Phase Test (Screening)

PAGE AITS_MAT(PT_03) - 8

NARAYANA IIT ACADEMY

(Andhra Pradesh, Kota and Delhi)

Max. Marks : 150 AITS MATHEMATICS Time: 2hrs.

Note : (1) Each question answered correctly is given +3 marks

(2) Each question answered incorrectly is given ( 1 mark

(3) 1 ( V questions are comprehensive containing 4 questions each1.A circle touches the y-axis and also touches the circle with center at (4, 0) and radius 3. The locus of the center of the circle is (a) a circle (b) an ellipse (c) a parabola(d) a hyperbola

2.The locus of a point P(h, k) moving under the condition that the line y = kx + h is a tangent to the parabola y2 = 4ax is (a) a circle (b) an ellipse (c) a hyperbola (d) a parabola

3.Let P(1, 4) and Q is a variable point on the lines |y| = 5, If PQ ( 3, then the number of positions of Q with integral coordinates is (a) 5(b) 6(c)8(d) 7

4.Let A(-3, 2), B(4, 3) and P(2a + 3, 2a - 3) is a variable point such that PA + PB is the minimum. Then a is (a) 0(b)

(c)

(d) 3

5.Equation of the image of the lines y = |x 1| by the line x = 3 is (a) y = |x 5| (b) |y| = x 5 (c) |y| + 4 = x(d) y = |x 5

6.If (, (, ( ((< ( < () are the points of discontinuity of the function , then the values of a for which the point ((, () and ((, a2) lie on the same side of the line 4x 3y + 4 = 0 are given by

(a) ((, 2)(b) (2, 2)

(c) (2, ()(d) R

7a, b, c, d, e, f, ( R and (a2+b2 + c2 + d2)x2 2(ab + bc + cd + de)x + (b2 + c2 + d2 + e2) ( 0, then the lines ax + by + c = 0 and cx + dy + e = 0 are (a) perpendicular (b) parallel (c) coincident (d) intersecting

8.If the normal at three points (al2, 2al), (am2, 2am), (an2, 2an) of the parabola y2 = 4ax are concurrent, then the common roots of the equations lx2 + mx + n = 0 and a2(b2 c2) x2 + b2(c2 a2)x + c2(a2 b2) = 0 is (a) independent of a, b, c but dependent on l, m, n

(b) independent of l,m,n but dependent on a,b, c(c) independent of a, b, n but dependent on c, l, m

(d) independent of a, b, c, l, m, n,9.If f(x) = and f(x) ( ((, (], then a ray of light parallel to the axis of the parabola y2 = 8x, after reflection from the internal surface of the parabola will necessarily pass through the point (a) ((, ()(b) ((, ()(c) ((, ()(d) ((, ()

10.The locus of the foot of the perpendicular from the origin to chords of the circlex2 + y2 2x 4y 4 = 0 which subtend a right angle at the origin is

(a)x2 + y2 x 2y 2 = 0(b)2(x2 + y2) 2x 4y + 3 = 0

(c)x2 + y2 2x 4y + 4 = 0(d)x2 + y2 + x + 2y 2 = 0

11.If x 3y + 1 = 0 is a diameter of a circle circumscribing the rectangle ABCD. A is (2, 5) and B is (6, 5). Then which of the statement is not true

(a) Area of the rectangle ABCD is 64 sq. units

(b) Centre of the circle is (2, 1)

(c) The other two vertices of the rectangle are (2, 3) and (6, 3)

(d) Equation of the sides are x = 2, y = 3, x = 5 and y = 6

12.If the focus of parabola (y b)2 = 8(x a) always lies between the line x + y = 2 and x + y = 6, then a + b lies in interval

(a)(0, 2)

(b)(0, 1)

(c)(0, 4)

(d)(1, 4)

13.The straight line y = x + (( > 0) touches the parabola y2 = 16 (x + 4), then the minimum value taken by ( is.

(a)4

(b)8

(c)16

(d)32

14.If two distinct chords are drawn from the point (4, 2) on the parabola x2 = 8y are bisected on the line y = mx, then the set of values of m is given by

(a)

(b)R

(c)(0, ()

(d)

15.A chord of slope m of the circle x2 + y2 = 2 touches the parabola y2 = 8x, then set of values of m is given by

(a)

(b)(1, 1)

(c)

(d)

16.The point on x + y = 2, which is nearest to the parabola y2 = 4(x 6) is

(a)

(b)

(c)

(d)

17.If the area of the triangle with vertices at ((, (), ((, () and ((, () is A, then the area of the triangle with vertices at and is

(a)A

(b)

A

(c)

A

(d)

A

18.If O is the origin and OA, OB are the tangents from the origin to the circle x2 + y2 4x + 2y + 4 = 0, the circumcentre of the triangle OAB is

(a)

(b)

(c)

(d)

19.The range of values of b such that the angle ( between the tangents drawn from (0, b) to the circle x2 + y2 = 4 lies in is (where ( is the angle which contains the circle)

(a)

(b)

(c)

(d)(4, 4)

20.The equation of the chord of the circle x2 + y2 6x + 2y + c = 0 passing through the point (1, 2) farthest from the center is

(a)2x 3y 4 = 0(b)2x + 3y + 4 = 0

(c)2x + 3y 4 = 0(d)2x 3y + 4 = 0

21.If , the family of lines ax + by + c = 0 is concurrent at the point

(a)

(b)

(c)

(d)

22.A curve y = f(x) passes through the point (6, 8) and the normal to the curve at that point happens to be a tangent to the circle x2 + y2 = 100. The value of is

(a)

(b)

(c)

(d)

23.If the straight lines (m + 1)x + ay + 1 = 0, (m + 2)x + by + 1 = 0, and (m + 3)x + cy + 1 =0, m ( 0, are concurrent, then a, b, c are in

(a)A.P. only for m = 1(b)H.P. for all m

(c)G.P. for all m(d)A.P. for all m

24.Vertices of a variable triangle are (5, 12), (13 cos(, 13 sin (), and (13 sin (, 13 cos (). Then locus of its orthocentre is

(a)x y + 7 = 0

(b)x y 7 = 0

(c)x + y 7 = 0

(d)x + y + 7 = 0

25.A straight line L with positive slope passes through the points (4, 4) and cuts the co-ordinate axes in fourth quadrants at points A and B. As L varies, the absolute minimum value of OA + OB is (0 is origin)

(a)0

(b)16

(c)8

(d)18

26.The equation represents

(a)a circle

(b)a pair of straight lines

(c)an ellipse with foci ( 4, 0)

(d) a line segments joining points (4, 1) and (4, 1)

27.If the vertices A and B of a triangle ABC are given by (1, 3), and (2, 7) and C moves along the line L : 3x + 2y + 1 = 0, the locus of the centroid of the triangle ABC is a straight line parallel to

(a)AB

(b)BC

(c)CA

(d)L

28.If the parabola y = ax2 + 2cx + b touch the x-axis. Then the line ax + by +c = 0 (a) passes through a fixed point

(b) has a fixed direction (c) makes a triangle with co-ordinate axes of constant area(d) data insufficient

29.Centroid of a triangle is origin. Circumcentre of the triangle is the fixed point from where all the lines ax + by + c = 0 pass, where a, b, c are distinct non-zero real parameters following the relation a3 + b3 + c3 = 3abc. The orthocentre of the triangle will be (a)

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

30.A point is taken on the director circle of the circle x2 + y2 = 8 and tangents are drawn from it to the parabola y2 = 16x. the locus of the middle points of chord of contact will be (a) y2 + (y2 8x)2 = 1024(b) 64y2 + (y2 8x)2 = 1024(c) y2 (y2 8x)2 = 1024(d) 64y2 (y2 8x)2 = 1024

P-1.Find the point P, on the circle x2 + y2 6x 8y + 16 = 0, where O is the origin and OX is the positive side of the x-axis.

31.(POX is minimum

(a)

(b)

(c)(0, 4)(d)

32.OP is maximum

(a)

(b)

(c)

(d)

33.OP is minimum

(a)

(b)

(c)

(d)

34.(POX is maximum

(a)(0, 4)(b)

(c)

(d)

P-2.Consider two concentric circle C1 : x2 + y2 4 = 0 and C2 : x2 + y2 9 = 0. A parabola is drawn through the points where C1 meets the y-axis and having arbitrary tangent of C2 as its directrix. C is the curve of the locus of the focus of drawn parabola

35.The curve C is

(a)Circle

(b)Parabola

(c)Ellipse

(d)Hyperbola

36.The number of common tangents to the circle x2 + y2 6x 8y 24 = 0 and C2 is

(a)3

(b)2

(c)1

(d)0

37.If the centroid of an equilateral triangle is the center of the curve C and its one vertex is (1, 1), then the equation of its circumcircle is

(a)x2 + y2 = 4

(b)x2 + y2 = 1

(c)x2 + y2 = 3

(d)x2 + y2 = 2

38.The point ((, () lies inside the curve C. Then, (2 lies in interval

(a)

(b)

(c)(0, ()

(d)

P-3.If and

Then the point P((, () is translated parallel to the line x y = 4 by a distance units, if the new position Q is in the third quadrant, then

39.The co-ordinate of P((, () is

(a)(2, 1)

(b)(1, 2)

(c)(2, 2)

(d)(0, 2)

40.The co-ordinates of Q is

(a)(2, 4)

(b)(2, 1)

(c)(2, 3)

(d)(3, 2)

41.The locus of the point A such that (PAQ is a right angle.

(a)x2 + y2 2y + 7 = 0 (b)x2 + y2 + 2y 7 = 0

(c)x2 + y2 + 2y + 7 = 0(d)x2 + y2 2y 7 = 0

42.Mr. Mishra starts walking from the point B (3, 3) and will reach the point C(O, 4) touching the line PQ at D. so that he will travel in the shortest distance, then D is

(a)

(b)

(c)

(d)

P-4.The parabola described parameterically by x = sin 2t, y = sin t + cos t, then

43.A line L passing through the focus of the parabola intersects the parabola in two distinct points. If m be the slope of the line L, then

(a)m < 1 or m > 1(b)1 < m < 1

(c)m < 2

(d)

44.The circle x2 + y2 + 2kx = 0, k(R. touches the directrix and latus-rectum of the parabola, then the set of the values of k is given by

(a)

(b)

(c)

(d)

45.The locus of the point whose chord of contact w.r.t the parabola passes through the focus of the parabola is

(a)x + 5/4 = 0

(b)x + 6/5 = 0

(c)x + 8/5 = 0

(d)x + 2 = 0

46.If ( and ( be the length of the line segments of focal chord of parabola, then is

(a)2

(b)4

(c)

(d)

P-5.Suppose f(x, y) = 0 is the equation of the circle such that f(x, 2) = 0 has equal roots (each equal to 1) and f(2, x) = 0 also has equal roots (each equal to 1), then

47.The center of the circle is

(a)(1, 1)

(b)(1, 1)

(c)(1, 1)

(d)(1, 1)

48.Area of an equilateral triangle inscribed in the circle f(x, y) = 0 is

(a)

sq. units(b)

sq. units

(c)

sq. units

(d)

sq. units

49.The equation of the smallest circle passing through the intersection of the line x y = 1 and the circle f(x, y) = 0

(a)x2 + y2 + x + y = 0(b)x2 + y2 x + y = 0

(c)x2 + y2 x y = 0(d)x2 + y2 + x y = 0

50.If the tangent at the point A on the circle f(x, y) = 0 meets the straight line 3x + 2y 6 = 0 at a point B on the x-axis, then the length of AB is

(a)2

(b)

(c)1

(d)

NARAYANA IIT ACADEMY

(Andhra Pradesh, Kota and Delhi)

AITS MATHEMATICSANSWERS

12345678910

CC ABABCDCA

11121314151617181920

DCBADCCACD

21222324252627282930

ABDABDDC C B

31323334353637383940

BCCACBDAAC

41424344454647484950

BBDCABDABC

Hints and Solution

1.CC1 = h + 3(h 4)2 + k2 = (h + 3)2 k2 8h + 16 = 6h + 9 ( k2 14h + 7 = 0 Required locus of C(h, k) is y2 = 14

2.

( hk = a hence, locus of P(h, k) is a hyperbola

3.Q = (x, (5), so PQ ( 3 ( (x 1)2 + (4 5)2 ( 9 ( 1 + (x 1)2 ( 9 [ 81 + (x 1)2 ( 9 is absurd](x 1)2 ( 8

( 1 ( x ( 1 + , but x is an integer.( x = 1, 0, 1, 2, 3

4.PA + PB ( AB (by triangle inequality) so, PA + PB is minimum if PA + PB = AB i.e. A, P, B are collinear

( 3(3 2a +3) + 2(2a + 3 4) + 1(8a 12 6a 9) = 0 ( 18 + 6a + 4a 2 + 2a 21 = 0 12a 41 = 0 ( a = 41/12

5.

6.Points ( = 0, ( = 1 and ( = 2

(0, 1) and (2, a2) are

same side of line 4x 3y + 4 = 0

12 3a2 > 0

a2 < 4 i.e. 2 < a < 2

7.(ax b)2 + (bx c)2 + (cx d)2 + (dx e)2 ( 0

( a, b, c, d, e are in G.P.

8.1 is the common root of the both the equations.

9.

Focus of y2 = 8x is (2, 0)

10.Equation to the chord is xx1 + yy1 =

Where M(x1, y1) is a foot of ( or from the origin homogenize the equation to the circle given, as we get

(x2 + y2) - (2x + 4y) (xx1 + yy1). 4(xx1 + yy1)2 = 0.

This represents a pair of perpendicular lines through the origin i.e.

giving the locus of (x1, y1) is

2(x2 + y2) 2x 4y 4 = 0

i.e.x2 + y2 x 2y 2 = 0

11.Area of the rectangle = 8 8 = 64 sq. units

12.Co-ordinates of focus will be S(a + 2, b), S lies to the opposite sides of origin w.r.t. line x + y 2 = 0 and same side as origin w.r.t line x + y 6 = 0 hence, 0 < a + b < 4

13.Tangent to y2 = 16(x + 4) is

y = (x + 4) +

(c = 4 +

Hence, the minimum value of c is 8

14.Any point on the line y = mx can be taken (t, mt).

Equation of chord of the parabola with (t, mt) as mid-point is

x.t 4(y + mt) = t2 8mt

It passes through (4, 2),

4t 4(2 + mt) = t2 8mt

t2 4(m + 1)t + 8 = 0

D > 0

For two distinct chords

16(m + 1)2 4.8 > 0

15.Equation of tangent to the parabola with slope m is

This line to be chord of the circle x2 + y2 = 2,

If (4 < (m4 + m2).2

m4 + m2 2 > 0

m2 1 > 0i.e.

16.Any point on x + y = 2 is ((, 2 (). A lines through ((, 2 - () perpendicular to x + y = 2 is

y (2 () = x ( (x y = 2( 2

.(1)

(1) should be a normal to y2 = 4(x 6).

Equation of normal is y + t(x 6) = 2t + t3

..(2)

from (1) & (2)

Hence point is

17.

18.Equation of chord of contact AB, is 2x y - 4 = 0

Required Equation of circle is

x2 + y2 4x + 2y + 4 + ((2x y - 4) = 0

..(1)

(1) passes through (0, 0) ( .. ( = 1

Equation of circumcircle is x2 + y2 2x + y = 0

19.(

i.e.

20.

(CM)2 = (CP)2 (PM)2

which is maximum when PM is minimum

i.e. coincides with M (i.e. middle point of chord)

Hence equation is 2x - 3y + 4 = 0

21.(a 2b + 3c)2 = 0

22.As (6, 8) lies on the circle, the normal to y = f(x) is the tangent to the circle at (6, 8), so that circle intersect orthogonally at (6, 8)

( = the reciprocal of the slope of the circle at (6, 8) =

23.

24.Circumcentre is (0, 0)

Centroid G is

G divides HO internally 2 : 1

(

..(1)

..(2)

from (1) and (2),

h k + 7 = 0

Locus of orthocentre H(h, k) is

x y + 7 = 0

25.Equation of line is y + 4 = m(x 4), as m > 0

mx y = 4m + 4

(

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27.Let (h, k) be the centroid of ( ABC with C as ((, ()

( ( = 3h 3

&

( ( = 3k + 4

since C((, () lies on L.

3(3h 3) + 2(3k + 4) + 1 = 0

9h + 6k = 0

Locus of (h, k) is 3x + 2y = 0 which is parallel to L.

28.Solving equation of parabola with x-axis (i.e. y = 0). we get ax2 + 2cx + b = 0, which should have two equal values of x, as x-axis touch the parabola. discriminant = 0 ( 4c2 4ab = 0 ( c2 = ab now area of

now c2 = ab, so area =

29.a3 + b3 + c3 = 3abc ( a + b + c = 0 as a2 + b2 + c2 ( ab + bc + ca (a, b, c distinct) Now ax + by +c = 0 is equation of line and on comparing with the relation x = 1, y = 1. so circumcentre is (1, 1). Centroid is (0, 0), let orthocentre H (x1, y1)

x1 = -2 y1 = -2 so orthocentre is (-2, -2)

30.Director circle of the given circle x2 + y2 = a2 will be x2 + y2 =

( x2 + y2 = 16 now point on it will be (4 cos(, 4sin()Now chord of contact for the parabola y2 = 16x, T = 0y ( 4 sin( = 8(x + 4 cos () . . . . . (i) equation of the chord whose middle point is (h, k). T = S1 ky 8(x + h) = k2 16h8x ky + k2 8h = 0 . . . . . (ii) comparing the coefficient of the two equation of the same line

so 64y2 + (y2 8x)2 = 1024

31.OC = 5, CP = 3, OP = 4, C ( (3, 4)

OL = 3

OM = OL + LM = OL + HP

4cos( = 3 + 3sin(

(4 3 tan( = 3 sec (

16 + 9 tan2( 24 tan ( = 9 sec2(

(tan ( = 7/24

32.Max. value of OP = OQ = OC + radius = 8,

P( (OQ cos (, OQ sin ()

33.

34.P(0, 4)

P-2.Let the focus of parabola be (x1, y1)

SP = distance of P from directrix

Here P are (0, 2) and (0, 2)

Eq. Of directrix is

x cos ( + y sin ( = 3

(

..(1)

and x2 + (y1 + 2)2 = (2 sin ( + 3)2

..(2)

Subtract, we get 8y1 = 24 sin (

( sin ( = y1/3

Adding, (1) and (2),

(

(

(

35.Locus of focus is

36.Distance between their centers = 5

|r1 r2| < 5 < r1 + r2

Hence, number of common tangents = 2

37.(x 0)2 + (y 0)2 =

38.(14(2 45 < 0

39.

= 1 + 1 = 2

and

= 1 + 0 = 1

40.

( (2, 3)

41.(PA2 + AQ2 = PQ2

Locus of A(h, k) is x2 + y2 + 2y 7 = 0

42.Equation of PQ is y = x 1

Image of B(3, 3) w.r.t line y = x 1 is B((4, 4)

For the shortest distance B(DC are collinear.

Equation of B(C is O y + 2x = 4

43.Equation of the parabola is y2 = x + 1

(1)

focus is

Any line through the focus is

..(2)

Solving (1) and (2)

If m ( 0

(D > 0

= m2 + 1 > 0 for all m

But if m = 0, then x does not have two real values distinct.

, except m = 0

44.

45.Equation of chord of contact from the point P(h, k) is

..(1)

(1) passes through focus

Required locus is

46.Semi-latus-rectum is the harmonic mean of line segments of focus chords i.e.

P-5

Equation of circle f(x, y) = 0 is x2 + y2 2x + 2y + 1 = 0

47.C ( (1, 1)

48.

area of an equilateral

sq. units

49.Equation of circle is

x2 + y2 2x + 2y + 1 + (((x y 1) = 0

x2 + y2 + (( 2)x + (2 ()y + 1 ( = 0

For smallest circle x y = 1 is a diameter of the required circle

(( = 1

Hence circle is x2 + y2 x + y = 0

50.Point B is (2, 0). Length of tangent =

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NARAYANA IIT ACADEMYSouth Centre: 47 B, Kalu Sarai, N. Delhi-16. North West Centre: 2/40, Adj. Central Mkt., W. Punjabi Bagh, N. Delhi-26.NARAYANA IIT ACADEMYSouth Centre: 47 B, Kalu Sarai, N. Delhi-16. North West Centre: 2/40, Adj. Central Mkt., W. Punjabi Bagh, N. Delhi-26.

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