HTTP://WWW.NARAYANADWARKA.BLOGSPOT.IN JEE–Advanced Maths – Assignment Only One Correct Answer Type 1. The locus of the orthocenter of the triangle formed by the lines (1+P)x–Py + P(1+P) = 0, (1+q)x – qy+q(1+q) = 0 and y = 0, where p ≠q, is (a) a hyperbola (b) a parabola (c) an ellipse (d) a straight line 2. If two different tangents of y 2 = 4x are the normals to x 2 = 4by, then (a) 1 b 2 2 (b) 1 b 2 2 (c) 1 b 2 (d) 1 b 2 3. Minimum distance between the curves 2 y x 1 and 2 x y 1 is equal to (a) 3 2 4 (b) 5 2 4 (c) 7 2 4 (d) 1 2 4 4. Minimum distance between the curves 2 y 4x and 2 2 x y 12x 31 0 is equal to (a) 21 (b) 26 (c) 5 (d) 28 5. Sides of a equilateral ABC touch the parabola 2 y 4ax, then point A, B and C lie on (a) 2 2 y x a 4ax (b) 2 2 y 3x a ax (c) 2 2 y 3x a 4ax (d) 2 2 y x a ax 6. Length of the focal chord of the ellipse 2 2 2 x y 1, a b that is inclined at an angle with the x–axis, is equal to (a) 2 2 2 2 2 2b a a sin b cos (b) 2 2 2 2 2 2b a a cos b sin (c) 2 2 2 2 2 2a b a sin b cos (d) 2 2 2 2 2 2a b a cos b sin 7. Eccentricity of the ellipse 2 2 5x 6xy 5y 8 (a) 1 2 (b) 3 2 (c) 2 3 (d) 1 3 8. PQ is a chord of the ellipse 2 2 2 2 x y 1. a b If O is the centre of the ellipse and eccentric angle of the points P and Q differ by , 2 then area of triangle OPQ is (a) ab (b) 2ab (c) ab / 2 (d) ab / 4
12
Embed
JEE Advanced Maths Assignment - NARAYANA IIT ...// JEE–Advanced Maths – Assignment Only One Correct Answer Type 1. The locus of the orthocenter of the triangle formed by the lines
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
HTTP://WWW.NARAYANADWARKA.BLOGSPOT.IN
JEE–Advanced
Maths – Assignment Only One Correct Answer Type
1. The locus of the orthocenter of the triangle formed by the lines (1+P)x–Py + P(1+P) = 0,
(1+q)x – qy+q(1+q) = 0 and y = 0, where p ≠q, is
(a) a hyperbola (b) a parabola
(c) an ellipse (d) a straight line
2. If two different tangents of y2 = 4x are the normals to x
2 = 4by, then
(a) 1
b2 2
(b) 1
b2 2
(c) 1
b2
(d) 1
b2
3. Minimum distance between the curves 2y x 1 and
2x y 1 is equal to
(a) 3 2
4 (b)
5 2
4
(c) 7
24
(d) 1
24
4. Minimum distance between the curves 2y 4x and
2 2x y 12x 31 0 is equal to
(a) 21 (b) 26
(c) 5 (d) 28
5. Sides of a equilateral ABC touch the parabola 2y 4ax, then point A, B and C lie on
(a) 22y x a 4ax (b)
22y 3 x a ax
(c) 22y 3 x a 4ax (d)
22y x a ax
6. Length of the focal chord of the ellipse 2
2 2
x y1,
a b
that is inclined at an angle with
the x–axis, is equal to
(a) 2
2 2 2 2
2b a
a sin b cos (b)
2
2 2 2 2
2b a
a cos b sin
(c) 2
2 2 2 2
2a b
a sin b cos (d)
2
2 2 2 2
2a b
a cos b sin
7. Eccentricity of the ellipse 2 25x 6xy 5y 8
(a) 1
2 (b)
3
2
(c) 2
3 (d)
1
3
8. PQ is a chord of the ellipse 2 2
2 2
x y1.
a b If O is the centre of the ellipse and eccentric
angle of the points P and Q differ by ,2
then area of triangle OPQ is
(a) ab (b) 2ab
(c) ab / 2 (d) ab / 4
HTTP://WWW.NARAYANADWARKA.BLOGSPOT.IN
9. An ellipse with major and minor axes of length 10 3 and 10 respectively, slides along
the co-ordinate axes and always remains confined in the first quadrant. The locus of the
centre of the ellipse will be the arc of a circle. The length of this arc will be equal to
(a) 10 units (b) 5 units
(c) 5
4 units (d)
5
3 units
10. Consider a circle 2 2 2x y d and an ellipse
2 2
2 2
x y1,
a b d > maxi. {a,b} from a
variable point P on the circle, tangents PA and PB are drawn to the ellipse. Locus of the
mid–point of chord AB is
(a) 2 2
2 2 4
2 2
x yx y d
a b
(b)
22 2
2 2 2
2 2
x yx y d
a b
(c) 2 2 2 2 4b x a y d (d)
2 2 2 2 4a x b y d
11. PQ is variable chord of the ellipse 2 2
2 2
x y1.
a b if PQ subtend a right angle at the centre
of the ellipse then 2 2
1 1,
OP OQ (O being the origin) is equal to
(a) 2 2
1 1
a b (b)
2 2
2 1
a b
(c) 2 2
1 2
a b (d)
2 2
1 12
a b
12. Consider any chord of the hyperbola xy=c2 that is parallel to the line y = x. Circles are
drawn having this chord as diameter. All these circles will pass through two fixed points
whose co–ordinates are
(a) c 2,c 2 , c 2, c 2 (b) c 2,c 2 , c 2, c 2
(c) c,c , c, c (d) c,c , c,c
13. The tangent at a point ‘P’ on the hyperbola 2 2
2 2
x y1,
a b meets one of its directix at the
point Q. If the line segment PQ subtends an angle at the corresponding focus, than is
always equal to
(a) 4
(b)
2
(c) 3
(d)
6
14. If tangent and normal to the hyperbola xy = c2, at any point ‘P’ cuts off intercept a1 and
a2 on the x–axis respectively and b1 and b2 on the y–axis, then 1 2 1 2a a b b is always
equal to
(a) –1 (b) 1
(c) 0 (d) none of these
15. Locus of the mid–point of the chord of the hyperbola 2 2 2x y a , that touch the
parabola 2y 4ax is
(a) 2x x a y (b) 2y x a y
HTTP://WWW.NARAYANADWARKA.BLOGSPOT.IN
(c) 3 2x x a y (d) 3 2y x a x
16. Locus of the point of intersection of tangent drawn to the hyperbola 2xy c at the
extremities of any normal chord is
(a) 2
2 2 2x y c xy 0 (b) 2
2 2 2x y c xy 0
(c) 2
2 2 2x y 4c xy 0 (d) 2
2 2 2x y 4c xy 0
17. A water jet from a fountain reaches its maximum height of 4 m at a distance 0.5 m from
the vertical passing through the point O of water outlet. The height of the jet above the
horizontal OX at a distance of 0.75m from the point O is
(a) 5m (b) 6m
(c) 3m (d) 7m
18. Equation of a normal to the curve y = x2–6x + 6 which is perpendicular o the straight line
joining the origin to the vertex of the parabola is
(a) 4x – 4y – 11 = 0 (b) 4x – 4y + 1 = 0
(c) 4x – 4y – 21 = 0 (d) 4x – 4y + 21 = 0
19. A circle drawn on any focal chord of the parabola y2=4ax as diameter cuts parabola at
two points ‘t’ and ‘t’ (other than the extrimity of focal chord) the
(a) tt = –1 (b) tt = 2
(c) tt = 3 (d) none of these
20. Two parabolas with the same axis, focus of each being exterior to the other and the latus
rectum being 4a and 4b. The locus of the middle points of the intercepts between the
parabolas made on the lines parallel to the common axis is a
(a) straight line if a = b (b) parallel line if a ≠ b
(c) parabola for all a, b (d) ellipse if b > a
21. If three distinct normal can be drawn to the parabola y2 – 2y = 4x – 9 from the point
(2a, 0) then range of values of a is
(a) No real values possible (b) (2, )
(c) (– , 2) (d) none of these
22. If the curves 2
2xy 1
4 and
22
2
xy 1
a for suitable value of a cut on four concylic
points, the equation
(a) 2 2x y 2 (b)
2 2x y 1
(c) 2 2x y 4 (d) none of these
23. Angle subtended by common tangents of two ellipse 2
4 x 4 +25y2
= 100 and
2 24 x 1 y 4 at origin is
(a) 3
(b)
4
(c) 2
(d) none of these
24. If PQR be an equilateral triangle inscribed in the auxillary circle of the ellipse 2 2
2 2
x y1
a b (a>b) and PQR be corresponding triangle inscribed within the ellipse then
centriod of the triangle PQR lies at
(a) centre of ellipse (b) focus of ellipse
(c) between focus and centre of major axis (d) none of these
HTTP://WWW.NARAYANADWARKA.BLOGSPOT.IN
25. The normal at a variable point P on the ellipse 2 2
2 2
x y1
a b , a>b of eccentricity ‘e’ meets
the axes of the ellipse Q and R then the locus of the mid point QR is coinc with a
eccentricity e such that
(a) e is independent of e (b) e = 1
(c) e = e (d) e = 1
e
26. If a variable line x cos ysin P, which is a chord of the hyperbola
2 2
2 2
x y1 b a
a b , subtends a right angle at the centre of the hyperbola then it always
touches a fixed circle whose radius is
(a) ab
b 2a (b)
a
a b
(c) 2 2
ab
b a (d)
ab
b a b
27. Let any double ordinate PNP of the hyperbola 2 2x y
125 16
be produced both sides to
meet the asymptotes in Q and Q, then PQ. PQ is equal to
(a) 25 (b) 16
(c) 41 (d) none of these
28. The equation of the line of latus rectum of the rectangular hyperbola xy = c2 is
(a) x y 2c (b) x y 2 2c
(c) x y 2c (d) x y 0
29. The line parallel to the normal to the curve xy = 1 is/are
(a) 3x 4y 5 0 (b) 3x 4y 5 0
(c) 4y 3x 5 0 (d) 3y 4x 5 0
30. The line 22px y 1 p 1 p 1 for different values of p touches
(a) An ellipse of eccentricity 2
3 (b) An ellipse of eccentricity
3
2
(c) Hyperbola of eccentricity 2 (d) none of these
31. If , are the eccentric angels of the ends of a focal chord of the ellipse 2 2
2 2
x y1,
a b
then the eccentricity of the ellipse is
(a)
sin sin
sin
(b)
sin sin
sin
(c)
cos cos
cos
(d)
cos cos
cos
32. If chords of contact of tangents from two points 1 1x , y and 2 2x , y to the ellipse
2 2
2 2
x y1
a b are at right–angle, then 1 2
1 2
x x
y y
(a) 2
2
a
b (b)
2
2
b
a
HTTP://WWW.NARAYANADWARKA.BLOGSPOT.IN
(c) 4
4
a
b (d)
4
4
b
a
33. PM and PN are the perpendiculars from any point P on the rectangular hyperbola xy = c2
to the asymptotes. The locus of the mid–point of MN is a hyperbola with eccentricity
(a) 2 (b) 2
(c) 1
2 (d) 2 2
34. An ellipse has eccentricity 1
2 and one focus at
1s ,1
2
. Its one directix is the common
tangent, (nearer to S) to the circle 2 2x y 1 and
2 2x y 1. The equation of the
ellipse in standard form is
(a) 2
219 x 12 y 1 1
3
(b)
221
12 x 9 y 1 13
(c)
2
2
1x
y 121
12 9
(d) 2
213 x 4 y 1 1
2
35. If 1p and 2p are the perpendiculars from the origin on the straight lines
x sec ycosec 2a and x cos ysin a cos 2 , then
(a) 2 2 2
1 24p p a (b) 2 2 2
1 2p 4p a
(c) 2 2 2
1 2p p 4a (d) 2 2 2
1 2p p a
36. If c is the centre and A, B are two points on the conic 2 24x 9y 8x 36y 4 0 such
that ACB ,2
then
2 2CA CB is equal to
(a) 13
36 (b)
36
13
(c) 16
33 (d)
33
16
37. A point moves such that the sum of the squares of its distances from the two sides of
length a of a rectangle is twice the sum of the squares of its distances from the other two
sides of length b. the locus of the point can be
(a) a circle (b) an ellipse
(c) a hyperbola (d) a pair of lines
38. If the tangent at the point 1 1P x , y to the parabola 2y 4ax meets the parabola
2y 4a x b at Q and R, then the mid–point of QR is
(a) 1 1x b, y b (b) 1 1x b, y b
(c) 1 1x , y (d) 1 1x b, y
39. If , are the eccentric angels of the extremities of a focal chord of an ellipse, then the
eccentricity of the ellipse is
(a)
cos cos
cos
(b)
sin sin
sin
HTTP://WWW.NARAYANADWARKA.BLOGSPOT.IN
(c)
cos cos
cos
(d)
sin sin
sin
40. PQ and RS are two perpendicular chords of t rectangular hyperbola xy = c2. If O is the
centre of the hyperbola, then the product of the slopes of OP, OQ, OR and OS is equal to
(a) –1 (b) 1
(c) 2 (d) 4
41. Let f be the focus of the parabola. From the end point (P) of focal chord PF
perpendicular PM is drawn to directix. From P a line is drawn through the mid–point (R)
of FM, then the angle between PR and FM is
(a) 45 (b) 60
(c) 90 (d) none of these
42. A normal drawn to parabola 2y 4ax meet the curve again at Q such that angle subtend
by PQ at vertex is 90 then coordinate of P can be
(a) 8a, 4 2a (b) 8a,4a
(c) 2a, 2 2a (d) none of these
43. If the focus of parabola (y–k)2=4(x–h) always lies between the line x + y = 1 and
x + y = 3 then
(a) 0 h k 2 (b) 0 h k 1
(c) 1 h k 2 (d) 1 h k 3
44. If xy = m2–4 be a rectangular hyperbola whose branches lies only in the 2
nd and 4
th
quadrant then
(a) m 2 (b) m 2
(c) m 2 (d) not possible
45. A tangent to the ellipse 2 2x y
125 16
at any point P meet the line x = 0 at a point Q. Let
R b the image of Q in the line y =x, then circle whose extremities of a diameter are Q
and R passes through a fixed point. The fixed point is
(A) (3,0) (B) (5,0)
(C) (0,0) (D) (4,0)
46. Number of points on the ellipse 2 2x y
150 20
from which pair of perpendicular tangents
are drawn to the ellipse 2 2x y
116 9
is
(A) 0 (B) 2
(C) 1 (D) 4
47. An ellipse is drawn with major and minor axes of lengths 10 and 8 respectively. Using
one focus and centre, a circle is drawn that is tangent to the ellipse, with no part of the
circle being outside the ellipse. The radius of the circle is :
(A) 2 (B) 3
(C) 3 (D) 4
48. A circle has the same centre as an ellipse and passes through the foci F1 and F2 of the
ellipse, such that the two curves intersect in 4 points. Let P be any one of their point
of intersection. If the major axis of the ellipse is 17 and the area of the triangle PF1F2
is 30, then the distance between the foci is
(A) 11 (B) 12
(C) 13 (D) 15
HTTP://WWW.NARAYANADWARKA.BLOGSPOT.IN
49. Any ordinate MP of an ellipse 2 2x y
125 9
meets the auxiliary circle in Q, then locus
of point of intersection of normals at P and Q to the respective curves, is
(A) 11 (B) 12
(C) 13 (D) 15
50. Number of distinct normal lines that can be drawn to ellipse 2 2x y
1169 25
from the
point P(0,6) is
(A) one (B) two
(C) three (D) four
51. If PQ is focal chord of ellipse 2 2x y
125 16
which passes through S (3,0) and PS = 2
then length of chord PQ is
(A) 8 (B) 6
(C) 10 (D) 4
52. If P is a moving point in the xy-plane is such a way that perimeter of triangle PQR is
16 {where Q (3, 5) , R (7,3 5 )} then maximum area of triangle PQR is
(A) 6 sq. unit (B) 12 sq. unit
(C) 18 sq. unit (D) 9 sq. unit
53. If f(x) is a decreasing function then the set of values of ‘k’, for which the major axis
of the ellipse
2 2
2
x y1
f k 11f k 2k 5
is the x-axis, is
(A) k 2,3 (B) k 3,2
(C) k , 3 2, (D) k , 2 3,
54. The equation to the locus of the middle point of the portion of the tangent to the
ellipse 2 2x y
116 9
included between the co-ordinate axes is the curve
(A) 9x2
+ 16y2 = 4 x
2y
2 (B) 16x
2 + 9y
2 = 4 x
2y
2
(C) 3x2
+ 4y2 = 4 x
2y
2 (D) 9x
2 + 16y
2 = x
2y
2
55. From a point P(1,2) pair of tangent’s are drawn to a hyperbola ‘H’ in which one
tangent to each are of hyperbola. H are 3x y 5 0 and 3x y 1 0 3x y 0
then eccentricity of H is
(A) 2 (B) 2
3
(C) 2 (D) 3
56. If a variable lines has its intercepts on the coordinates axes e, e where e e
,2 2
are the
eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches
the circle 2 2 2x y r , where r =
(A) 1 (B) 2
(C) 3 (D) can not be decided
57. If angle between asymptote’s of hyperbola 2 2
2 2
x y1
a b is 120 and product of
perpendiculars drawn from foci upon its any tangent is 9, then locus of point of
intersection of perpendicular tangent of the hyperbola can be
(A) 2 2x y 6 (B) 2 2x y 9
HTTP://WWW.NARAYANADWARKA.BLOGSPOT.IN
(C) 2 2x y 3 (D) 2 2x y 18
58. C be a curve which is locus of point of intersection of lines x = 2 + m and my=4–m.
A circle 2 2
s x 2 y 1 25 intersects the curve C at four points, P, Q, R and S.
If O is centre of the curve C, then OP2 + OQ
2 + OS
2 is
(A) 50 (B) 100
(C) 25 (D) 25/2
59. The combined equation of the asymptotes of the hyperbola 2 22x 5xy 2y 4x 5y 0 is