JEE (MAIN / ADVANCE ) MATHS by SUHAG KARIYA ... of the triangle formed by the line x + y = 3 and the angle bisectors of the line pair x y 4y 4 02 2 is (code-V2T13PAQ15

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It is 2D Sheet 1, Overall Sheet 11, Single Correct TypeQue. 1. The shortest distance from the line 3x 4y 25 to the circle 2 2x y 6x 8y is equal to

(a) 7/5 (b) 9/5 (c) 11/5 (d) 32/5 (code-V2T1PAQ7)

Que. 2. The graph of y x against (y + x) is as shown

Which one of the following shows the graph of y against x ?

(y+x)

(y x)

(a) x

y

(b) x

y

O(c)

x

y

O(d)

x

y

O (code-V2T3PAQ3)

Que. 3. If H represent the harmonic mean between the abscissae, and K that between the ordinates of thepoints, in which a circle 2 2 2x y c is cut by a chord x my , where and m are the directionconsines of the unit vector in the xy plane, then H mK has value equal to

(a) 2c2

(b) 2c

2

(c)

22c

(d)

2c22

(code-V2T12PAQ6)

Que. 4. Area of the triangle formed by the line x + y = 3 and the angle bisectors of the line pair2 2x y 4y 4 0 is (code-V2T13PAQ15)

(a) 1/2 (b) 1 (c) 3/2 (d)2Que. 5. The shaded area enclosed by 2f (x) 12 ax x coordinate axes

and the ordinate at x 3 is 45 square units. If m and n are the x-axisintercepts of the graph of y f (x) then the value of (m n a) equals (a) 0 (b) 4 (c) 6 (d) 8 (code-V2T13PAQ18)

Que. 6. The vertices of a triangle ABC are 2 2 2A p , p , B q ,q ,C r , r . The area of the triangle ABC is

(a) 1 p q q r r p2

(b) 1 p q q r r p2

(code-V2T14PAQ2)

(c) 1 p q q r r p2

(d) 1 p q q r p r2

Que. 7. The least integral value of k for which 2 1 1k 2 x 8x k 4 sin sin12 cos cos12 for all

x R, is (code-V2T14PAQ3)

(a) – 7 (b) – 5 (c) – 3 (d) 5

Que. 8. The equation 3x t 9 and 33ty 6

4 represents a straight line where t is a parameter. The y-

intercept of the line is (code-V2T14PAQ9)(a) 3/ 4 (b) 9 (c) 6 (d) 1

Que. 9. If 2

4x1

and

2

2

2 2y1

where is real parameter then 2 2x xy y lies between [a,b] then

(a+b) is (code-V2T14PAQ10)(a) 8 (b) 10 (c) 13 (d) 25




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Que. 10. Let C1 and C2 are circles difined by 2 2x y 20x 64 0 and 2 2x y 30x 144 0. The length ofthe shortest line segment PQ this is tangent to C1 at P and to C2 at Q is (code-V2T14PAQ14)

(a) 15 (b) 18 (c) 20 (d) 24Que. 11. A variable line moves in such way that the product of the perpendiculars form (a,0) and (0,0) is

equal to k2. The locus of the feet of the perpendicular from (0,0) upon the variable line is a circle, thesquare of whose radius is Given : | a | 2 | k | (code-V2T14PAQ18)

(a) 2

2a k4 (b)

2 2a k4 (c)

22 ka

4 (d)

2 2a k2

Que. 12. The tangent and normal at the extremities of the latus rectum of a parabola 2y 4x form a quadri-lateral whose area is (code-V2T14PAQ23)

(a) 4 2 (b) 8 (c) 8 2 (d) 16

Que. 13. If the lines

x sin y cos 0

x cos y sin 0

x sin y cos 0

pass thorugh the same point where R then lies in the

interval (code-V2T17PAQ2)

(a) 1,1 (b) 2, 2 (c) 2,2 (d) ,

Que. 14. The range of values of m for which the line y mx and the curve 2

xyx 1

enclose a region, is

(a) (–1,1) (b) (0,1) (c) [0,1] (d) (1, ) (code-V2T17PAQ3)

Que. 15. A(1,0) and B(0,1) and two fixed points on the circle 2 2x y 1. C is a varible point on this circle.As C moves, the locus of the orthocentre of the triangle ABC is (code-V2T17PAQ5)

(a) 2 2x y 2x 2y 1 0 (b) 2 2x y x y 0

(c) 2 2x y 4 (d) 2 2x y 2x 2y 1 0

Que. 16. Mr. Suhag Kariya lives at origin on the cartesian plain and has his office at (4,5). His friend Mr.Vivek Jain lives at (2,3) on the same plane. Mr. Suhag Kariya can go to his office travelling one blockat a time either in the +y or +x direction. If all possible paths are equally likely then the probabilitythat Mr. Suhag Kariya passed his friends house is

(a) 1/2 (b) 10/21 (c) 1/4 (d) 11/21 (code-V2T20PAQ5)

Que. 17.If the left hand side of the equation 2 2x y x 3y sec 0 can be factorised into two linearfactors then the value is (code-V1T4PAQ3)

(a) 3

(b)76

(c) 43

(d) 56

Que. 18. Let x, y, z, t be real numbers 2 2 2 2x y 9; z t 4 and xt yz 6 then the greatest value of P xz,is (code-V1T5PAQ5)

(a) 2 (b) 3 (c) 4 (d) 6Que. 19. If the two vertices of a trianlge are (7,2) and (1,6) and its centroid is (4,6) then the corrdinate

of the third vertex are (a,b). The value of (a + b), is (code-V1T7PAQ2)

(a) 13 (b) 14 (c) 15 (d) 16




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Que. 20. Number of values of ‘a’ for which the lines 2x y 1 0ax 3y 3 03x 2y 2 0

are concurrent (code-V1T7PAQ4)

(a) 0 (b) 1 (c) 2 (d) infiniteQue. 21. Number of straight lines parallel to the line 3x 6y 7 0 and have intercept of length 20

between the coordinate axes (code-V1T7PAQ11)

(a) 1 (b) 2 (c) 4 (d) Infinite

Que. 22. A circle has radius of 210log a and a circumference of 4

10log b . The value of alog b is equalto (code-V1T10PAQ3)

(a) 1

4 (b) 1

(c) (d) 2

Que. 23. The points (x,y) lies on the line 2x 3y 6. The smallest value of the quantity 2 2x y , is

(a) 6 1313

(b) 6 (c) 1 132 (d) 13 (code-V1T12PAQ6)

Que. 24. If ( 2,7) is the highest point on the graph of 2y 2x 4ax k, then k equals(a) 31 (b) 11 (c) –1 (d) – 1/3 (code-V1T13PAQ2)

Que. 25. If the point P u, v is on the graph of 2y ax bx c,a 0, which of the following isalso on the graph ? (code-V1T13PAQ9)

(a) b u, va

(b) b u, va

(c)b u, va

(d) b u, va

Que. 26. Locus of all point P(x,y) satisfying 2 2x y 3xy 1 consists of union of (code-V2T13PAQ19)

(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. 27. The length of a line segament AB is 10 units. If the coordinates of one extremity are (2, –3)and the abscissa of the other extremity is 10 then the sum of all possible values of the ordinate of theother extremity is (code-V1T15PAQ2)(a) 3 (b) – 4 (c) 12 (d) – 6

Que. 28. The value of k for which the points A k 1,2 k ;B 1 k, k and C 2 k,3 k are collinear is

(a) 0 (b) 12 (c) 1 (d) 2 (code-V1T15PAQ3)

Que. 29. A point P(x,y) moves so that the sum of the distances from P to the coordinate axes is equal tothe distance from P to the point A(1,1). The equation of the locus of P in the first quadrant is(a) (x 1)(y 1) 1 (b) (x 1)(y 1) 2 (c) (x 1)(y 1) 1 (d) (x 1)(y 1) 2 (code-V1T15PAQ7)

Que. 30. If x, y R satisty the equation 2 2x y 4x 2y 5 0, then the value of the expression

2x y 4 xy

x xy

is (code-V1T15PAQ9)

(a) 2 1 (b) 2 12 (c) 2 1

2 (d)

2 12




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Que. 31. The ordinate of a point P on the line 6x y 9, which is closest to the point (–3, 1) can beexpresed in the form a/b. Where a,b N and are in lowest form, the value (a+b) equals(a) 86 (b) 44 (c) 65 (d) 100 (code-V1T17PAQ4)

Que. 32. Consider a circle 2 2x y ax by c 0 lying completely in first quadrant. If m1 and m2 arethe maximum and minimum values of y/x for all ordered pairs (x,y) on the circumference of thecircle then the value of 1 2m m is (code-V1T17PAQ5)

(a) 2

2a 4cb 4c

(b) 22ab

b 4c(c) 2

2ab4c b (d) 2

2abb 4ac

Que. 33. Let A(a,0) and B b,0 be fixed distinct points on the x -axis, none of which coincides withthe origin O(0,0), and let C be a point on the y-axis. Let g be a line through the origin O(0,0) andperpendicular to the line AC. The locus of the point of intersection of the lines g and BC if C variesalong the y-axis, is (Provided 2c ab 0 ) (code-V1T17PAQ6)

(a) 2 2x y x

a b (b)

2 2x y ya b (c)

2 2x y xb a (d)

2 2x y yb a

Que. 34. A rectangular billiard table has vertices at P(0,0), Q(0,7), R(10,7) and S (10,0). A small bil-liard ball starts at M(3,4) and moves in a straight line to the top of the table, bounces to the right sideof the table, then comes to rest at N(7,1). The y-cordinate of the pont where it hits the right side, is(a) 3.7 (b) 3.8 (c) 3.9 (d) 4 (code-V1T17PAQ7)

Que. 35. A triangle formed by 3 lines denoted by equation 3 2 2 35x 11x y 6xy y 0 will always be(a) acute angled (b) abtuse angled (c) right angled (d) none (code-V1T19PAQ1)

Que. 36. A point is selected at random inside an equilateral triangle. From this point perpendiculars aredropped to each side. The sum of these perpendiculars is (code-V1T19PAQ2)(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) maximum when the point is centroid of the triangle

Que. 37. One diagonal of a square is the portion of the variable line 2x 1 y ; 0; 1

which is intercepted between the axes. If the area of the square is 174 then the number ofvertices of

the square whose both thecoordinates are integers, is (code-V1T19PAQ5)(a) one (b) two (c) four (d) none

Que. 38. Let nC be a circle of radius n2

n

centeredat the origin for n 0,1, 2,......... and AAn be the area of

the region that is inside the circle C2n and out side the circle 2n 1C for n = 0, 1, 2........ The value of the

sum nn 0

A

equals. (code-V1T19PAQ6)

(a) 35

(b) 913

(c) 715

(d) 1013

Que. 39. A circle of radius 2 has its centre at (2,0) and another cricle of radius 1 has its centre at (5,0). A lineis tangent to the two circles at points in the first quadrant. The y-intercept of the tangent line is(a) 2 (b) 2 2 (c) 3 2 (d) 4 2 (code-V1T20PAQ1)




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Que. 40. The smallest distance between the circle 2 25 x y 3 1 and the line 5x 12y 4 0, is

(a) 1

13 (b) 2

13 (c) 3

15 (d) 4

15 (code-V1T20PAQ2)

Que. 41. A convex quadrilateral is drawn such that each of its vertices (x,y) satisty the equaation 2 2x y 73

and xy 24. The are of the quadrilateral is (code-V1T20PAQ4)

(a) 64 (b) 55 (c) 55 2 (d) 110

Que. 42. If the vertices of a ABC are A 5,0 , B 3, 4 and C 5,2 5 then the coordinates of theorthocentre is (code-V1T20PAQ6)

(a) 8,4 2 5 (b) 8 5, 5 (c) 8 5, 4 2 5 (d) 8 5, 2 5

Que. 43. A particle P moves from the point A(0,4) to the point B 10, 4 . The particle P can travel the upper

half plane x, y | y 0 at the speed of 2 m/s and travel the lower half plane x, y | y 0 at thespeed of 2 m/s. The coordinates of a point on the x-axis, if the sum of the squares of the travel timesof the upper and lower half planes is minimum, is (code-V1T20PAQ8)(a) (1,0) (b) (2,0) (c) (4,0) (d) (5,0)

Comprehesion Type# 1 Paragraph for Q. 1 to Q. 3 (code-V2T2PAQ1,2,3)

Consider a variable line L which passes through the point of intersection ‘P’ of the lines3x 4y 12 0 and x 2y 5 0 , meeting the coordinate axes at the points A and B.

Que. 1. Locus of the middle point of the segment AB has the equation(a) 3x 4y 4xy (b) 3x 4y 3xy (c) 4x 3y 4xy (d) 4x 3y 3xy

Que. 2. Locus of the feet of the perpendicular from the origin on the variable line ‘L’ has the equation

(a) 2 22 x y 3x 4y 0 (b) 2 22 x y 4y 3x 0

(c) 2 2x y 2x y 0 (d) 2 2x y x 2y 0 Que. 3. Locus of the centroid of the varible triangle OAB has the equation (where ‘O’ is the origin)

(a) 3x 4y 6xy 0 (b) 4x 3y 6xy 0 (c) 3x 4y 6xy 0 (d) 4x 3y 6xy 0 # 2 Paragraph for Q. 4 to Q. 6 (code-V2T4PAQ7,8,9)

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. linethrough P intersects C at Q and R,S is the midpoint Of QR.

Que. 4. 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 theproduct (PQ)(PR)?

(a) 2 2d r (b) 2PT , where T is a point on C and PT is tangent to C

(c) 2PS QS RS (d) 2PS

Que. 6. Let XYZ be an equilateral triangle inscribed in C. If , , denote thedistances of D from vertices

X, Y, Z respectively, the value of prodict , is

(a) 0 (b) 8

(c) 3 3 3 3

6 (d) None of these




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# 3 Paragraph for Q. 7 to Q. 9 (code-V2T6PAQ1,2,3)

The base of an isoceles triangle is equal to 4, the base angle is equal to o45 . A straight line cuts theexternsion of the base at point M at the angle and bisects the lateral side of the triangle which isnearest to M.

Que. 7. The area ‘A’ of the quadrilateral which the straight line cuts off from given triangle is

(a) 3 tan1 tan

(b) 3 2 tan1 tan

(c) 3 tan1 tan

(d) 3 5 tan1 tan

Que. 8. The range of values of ‘A’ for differnt values of , lie in the interval,

(a) 5 7,2 2

(b) 4,5 (c) 94,2

(d) 3,4

Que. 9. The length of portion straight line inside the triangle may lie in the range :

(a) 2, 4 (b) 3 , 32

(c) 2, 2 (d) 2, 3


Let C be curve difined by 2a bxy e . The curve C passing through the point P(1,1) and the slope of the

tangent at P is (–2). Also C1 and C2 are the circles 2 2(x a) (y b) 3 2 2(x 6) (y 11) 27 re-spectively.

Que. 10. The value of 2 2a b is equal to(a) 2 (b) 8 (c) 18 (d) 32

Que. 11. The length of the shortest line segment AB which is tangent to C1 at A and to C2 at B is(a) 9 3 (b) 10 3 (c) 11 (d) 12

Que. 12. If f is a real valued derivable function satisfying x f (x)fy f (y)

with f '(1) 2. Then the value of the

integral a

b

f (x)d n x is equal to

(a) 0 (b) 2 2e e

2

(c) 2 2e e2

(d) 2

# 5 Paragraph for Q. 13 to Q. 15 (code-V2T7PAQ7,8,9)Given the continuous function

2

2

2

x 10x 8, x 2y f (x) ax bx c 2 x 0,a 0

x 2x x 0

If a line L touches the graph of y f (x) at three points thenQue. 13. The gradient of the line ‘L’ is equal to

(a) 1 (b) 2 (c) 4 (d) 6Que. 14. The value of a b c is equal

(a) 5 2 (b) 5 (c) 6 (d) 7Que. 15. If y f (x) is differentiable at x 0 then the value of b

(a) is –1 (b) is 2 (c) is 4 (d) ican not be determined




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# 6 Paragraph for Q. 16 to Q. 18 (code-V2T13PAQ1,2,3)Let ABCD is a square with sides of unit length. Points E and F are taken on sides AB and AD

respectively so that AE = AF. Let P be a point inside the square ABCD.Que. 16. The maximum possible area of quadrilateral CDFE is

(a) 18 (b)

14 (c)

58 (d)

38

Que. 17. The value of 2 2 2 2PA PB PC PD is equal to(a) 3 (b) 2 (c) 1 (d) 0

Que. 18. Let a line passing through point A divides the square ABCD in to two parts so that area of oneportion is double the other, then the length of portion of line inside the square is

(a) 103

(b) 133

(c) 113

(d) 23


Consider a triangle PQR coordinates of its vertices P 8,5 ;Q 15, 19 and R 1, 7 . The bisec-tor of the interior angle of P has the equation which can be written in the form ax 2y c 0 .

19. The distance between the orthocentre and the circumcentre of the triangle PQR is

(a) 1122

(b) 1152

(c) 1104

(d) 3114

20. Radius of the incircle of the triangle PQR is(a) 4 (b) 5 (c) 6 (d) 8

21. The sum of the coefficients (a + c)(a) 129 (b) 78 (c) 89 (d) 99

# 8 Paragraph for Q. 22 to Q. 24 (code-V1T8PAQ7,8,9)Consider the family of lines passing through the intersection of the lines

1U ; 3x 4y 7 0 and 2U : 4x 3y 1 0 22. A member of the family which bisects the angle between them and is closer to the origin, is

(a) x 7y 6 0 (b) 7x y 8 0 (c) 7x y 6 0 (d) y 7x 4 0 23. A member of this family with gradient minus 2 has y-intercept equal to

(a) 2 (b) –3 (c) 1 (d) – 224. A member of this family whose slope is not difined is

(a) y 1 0 (b) x 1 (c) 3x 4 (d) x 1 0 # 9 Paragraph for Q. 25 to Q. 27 (code-V1T16PAQ1,2,3)

Consider 3 non collinear point A (9,3); B(7,–1) and C(1,–1). Let P(a,b) be the centre and ‘R’ is the

radius of the circle ‘S’ pasing through A,B,C. Also H x, y are the coordinates of the orthocentre ofthe triangle ABC whose are be denoted by .

25. If D, E and F area the middle points of BC, CA and AB respectively then the area of the triangle DEFis(a) 12 (b) 6 (c) 4 (d) 3

26. The value a b R equals(a) 3 (b) 12 (c) 13 (d) None

27. The ordered pair x, y is(a) (9,5) (b) (–9,6) (c) (9,–6) (d) (9,–5)




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Consider a circle 2 2x y 4 and a point P(4,2). denotes the angle enclosed by the tangents fromP on the circle and A,B, are the points of contact of the tangents from P on the circle.

Que. 28. The value of lies in the interval

(a) o0,15 (b) o o15 ,30 (c) o o30 ,45 (d) o o45 ,60

Que. 29. The intercept made by a tangent on the x - axis is(a) 9/4 (b) 10/4 (c) 11/4 (d) 12/4

Que. 30. Locus of the middle points of the portion of the tangent of the circle terminated by the coordinateaxes is(a) 2 2 2x y 1 (b) 2 2 2x y 2 (c) 2 2 2x y 3 (d) 2 2 2cx y 4


Consider a family of lines 4a 3 x a 1 y 2a 1 0 where a R

Que. 31. The locus of the foot of the perpendicular from the origin on each member of this family, is

(a) 2 22x 1 4 y 1 5 (b) 2 22x 1 y 1 5

(c) 2 22x 1 4 y 1 5 (d) 2 22x 1 4 y 1 5

Que. 32. A member of this family with positive gradient making an angle of / 4 with the line 3x 4y 2, is(a) 7x y 5 0 (b) 4x 3y 2 0 (c) x 7y 15 (d)5x 3y 4 0

Que. 33. Minimum area of the triangle which a member of this family with negative gradient can make withthe positive semi axes, is(a) 8 (b) 6 (c) 4 (d) 2


Consider 3 circles2 2

12 2

22 2

3

S : x y 2x 3 0S : x y 1 0

S : x y 2y 3 0

34. The radius of the circle which bisect the circumferences of the circles 1 2 3S 0; S 0; S 0 is

(a) 2 (b) 2 2 (c) 3 (d ) 10

35. If the circle S = 0 is orthogonal to 1 2 3S 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) 1 (c) 2 (d) 3

36. The radius of the circle touching 1 2S 0 and S 0 at (1,0) and passing through (3,2) is

(a) 1 (b) 12 (c) 2 (d) 2 2# 13 Paragraph for Q. 37 to Q. 39 (code-V1T17PAQ14,15,16)

An altutude BD and a bisector BE are drawn in the trianlge ABC from the vertex B. It is known thatthe length of side AC = 1, and the magnitudes of the angles BEC, ABD, ABE, BAC form an arithmeticprogression.

37. The area of circle circumscribing ABC is

(a) 8

(b) 4

(c) 2

(d)




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38. Let ‘O’ be the circumcentre ABC, the radius of circle inscribed in BOC is

(a) 1

8 3 (b) 1

4 3 (c) 1

2 3 (d) 12

39. Let B' be the image of point B with respect to side AC of ABC , then the length BB ' is equal to

(a) 34

(b) 24

(c) 12 (d) 3

2

Assertion & Reason TypeIn this section each que. contains STATEMENT-1 (Assertion) & STATEMENT-2(Reason).Each

question has 4 choices (A), (B), (C) and (D), out of which only one is correct.Bubble (A) STATEMENT-1 is true, STATEMENT-2 is True; STATEMENT-2 is a correct

explanation for STATEMENT-1.Bubble (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct

explanation for STATEMENT-1.Bubble (C) STATEMENT-1 is True, STATEMENT-2 is False.Bubble (D) STATEMENT-1 is False, STATEMENT-2 is True.

Que. 1. Consider the following statements (code-V2T3PAQ11)

Statement 1: The equation 2 2x 2y 2 3x 4y 5 0 represents two real lines on the cartesian plane.

because

Statement 2: A general equation of degree two 2 2ax 2hxy by 2gx 2fy c 0 denotes a line pair

if 2 2 2abc 2fgh af bg ch 0

Que. 2. Consider the folloiwing statements (code-V2T3PAQ13)

Statement 1: The area of the triangle formed by the points A(20,22);(B(21,24) and C(22, 23) is thesame as the area of the triangle formed by the point P(0,0);Q(1,2) and R(2,1).

becauseStatement 2: The area of the triangle is invariant w.r.t. the translation of the coordinate axes.

Que. 3. Statement 1: The circle 2 21C : x y 6x 4y 9 0 bisects the circumference of the circle

2 22C : x y 8x 6y 23 0.

becauseStatement 2: Centre of the circle C1 lies on the circumference of C2. (code-V2T6PAQ4)

Que. 4. Passing through a point A(6,8) a variable secant line L is drawn to the circle2 2S: x y 6x 8y 5 0. Form the point of intersection of L with S, a pair of tangent lines are drawn

which intersect at P. (code-V2T6PAQ6)

Statement 1 : Locus of the point P has the equation 3x 4y 40 0.

becauseStatement 2 : Point A lies outside the circle.

Que. 5. Statement 1 : The equation 2 2x 2 y 3 1 does not represent a circle. (code-V2T15PAQ5)

bacause

Statement 2 : 2 2x 2 y 3 1 represents no real locus.




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Que. 6. Consider the curves 2

2 2 31 2

yC : x a and C : xy c3

(code-V2T16PAQ12)

Statement 1: C1 and C2 are orthogonal curves.becauseStatement 2: C1 and C2 intersect at right angles everywhere wherever they intersect.

Que. 7. Consider a general expression of degree two variables as 2 2f (x, y) 5x 2y 2xy 6x 6y 9

Statement 1: f (x, y) can be resolved into two linear factor over real coefficients.because (code-V1T6PAQ4)

Statement 2: Discriminant of f (x,y) i.e. 2 2abc 2fgh 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 respectively. (code-V1T6PAQ5)

Statement 1:Area of ABC is four times the area of DEF because Statement 2: Ratio of the areas of two similiar triangle is the ratio of proportional sides.

Que. 9. Statement 1: If the diagonals of the quadrilateral formed by the lines px qy r 0,

p 'x q ' y r 0,px qy r ' 0,p ' x q ' y r ' 0 are at right angles, then 2 2 2 2p q p ' q ' .

because (code-V1T8PAQ10)

Statement 2: Diagonals of a rhombus are bisected and prependicular to each other.

Que. 10. Statement 1: The joint equation of lines 2 2 2 2y x and y x is y x i.e., x y 0 because (code-V1T8PAQ12)

Statement 2: The joint equation of lines ax by 0 and cx dy 0 is (ax by)(cx by) 0 wherea,b,c,d are constant.

Que. 11. Given a ABC whose vertices are 1 1 2 2 3 3A x , y ;B x , y ;C x , y . Let there exists a point

P(a,b) such that 1 2 3 1 2 36a 2x x 3x ; 6b 2y y 3y (code-V1T16PAQ11)

Statement 1: Area of triangle PBC must be less than the area of ABCbecauseStatement 2: P lies inside the triangle ABC

Que.12. Let points A, B, C are represented by i ia cos ,a sin i 1, 2,3 and

1 2 2 3 3 13cos cos cos .2

. (code-V1T18PAQ10)

Statement 1: Orthocentre of ABC is at originbecauseStatement 2 : ABC is equilateral triangle.

Que. 13. Let C1 denotesa family of circles with centre on x-axis and touching the y-axis at the origin.and C2 denotes a family of circles with centre on y-axis and touching the x-axis at the origin.Statement 1: Every member of C1 intersects any member of C2 at right anglesat the point

other than origin.because (code-V1T19PAQ7)Stastement 2: If two circles interesect at 90o at one point of their intersection, then they must

intersectat 90o on the other point of intersection also.




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Que. 14. Consider the lines 1 2 3x 2y x 3y: 1 0; : 1 0; : 5x 3y 1 03 3 2 4

Statement 1: The lines 1 2 3, and are concurrent. (code-V1T19PAQ8)

because

Statement 2: The area of the trianlge formed by the points 1 2 1 3, , ,3 3 2 4

and (5,–3) vanishes.

Que. 15. Consider the lines 1 2L : 3x 4y 2 and L : 5x 12y 7 (code-V1T10PAQ11)

Statement 1: Every point on the line 64 8y 61 is equidistant from 1L and 2L .because

Statement 2: Is the bisector of the angle between 1L and 2L .which contains the origin in its region.

Que. 16. Consider the line L : 3x y 4 0 and the points A( 5,6) and B 3, 2 (code-V1T10PAQ13)

Statement 1: There is exactly one point on the line L which is equidistant form the point A and B.becauseStatement 2: The point A and B are on different sides of the line.

More than One Correct TypeQue. 1. Three distinct lines are drawn in a plane. Suppose there exist exactly n circles in the plane tangent

to all the three lines, then the possible values of n is/are(a) 0 (b) 1 (c) 2 (d) 4 (code-V2T1PAQ13)

Que. 2. Consider the points O, 0,0 ,A 0,1 and B 1,1 in the x-y plane. Suppose that point C(x,1) andD(1,y) are chosen such that 0 < x < 1 and such that O,C and D are collinear. Let sum of the area oftriangles OAC and BCD be denoted by ‘S’ then which of the following is/are correct ?

(a) Minimum value of S is irrational lying in (1/3, 1/2)(b) Minimum value of S irrational in (2/3, 1)(c) The vlaue of x for minimum value of S lies in (2/3, 1)(d) The value of x for minimum value of S lies in (1/3, 1/2) (code-V2T1PAQ14)

Que. 3. If is the angle between the pair of tangents drawn from (c, 0) to the corcle 2 2x y 1 then whichof the following conclusion(s) is/are true ?

(a) If 5 , c 1, 6 26

(b) If 3 , c 1, 5 1

5

(c) If , c 1, 22

(d) If , c 1, 2

3

(code-V2T2PAQ13)

Que. 4. If is eliminated from the equation a sec x tan y and bsec0 y tan x (a and b areconstant) then the eliminant denotes the equation of (code-V2T15PAQ12)

(a) The director circle of the hyperbola 2 2

2 2

x y 1a b

(b) anuxiliary circle of the ellipse 2 2

2 2

x y 1a b

(c) Director circle of the ellipse 2 2

2 2

x y 1a b

(d) Director circle of the circle 2 2

2 2 a bx y .2




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Que. 5. If the vertices P,Q,R of a triangle PQR are rational points, which of the following points of thetriangle PQR is/are always rational point(s) (code-V2T17PAQ15)

(a) centriod (b) incentre (d) circumcentre (d) orthocentre

Que. 6. The origin, the intersection of the lines 2 22x 5xy 3y 3x 5y 2 0 and the points in whichthese lines are cut by the line 3x 5y 2, are the vertices of a (code-V2T18PAQ11)

(a) parallelogram (b) rectangle (d) rhombus (d) squareQue. 7. The equations to the lines through the point of intersection of 3x y 20 and x 2y 5 0 which

are at a distance 5 from the origin, is/are (code-V1T7PAQ12)

(a) 4x 3y 25 (b) 3x 4y 25 (c) 4x 3y 25 (d) 4x 3y 25

Que. 8. A circle centred at ‘O’ has radius 1 and contains the points the A. Segment AB is tangent to thecircle at A and AOB . If point C lies on OA and BC bisects the angle ABO then OC equals

(a) sec sec tan (b) cos1 sin

(c) 1

1 sin (d) 2

1 sincos

(code-V1T12PAQ12)

Que. 9. Let a, b,c Q satisfying a b c. Which of the following statement(s) hold true for the quadratic

polynomial 2f (x) a b 2c x b c 2a x c a 2b ? (code-V1T14PAQ6)

(a) The mouth of the parabola y = f(x) opens upwards.(b) Both roots of the equation f(x) = 0 are rational.(c) x-coordinate of vertex of the graph is positive.(d) Product of the roots is always negative.

Que. 10. If 2sin x 2x b 2, for all real values of x 1 and 0, 2 2, , then possiblereal values of ‘b’ is/are (code-V1T14PAQ10)(a) 2 (b) 3 (c) 4 (d) 5

Que. 11. If 27 ay log 2x 2x a 3 is difined x R, then possible integral value(s) of a is/are

(a) –3 (b) –2 (c) 4 (d) 5 (code-V1T14PAQ11)

Que. 12. If the equation 2 2ax 2cxy by d represents two real and distinct straight lines then thenecessary and suffcient conditions can be (code-V1T17PAQ12)

(a) d is zero and 2c ab (b) 2c ab and d R {0}

(c) 2c 4ab and d R (d) 2d 0 and c ab

Que. 13. If 2 2 24a c b 4ac then the variable line ax + by + c = 0 always passes through one or theother of the two fixed ponts. The coordinates of the fixed point can be (code-V1T17PAQ13)

(a) ( 2, 1) (b) (2, 1) (c) ( 2,1) (d) (2,1)

Que. 14. If the lines (code-V1T17PAQ14)

21

22

23

u : x y 1 0u : x y 1 0

u : x y 0

Passes through the same point then the value(s) of equals (code-V1T17PAQ15)

(a) 1 (b) 2 (c) 2 (d) 2




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Que. 15. Let 1 2L :3x 4y 1& L :5x 12y 2 0 be two given lines. Let image of every point on L1

with reespect to a line L lies on L2, then possible equations of L can be (code-V1T17PAQ16)

(a) 14x 112y 23 0 (b) 64x 8y 3 0

(c) 11x 4y 0 (d) 52y 45x 7

Que. 16. Let A(1,1) and B(3,3) be two fixed points and P be a variable point such that area of PABremains constant equal to 1 for all positions of P, then locus of P is given by (code-V1T17PAQ17)

(a) 2y 2x 1 (b) 2y 2x 1 (c) y x 1 (d) y x 1

Match Matrix TypeQue. 1. Column - I Column - II (code-V2T19PBQ2)

A. The lines y = 1; x – 6y + 4 = 0 and x+6y–9=0 P. a cyclic quadraliteralconstitute a figure which is

B. The points A a,0 , B 0, b ,C c,0 and D 0,d Q. a rhombusare such that ac bd and a, b,c,d are all non-zero.The points A,B,C and D always constitute

C. The figure formed by the four lines R. a square

ax by c 0 a b , is

D. The line pairs 2x 8x 12 0 and 2y 14y 45 0 S. a trapeziumconstitute a figure which is

Que. 2. Column - I Column - II (code-V1T8PBQ2)

A. Four lines x 3y 10 0, x 3y 20 0 P. a quadrilateral which is neither3x y 5 0 and 3x y 5 0 form a a parallelogram nor a trapezium norfigure which is a kite

B. The point A(1,2), B(2,–3), C(–1,–5) and Q. a parallelogramD(–2,4) in order are the vertices of

C. The lines 7x 3y 33 0,3x 7y 19 0 R. a rectangle of area 10 sq. units3x 7y 10 0 and 7x 3y 4 0 form afigure which is

D. Four lines 4y 3x 7 0,3y 4x 7 0, S. a square4y 3x 21 0,3y 4x 14 0 form a figure which is

Que. 3. Column - I Column - II (code-V1T17PBQ1)

A. The four lines 3x 4y 11 0;3x 4y 9 0; P. a quadrilateral which is neither a4x 3y 3 0 and 4x 3y 17 0 enclose a parallelogram nor a trapezium norfigure which is a kite.

B. The lines 2x y 1, x 2y 1, 2x y 3 and Q. a parallelogram which is neither ax 2y 3 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 2 22x 7xy 3y 5x 10y 25 0, A and Bare the points in which these lines are cut by the linex 2y 5 0 , then the points O,A,P,B (in some order)are the vertices of S. a square.




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Que. 4. Set of family of lines are discribed in column - I and their mathematical equation are given incolumn - II. Match the entry of column - I with suitable entry of column - II. (m and n are parameters).

Column - I Column - II (code-V1T20PBQ1)

A. having gradient 3 P. mx y 3 2m 0

B. having y intercept three times the x-intercept Q. mx y 3m 0

C. having x-intercept (–3) R. 3x y 3a

D. concurrent at (2,3) S. 3x y a 0

Subjective Type ( Up to 4 digit)Que. 1. A rhombus ABCD has sides of length 10. A circle with centre ‘A’ passes through C (the opposite

vertex) 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-V2T2PDQ1)

Que. 2. The circles, which cut the family of circles passing through the fixed pointsA (2,1) and B (4,3) orthogonally, pass through two fixed points 1 1 2 2x , y and x , y , which may be

real or imaginary. Find the value of 3 3 3 31 2 1 2x x y y . (code-V2T8PDQ3)

Que. 3. Consider 3 lines (code-V1T7PDQ3)

1

2

3

L : 5x y 4 0L : 3x y 5 0L : x y 8 0

If these lines enclose a triangle ABC and sum of the square of the tangent ofthe interior angles can beexpressed in form p/q where p and q are relatively prime numbers, compute the value of(p + q).

Que. 4. If the expression 2 2f (x, y) 9x 16y 777x 992y k can be resolved into two linear factors thenfind the value of k. (code-V1T12PDQ1)

Que. 5. Point P is 2/3 of the way from the point F( 5,3) to G(4,15). Line L is perpendicular to the lineFG and passes through the point P. If the equation of the line L is ax + by = c, where a, b and c arerelatively prime integer and a > 0 then find the vlaue of 8a 9b 10c . (code-V1T17PDQ1)

Que. 6. Consider the circle whose centre is in the first quadrant and which is tangent to both the co-ordinate axes and the line L, whose equation is 3x 4y 120. If the co-ordinates of the point of

tangency of the circle with the line L are 1 1p ,q and 2 2p ,q and (a,b) and (c,d) are the coordinates

of the centres of the two corcles. Find 1 2 1 2p p q q a b c d (code-V1T17PDQ2)

Que. 7. If the equation of the diagonals of the parallelogram formed by the lines2x y 7 0; 2x y 5 0; 3x 2y 5 0 and 3x 2y 4 0 are ax by 5 0 and

px qy 1 0. Find the value of a b p q . (code-V1T17PDQ3)




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[SOLUTION]Single Correct Type

Que. 1. A. Centre : 3, 4 and r 5 perpendicular distance from (3, –4) on 3x 4y 25 0 is

9 16 25 32 32 7p d 5 .5 5 5 5

Que. 2. C. y x k k 1 ; y x k y x y 1 k x 1 ky x

1 ky x1 k

where 1 k 11 k

.

Que. 3. A. Solving line and circle 22 2 2 2m x x m c 2 2 2 2 2 2m x 2 x m c 0 1x

2x

2 2m 1 given 2 2 2 2 2 21 2

1 2

2 m c2x x m cH Hx x 2

................ (1)

2 2 2c||| y mK

..................... (2)

1 1

A(x y )

2 2

B(x y )

x + my =

(1) + (2) 2 2 2 2 22 c m cH mK 2

where 2 2m 1 .

Que. 4. A. 22 2 2 1.1 1x y 4y 4 0 x y 2 0 x y 2 x y 2 0 Area .2 2

x + y = 2x + y = 3

x

y

x y + 2 = 0(0,3)

(1,2)

( 2,0)

(0,2)

Que. 5. D. 3

2

0

x ax 12 dx 45 gives a = 4 Hence 2f (x) 12 4x x (2 x)(6 x) hence

m 2 and n 6 m n a 6 2 4 8.

Que. 6. D.

2 2 2

2 2 2

2 2 2

p p 1 p q p q 0 p q 1 01 1 1D q q 1 q r q r 0 p q q r q r 1 02 2 2

r r 1 r r 1 r r 1

1 1p q q r p q q r p q q r p r .2 2




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Que. 7. D. 1 1sin sin12 sin sin 12 4 12 4 1 1cos cos12 cos cos 4 12 4 12

2k 2 x 8x k 4 0 If k 2 then .8x 4 0 (not possible) then of k 2 then k 2 0 and

64 4 k 2 k 4 0 216 k 2k 8 2k 2k 24 0 k 6 k 4 0 k 5.

40

2

Que. 8. A.

4 y 6x 9 ; put x 0, 4 y 6 27 y 3 / 4.

3

Que. 9. A. tan x 2sin 2 and y 2cos 2 2 2E x xy y 4 4sin 2 cos 2 4 2sin 4

E 2,6 a b 8.

Que. 10. C. Centres are (10,0) and (–15,0) 1 2r 6; r 9 d 25 1 2r r d Circles are separted

221 2PQ d r r 625 225 20

Q

l

1C2C

1rl

A (10,0)

A ( 15,0)

P1rd

2r

Que. 11. A. Let the equation of the variable line is 2 21 1 1 1xx yy x y 0

2 22 21 1 1 21 1

1 2 2 2 2 21 1 1 1

2 2 2 2 2 21 1 1

ax x yx yp p kx y x y

i.e., x y ax k locus x y ax k 0

y

x

( )1 1x , y

(a,0)O

]

2

2 2ar k4

+ve sign 2

2 2ar k4

(not possible as 2r becomes –ve)

–ve sign2

2 2ar k .4

Varible line 1

1

xmy

= -

2p1p

(0,0) (0,0)

1 1(x , y )

Que. 12. B.




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Que. 13. B.sin cos

D 1 cos sin 01 sin cos

2 2cos sin sin cos sin cos sin cos 0

2 2D sin .cos sin cos .sin cos sin 2 cos 2

sin 2 cos 2 2, 2 .

Que. 14. B. Solving 22

x 1mx x 1 or x 0x 1 m

2 1x 1 0m

for a region m 1 0

m

m 0,1 Note : form = 0 or 1 the line does not enclose a region.

Que. 15. A. Let C cos ,sin ;H h, k is the orthocentre of the ABC

(h,k) (0,0)G2 1 Circumcentre

cos 1 sin 1,3 3

+ + 2 2

2 2

h 1 cosk 1 sin

x 1 y 1 1

x y 2x 2y 1 0

Que .16. B.9!n(S) 126

4!.5!n(A) 0 to F and F to P

5! 4!. 10.6 602!.3! 2!.2!

60 10P(A)126 21

10

2345

1 2 3 4

P(4,5)

Que. 17. (C) a 1; b 1; c sec1 3h 0; g ; f2 2

2 2 2using abc 2fgh af bg ch 09 1 4sec 0 sec 2 sec 24 4 3

Que. 18. (B) Let x 3cos ; y 3sin z 2cos ; t 2sin

o6cos .sin 6sin cos 6 6sin 1 90 90

x 3cos ; y 3sin z 2sin ; t 2sin p xz 6sin cos 3sin 2

maxp 3.




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Que. 19. (D)2y 8x x 15 (x 5)(3 x) y 0 (x 5)(3 x) (x 5)(x 3) 0 x 5 or x 3.

Que. 20. (D) all values of ‘a’.

Que. 21. (D) Slope of the given line is – 1/3 let one line is x y b b 11 slopea b a a 3

3b a ........(1) also given 2 2a b 100 ........(2) (1) and (2) b 10 b 10;a 3 10

b 10;a 3 10 None a and b must be of same sign (b)

Que. 22. (C) 10C 4log b 2 r 1010 10 10 a

10

log b4 log b 2 .2 log a (as r 2 log a) log b .log a

Que. 23. A. Let x r cos ; y r sin 2r cos 3r sin 6 6r ;

2 cos 3r sin

to find

2 2min

minx y i.e. r for r to be minimum 2cos 3sin must be maximum i.e., 13

min6 6 13r .

1313

Que. 24. C. 2y 2x 4ax k; abscissa corresponding to the vertex is b2a

i.e. 4a 2 a 2

4

now y( 2) 7 7 8 16 k k 1. Que. 25. B. Verify each alternative.

Que. 26. A. 33 3x y 1 3(x)(y)( 1) 0 2 2 2(x y 1) (x y) (y 1) (x 1) 0

x y 1 or x y 1 a line x y 1 or a point 1, 1 (A).

Que. 27. D. 2 264 k 3 100 k 3 36 k 3 6 or 6 k 3 or 9. A B(2, 3) (10, k)

10

Que. 28. C. k 1 2 k 1 2k 2 0

D 1 k k 1 0; 1 2k 3 0 0k 2 3 k 1 k 2 3 k 1

1 6k 2 4k 0; 2k 2; l 1.

Que. 29. B. 2 2 2x y x 1 y 1 2xy 2 2x 2y x y xy 1

x y xy 1 2 x 1 y 1 2.

Que. 30. D. 2 2f (z, y) (x 2) (y 1) 0 x 2 and y 1

2 22 1 4 2 2 1 2 1E .

2 2 22 2 1

Que. 31. D. Slope of 8 6hAPh 3

hence 8 6h ( 6) 1h 3

h 45 / 37 hence coordinate

9 6 (45 / 37) 63 / 37 a b 100.




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Que. 32. C. Substituting y = mx in the equation of circle we get

2 2 2

2 2

2 2

2 2 2 2

x m x ax bmx c 0

x 1 m a bm x c 0

D 0 a bm 4c 1 m 0

a b m 2abm 4c 4cm 0

y / x denotes the slope ofthe tangent from the originon the circle

(0,0)

2y m x=

1y m x=y

x

2 2 2m b 4c 2abm a 4c 0 1

2

mm 1 2 2 2

2ab 2abm m .b 4c 4c b

Que. 33. C. Equation of the line g is ay xc

as (h,k) lies on it,

hence ak hc

.......... (1)

Now equation of BC x y 1b c (h,k) lies on it

h k 1b c ............... (2)

Substituting ahck

in (2) form (1) 2 2h k 1

hb ab locus of P is

2 2x y xb a .

Que. 34. A.a 1 7 atan

3 10 b

also 7 4 3tanb 3 b 3

hence 3 a 1 7 a

b 3 3 10 b

form 1 st two relations

9 ab b 3a 3 3a 6 ab b .................. (1)form last two 10a ab 10 b 21 3a

13a ab b 31 or ab b 13a 31 ............ (2)hence from (1) and (2) 3a 6 13a 31 10a 37 a 3.7.

Que. 35. D. Homogeneous equation of degree 3 3 lines through lines concurrent.

Que. 36. B. 21 2 3

1 3a p p p a2 4

1 2 33ap p p2

length of altitude o Altitudesin 60a

Length of altitude 3a2

Que. 37. A.x y 1

1

Area of square is 22 21 17 1 17d 1

2 4 2 4

22 2 2 1 17 24 4 15 0 24 10 6 15 0 2 5 2 3 0

5 3or2 2

But 502

2 vertics are 3 5,0 and 0,2 2

Remaining 2 vertices

are AB CD5 3m m3 5

9 25 34AB

4 2

34BM4

using parametric coordinates

y

xA(a,0)

B(b,0)

m = c/a

(0,c)C

m = a/c g

P(h,k)

(0,7)Q

y(b,7) 10 b R(10,7)

7 a

(10,a)

a 1

xS(100)3

(7,1) 3

(3,4)

(0,0)P

o90

B CQ

A

h hp

1p

3p2p




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of B and D are 3 34 5 5 34 3. ; .4 4 34 4 4 34 1 1D , B 2,2

2 2

Que. 38. B. n 0 1 2 3 nn n 0lim A A A A A ........... A

Now

22 2

0 0 12A r r 13

4 6 8 102 2 4 2

2 2 3 3 2 52 2 2 2A r r A r r3 3 3 3

and so on

Hence

2 4 6

n 22 2 2 9A 1 ............ .3 3 3 131 2 / 3

Que. 39. B.

2 2

2

1 1 5a 8

2 a 2equation of line : y m(x 8)

6m 2 9m 1 m1 m

1m (reject +ve sign as slope is ve)8

1m y intercept 8 2 2.8

Que. 40. B.

0 0

2 2

2 2

The distance form a point to a line Ax By C

Ax By c 0 isA B

The centre of the circle is (5, 3),so the distance from this point to the line5x 12y 4 0 is

25 36 45.5 12.( 3) 4 15311695 12




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Que. 41. D.

2 2

2

x y 73; xy 24

x y 73 48 121x y 11

and x y 5x,y 8,3 , 8, 3 ,]

3,8 , 3, 8 Area 110.

Que. 42. C. Clearly, coordinates of circumcentre (0) is (0,0) OA OB OC and centroid is

8 5 4 2 5,3 3

since centriod divides the line segment joining orthocentre and circum centre in

the ratio 2:1, hence coordinates of orthocentre are 8 5,4 2 5 .

Que. 43. B .

22 22

22 2

2

Let the point on the x-axis is (c,0) Sum of the squares of travel times is

10 c 16c 16T1 2

116 c 20c 5c 16 c 5c 454 4

5 c 4c 36 T is minimum is c 2.4

Comprehesion Type# 1 Paragraph for Q. 1 to Q. 3

1. - A. 2. - B. 3 - C.Point of intersection the line 3x 4y 12 0 x 2y 5 0 is x = 2 and y = 3/2

(i). Equation of AB is

x y 2 31 1 4k 3h 4kh 3x 4y 4xy 0.2h 2k 2h 4k

(ii).

2 2

k k (3/ 2) k 2k 3. 1 . 2h h 2 h h 2

2h(h 2) k(2k 3) 02(x y ) 4x 3y 0.

B P(2,3/2)

(h,k)

O A x

y

B(0,2k)

P(2,3/2)

(h,k)

O (2a,0)A x

y




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(iii). Here, a 3h, and b 3k equation of AB is

x y 2 11 1 3x 4y 6xy 0.3h 3h 3h 2k

# 2 Paragraph for Q. 4 to Q. 6 4. - D. 5. - D. 6 - A.

(i). Locus of S is a part of circle with OP as diameter passing inside the circle ‘C’

N

Q

O

C

PM

RS(h,k)

(ii). 2 2 2P R PQ PT PN PM d r d r d r

22 2 2PS SR PS SQ PS SQ SQ SR PS SQ SR PQ PR PS

(iii). Using Ptolemy’s theorem

YD XZ XY ZD YZ XD XZ ZD XD

XY YZ ZX (A).

Alternatively :

2 22 YD XY12 2 YD

form

XYD ;

2 22 YD YZ12 2 YD

form YZD

2 2 2 22 2YD YD XY .........(1); YD YD YZ .........(2)

2 2(1) (2)

# 3 Paragraph for Q. 7 to Q. 97. - D. 8. - D. 9 - C.

Equation of line PM : y 1 tan x 1 Intersection point ‘Q’

of AC and MP. 4 x 1 tan x 1

3 tan 1 3 tanQ ,1 tan 1 tan

Area of APQ modulus of

1 1 1

5 1 tan 21 2 2 12 1 tan

3 tan 1 3tan 11 tan 1 tan

B (a,b)

P(2,3/2)

G(h,k)

O (a,0)Ax

y

X

Y Z

o60o60

D

o30 o30

(1,1)y = x

A(2,2)

y = 4 x

Q

B (0,0)M

C(4,0)

o90

o45 o45P




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(i). Area of quadrilateral BPQC Area, 5 1 tan 21 1 tan 3 5tanA 4 22 1 tan 1 tan 1 tan

(ii).2A 5

1 tan

note that, 0, 1 tan 1, 2 A 3,4 .

4

(iii).

2 22

23 tan 1 3tan 4 4PQ 1 11 tan 1 tan 1 sin 2cos sin

2sin 2 0,1 PQ 2,4 PQ 2,2 .# 4 Paragraph for Q. 10 to Q. 12

10. - A. 11. - C. 12 - B.

(i). 2a bxy e , passes through (1,1) a b1 e a b 0 also (1,1)

dy 2dx

2a bxe .2bx 2

a be .2b(1) 2 b 1and a 1 (a,b) (1, 1) 2 2a b 2 .

(ii). Hence

22 2

1 1 222 2

2

C : (x 1) (y 1) 3 C & Care separatedC : x 6 y 11 3 3

(1, 1) (6,11)

A

B

l

1C2C

2r 3=

1r 3=

1r

13

222 2 21 2AB d r r 169 4 3 121 AB 11.

(iii). Again

h 0

f x hf (x) 1

f (x)f (x h) f (x)f '(x) lim ; f (1) 1h h

h 0

hf 1 1f (x) xlim hx

x

2f '(1).f (x) 2f (x) as f (1) f (1) but f (1) 0 f (1) 1x x

f '(x) 2f (x) x

n(f (x)) 2 n x C x 1,

2f (1) 1 C 0 f (x) x ea 1 e 2 2 2

2

b 1 1/ e 1/ e

x e eI f (x)d n x x d n x xdx2 2

.

# 5 Paragraph for Q. 13 to Q. 1512. - C. 14. - D. 15 - B.

2

2

2

x 10x 8, x 2f (x) ax bx c 2 x 0,a 0

x 2x x 0

For continuous at x 0 c 0

Continuous at x 2 4 20 8 4a 2b 8 4a 2b 2a b 4 ............... (1)




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Now let the line y mx p is tangent to all the 3 cuves solving 2y mx p and y x 2x 2 2 2x 2x mx p x (2 m)x p 0 D 0 (2 m) 4p 0 ................ (2)

again solving 2y mx p and y x 10x 8 2 2x 10x 8 mx p x (10 m)x 8 p 2 2 2 2(10 m) 4(8 p) 0 (10 m) 32 4p 0 (10 m) (2 m) 32

2(100 20m) (4 4m) 32 (m cancels out) 96 16m 32 64 16m m 4 and p 1

hence equation of line tangent to 1st and last curves is y 4x 1 ................ (3)now solving this with 2y ax bx (as c 0) 2 2ax bx 4x 1 ax (b 4)x 1 0 D 0

2(b 4) 4a Also b 2a 4 (form1) 24a 4a a 0 a 1 and b 6

x 0f '(0 ) lim 2ax b b;

x 0

f '(0 ) lim 2x 2 2

b 2 .

# 6 Paragraph for Q. 16 to Q. 1816. - C. 17. - D. 18 - B.

(i). Area of CDFE 21 1A 1 x 1 x2 2

2 22 x 1 x 1 x x2 2

max

1 11 5 12 4A at x2 8 2

y

x

D(0,1) C(1,1)

(0,0)A

E(x,0) B(1,0)

F(0,x)

(ii). 2 2 2 2 2 2 2 2 2 2 2 2PA PB PC PD 0

y

x

D(0,1) C(1,1)

(0,0)A

B(1,0)

(iii). 2

2AQ

1 1 2 2 13y 1 (1) y L 1 .2 3 3 3 3

y

x

D(0,1) C(1,1)

(0,0)A

B(1,0)

Q(1,y)




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# 7 Paragraph for Q. 19 to Q. 2119. - A. 20. - B. 21 - C.

Triangle is right R = orthocentre ; M = circumcentre.

(i)223 1RM 1 12

2 2

M15

R(1, 7)Q( 15, 19

20

35 D

P ( 8,15)

232 7

(ii) Incentre 20 8 15 15 25(1) 160 225 25 360x 6

20 15 25 60 60

20 5 15 19 25 7 100 285 175 360y 660 60 60

hence incentre (–6 , –6) (can be

used to determine the equation of PD) 20.15r 5.

s 2.30

(iii) Coordinates of D using section formulea are 235,2

and PR11m2

equation PD is 11x 2y 78 0 a c 89. # 8 Paragraph for Q. 22 to Q. 24

22. - A. 23. - B. 24 - D.Interection point is (–1, –1). Now proceed.

# 9 Paragraph for Q. 25 to Q. 2725. - D. 26. - B. 27 - D.

(i) Area of 9 3 1

1 1DEF area( ABC) 7 1 1 3.4 4

1 1 1

(ii) Since P(a,b) is the circumcentre

2 2 2 2

2 2 2 2

a 7 b 1 a 9 b 3 .............(1)

a 7 b 1 a 1 b 1 .............(2)

Solving (1) and (2) a 4 and b 3

2 2R 4 7 3 1 5 a b c 4 3 5 12.

(iii)H CG

12

( y)x , (17/3,1/3) (4,3)

17 2 4 x 1 2 3 yx 17 8 9 y 53 3 3 3

point H 9, 5 .




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# 10 Paragraph for Q. 28 to Q. 3028. - D. 29. - B. 30 - A.

Tangent

2

2 2

2

y 2 m x 4

mx y 2 4m 0

2 4mp 21 m

1 2m 1 m

3m 4m 04m 0 or m3

O2

B

A

Y

X

4

P(4,2)

2 5

Hence equation of tangent is y = 2 and (with infinite intercept on x - axis)

or 4y 2 x 4 3y 6 4x 16 4x 3y 10 03

x - intercept 104

Ans. (ii) (B).

Variable line with mid point (h,k) x y 1,2h 2k

it touches the circle 2 2x y 4

2 2

2 2

1 1 1 124h 4k 41 1

4h 4k

locus is 2 2x y 1 Ans. (iii) (A)

# 11 Paragraph for Q. 31 to Q. 3331. - D. 32. - A. 33 - C.

Given 4a 3 x a 1 y 2a 1 0 3x y 1 a 4x y 2 0 family of lines passesthrough the fixed point P which is the intersection of 3x y 1 and 4x y 2 Solving P(1,2), now

(i)k k 2. 1h h 2

2 2locus is x x 1 y y 2 0 x y 2y x 0

2

2 2 21 5x y 1 2x 1 4 y 1 52 4

y

O x

(h,k)(1,2)

(ii) We have y 2m(x 1) ...........(1) this makes an angle of / 4 with 3x 4y 2 with slope

3/ 4

m 3/ 4 4m 31; 1 4m 3 4 4m1 3m / 4 4 3m

(with + ve sign) m 7

with – ve sign 14m 3 4 3m 7m 1 m rejected7

hence the line is

y 2 7 x 1 7x y 5 0.




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(iii) Again y 2 m(x 1) 2x 0; y 2 m; y 0, x 1m

22A 2 m 1 m 0m

4 42A 2 m 2 4 mm m

let m M M 0 42A 4 MM

224 M 4

M

228 MM

area is minimum if M = 2 m 2 min min2A | 8 A | 4.

# 12 Paragraph for Q. 34 to Q. 3634. - C. 35. - D. 36 - C.

(i)

22 2 2 2

2 22 2

2

r a b 1 a 1 b 4 2a 4 0 a 2

and a 1 b 4 a b 1 4 2a 2b b 2

r 9 r 3.

Ans.

(ii) 1 2 2 3S S 0 x 1 S S y 1 Radical centre 1,1 radius T 1L S 1

equation of circle is 2 2x 1 y 1 1 radius 1 and a 1; b 1 a b r 3.

(iii) 2 2

2 2

family of circles touches the line x 1 0 at 1,0 is

x 1 y 0 x 1 0 passing through 3,24 4 2 0 4

x y 6x 5 0 radius 9 5 2.




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# 13 Paragraph for Q. 37 to Q. 3937. - B. 38. - B. 39 - D.

Angles BEC, ABD, ABE and BAC are in A.P.let BEC 3 ABDABE and BAC 3Now, 3 3[using exterior angle theorem]

o o o

77 ,24 24

and From ABD

3 2 22 2

B 2 ,4 2

A , C ABC si 30 90 60 triangle6 3

(i) Area of circle circumscribing 21ABC

2 4

(ii) BOC is equilateral

23 114 2r

1 3s 4 32 2

(iii) 1 3 3BD OB sin sin BB' 2BD .3 2 3 4 2

Assertion & Reason Type

Que. 1. D. The given equation is 2 2x 3 2 y 1 0 hence it denotes only a point P 3,1 or two

imaginary lines through P 3,1 as 0.

Que. 2. A. Let x 20 and y 22 now, , x 20 X; y 22 Y if 20,22 (0,0) (21,24) (1,2)

(22,23) (2,1) .Que. 3. B. 1 2C :centre (3, 2); C : centre (4,3) radical axis of C1 and C2

is 1 2C C 0 2x 2y 14 0

x y 7 0 ................(1) since (1) passes through the centre of 2C (4,3) hence S-1 is correct.

2C

1C also (3,2) lies on 2C hence S-2 is correct but that is not becorrect explanation S - 1.




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Que. 4. D. Locus of P is polar of A (6,8) w.r.t. the circle S = 0 i.e., 2 2 2x y 6x y 6x 8y 5 0

1 1 1 1xx yy g x x f y y c 0 6x 8y 3(x 6) 4(y 8) 5 0 3x 4y 18 32 5 0

3x 4y 45 0 S - 1 is incorrect. Also the point A (6,8) lies outside S. Hence S-1 is false and S-2 is true.

Que. 5. D.

Que. 6. A.1 1

11 1

x y 1

3x2y dy dyC : 2x 0 m3 dx dx y

1 1

2 3 12 2

x y 1

ydy dyC : 3xy y 0 mdx dx 3x

1 2 1 2m .m 1 C and C are orthogonal.

Que. 7. (D) 2 2f (x, y) (2x y) (x y 3) S -1 is flase, It represent a point (1,2).Que. 8. (A) D,E,F are the middle points of the sides of the triangle.

D AE B 4F C

similiar triangles

Que. 9. (A.).The quadrilateral is obvioulsy a parallelogram and if the diagonals are at right angles, is mustbe a rhombus. Hence, the distance between the pairs of opposite sides must be the same

i.e.

2 2 2 2

2 2 2 2

r r ' r r 'p q p ' q '

p q p ' q '

Que. 10. (D). The joint equation of y = x and y x is x y x y 0 i.e. 2 2x y 0.

Que. 11. A. 1 2 3 1 2 32x x 3x 2y y 3yP ,6 6

hence P lies inside the triangle

( )1 1

Ax , y ( )2 2

Bx , y

( )3 3C x , y

P (a,b)

3

3

1 D 2

1 2 1 22x x 2 y y,3 3+ +

area of PBC area of ABC

Que. 12. A. 2 2 21 2 2 3 3 1 1 2 32cos cos 2cos cos 2cos cos cos cos cos

2 2 21 2 3 1 2 2 3 3 1sin sin sin 2sin sin 2sin sin 2sin sin 0

21 2 3 1 2 3cos cos cos sin sin sin 0

2 3 2 3cos cos cos 0 &sin sin sin 0

centroid and circumcentre of ABC is at origin ABC is equiliateral Orthocentre of ABC is also origin.

O

E

C

A

B

F

D




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Que. 13. A.

Que. 14. A Concurrency of lines 1/ 3 2 / 3 11/ 2 3 / 4 1

5 3 1

S-2 is correct.

Que. 15. C. 1L : 3x 4y 2 0 , 2L : 5x 12y 7 0 0,0 containing bisector is

3x 4y 2 5x 12y 75 13

39x 52y 26 25x 60y 35 14x 112y 9 0

None (0,0) containing bisector is 3x 4y 2 5x 12y 75 13

Que. 16. B. The points A and B may be on same side also

More than One Correct TypeQue. 1. A,C,D. Case - I : If lines form a triangle then n = 4 i.e., 3 excircles and 1 incircle

O

y

x

Case - II : If lines are concurrent or all 3 parallel then n = 0Case - III : If two are parallel and third cuts then n = 2 hence A,C,D.

Que. 2. A,C. S Area of OAC area of BCD 1 x y 11.x 0 x 1

2 2

x 1 y 1xS2 2

....(1)

Now 's CBD and OCA are similar




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2

22 2

2

y 1 1 x 1 x 1y 11 x x x

x 1 1/ x 1 x 1x xS2 2 2 2x

x x 1 2x 2x 1 1x 12x 2x 2x

1x 1 22x

y

x(0,0)O

(0,1)A B(1,1)(x,1)C

D(1,y)

A is minimum if 1x2x

i.e, 1x2

which lies in (2/3, 1) and minA 2 1 which lies in

1/ 3,1/ 2 A & C.

Que. 3. B,C,D. (A). If 5 5 ;6 2 12

now, 1sin ;2 c

o

1 2 2 4c 6 2sin 75 6 23 1

(A) is not correct.

(B). o o

3 1 1 4;c 5 12 10 sin 54 cos36 5 1

B is correct.

(C). 1,c 2

2 4 sin / 4

(C) is correct.

(D). o

1,c 22 6 sin 30 (D) is correct.

Que. 4. C,D. a sec y x tan bsec x y tan by squaring and adding

2 2 2 2 2 2 2a b sec x 1 tan y 1 tan 2 2 2 2x y a b (C) and (D).Que. 5. A,C,D.Que.6. A. 2x y 1 x 3y 2 0 hence the lines are

2x y 1 0 1 5P ,x 3y 2 0 7 7

equation of the two lines lines join-

ing origin and the point of intersection of 3x 5y 2 and f (x, y) 0

is 22 2 3x 5y 3x 5y 2 3x 5y

2x 5xy 3y 02 4

2 22 22x 5xy 3y 3x 4y 3x 5y 0

2 22x 5xy 3y 0

Y

XO

c = 1(c,0)

1




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Que. 7. C,D.

Que. 8 . A,C,D. Using property of anlge bisector sec xtan 1 x

x tan sec xsec

secxsec tan

1x1 sin

Que. 9. A,B,C. 2f (x) Ax Bx C A a b 2c a c b c 0 A 0 mouth opensupwards now x= 1 is obvious solution terefore both roots are rational.

ve ve

b a c a 0 B 0;

0 c b a

vertex B 0

2A hence abscissa ‘a’ of the vertex > 0

O

y

x1

(D) need not be correct as with a 5,b 4,c 2,P 0 and a 6, b 3,c 2, P 0 (A), (B) and (C) are correct.

Que. 10. C,D. Abscissa coresponding to the vertex is given by 1x 1

sin

is the vertex

the graph of 2f (x) sin x 2x b as shown x 1 minimum of 2f (x) (sin )x 2x b 2 must be greater then zero but minimum is at x =1 i.e.,sin 2 b 2 0; b 4 sin , (0, );b 4 as sin 0 in (0, )

O

y

x1 x=cosec

vertex

Hence (C) and (D) are correct.Alternatively : 2f (x) (sin )x 2x b 2 f (x) 0 x 1 now vertex 1Case - I : (1) D 0, (2) F(1) 0

D 0 4 4sin (b 2) 2cos ec 2 b

b cosec 2 b 3 ........(1)

O

y

x1

and f (1) 0 sin b 4 0b 4 sinb 4 ..........(2)




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ES.C

OM Case - II : f (1) 0 and D 0 f (1) 0

b 4 D 0 b 3b 4 (C) and (D) are the correct answers

O

y

x1

Que. 11. B,C,D. 22x 2x a 3 must be positive hence D 0

i.e., 54 8(a 3) 0 1 2a 6 0 2a 5 a ..........(1)2

Also base of the logrithm

7 a 0 and 7 a 1 a 7 & a 6 ...........(2) from (1) and (2) 5a ,6 6,72

.

Que. 12. A.2 2ax 2cxy by d 0 2abd 0 0 0 dc 0 2d ab c 0 2d c ab 0

either d = 0 and 2c ab or 2c ab

Que. 13. B,D. 2 2 24a c 4ac b 2a c b Or b now ax by c 0 ax by b 2a 0

a x 2 b y 1 0 (2, 1) (B)

or ax by b 2a 0 a x 2 b y 1 0 (2,1) (D).

Que. 14. B,D. For concurrency

2

2

1 11 1 01 1

2 4 2 2

2 4 2 2

2 4 2

1 1 1 1 0

1 1 1 0

1 2 2 0

let 2 2 2t t t 1 2t 2 0 t t 1 t 1 2 t 1 0 t t 2 0 t 2 t 1 0 t 2

2 2; 2.

Que. 15. A,B. L must be angle. bisector of 1 2L & L

D

1L

L

2L

A

P

L is given by 1 2 1 23x 4y 1 5x 12y 2 P DA P DA P AD P AD.

5 13




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Que. 16. C,D.1 12 2 h 1 h2 2

P wil lie on line parallel to AB at a perpendicular distanceof

12 from AB. locus of P is

5 3 3 5y x or y x y x 1 or y x 12 2 2 2

P

B(3,3)

(1,1)A (2,2)

1 1 1 1 3 52 22 22 2 2 2

+ + =

1 1 5 32 22 2 2 2

+ =

Match Matrix TypeQue. 1. A - P,S. B - P. C - Q. D - P,Q,R.

A. Ovioulsy trapezium a 37

a bb 37

Here isoscelse trapazium, hence cyclic quadrilateral

also P,S.

B.

btanb a cac bd

ac d tanc

hence cyclic quadrilateral P..




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C.c cax by c 0 if y 0, x if x 0, ya b

rhombus Q.

D. 2x 6 x 2 0 x 6 and x 2 y 14y 45 0 y 9 y 5 0 a square.

Que. 2. A - Q,R,S. B - P. C. - Q,S. D - Q.

A.1 2

10 10h 10 h 1010 10

a square of side 10 are 10 Q,R,S.

3x y 0- = 3x y5 0- =

x 3y 10 0+ - =

x 3y 20 0+ - =

B. AB DC5 9m 5 m 91 1

not a parallelogram P..

||| y (C) and (D) can be checked.

D( 2, 4)-A(1, 2)

B(2, 3)-

C( 1, 5)- -

y

x




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Que. 3. A - S. B - R. C - Q.

A. 1

2

20d 45 square20d 45

3x 4y+11=0

3x 4y

4x + 3y +3 = 04x + 3y 17 = 0

B.1

2

2d52d5

interior not 90o rhombus

2x + y 1 = 0

2x + y = 0

x + 2y = 0x + 2y = 0

C. 2 22x 7xy 3y 5x 10y 25 0 x 3y 5 2x y 5 the point of intersection is (4,3)

homogenising f(x,y) = 0 and x 2y 5 0 we get the homogeneous equation 2 22x 7xy 3y 0

hence OAPB is a parallelogram

O

y

x

A(3,1)

(1,2)(4,3)P x 3y+5=0

x+2y 5=0

2x y 5=0

Que. 4. A - S. B - R. C - Q. D - P.Can be easily analysed.

Subjective TypeQue. 1. (75 sq. unit) Let radius of circle with centre A = R and radius

of the circle with centre B = r now AC = R ; BD = r Area of the

rhombus 1 2d d R.r2 2

now R – r = 10 ................ (1)

also in right 2 2

2 2R r 100 R r 4002 2

................(2)

Squaring (1), 2 2 RrR r 2Rr 100; 2Rr 300 75 sq. units.2




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Que. 2. (40).The equation of circle taking AB as diameter (x 2)(x 4) (y 1)(y 3) 0 ........... (1)The equation of the line joining the points A and B is x y 1 0 ..................... (2)The equation of members of family of circle passing through A and B is given by

S x 2 x 4 y 1 y 3 x y 1 0 where is parameter, , R

2 2S x y 6 x 4 y 11 0 ............ (1), Let the circles which cuts the members of

circles be 2 21S x y 2gx 2fy c 0 .................. (2), Applying condition of orthogonality for (1)

and (2), we get 6 42g 2f c 11 .2 2

i.e., ( 6g 4f c 11) g f 1 0 . This will

also hold for all R we have 6g 4f c 11 0 and g f 1 0 solving these equations

for g and f in terms of c, we get c 15 c 5g and f10 10

substituting the values of g and f in terms of

c in (2), we get the circles cutting the circle of system (1) orthogonally as

2 2 2 2c 15 c 5 cx y 2 x 2 y c 0 or x y 3x y (x y 5) 010 10 5

which represents

equatoin of family of circle passing through two fixed points whose coordinates obtained by solvingequation 2 2i.e., solving x y 3x y 0 and x y 5 0 2x 6x 10 0(D 0) 1 2x x 6;

1 2x x 10 ||| y 21 2 1 2y 4y 5 0 y y 4; y y 5 3 3 3 3

1 2 1 2x x y y

3 31 2 1 2 1 2 1 2 1 2 1 2x x 3x x x x y y 3y y (y y ) 216 30(6) 64 60 36 4 40

Que. 3. (465) Arranging the lines in descensing order 1 2 3m 5; m 3; and m 1

2 3 3 11 2

1 2 2 3 3 1

m m m mm m 2 1 3 1 1 5 3tan a ; tan B 2; tan C1 m m 1 15 8 1 m m 1 3 1 m m 1 5 2

2 1 9 1 256 144 401tan A 4 p q 465.64 4 64 64

Que .4. (1394.25)777a 9, b 16, h ;f 496;c k

2

condition; 2

2 2 22 2 777abc at bg 0 144k 9 496 16 0 or 144k 4 777 9 4962

2 2 2 236k 777 9 248 36k 9 259 248 36k 9 507 11 K 1394.25.

Que. 5. (0530.00) FG 81 144 15 slope of 12 4FG9 3

slope of 3L4

FP 2 5 8 3 30P is 1; 11 P 1,11PG 1 3 3

equation P is 3y 11 x 1

4

15 L

GF 2 1P

10 5(4,15)( 5,3)




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4y 44 3x 3 3x 4y 47 a 3; b 4; c 47 8a 9b 10c 24 36 470 530. Que. 6. (0210.00) Circle touches both the coordinates axes and also the line L

1 1 2 2

2 1

2 1

1 1 2 2

centre can be r , r or r , r

3r 4r 120 r 7r 120 5r5

2r 120 or 12r 120r 60 or r 10

r , r 10,10 and r , r 60,60

Normal for circle1 with centre 10,10 is

1 1r , r

1 1r , r

L: 3x + 4y = 120x

y

4y 10 x 10 3y 30 4x 40 4x 3y 10 or 16x 12y 403

................(1)

and tangent line L is 3x 4y 120 or 9x 12y 360 ................(2)

solving (1) and (2) 16x 12y 409x 12y 360

25x 400 x 16 and y 18 Normal for circle 2 with centre (60,60) is

4y 60 x 60 3y 180 4x 240 4x 3y 603

or 16x 12y 2409x 12y 360

..................... (3) solving with equation (2)

25x 600 x 24 and y 12 16,18 ; 24,12

1 2 1 2p p q q a b c d 16 24 18 12 10 10 60 60 210 .

Alternatively :600r 10

s 60

Que. 7. (40). Equation of the diagonal AC 1 3 2 4u u u u

1 4 2 3

2x y 7 3x 2y 5 3x 2y 4 2x y 5

5 2x y 7 3x 2y 5 3x 2y 4 2x y 2011x 19y 35 7x 14y 20 18x 33y 15 0

6x 11y 5 0 a 6 and b 11equation of the other diagonal ie. BD is u u u u 2x y 7 3x 2y 4

2x y 5 3x 2y 5 4 2x y 7 3

x 2y

5 2x y 5 3x 2y 25 29x 10y 2825x 5y 25 54x 15y 3 0 18x 5y 1 0

p 18 and q 5a b p q 6 11 18 5 40.

AB

C

D

1u 2x y 7 0= + =

2u 2x y 5 0= =

4u 3x2y 4 0

= ++ =

3u 3x2y 5 0

=+ =

Documents

JEE (MAIN / ADVANCE ) MATHS by SUHAG KARIYA ... of the triangle formed by the line x + y = 3 and the angle bisectors of the line pair x y 4y 4 02 2 is (code-V2T13PAQ15