THIS FILE CONTAINS (COLLECTION # 1) Very Important Guessing Questions For IIT JEE 2010 With Detail Solution

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1 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: of 9 THIS FILE CONTAINS (COLLECTION # ) Ver Important Guessing Questions For IIT JEE 00 With Detail Solution Junior Students Can Keep It Safe For Future IIT JEEs Two Dimensional Geometr (D) The Point Straight Lines Circles Parabola Ellipse Hperbola Here Solutions Given Are Long (Genuine) Method But In 5 Das Class I Will Give SHORT TRICKS, Which Are Must For IIT / AIEEE Just To Save Your Time Inde For Collection # Question (Page to 9) Single Correct Answer Tpe Question Comprehension Tpe Quetions Assertion Reason Tpe Question More Correct Answers Tpe Questions Subjective (Up to Digits) Detiail Solution B Genuine Method (But In) Classroom I Will Give Short Tricks ) For Collection # Question (Page to 56) Same As Above Teko Classes,Bhopal Ph.: (0755)

2 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: of 9 Single Correct Tpe Que.. The shortest distance from the line + = 5 to the circle + = 6 8 is equal to (a) 7/5 (b) 9/5 (c) /5 (d) /5 (code-vtpaq7) Que.. The graph of ( ) against ( + ) is as shown ( ) Which one of the following shows the graph of against? (+) (a) (b) O (c) O (d) O Teko Classes,Bhopal Ph.: (0755) (code-vtpaq) Que.. If H represent the harmonic mean between the abscissae, and K that between the ordinates of the points, in which a circle + = c is cut b a chord l + m = δ, where l and m are the direction consines of the unit vector in the plane, then l H + mk has value equal to c c c c (a) δ (b) δ (c) δ (d) δ (code-vtpaq6) δ δ δ δ Que.. Area of the triangle formed b the line + = and the angle bisectors of the line pair + = 0 is (code-vtpaq5) (a) / (b) (c) / (d) Que. 5. The shaded area enclosed b f () = + a coordinate aes and the ordinate at = is 5 square units. If m and n are the -ais intercepts of the graph of = f () then the value of (m + n + a) equals (a) 0 (b) (c) 6 (d) 8 (code-vtpaq8) Que. 6. The vertices of a triangle ABC are p q q r r p A p, p, B q,q,c r, r. The area of the triangle ABC is (a) ( + )( + )( + ) (b) p q q r r p (c) ( + )( )( ) (d) ( + )( + )( ) p q q + r r + p (code-vtpaq) p q q r p r Que. 7. The least integral value of k for which ( k ) 8 k sin ( sin) cos ( cos) > + for all R, is (code-vtpaq) (a) 7 (b) 5 (c) (d) 5 t Que. 8. The equation = t + 9 and = + 6 represents a straight line where t is a parameter. The - intercept of the line is (code-vtpaq9) (a) / (b) 9 (c) 6 (d) λ λ Que. 9. If = and = + λ where λ is real parameter then + lies between [a,b] then + λ (a+b) is (code-vtpaq0) (a) 8 (b) 0 (c) (d) 5

3 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: of 9 Que. 0. Let C and C are circles difined b = 0 and = 0. The length of the shortest line segment PQ this is tangent to C at P and to C at Q is (code-vtpaq) (a) 5 (b) 8 (c) 0 (d) Que.. A variable line moves in such wa that the product of the perpendiculars form (a,0) and (0,0) is equal to k. The locus of the feet of the perpendicular from (0,0) upon the variable line is a circle, the square of whose radius is ( Given : a < k ) (code-vtpaq8) (a) a k + (b) a + k (c) a a + (d) k Que.. The tangent and normal at the etremities of the latus rectum of a parabola = form a quadrilateral whose area is (code-vtpaq) (a) (b) 8 (c) 8 (d) 6 + k Que.. If the lines λ + sin α + cos α = 0 + cos α + sin α = 0 pass thorugh the same point where sin α + cos α = 0 α R then λ lies in the interval (code-vt7paq) (b), (a) [,] (c) [,] (d) [, ] Que.. The range of values of m for which the line = m and the curve = enclose a region, is + (a) (,) (b) (0,) (c) [0,] (d) (, ) (code-vt7paq) Que. 5. A(,0) and B(0,) and two fied points on the circle + =. C is a varible point on this circle. As C moves, the locus of the orthocentre of the triangle ABC is (code-vt7paq5) (a) + + = 0 (b) + = 0 (c) + = (d) = 0 Que. 6. Mr. Shuag Karia lives at origin on the cartesian plain and has his office at (,5). His friend Mr. Vivek Jain lives at (,) on the same plane. Mr. Shuag Karia can go to his office travelling one block at a time either in the + or + direction. If all possible paths are equall likel then the probabilit that Mr. Shuag Karia passed his friends house is (a) / (b) 0/ (c) / (d) / (code-vt0paq5) Que. 7.If the left hand side of the equation + + secθ = 0 can be factorised into two linear factors then the value θ is (code-vtpaq) (a) π (b) 7 π 6 (c) π (d) 5 π 6 Que. 8. Let,, z, t be real numbers + = 9; z + t = and t z = 6 then the greatest value of P = z, is (code-vt5paq5) (a) (b) (c) (d) 6 Que. 9. If the two vertices of a trianlge are (7,) and (,6) and its centroid is (,6) then the corrdinate of the third verte are (a,b). The value of (a + b), is (code-vt7paq) (a) (b) (c) 5 (d) 6 Teko Classes,Bhopal Ph.: (0755)

4 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: of 9 Que. 0. Number of values of a for which the lines + = 0 are concurrent (code-vt7paq) a + = 0 + = 0 (a) 0 (b) (c) (d) infinite Que.. Number of straight lines parallel to the line = 0 and have intercept of length 0 between the coordinate aes (code-vt7paq) (a) (b) (c) (d) Infinite Que.. A circle has radius of log0 ( a ) and a circumference of 0 log b. The value of loga b is equal to (code-vt0paq) (a) π (b) π (c) π (d) π Que.. The points (,) lies on the line + = 6. The smallest value of the quantit +, is (a) 6 (b) 6 (c) (d) (code-vtpaq6) Que.. If (,7) is the highest point on the graph of = a + k, then k equals (a) (b) (c) (d) / (code-vtpaq) Que. 5. If the point P ( u, v) = is on the graph of = a + b + c,a 0, which of the following is also on the graph? (code-vtpaq9) b (a) u, v a b (b) u, v a b (c) + u, v a b (d) + u, v a Que. 6. Locus of all point P(,) satisfing + + = consists of union of (code-vtpaq9) (a) a line and an isolated point (b) a line pair and an isolated point (c) a line and a circle (d) a circle and an isolated point. Que. 7. The length of a line segament AB is 0 units. If the coordinates of one etremit are (, ) and the abscissa of the other etremit is 0 then the sum of all possible values of the ordinate of the other etremit is (code-vt5paq) (a) (b) (c) (d) 6 Que. 8. The value of k for which the points A ( k +, k ); B( k, k) and C( + k, k) are collinear is (a) 0 (b) (c) (d) (code-vt5paq) Que. 9. A point P(,) moves so that the sum of the distances from P to the coordinate aes is equal to the distance from P to the point A(,). The equation of the locus of P in the first quadrant is (a) ( + )( + ) = (b) ( + )( + ) = (c) ( )( ) = (d) ( )( ) = (code-vt5paq7) Que. 0. If, R satist the equation = 0, then the value of the epression + + is (code-vt5paq9) (a) + (b) + (c) (d) + Teko Classes,Bhopal Ph.: (0755)

5 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 5 of 9 Que.. The ordinate of a point P on the line 6 + = 9, which is closest to the point (, ) can be epresed in the form a/b. Where a,b N and are in lowest form, the value (a+b) equals (a) 86 (b) (c) 65 (d) 00 (code-vt7paq) Que.. Consider a circle + + a + b + c = 0 ling completel in first quadrant. If m and m are the maimum and minimum values of / for all ordered pairs (,) on the circumference of the circle then the value of ( m m ) (a) a b c c + is (code-vt7paq5) ab (b) b c ab (c) c b ab (d) b ac Que.. Let A(a,0) and B( b,0 ) be fied distinct points on the -ais, none of which coincides with the origin O(0,0), and let C be a point on the -ais. Let g be a line through the origin O(0,0) and perpendicular to the line AC. The locus of the point of intersection of the lines g and BC if C varies along the -ais, is (Provided c + ab 0 ) (code-vt7paq6) (a) + = (b) + = (c) + = (d) + = a b a b b a b a Que.. A rectangular billiard table has vertices at P(0,0), Q(0,7), R(0,7) and S (0,0). A small billiard ball starts at M(,) and moves in a straight line to the top of the table, bounces to the right side of the table, then comes to rest at N(7,). The -cordinate of the pont where it hits the right side, is (a).7 (b).8 (c).9 (d) (code-vt7paq7) Que. 5. A triangle formed b lines denoted b equation = 0 will alwas be (a) acute angled (b) abtuse angled (c) right angled (d) none (code-vt9paq) Que. 6. A point is selected at random inside an equilateral triangle. From this point perpendiculars are dropped to each side. The sum of these perpendiculars is (code-vt9paq) (a) half the sum of the sides of the triangle (b) equal to the altitude of the triangle. (c) least when the point is the centroid to the triangle (d) maimum when the point is centroid of the triangle Que. 7. One diagonal of a square is the portion of the variable line λ + ( λ ) = λ λ; λ > 0; λ which is intercepted between the aes. If the area of the square is 7 then the number ofvertices of Que. 8. the square whose both thecoordinates are integers, is (code-vt9paq5) (a) one (b) two (c) four (d) none Let Cn be a circle of radius n n centeredat the origin for n = 0,,,... and A n be the area of the region that is inside the circle C n and out side the circle C n + for n = 0,,... The value of the sum Anequals. (code-vt9paq6) n= 0 (a) π 5 (b) 9 π (c) 7 π 5 (d) 0 π Que. 9. A circle of radius has its centre at (,0) and another cricle of radius has its centre at (5,0). A line is tangent to the two circles at points in the first quadrant. The -intercept of the tangent line is (a) (b) (c) (d) (code-vt0paq) Teko Classes,Bhopal Ph.: (0755)

6 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 6 of 9 Que. 0. The smallest distance between the circle (a) (b) = and the line 5 + = 0, is (c) 5 (d) 5 (code-vt0paq) Que.. A conve quadrilateral is drawn such that each of its vertices (,) satist the equaation + = 7 and =. The are of the quadrilateral is (code-vt0paq) (a) 6 (b) 55 (c) 55 (d) 0 Que.. If the vertices of a A 5,0,B, and C ABC are 5, 5 then the coordinates of the orthocentre is (code-vt0paq6) (a) ( 8, + 5) (b) ( 8 5, 5) (c) ( 8 + 5, + 5 ) (d) ( 8 5, 5 ) Que.. A particle P moves from the point A(0,) to the point B( 0, ). The particle P can travel the upper half plane {, 0} at the speed of m/s and travel the lower half plane {, 0} at the speed of m/s. The coordinates of a point on the -ais, if the sum of the squares of the travel times of the upper and lower half planes is minimum, is (code-vt0paq8) (a) (,0) (b) (,0) (c) (,0) (d) (5,0) Comprehesion Tpe # Paragraph for Q. to Q. (code-vtpaq,,) Consider a variable line L which passes through the point of intersection P of the lines + = 0 and + 5 = 0, meeting the coordinate aes at the points A and B. Que.. Locus of the middle point of the segment AB has the equation (a) + = (b) + = (c) + = (d) + = Que.. Locus of the feet of the perpendicular from the origin on the variable line L has the equation (a) ( + ) = 0 (b) + = 0 (c) + = 0 (d) + = 0 Que.. Locus of the centroid of the varible triangle OAB has the equation (where O is the origin) (a) = 0 (b) + 6 = 0 (c) + 6 = 0 (d) = 0 # Paragraph for Q. to Q. 6 (code-vtpaq7,8,9) Let C be a circle of radius r with centre at O. Let P be a point outside C and D be a point on C.A. line through P intersects C at Q and R,S is the midpoint Of QR. Que.. For different choices of line through P, the curve on which S lies, is (a) a straight line (b) an arc of circle with P as centre (c) an arc of circle with PS as diameter (d) an arc of circle with OP as diameter Que. 5. Let P is situated at a distance d form centre O, then which of the folloiwng does not equal the product (PQ)(PR)? (a) d r (b) PT, where T is a point on C and PT is tangent to C (c) ( PS) ( QS)( RS) (d) ( PS) Que. 6. Let XYZ be an equilateral triangle inscribed in C. If α, β, γ denote thedistances of D from vertices X, Y, Z respectivel, the value of prodict ( β + γ α)( γ + α β)( α + β γ ), is (a) 0 (b) αβγ 8 (c) α + β + γ αβγ 6 (d) None of these Teko Classes,Bhopal Ph.: (0755)

7 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 7 of 9 # Paragraph for Q. 7 to Q. 9 (code-vt6paq,,) The base of an isoceles triangle is equal to, the base angle is equal to o 5. A straight line cuts the eternsion of the base at point M at the angle θ and bisects the lateral side of the triangle which is nearest to M. Que. 7. The area A of the quadrilateral which the straight line cuts off from given triangle is (a) + tan θ (b) + tan θ (c) + tan θ (d) + 5tan θ + tan θ + tan θ tan θ + tan θ Que. 8. The range of values of A for differnt values of θ, lie in the interval, (a), (b) (,5 ) (c), (d) (,) Que. 9. The length of portion straight line inside the triangle ma lie in the range : (a) (, ) (b), # Paragraph for Q. 0 to Q. (code-vt7paq,5,6) (c) (, ) (d) (, ) a+ b Let C be curve difined b = e. The curve C passing through the point P(,) and the slope of the tangent at P is ( ). Also C and C are the circles ( a) + ( b) = ( 6) + ( ) = 7 respectivel. Que. 0. The value of a + b is equal to (a) (b) 8 (c) 8 (d) Que.. The length of the shortest line segment AB which is tangent to C at A and to C at B is (a) 9 (b) 0 (c) (d) Que.. If f is a real valued derivable function satisfing f () f = f () with f '() =. Then the value of the a l is equal to integral f ()d( n ) b (a) 0 e e e e (b) (c) (d) # 5 Paragraph for Q. to Q. 5 (code-vt7paq7,8,9) Given the continuous function , = f () = a + b + c < < 0, a 0 If a line L touches the graph of = f () at three points then Que.. The gradient of the line L is equal to (a) (b) (c) (d) 6 Que.. The value of ( a + b + c) is equal (a) 5 (b) 5 (c) 6 (d) 7 Que. 5. If = f () is differentiable at = 0 then the value of b (a) is (b) is (c) is (d) ican not be determined Teko Classes,Bhopal Ph.: (0755)

8 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 8 of 9 # 6 Paragraph for Q. 6 to Q. 8 (code-vtpaq,,) Let ABCD is a square with sides of unit length. Points E and F are taken on sides AB and AD respectivel so that AE = AF. Let P be a point inside the square ABCD. Que. 6. The maimum possible area of quadrilateral CDFE is (a) 8 (b) (c) 5 8 (d) 8 Que. 7. The value of ( PA) ( PB) ( PC) ( PD) + is equal to (a) (b) (c) (d) 0 Que. 8. Let a line passing through point A divides the square ABCD in to two parts so that area of one portion is double the other, then the length of portion of line inside the square is (a) 0 (b) (c) (d) # 7 Paragraph for Q. 9 to Q. (code-vt8paq,5,6) Consider a triangle PQR coordinates of its vertices P( 8,5 );Q( 5, 9) and R (, 7 ). The bisector of the interior angle of P has the equation which can be written in the form a + + c = The distance between the orthocentre and the circumcentre of the triangle PQR is (a) (b) 5 (c) 0 (d) 0. Radius of the incircle of the triangle PQR is (a) (b) 5 (c) 6 (d) 8. The sum of the coefficients (a + c) (a) 9 (b) 78 (c) 89 (d) 99 # 8 Paragraph for Q. to Q. (code-vt8paq7,8,9) Consider the famil of lines passing through the intersection of the lines U ; = 0 and U : + = 0. A member of the famil which bisects the angle between them and is closer to the origin, is (a) 7 6 = 0 (b) = 0 (c) = 0 (d) = 0. A member of this famil with gradient minus has -intercept equal to (a) (b) (c) (d). A member of this famil whose slope is not difined is (a) + = 0 (b) = (c) = (d) + = 0 # 9 Paragraph for Q. 5 to Q. 7 (code-vt6paq,,) Consider non collinear point A (9,); B(7, ) and C(, ). Let P(a,b) be the centre and R is the radius of the circle S pasing through A,B,C. Also H (, ) are the coordinates of the orthocentre of the triangle ABC whose are be denoted b. 5. If D, E and F area the middle points of BC, CA and AB respectivel then the area of the triangle DEF is (a) (b) 6 (c) (d) 6. The value ( a + b + R) equals (a) (b) (c) (d) None 7. The ordered pair, is (a) (9,5) (b) ( 9,6) (c) (9, 6) (d) (9, 5) Teko Classes,Bhopal Ph.: (0755)

9 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 9 of 9 # 0 Paragraph for Q. 8 to Q. 0 (code-vt7paq,,) Consider a circle + = and a point P(,). θ denotes the angle enclosed b the tangents from P on the circle and A,B, are the points of contact of the tangents from P on the circle. Que. 8. The value of θ lies in the interval o (a) o o o o 0,5 (b) ( 5,0 ) (c) ( 0,5 ) (d) ( o o 5,60 ) Que. 9. The intercept made b a tangent on the - ais is (a) 9/ (b) 0/ (c) / (d) / Que. 0. Locus of the middle points of the portion of the tangent of the circle terminated b the coordinate aes is (a) + = (b) + = (c) + = (d) c = = # Paragraph for Q. to Q. (code-vt7paq8,9,0) Consider a famil of lines ( a + ) ( a + ) ( a + ) = 0 where a R Que.. The locus of the foot of the perpendicular from the origin on each member of this famil, is (a) ( ) + ( + ) = 5 (b) + + = 5 (c) ( + ) + ( ) = 5 (d) + = 5 Que.. A member of this famil with positive gradient making an angle of π / with the line =, is (a) 7 5 = 0 (b) + = 0 (c) + 7 = 5 (d) 5 = 0 Que.. Minimum area of the triangle which a member of this famil with negative gradient can make with the positive semi aes, is (a) 8 (b) 6 (c) (d) # Paragraph for Q. to Q. 6 (code-vt7paq,,) Consider circles S : + + = 0 S : + = 0 S : + + = 0. The radius of the circle which bisect the circumferences of the circles S = 0; S = 0; S = 0 is (a) (b) (c) (d ) 0 5. If the circle S = 0 is orthogonal to S = 0; S = 0 and S = 0 and has its centre at (a,b) and radius equals to r then the value of ( a + b + r) equals (a) 0 (b) (c) (d) 6. The radius of the circle touching S = 0 and S = 0 at (,0) and passing through (,) is (a) (b) (c) (d) # Paragraph for Q. 7 to Q. 9 (code-vt7paq,5,6) An altutude BD and a bisector BE are drawn in the trianlge ABC from the verte B. It is known that the length of side AC =, and the magnitudes of the angles BEC, ABD, ABE, BAC form an arithmetic progression. 7. The area of circle circumscribing ABC is (a) 8 π (b) π (c) π (d) π Teko Classes,Bhopal Ph.: (0755)

10 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 0 of 9 8. Let O be the circumcentre ABC, the radius of circle inscribed in BOC is (a) 8 (b) (c) (d) 9. Let B' be the image of point B with respect to side AC of ABC, then the length BB ' is equal to (a) (b) (c) Assertion & Reason Tpe In this section each que. contains STATEMENT- (Assertion) & STATEMENT-(Reason).Each question has choices (A), (B), (C) and (D), out of which onl one is correct. Bubble (A) STATEMENT- is true, STATEMENT- is True; STATEMENT- is a correct eplanation for STATEMENT-. Bubble (B) STATEMENT- is True, STATEMENT- is True; STATEMENT- is NOT a correct eplanation for STATEMENT-. Bubble (C) STATEMENT- is True, STATEMENT- is False. Bubble (D) STATEMENT- is False, STATEMENT- is True. Que.. Consider the following statements (code-vtpaq) Statement : The equation = 0 represents two real lines on the cartesian plane. because Statement : A general equation of degree two a + h + b + g + f + c = 0 denotes a line pair if abc + fgh af bg ch = 0 Que.. Consider the folloiwing statements (code-vtpaq) Statement : The area of the triangle formed b the points A(0, );(B(, ) and C(, ) is the same as the area of the triangle formed b the point P(0,0);Q(,) and R(,). because Statement : The area of the triangle is invariant w.r.t. the translation of the coordinate aes. Que.. Statement : The circle C : = 0 bisects the circumference of the circle C : = 0. because Statement : Centre of the circle C lies on the circumference of C. (code-vt6paq) Que.. Passing through a point A(6,8) a variable secant line L is drawn to the circle S: = 0. Form the point of intersection of L with S, a pair of tangent lines are drawn which intersect at P. (code-vt6paq6) Statement : Locus of the point P has the equation + 0 = 0. because Statement : Point A lies outside the circle. Que. 5. Statement : The equation bacause Statement : + + = does not represent a circle. (code-vt5paq5) + + = represents no real locus. (d) Teko Classes,Bhopal Ph.: (0755)

11 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: of 9 Que. 6. Consider the curves C : = a and C : = c (code-vt6paq) Statement : C and C are orthogonal curves. because Statement : C and C intersect at right angles everwhere wherever the intersect. Que. 7. Consider a general epression of degree two variables as f (, ) = Statement : f (, ) can be resolved into two linear factor over real coefficients. because (code-vt6paq) Statement : Discriminant of f (,) i.e. abc + fgh af ch = 0. Que. 8. Let triangle ABC be an acute triangle and O be its circumcentre. D, E and F are the foot of the perpendiculars dropped from O to BC, CA and AB respectivel. (code-vt6paq5) Statement :Area of ABC is four times the area of DEF because Statement : Ratio of the areas of two similiar triangle is the ratio of proportional sides. Que. 9. Statement : If the diagonals of the quadrilateral formed b the lines p + q + r = 0, p' + q ' + r = 0,p + q + r ' = 0,p' + q ' + r ' = 0 are at right angles, then p + q = p' + q '. because (code-vt8paq0) Statement : Diagonals of a rhombus are bisected and prependicular to each other. Que. 0. Statement : The joint equation of lines = and = is = i.e., + = 0 because (code-vt8paq) Statement : The joint equation of lines a + b = 0 and c + d = 0 is (a + b)(c + b) = 0 where a,b,c,d are constant. Que.. Given a ABC A, ;B, ;C,. Let there eists a point whose vertices are ( ) ( ) ( ) P(a,b) such that 6a = + + ; 6b = + + (code-vt6paq) Statement : Area of triangle PBC must be less than the area of ABC because Statement : P lies inside the triangle ABC Que.. Let points A, B, C are represented b a cos θ,a sin θ i =,, and cos( θ θ ) + cos( θ θ ) + cos ( θ θ ) =.. (code-vt8paq0) Statement : Orthocentre of ABC is at origin because Statement : ABC is equilateral triangle. Que.. Let C denotesa famil of circles with centre on -ais and touching the -ais at the origin. and C denotes a famil of circles with centre on -ais and touching the -ais at the origin. Statement : Ever member of C intersects an member of C at right anglesat the point other than origin. because (code-vt9paq7) Stastement :If two circles interesect at 90 o at one point of their intersection, then the must intersectat 90 o on the other point of intersection also. Teko Classes,Bhopal Ph.: (0755) i i

12 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: of 9 Que.. Consider the lines l : + = 0; l : + = 0; l : 5 + = 0 Statement : The lines l, l andl are concurrent. (code-vt9paq8) because Statement : The area of the trianlge formed b the points,,, and (5, ) vanishes. Que. 5. Consider the lines L : + = and L :5 = 7 (code-vt0paq) Statement : Ever point on the line 6 8 = 6 is equidistant from L and L. because Statement : Is the bisector of the angle between L and L.which contains the origin in its region. Que. 6. Consider the line L : = + + = 0 and the points A( 5, 6) and B(, ) (code-vt0paq) Statement : There is eactl one point on the line L which is equidistant form the point A and B. because Statement : The point A and B are on different sides of the line. More than One Correct Tpe Que.. Three distinct lines are drawn in a plane. Suppose there eist eactl n circles in the plane tangent to all the three lines, then the possible values of n is/are (a) 0 (b) (c) (d) (code-vtpaq) Que.. Consider the points O, ( 0,0 ), A( 0, ) and B, in the - plane. Suppose that point C(,) and D(,) are chosen such that 0 < < and such that O,C and D are collinear. Let sum of the area of triangles OAC and BCD be denoted b S then which of the following is/are correct? (a) Minimum value of S is irrational ling in (/, /) (b) Minimum value of S irrational in (/, ) (c) The vlaue of for minimum value of S lies in (/, ) (d) The value of for minimum value of S lies in (/, /) (code-vtpaq) Que.. If θ is the angle between the pair of tangents drawn from (c, 0) to the corcle + = then which of the following conclusion(s) is/are true? 5 π (a) If θ, π c (, 6 + ) 6 π (c) If θ, π c (, ) π (b) If θ, π c (, 5 ) 5 π (d) If θ, π c (, ) (code-vtpaq) Que.. If θ is eliminated from the equation a secθ tan θ = and bsec0 + tan θ = (a and b are constant) then the eliminant denotes the equation of (code-vt5paq) (a) The director circle of the hperbola a = (b) anuiliar circle of the ellipse b a + = b (c) Director circle of the ellipse a + = (d) Director circle of the circle b a + b + =. Teko Classes,Bhopal Ph.: (0755)

13 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: of 9 Que. 5. If the vertices P,Q,R of a triangle PQR are rational points, which of the following points of the triangle PQR is/are alwas rational point(s) (code-vt7paq5) (a) centriod (b) incentre (d) circumcentre (d) orthocentre Que. 6. The origin, the intersection of the lines = 0 and the points in which these lines are cut b the line 5 =, are the vertices of a (code-vt8paq) (a) parallelogram (b) rectangle (d) rhombus (d) square Que. 7. The equations to the lines through the point of intersection of + = 0 and 5 = 0 which are at a distance 5 from the origin, is/are (code-vt7paq) (a) + = 5 (b) = 5 (c) = 5 (d) + = 5 Que. 8. A circle centred at O has radius and contains the points the A. Segment AB is tangent to the circle at A and AOB = θ. If point C lies on OA and BC bisects the angle ABO then OC equals θ θ θ (b) cos θ sin θ (a) sec ( sec tan ) (c) + sin θ sin θ (d) cos θ (code-vtpaq) Que. 9. Let a, b,c Q + satisfing a > b > c. Which of the following statement(s) hold true for the quadratic polnomial f () = a + b c + b + c a + c + a b? (code-vtpaq6) (a) The mouth of the parabola = f() opens upwards. (b) Both roots of the equation f() = 0 are rational. (c) -coordinate of verte of the graph is positive. (d) Product of the roots is alwas negative. Que. 0. If ( sin α) + b, for all real values of and α 0, π π, π, then possible real values of b is/are (code-vtpaq0) (a) (b) (c) (d) 5 Que.. If log ( 7 a a ) = is difined R, then possible integral value(s) of a is/are (a) (b) (c) (d) 5 (code-vtpaq) Que.. If the equation a + c + b = d represents two real and distinct straight lines then the necessar and suffcient conditions can be (code-vt7paq) (a) d is zero and c > ab (b) c = ab and d R {0} (c) c = ab and d R (d) d = 0 and c = ab Que.. If a + c = b ac then the variable line a + b + c = 0 alwas passes through one or the other of the two fied ponts. The coordinates of the fied point can be (code-vt7paq) (a) (, ) (b) (, ) (c) (,) (d) (,) Que.. If the lines (code-vt7paq) u : λ = 0 u : λ + = 0 u : + λ = 0 Passes through the same point then the value(s) of λ equals (code-vt7paq5) (a) (b) (c) (d) Teko Classes,Bhopal Ph.: (0755)

14 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: of 9 Que. 5. Let L : + = & L :5 + = 0 be two given lines. Let image of ever point on L with reespect to a line L lies on L, then possible equations of L can be (code-vt7paq6) (a) + = 0 (b) 6 8 = 0 (c) = 0 (d) 5 5 = 7 Que. 6. Let A(,) and B(,) be two fied points and P be a variable point such that area of PAB remains constant equal to for all positions of P, then locus of P is given b (code-vt7paq7) (a) = + (b) = (c) = + (d) = Match Matri Tpe Que.. Column - I Column - II (code-vt9pbq) A. The lines = ; 6 + = 0 and +6 9=0 P. a cclic quadraliteral constitute a figure which is B. The points A ( a,0 ),B( 0,b ),C( c,0 ) and D( 0, d) Q. a rhombus are such that ac = bd and a,b,c,d are all non-zero. The points A,B,C and D alwas constitute C. The figure formed b the four lines R. a square a ± b ± c = 0 ( a b ), is D. The line pairs 8 + = 0and + 5 = 0 S. a trapezium constitute a figure which is Que.. Column - I Column - II (code-vt8pbq) A. Four lines + 0 = 0, + 0 = 0 P. a quadrilateral which is neither + 5 = 0 and 5 = 0 form a a parallelogram nor a trapezium nor figure which is a kite B. The point A(,), B(, ), C(, 5) and Q. a parallelogram D(,) in order are the vertices of C. The lines 7 + = 0, = 0 R. a rectangle of area 0 sq. units 7 0 = 0 and 7 + = 0 form a figure which is D. Four lines 7 = 0, + 7 = 0, S. a square = 0, + = 0 form a figure which is Que.. Column - I Column - II (code-vt7pbq) A. The four lines + = 0; 9 = 0; P. a quadrilateral which is neither a + + = 0 and + 7 = 0 enclose a parallelogram nor a trapezium nor figure which is a kite. B. The lines + =, + =, + = and Q. a parallelogram which is neither a + = from a figure which is rectangle nor a rhombus C. If O is theorigin, P is the intersection of the R. a rhombus which is not a square. lines = 0, A and B are the points in which these lines are cut b the line + 5 = 0, then the points O,A,P,B (in some order) are the vertices of S. a square. Teko Classes,Bhopal Ph.: (0755)

15 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 5 of 9 Que.. Set of famil of lines are discribed in column - I and their mathematical equation are given in column - II. Match the entr of column - I with suitable entr of column - II. (m and n are parameters). Column - I Column - II (code-vt0pbq) A. having gradient P. m + m = 0 B. having intercept three times the -intercept Q. m + m = 0 C. having -intercept ( ) R. + = a D. concurrent at (,) S. + a = 0 Subjective Tpe ( Up to digit) Que.. A rhombus ABCD has sides of length 0. A circle with centre A passes through C (the opposite verte) likewise, a circle with centre B passes through D.If the two circles are tangent to each other, find the area of the rhombus. (code-vtpdq) Que.. The circles, which cut the famil of circles passing through the fied points A (,) and B (,), and,, which ma be = orthogonall, pass through two fied points real or imaginar. Find the value of ( ) (code-vt8pdq) Que.. Consider lines (code-vt7pdq) L : 5 + = 0 L : + 5 = 0 L : = 0 If these lines enclose a triangle ABC and sum of the square of the tangent ofthe interior angles can be epressed in form p/q where p and q are relativel prime numbers, compute the value of(p + q). Que.. If the epression f (, ) = k can be resolved into two linear factors then find the value of k. (code-vtpdq) Que. 5. Point P is / of the wa from the point F( 5,) to G(,5). Line L is perpendicular to the line FG and passes through the point P. If the equation of the line L is a + b = c, where a, b and c are relativel prime integer and a > 0 then find the vlaue of ( 8a + 9b + 0c ). (code-vt7pdq) Que. 6. Consider the circle whose centre is in the first quadrant and which is tangent to both the coordinate aes and the line L, whose equation is + = 0. If the co-ordinates of the point of tangenc of the circle with the line L are ( p,q ) and ( ) of the centres of the two corcles. Find ( p p q q a b c d) p,q and (a,b) and (c,d) are the coordinates (code-vt7pdq) Que. 7. If the equation of the diagonals of the parallelogram formed b the lines + 7 = 0; 5 = 0; + 5 = 0 and + + = 0 are a + b 5 = 0 and p q = Find the value of a + b + p + q. (code-vt7pdq) Teko Classes,Bhopal Ph.: (0755)

16 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 6 of 9 [SOLUTION] Single Correct Tpe Que.. A. Centre : (, ) and r = 5 perpendicular distance from (, ) on + 5 = 0 is p = = d = 5 = = > = + = + + Que.. C. k ( k ); k ( ) ( k) ( k) + k = k where + k <. k Que.. A. Solving line and circle m + δ = m c l ( l ) l ( l + m = ) given ( m c ) δ δ m c H = = lh = + lδ δ m + δ + δ m c = 0... () mk = δ c l l... () δ () + () A ( ) δ c ( l + m ) c l H + mk = = δ where ( + m = ) δ δ l. Que.. A. l + m = δ B ( ). + = 0 = = 0 Area = =. (0,) + = 0 (0,) (,) (,0) + = + = Que. 5. D. Que. 6. D. + a + d = 5 gives a = Hence f () = + = ( + )(6 ) hence 0 m = and n = 6 m + n + a = 6 + = 8. p p p q p + q 0 p q 0 D = q q = q r q + r 0 = p + q q + r q r 0 r r r r r r = ( p + q)( q + r) ( p q) ( q r) ( p q)( q r)( p r ). + + = + + Teko Classes,Bhopal Ph.: (0755)

17 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 7 of 9 sin sin sin sin cos cos = cos cos π = π Que. 7. D. = ( π ) = π k k + > 0 If k = then.8 + > 0 (not possible) then of k then k > 0 and 6 ( k )( k ) 0 + < 6 < k + k 8 k k 0 k + 6 k > 0 k = 5. + > 6 0 Que. 8. A. ( ) 6 9 = ; put = 0, ( 6) = 7 = /. Que. 9. A. λ = tan θ = sin θ and = cos θ E = + = sin θ cos θ = sin θ E [, 6] a + b = 8. Que. 0. C. Centres are (0,0) and ( 5,0) r = 6; r = 9 d = 5 r + r < d Circles are separted PQ = l = d r + r = 65 5 = 0 C A (0,0) r P l l r Q d r C A ( 5,0) Que.. A. Let the equation of the variable line is ( ) + + = 0 + a + pp = = k + + i.e., + a = k locus + a ± k = 0 O (, ) (a,0) ] a r = ( ± k ) +ve sign r a = k (not possible as r becomes ve) ve sign a r = k. (, ) p Varible line m = - p Que.. B. (0,0) (0,0) Teko Classes,Bhopal Ph.: (0755)

18 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 8 of 9 λ sin α cos α Que.. B. D = cos α sin α = 0 = λ cos α + sin α sin α[ cos α sin α ] + cos α[ sin α cos α ] = 0 sin α cos α [ ] D sin.cos sin cos.sin cos sin cos = λ α α + α α α α = λ α + α λ = sin α + cos α λ,. Que.. B. Solving m = + = or = 0 = > 0 for a region m < 0 + m m m m ( 0,) Note : form = 0 or the line does not enclose a region. Que. 5. A. Let C( cos,sin );H( h,k) θ θ is the orthocentre of the ABC Circumcentre (h,k) G (0,0) cos + sin +, h = + cos θ k = + sin θ + = θ θ + + = 0 Que.6. B. 9! n(s) = = 6!.5! n(a) = 0 to F and F to P 5!! =. = 0.6 = 60!.!!.! 60 0 P(A) = = P(,5) Que. 7. (C) a = ; b = ; c = secθ h = 0; g = ; f = using abc + fgh af bg ch = 0 9 π secθ + = 0 secθ = secθ = θ = Que. 8. (B) Let = cos θ ; = sin θ z = cos φ ; t = sin φ o 6 cos θ.sin φ 6sin θ cos φ = 6 6sin ( θ φ ) = θ = 90 + θ φ θ = 90 = cos θ ; = sin θ z = sin θ ; t = sin θ p = z = 6sin θ cosθ = sin θ pma =. Teko Classes,Bhopal Ph.: (0755)

19 Que. 9. (D) IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 9 of 9 = 8 5 = ( 5)( ) < 0 ( 5)( ) ( 5)( ) > 0 > 5 or <. Que. 0. (D) all values of a. Que.. (D) Slope of the given line is / let one line is + = slope = b b = a b a a b = a...() also given a + b = 00...() () and () b = ± 0 b = 0;a = 0 b = 0;a = 0 None a and b must be of same sign (b) 0 Que.. (C) C = log0 b = π r log b log0 b = π. log0 a (as r = log0 a) loga b. log a = π = π Que.. A. Let = r cos θ ; = r sin θ r cos θ + r sin θ = 6 i.e. r 6 6 r min = =. + min for r to be minimum cos sin min Que.. C. a + k; abscissa corresponding to the verte is now ( ) = 7 7 = k k =. Que. 5. B. Verif each alternative. Que. 6. A. + + ()()( ) = r = ; cos θ + r sin θ to find θ + θ must be maimum i.e., b i.e. a = ( ) 0 + = or = = a line + = or a point (, ) (A). Que. 7. D. a = a = + + = + = + = = + A B 6 k 00 k 6 k 6 or 6 k or 9. 0 (, ) (0, k) Que. 8. C. k + k k 0 D = k k = 0; k 0 = 0 k + k k + k [ 6k + + k] = 0; + k = ; l = +. Que. 9. B. + = + = + + = = ( + )( + ) =. Que. 0. D. f (z, ) = ( ) + ( ) = 0 = and = + ( + ) E = = =. Que.. D. Slope of 8 6h AP = h + = 9 6 (5 / 7) = 6/ 7 a + b = 00. hence 8 6h ( 6) = h + h = 5 / 7 hence coordinate Teko Classes,Bhopal Ph.: (0755)

20 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 0 of 9 Que.. C. Substituting = m in the equation of circle we get + m + a + bm + c = 0 + m + a + bm + c = 0 D = 0 a + bm c + m = 0 a + b m + abm c cm = 0 m b c + abm + a c = 0 Que.. C. Equation of the line g is a hence k = h... () c Now equation of BC + = (h,k) lies on it b c / denotes the slope of the tangent from the origin on the circle m h k + =... () b c (0,0) ab ab m m. + = = m b c c b a = as (h,k) lies on it, c (0,c)C ah h k Substituting c = in () form () + = locus of P is k hb ab b Que.. A. a 7 a tan θ = = 0 b also 7 tan θ = = b b a 7 a hence = = form st two relations b 0 b 9 = ab b a = a + 6 = ab b... () form last two 0a ab 0 + b = a a ab + b = or ab b = a... () + =. a = m = m hence from () and () a + 6 = a 0a = 7 a =.7. Que. 5. D. Homogeneous equation of degree lines through lines concurrent. A Que. 6. B. a = a ( p + p + p ) = a p + p + p = = h h p p p o Altitude a length of altitude sin 60 = Length of altitude = p a B Q C Que. 7. A. λ + λ = 7 Area of square is 7 d = λ + λ = λ λ + = 7 λ λ 5 = λ = or λ = But (0,7)Q m = c/a A (a,0) m = a/c λ λ + λ = B (b,0) λ 5 λ + = 0 5 λ > 0 λ = vertics are 5,0 and 0, Remaining vertices (0,0)P (,) g P(h,k) (b,7) 0 b R(0,7) θ 7 a θ o 90 θ (7,) θ θ (0,a) a S(00) m = 5 m = 5 are AB CD AB = = BM = using parametric coordinates Teko Classes,Bhopal Ph.: (0755)

21 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: of of B and D are. ;. ± ± D, ( ) B (, ) ( + ) lim A = A + A + A + A A Now n = Que. 8. B. n ( 0 n ) n 0 A0 = π r0 r = π A = π r r = π A = π r r 5 = π and so on 6 π 9π n = π + + = = Hence A ( / ) Que. 9. B. 5 = a = 8 a equation of line : = m( 8) 6m = 9m = + m + m m = ± (reject +ve sign as slope is ve) 8 m = intercept = 8 =. 8 Que. 0. B. The distance form a point to a line A 0 + B0 + C A + B + c = 0 is A + B The centre of the circle is (5, ), so the distance from this point to the line5 + = 0 is ( ) = = Teko Classes,Bhopal Ph.: (0755)

22 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: of 9 Que.. D. + = 7; = + = = + = ± and = ± 5 ( ), 8,, 8,,],8,, 8 Area = 0. Que.. C. Clearl, coordinates of circumcentre (0) is (0,0) OA = OB = OC and centroid is , since centriod divides the line segment joining orthocentre and circum centre in the ratio :, hence coordinates of orthocentre are ( 8 + 5, + 5). Que.. B. Let the point on the -ais is (c,0) Sum of the squares of travel times is c 6 ( 0 c) T = c 0c 5 = c = c 5c = ( c c + 6 ) T is minimum is c =. Comprehesion Tpe # Paragraph for Q. to Q.. - A.. - B. - C. Point of intersection the line + = = 0 is = and = / (i). Equation of AB is B P(,/) (0,k) (h,k) + = + = k + h = kh + = 0. h k h k (ii). k k (/ ) k k. =. = h h h h h(h ) + k(k ) = 0 + = ( ) 0. B P(,/) (h,k) O A O (a,0)a Teko Classes,Bhopal Ph.: (0755)

23 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: of 9 (iii). Here, a = h, and b = k equation of AB is B P(,/) (a,b) + = + = + 6 = 0. h h h k G(h,k) (i). # Paragraph for Q. to Q. 6 O (a,0)a. - D D. 6 - A. Locus of S is a part of circle with OP as diameter passing inside the circle C C N Q O R S(h,k) M P (ii). P R PQ = PT = PN PM = d r d + r = d r ( PS SR )( PS SQ) PS SQ ( SQ SR ) PS ( SQ)( SR ) ( PQ)( PR ) ( PS) = + = = (iii). Using Ptolem s theorem ( YD)( XZ) = ( XY)( ZD) + ( YZ)( XD) = XZ( ZD + XD) { ( XY = YZ = ZX) } Q β = γ + α (A). Alternativel : γ + YD XYD ; = γ YD α + ( YD) ( XY) α = YD ( YZ) form YZD form α YD = α YD XY...(); γ YD = γ + YD YZ...() () () α γ β = α γ β = α + γ 7. - D D. 9 - C. # Paragraph for Q. 7 to Q. 9 Equation of line PM : = tan θ( ) Intersection point Q of AC and MP. = tan θ( ) + tan θ + tan θ Q, + tan + tan θ Area of 5( + tan θ) = + tan θ + tan θ + tan θ + tan θ + tan θ APQ = modulus of θ M Y β o 0 A(,) = o 90 Q (,) P o 5 B (0,0) X o 5 α o 60 o 0 D o 60 γ Z = C(,0) Teko Classes,Bhopal Ph.: (0755)

24 (i). (ii). IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: of 9 tan 5tan 5 tan Area of quadrilateral BPQC Area, θ + θ + θ A = = = + tan θ + tan θ + tan θ A 5 tan π 0, tan, A,. = note that, θ + θ + θ (iii). + tan θ + tan θ PQ = + = = + tan θ + tan θ θ + θ + sin θ sin θ 0, PQ, PQ,. ( cos sin ) # Paragraph for Q. 0 to Q A.. - C. - B. d a+ b a+ b (i). = e, passes through (,) = e a + b = 0 also d = = (a,b) = (, ) a + b =. a+ b e.b() = b and a (,) = a+ b e.b = C : C & C are separated + = (ii). Hence + + = C : ( 6) ( ) ( ) C A B C r = l (, ) (6,) r = r AB = l = d r + r = 69 = AB =. (iii). Again f ( h) + h f f () f () + f ( + h) f () f () f '() = lim ; f () = = lim h 0 h h 0 h h = f '().f () f () as f () f () but f () 0 f () = = = f '() f () = l n(f ()) = l n + C =, f () C 0 f () a e e e e I = f ()d ln = d l n = d = =. b /e /e # 5 Paragraph for Q. to Q. 5 = = =. - C.. - D. 5 - B , f () = a + b + c < < 0, a 0 For continuous at = 0 c = 0 Continuous at = = a b 8 = a b a b =... () Teko Classes,Bhopal Ph.: (0755)

25 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 5 of 9 Now let the line = m + p is tangent to all the cuves solving = m + p and = + + = m + p ( m) p = 0 D = 0 ( m) p = 0... () again solving = m + p and = = m + p + (0 m) + 8 p (0 m) (8 p) = 0 (0 m) + p = 0 (0 m) ( m) = (00 0m) ( m) = (m cancels out) 96 6m = 6 = 6m m = and p = hence equation of line tangent to st and last curves is =... () now solving this with = a + b (as c = 0) a + b = a + (b ) + = 0 D = 0 (b ) = a Also b = a + (form ) a = a a 0 a = and b = 6 f '(0 + ) = lim a + b = b; C D. 8 - B. (i). Area of CDFE A = ( ) f '(0 ) = lim = + = b =. # 6 Paragraph for Q. 6 to Q = = A ma + 5 = at = 8 F(0,) D(0,) C(,) A (0,0) E(,0) B(,0) (ii). PA PB + PC PD = α + γ α + δ + δ +β γ + β = 0 β D(0,) C(,) α A (0,0) γ δ B(,0) = = AQ = + = () L. (iii). D(0,) C(,) Q(,) A (0,0) B(,0) Teko Classes,Bhopal Ph.: (0755)

26 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 6 of 9 # 7 Paragraph for Q. 9 to Q A B. - C. Triangle is right R = orthocentre ; M = circumcentre. P ( 8,5) (i) RM = + = 7 M 5 5 D Q( 5, 9 0 R(, 7) (ii) () Incentre = = = = = = = = 6 hence incentre ( 6, 6) (can be used to determine the equation of PD) r = = = 5. s.0 (iii) Coordinates of D using section formulea are 5, and mpr = equation PD is = 0 a + c = 89. # 8 Paragraph for Q. to Q.. - A.. - B. - D. Interection point is (, ). Now proceed. # 9 Paragraph for Q. 5 to Q D B. 7 - D. (i) (ii) 9 Area of DEF = area( ABC) = 7 =. Since P(a,b) is the circumcentre a 7 + b + = a 9 + b...() ( ) + ( + ) = ( ) + ( + ) a 7 b a b...() Solving () and () a = and b = R = = 5 a + b + c = =. (iii) H G C, (7/,/) (,) ( ) 7 = = 7 8 = 9 = = 5 point H 9, 5. Teko Classes,Bhopal Ph.: (0755)

27 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 7 of 9 # 0 Paragraph for Q. 8 to Q D B. 0 - A. Tangent = m( ) m + m = 0 m p = = + m m = + m m m = 0 m = 0 or m = Y A 5 O B α θ P(,) X Hence equation of tangent is = and (with infinite intercept on - ais) = 6 = 6 0 = 0 - intercept or Variable line with mid point (h,k) + =, it touches the circle h k + = 0 = Ans. (ii) (B). = + = locus is + = Ans. (iii) (A) h k + h k. - D.. - A. - C. # Paragraph for Q. to Q. Given ( a + ) ( a + ) ( a + ) = 0 + a = 0 famil of lines passes through the fied point P which is the intersection of = and = Solving P(,), now (i) k k. = h h locus is ( ) + ( ) = 0 + = ( ) = ( ) + ( ) = 5 (,) (h,k) O (ii) We have m( )...() this makes an angle of π / with = with slope / m / m = ; = ± m = + m + m / + m (with + ve sign) m = 7 = = = hence the line is 7 with ve sign m m 7m m ( rejected) = 7( ) 7 5 = 0. Teko Classes,Bhopal Ph.: (0755)

28 (iii) Again = m( ) IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 8 of 9 0; m; 0, m m = = = = A = ( m) ( m < 0) A = m + = + m m m let m = M ( M > 0) A = + M + M = M + M = 8 + M area is minimum if M = m = A M min = 8 A min =. # Paragraph for Q. to Q C D. 6 - C. (i) r = a + b + = a + + b + a + = 0 a = r 9 r. and a + + b + = a + b + + a = b b = = = Ans. (ii) S S 0 S S = = = Radical centre (,) equation of circle is = radius LT = S = + = radius = and a = ; b = a + b + r =. (iii) famil of circles touches the line = 0 at, 0 is ( ) + ( 0) + λ ( ) = 0 passing through (, ) + + λ = 0 λ + + = = radius 9 5. Teko Classes,Bhopal Ph.: (0755)

29 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 9 of 9 # Paragraph for Q. 7 to Q B B. 9 - D. Angles BEC, ABD, ABE and BAC are in A.P. let BEC = α β ABD = α β ABE = α + β and BAC = α + β Now, α β = α + β + α + β [using eterior angle theorem] π 7π α 7 β β =, α = and From ABD π π α β + α + β = β + β = π π α + β = B = ( α + β ) =, π π = = 6 o o o A, C ABC si triangle (i) Area of circle circumscribing π ABC = π = (ii) BOC is equilateral r = = = s (iii) π π BD = OB sin = sin = BB' = BD =. Assertion & Reason Tpe Que.. D. The given equation is ( ) + ( ) = 0 hence it denotes onl a point P imaginar lines through P(,) as = 0. Que.. A. Let = 0 and = now, 0 X; Y (, ) (,). = = if, or two 0, (0,0) (, ) (, ) Que.. B. C : centre (,); C : centre (,) radical ais of C and C is C C = 0 + = = 0...() since () passes through the centre of C (,) hence S- is correct. C C also (,) lies on C hence S- is correct but that is not becorrect eplanation S -. Teko Classes,Bhopal Ph.: (0755)

30 IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: 0 of 9 Que.. D. Locus of P is polar of A (6,8) w.r.t. the circle S = 0 i.e., = g + + f + + c = ( + 6) ( + 8) + 5 = = = 0 S - is incorrect. Also the point A (6,8) lies outside S. Hence S- is false and S- is true. Que. 5. D. Que. 6. A. d d C : = 0 = = m d d d d C : + = 0 = = m = are orthogonal. d d m.m C and C Que. 7. (D) f (, ) = ( ) + ( + ) S - is flase, It represent a point (,). Que. 8. (A) D,E,F are the middle points of the sides of the triangle. D = A E = B similiar triangles F = C Que. 9. (A.). The quadrilateral is obviouls a parallelogram and if the diagonals are at right angles, is must be a rhombus. Hence, the distance between the pairs of opposite sides must be the same i.e. r r ' r r ' = p + q = p ' + q ' ( p + q ) ( p ' + q ' ) Que. 0. (D). The joint equation of = and Que.. A P, 6 6 = is + = 0 i.e. = 0. = hence P lies inside the triangle C (, ) B F A O D E C P (a,b) A D (, ) (, ) + +, B area of PBC < area of ABC Que.. A. ( cos cos cos cos cos cos cos cos cos ) ( sin θ + sin θ + sin θ + sinθ sinθ + sinθ sinθ + sinθ sinθ ) = 0 θ θ + θ θ + θ θ + θ + θ + θ + cos θ + cos θ + cos θ + sin θ + sin θ + sin θ = 0 cos θ + cos θ + cos θ = 0 & sin θ + sin θ + sin θ = 0 centroid and circumcentre of ABC is at origin ABC is equiliateral Orthocentre of ABC is also origin. Teko Classes,Bhopal Ph.: (0755)

31 Que.. A. IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: of 9 Que.. A Concurrenc of lines / / / / 5 S- is correct. + + = ( 0,0) Que. 5. C. L : + = 0, L : containing bisector is = = = 0 5 None (0,0) containing bisector is = 5 Que. 6. B. The points A and B ma be on same side also More than One Correct Tpe Que.. A,C,D. Case - I : If lines form a triangle then n = i.e., ecircles and incircle O Case - II : If lines are concurrent or all parallel then n = 0 Case - III : If two are parallel and third cuts then n = hence A,C,D. Que.. A,C. S = Area of OAC + area of BCD ( )( ) S =...() Now 's CBD and OCA are similar ( )( ). = + 0 < < Teko Classes,Bhopal Ph.: (0755)

32 = = + = IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: of 9 / S = = = = = + = + A is minimum if ( /,/ ) A & C. = i.e, 5π θ 5 π Que.. B,C,D. (A). If θ = = ; now, 6 c = 6 o sin 75 = = 6 + = ( + ) (0,)A (0,0)O C (,) D(,) B(,) = which lies in (/, ) and Amin = which lies in θ sin = ; c (A) is not correct. θ π (B). = ;c = = = = 5 B is correct. o o 0 sin 5 cos (C). θ π =,c = = (C) is correct. sin / ( π ) θ π (D). =,c = = o (D) is correct. 6 sin 0 Que.. C,D. a sec θ = + tan θ bsec θ = tan θ b squaring and adding Que. 5. A,C,D. ( a b ) sec ( tan ) ( tan ) + θ = + θ + + θ + = a + b (C) and (D). Que.6. A. ( )( + + ) = 0 hence the lines are = 0 5 P, 0 equation of the two lines lines joining origin and the point of intersection of 5 = and f (, ) = + + = is ( )( ) ( ) = = + 5 = 0 O Y π θ/ c = X (c,0) Teko Classes,Bhopal Ph.: (0755)

33 Que. 7. C,D. IIT JEE/AIEEE MATHS b SHUAAG SIR Bhopal, Ph. (0755) Question on D Collection # Page: of 9 Que. 8. A,C,D. Using propert of anlge bisector sec θ = tan θ secθ = sec θ + tan θ = + sin θ Que. 9. A,B,C. tan θ = sec θ sec θ f () = A + B + C A= a + b c = a c + b c > 0 A> 0 mouth opens upwards now = is obvious solution terefore both roots are rational. b a + c a < 0 B < 0; ve ve 0 c b a verte B = > 0 hence abscissa a of the verte > 0 A O (D) need not be correct as with a = 5,b =,c =, P < 0 and a = 6,b =,c =,P > 0 (A), (B) and (C) are correct. Que. 0. C,D. Abscissa coresponding to the verte is given b = > is the verte sin α the graph of f () = ( sin α) + b as shown minimum of f () = (sin α) + b must be greater then zero but minimum is at = i.e., sin α + b 0; b sin α, α (0, π);b as sin α > 0 in (0, π) O Hence (C) and (D) are correct. verte =cosec Alternativel : f () (sin ) b [ f () 0 ] Case - I : () D 0, () F() 0 D 0 sin α(b ) cos ecα + b b cosec α + b...() = α + now verte > O and f () 0 sin α + b 0 b sin α b...() Teko Classes,Bhopal Ph.: (0755)