentricity of 'S' is 1. () length of latus rectum of 'S' is 16 units. (c) eccentricity of 'S' is. length of latus rectum of 'S' is units. 5. Let normals drawn at points A, B (0, 0) and C to the paraola y = 4x e concurrent at point P (, 0). If tangents drawn at 'A' and 'C' to the paraola intersects at point 'D', then : (a) area of ABC is square units. () quadrilateral PABC is cyclic. (c) circumcentre of quadrilateral ADCP is cyclic. ABC lies outside the triangle. Following questions are assertion and reasoning type questions. Each of these questions contains two statements, Statement 1 (Assertion) and Statement (Reason). Each of these questions has four alternative answers, only one of them is the correct answer. Select the correct answer from the given options : (a) Both Statement 1 and Statement are true and Statement is the correct explanation of Statement 1. () Both Statement 1 and Statement are true ut Statement is not the correct explanation of Statement 1. (c) Statement 1 is true ut Statement is false. Statement 1 is false ut Statement is true. 6. Statement 1 : If the curve C 1 is given parametrically y the equations x = sin t + and y = 1 + sint for all real values of 't', then it represents the paraolic curve y y 4x + 9 = 0 Statement : The point ( + sin t, 1 + sin t ) lies on the curve (y 1) = 4 (x ) for all real values of 't'. 7. Statement 1 : Let tangents e drawn to y = 4 ax from a variale point 'P' moving on x + a = 0, then the locus of foot of perpendicular drawn from 'P' on the chord of contact is given y y + (x a) = 0 Statement : The intercept made y any tangent with finile non-zero slope of the paraola etween the directrix and point of tangency always sutends a right angle at focus.. Statement 1 : If normal drawn at any point 'P' on the paraola y = 4ax meets the curve again at 'Q', then the least distance of Q from the axis of paraola is 4 a Statement : If the normal at 't' point meets the curve again at 't 1 ' point, then t 1 t t and t1. 9. Statement 1 : Let perpendicular tangents of the conic y x 4y 4 0 intersects each other at point (, ), then ' ' must e and R Statement : Locus of the point of intersection of perpendicular tangents to a paraolic curve is the directrix of curve. 0. Statement 1 : Let a normal chord PQ e drawn for paraola y = 4x with point 'P' eing (4, 4). Circle descried with PQ as diameter passes through the focus F (1, 0) Statement : normal chord PQ sutends an angle of tan 1 (5) at origin. [ 16 ] Mathematics for JEE-01

4 Paraola Comprehension passage (1) ( Questions No. 1- ) Let the locus of the circumcentre of a variale triangle having sides x = 0, y = 0 and lx + my 1 = 0, where (l, m) lies on y x = 0, e curve 'C', then answer the following questions. 1. Curve 'C' is symmetric aout the line : (a) y + = 0 () y = 0 (c) x + = 0 x = 0. Length of smallest focal chord of curve 'C' is : (a) units (c) 1 unit () 1 unit 1 4 unit. From point 'P' if perpendicular pair of tangents can e drawn to the curve 'C', then 'P' can e : 1 (a), (c), () 1,, Comprehension passage () ( Questions No. 4-6 ) Let C 1 : y = x + ax + and C : y = cx + dx + 1 e two paraolic curves having vertex points at 'A' and 'B' respectively. If the projection of 'A' and 'B' on the x-axis is A' and B' respectively, as shown in the figure (1), and AA' = BB', OA' = OB', where 'O' is origin, t hen answer the following questions. 4. Which one of the following inequality is correct. (a) > 1 () ac < 0 (c) cd < 0 d 0 5. If and c are non-zero real numers, then value of a is equal to : (a) d c (c) d c d c cd o 6. In figure (1), if A' AB ' B ' BA' 10, then which one of the following equality holds true : (a) (5 d c)(5 a ) 1 () (5 a )(5 d c) 16 ad (c) (5 a )(5 d c) 16 a d (5 a )(5 d c) 4d Comprehension passage () ( Questions No. 7-9 ) Let paraolic curves 'C 1 ' and 'C ' e given y y + x + = 0 and y + x + = 0 respectively. Curve 'C' represents a circle with centre at 'C 0 ', where OP and OQ are tangents from origin 'O' to the circle 'C'. If circle 'C' touches oth the paraolic curves C 1, C, and have minimum area, then answer the following questions. 7. Equation of circle 'C' is : (a) 4x + 4y + (x + y) + 19 = 0 () x + y + 11(x + y) + 10 = 0 (c) 4(x + y ) + 11(x + y) + 9 = 0 4(x + y ) + 11(x + y) + 9 = 0. Area ( in square units ) of quadrilateral OPC 0 Q is given y : (a) 1 () 1 figure (1) (c) [ 164 ] Mathematics for JEE-01

5 9. A common tangent to the paraolic curves 'C 1 ' and 'C ' can e given y : (a) 4x + 4y + 7 = 0 () 4x + 4y + 5 = 0 (c) 4x + y + 7 = 0 x + 4y + 5 = 0 Comprehension passage (4) ( Questions No ) Let variale paraolic curves e drawn through the fixed diametric ends (0, r) and (0, r) of the circle x + y = r such that the directrix of variale paraolic curves always touch the circle x + y = R. If the path traced y the focus of the variale paraolic curves is represented y a conic section of eccentricity 'e', then answer the following questions. 10. If R ( r, r ), then eccentricity 'e' may e equal to : (a) () sin 4 (c) sin 1 cos 11. If r R > 0, then 'e' may e equal to : (a) tan () cosec 4 (c) sec cos 1. If r ( R, R ), then 'e' may e equal to : (a) 1 () sec 14. Let a tangent e drawn to paraola y y 4x + 5 = 0 at any point 'P' on it. If the tangent meets the directrix at 'Q' and the moving point 'M', divides QP externally in the ratio 1 :, then locus of 'M' passes through (, 0). The value of ' ' is equal to Let the paraola y = ax + x + touches the line x + y = 0 at point 'P'. If a line through 'P', parallel to x-axis, is drawn to meet y + 1 = x at 'Q' and 'R' and the area of OQR (where 'O' is origin) is 'A' square units, then value of 9 A is equal to Let the tangent at point P(, 4) to the paraola y = x meets the paraola y = x + 5 at 'A' and 'B'. If the midpoint of AB is point (, ), then ( ) is equal to Let PQ e the normal chord for the paraola y 4x y + 9 = 0. If PQ sutends an angle of 90º at the vertex of the paraola, then square of slope of the normal chord is equal to Let all the sides (or the extension of sides) of on equilateral triangle ABC touch the paraola y 4x = 0. If the vertices of ABC lie on the curve 'C' and curve 'C' passes through the point P(1, k), where 'P' lies aove the x-axis, then value of 'k' is equal to Let tangent and normal drawn to paraola at point P( t, 4 t), t 0, meets the axis of paraola at points 'Q' and 'R' respectively. If rectangle PQRS is completed, then locus of vertex 'S' of the rectangle is given y curve 'C'. Total numer of integral points inside the region of curve 'C' in the first quadrant is equal to... (c) sec 0. Let 'P' and 'Q' e the end points of the latus rectum of paraolic curve y 4y + x = 0 and point 'R' lies on the circle x + y 4x 4y + 7 = 0. If PR + RQ is minimum, then maximum numer of locations for point 'R' is / are Let three normals e drawn from point 'P' with slopes, and to the paraola y = 4x. If locus of 'P' with the condition k is a part of the paraolic curve y 4x = 0, then value of 'k' is equal to... [ 165 ] Mathematics for JEE-01

6 Paraola 1. Let points P ( 6, 4), Q (, 0), R(, 4) and S (, ) form a quadrilateral PQRS and a paraolic curve 'C' with axis of symmetry along y 4 0 passes through P, Q and S. With reference to curve 'C', match the following columns I and II. Column (I) Column (II) (a) Length of latus rectum of curve 'C', is : (p). () Length of doule ordinate of curve 'C' which (q) 5 6. sutends an angle of 90º at the vertex of curve is : (c) If 'F' is focus of curve 'C' and 'r' is the in-radius (r) 4. of QFS, then value of r is equal to : Circum-radius of QFS is : (s) Match the following columns (I) and (II) Column (I) Column (II) (a) Paraolic curve y = x + 5x + 4 meets the x-axis at (p) 1 'A' and 'B'. Length of tangent from origin to the circle passing through 'A' and 'B' is equal to : () Point P(, ) lies in the exterior region of oth (q) 1 the paraolic curves y = x. If 'P' is integral point, then ' ' can e equal to : (c) From point P (9, 6), if two normals of slope m 1 and (r) m are drawn to paraola y = 4x, then m 1 m is equal to If two distinct chords through the point (a, a) of a (s) paraola y = 4ax are isected y the line x + y = 1, then the length of latus rectum can e equal to : (t). Let the tangents from P(, ) to the paraolic curve x x + y 15 = 0 e PA and PB, where the chord of contact is AB. Match the possile nature of triangle PAB (in column II) with the conditions on and (in column I). Column (I) Column (II) (a) If 1 ; 5, then PAB may e : (p) Right-angled triangle. () If R ; 4, then PAB may e : (q) Acute-angled triangle. (c) If 15 ; 4, then PAB may e : (r) Otuse-angled triangle. If 15 ; 4, then PAB may e : (s) Scalene triangle. [ 166 ] Mathematics for JEE-01

7 1. ().. (c) 4. () 5. (c) Ex 6. (c) 7. (a). (a) 9. (a) 10. (a) 11. () 1. (a) 1. (a) 14. (a) 15. (c) 16. () 17. (c) 1. () 19. (c) 0. (a) 1. (, c). (a, ). (a,, c) 4. (a, ) 5. (a, c, d) (a). (a) 9. (a) 0. () 1. (). (c). (c) 4. () 5. (c) Ex 6. (c) (a) 10. (c) 11. (c) ( ) 14. ( 5 ) 15. ( ) 16. ( 0 ) 17. ( ) 1. ( 4 ) 19. ( 9 ) 0. ( ) 1. (a) r. (a) r. (a) q () p () p, q, r, s, t () p, s (c) r (c) r (c) r, s q q, r, s q, s [ 167 ] Mathematics for JEE-01

Drill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3

Drill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3 Drill Exercise -. Find the distance between the pair of points, (a sin, b cos ) and ( a cos, b sin ).. Prove that the points (a, 4a) (a, 6a) and (a + 3 a, 5a) are the vertices of an equilateral triangle.