Maths DPP-6 (Advanced)

MATHEMATICS TARGET : JEE (Advanced) 2014 TEST SYLLABUS

Class : XII & XIII Date : 30-04-2014 TEST : Part Test -3(PT-3) Course: VIJETA (JPAD) & VIJAY(JRAD) Syllabus : Conic Section & Solution of Triangle Date : 27-04-2014 DPP No. : 06

Note : Dear Students, DPP # 11 in Revision plan for JEE(Advanced) will be of Matrices & Determinant and Miscellaneous Revision DPP # 6 CONIC SECTION And SOLUTION OF TRIANGLE Total Marks : 152

Max. Time : 150 min.

Single choice Objective (–1 negative marking) Q.1 to Q.39

(3 marks 3 min.)

[117, 117]

Multiple choice objective (no negative marking) Q.40 to Q.42 Subjective Questions (–1 negative marking) Q.43 Match the Following (no negative marking) Q.44, 45

(5 marks, 4 min.) (4 marks, 5 min.) (8 marks, 8 min.)

[15, 12] [4, 5] [16, 16]

1.

Two sides of an equilateral triangle ABC touch the parabola y2 = 4x, then points A, B & C lie on the curve (A) y2 = (x + 1)2 + 4x (B) y2 = 3(x + 1)2 + x 2 2 (C) y = 3(x + 1) + 4x (D) y2 = (x + 1)2 + x

2.

If the equation of family of ellipse is

x2 2

cos 



y2 sin 

latus rectum is (A) 2y2(1 + x2) = (1 – x2)2 (C) 2y(1 – x2) = 1 + x2

3.

2

If tangent and normal at a point P to the ellipse

= 1, where

(B) 2x2(1 + y2) = (1 – y2)2 (D) 2x(1 – y2) = 1 + y2

x2 y 2  = 1 intersect major axis at points T and N respectively 25 16

in such a way that ratio of areas of PTN and PSS' is

4.

  <  < , then the locus of extremities of the 4 2

91 , then area of PSS' is (S and S' are focii of ellipse) 60

(A) 6 3 sq. units

(B) 12 3 sq. units

(C) 4 3 sq. units

(D) 3 3 sq. units

The line joining the orthocentre and the centroid of the triangle formed by a focal chord of the parabola with the tangents at its extremities is (A) Parallel to the axis of the parabola (B) Perpendicular to the axis of the parabola (C) Neither parallel nor perpendicular to the axis of the parabola (D) Nothing can be certainly said Page # 1

5.

All the vertices of a trapezium lie on the parabola y2 = 4x. Its diagonals pass through (1, 0) and have a length 25 units each. Then the area of the trapezium is 4

(A) 100 sq. units

6.

(B) 75/4 sq. unit

(C) 25/4 sq. units

(D) 65 sq. units

A line is drawn from A(–2, 0) to intersect paraboala y2 = 4x at P & Q in the first quadrant. If

1 1 1   , AP AQ 4

then the slope of the line is always (A) >

7.

(B) <

3

1 3

(C) <

1

(D)

3

3

Consider the parabola y2 = 4x. A(4, –4) & B(9, 6) be two fixed points, on the parabola. Then a point C on the parabola for which the area of ABC is maximum is 1  (A)  , 1 4 

(B) (4, 4)



(C) 3 , 2 3





(D) 3 , – 2 3



8.

If the bisector of APB, where PA and PB are the tangents to the parabola y2 = 4ax, is equally inclined to the co-ordinate axes, then the locus of P is (A) tangent at the vertex (B) directrix of the parabola (C) a circle with centre at origin and radius ‘a’ units (D) the latus rectum of the parabola

9.

From a given point A on the parabola y2 = 4ax, a focal chord AB and a tangent is drawn. Now two circles are drawn. One with AB as diameter and the other taking intercept of the tangent between point A and P (where the tangent meets the directrix) as diameter. Then the common chord of the two circles is (A) line joining focus S and P (B) line joining focus S and A (C) tangent line to parabola at A (D) None of these

10.

Normals AO, AA1 & AA2 one drawn to the parabola y2 = 8x from A(h, 0). If triangle OA1A2 is equilateral, then ‘h’ can be equal to (A) 24 (B) 26 (C) 28 (D) 30

11.

If the normal to the parabola at P meets it again at Q and if PQ and the normal at Q make angles  &  with the x-axis respectively, then tan ( tan  + tan ) has a value equal to (A) 0 (B) –1 (C) –2 (D) – 1/2

12.

Normals to the parabola y2 = 4ax at A(x1, y1) and B(x2, y2) meet on the parabola such that x1 + x2 = 4. Then |y1 + y2| = (A)

13.

(B) 2 2

2

(C) 4 2

(D) 6 2

If two different tangents of y2 = 4x are normals to x2 = 4by then (A) |b| >

(C) |b| >

1 2 2 1 2

(B) |b| <

(D) |b| <

1 2 2 1 2

Page # 2

14.

If the line segment joining the foci of the ellipse

x2 a2



y2 b2

= 1 (a > b) does not subtend a right angle at any

point on the ellipse, then the range of eccentricity of the ellipse is

15.

1  (A)  , 1 2 

 1  , 1 (B)   2 

 1 (C)  0 ,   2

 1   (D)  0 , 2 

The point P(, ) lying on the ellipse 4x2 + 3y2 = 12 in the first quadrant, such that the area enclosed by the lines y = x, y = , x = , and the x-axis is minimum is 3  (A)  , 1 2 

 2 2   (B) 1, 3   

 3    (C)  2 , 3   

 3 15    (D)  4 , 2   

16.

With a given point and a given line as focus and directrix, a series of ellipses are described. Then locus of the extremities of their minor axis is (A) an ellipse (B) parabola (C) hyperbola (D) straight line

17.

The auxilliary circle of a family of ellipse passes through origin and makes intercepts of 8 and 6 units on the x-axis and y-axis respectively. If the eccentricity of all such ellipses is

1 , then the locus of their focus is 2

(A)

x2 y2  = 25 16 9

(B) 4x2 + 4y2 – 32x – 24y + 75 = 0

(C)

x2 y2  = 25 9 16

(D) 2x2 + y2 = 2

18.

Let S be the focus, C the centre, and P be any point on the directrix of an ellipse of eccentricity 'e'. The line PC meets the ellipse at A. Then the angle between PS and the tangent at A is (A) tan–1(e) (B) /2 (C) tan–1(1 – e2) (D) /4

19.

If the ellipse

x2 a2



y2 b2

= 1 is inscribed in a rectangle whose length to breadth ratio is 2 : 1, then the area of

the rectangle is (A)

4 2 (a + b2) 7

(B)

4 2 (a + b2) 3

(C)

12 2 (a + b2) 5

(D)

8 2 (a + b2) 5

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20.

21.

S1 : S2 :

Diagonals of a parallelogram inscribed in an ellipse always intersect at its centre. Centre of an ellipse bisects every chord passing through it.

(A)

STATEMENT-1 is true, STATEMENT-2 is true and STATEMENT-2 is correct explanation for STATEMENT-1

(B)

STATEMENT-1 is true, STATEMENT-2 is true and STATEMENT-2 is not correct explanation for STATEMENT-1

(C)

STATEMENT-1 is true, STATEMENT-2 is false

(D)

STATEMENT-1 is false, STATEMENT-2 is true

(E)

Both STATEMENTS are false

Two tangents to the hyperbola

x2 a2

–

y2 b2

= 1 having slopes m1 & m2 cut the coordinate axes in four con-cyclic

points. Then m1m2 is equal to (A) 1

22.

(B) –1

(C)

a b

(D) –

If a variable line has intercepts of e1 and e2 on the co-ordinate axes, where

b a

e1 e & 2 are the eccentricities of 2 2

a hyperbola and its conjugate, then the line always touches a fixed circle x2 + y2 = r2 , where ‘r’ is (A) 1

23.

(B) 2

(C) 3

If tangents PQ and PR are drawn from a variable point P to hyperbola

(D) 4

x2 a2

–

y2 b2

= 1 (a > b), such that the

fourth vertex S of parallelogram PQSR lies on the circumcircle of PQR, then the locus of P is (A) x2 + y2 = b2 (B) x2 + y2 = a2 2 2 2 2 (C) x + y = a – b (D) x2 + y2 = a2 + b2 24.

The chord of contact of a point P with respect to a hyperbola and its auxiliary circle are at right angles, then P lies on (A) conjugate hyperbola (B) directrices (C) one of the asymptotes (D) None of these

25.

The polar of any point on an asymptote of a hyperbola with respect to the hyperbola is always (A) parallel to that asymptote (B) perpendicular to that asymptote (C) neither parallel nor perpendicular to the asmptotes (D) some times parallel, some times perpendicular to that asmptote

26.

Let ABC be such that BAC =

2 and AB = x. Further (AB)(AC) = 1. If x varies, then the longest possible 3

length of the angle bisector AD is (A) 2 (B) 1

(C) 1/2

(D) 2/3 Page # 4

27.

If the median AD of ABC makes an angle (A) 1

28.

(B) 2

(C) 1/3

(D) 2/5

ABC is an equilateral triangle of side 8 cm. If R, r, & h are the circumradius, inradius and altitude of ABC, then

R r has a value equal to h

(A) 4

29.

 with side BC, then |cot B – cot C| = 4

(B) 2

(C) 1

(D) 3

 r  r  If in a triangle 1 – 1 1 – 1  = 2, then the triangle is r r 2  3  

(A) right angled

(B) isosceles

(C) equilateral

(D) None of these

30.

If H is the orthocentre of an acute angle triangle whose circum-circle is x2 + y2 = 16, then circumdiameter of HBC is (A) 1 (B) 2 (C) 4 (D) 8

31.

S-1 :

If a, b, c are the sides of a triangle then the minimum value of

S-2 :

A.M.  G.M.  H.M.

(A)

STATEMENT-1 is true, STATEMENT-2 is true and STATEMENT-2 is correct explanation for STATEMENT-1

(B)

STATEMENT-1 is true, STATEMENT-2 is true and STATEMENT-2 is not correct explanation for STATEMENT-1

(C)

STATEMENT-1 is true, STATEMENT-2 is false

(D)

STATEMENT-1 is false, STATEMENT-2 is true

(E)

Both STATEMENTS are false

32.

2a 2b 2c   is 9. bc–a ca–b ab–c

In ABC, ABC = 120°, AB = 3cm and BC = 4cm. If perpendicular constructed to AB at A and to BC at C meet at D, then CD = (A) 3

(B)

8 3 3

(C) 5

(D)

10 3 3

33.

If AD, BE and CF are the medians of a ABC, then (AD2 + BE2 + CF 2) : (BC2 + CA2 + AB2) is equal to (A) 4 : 3 (B) 3 : 2 (C) 3 : 4 (D) 2 : 3

34.

If ABCD is a cyclic quadrilateral such that 12tanA – 5 = 0 and 5cosB + 3 = 0, then the equation whose roots are cosC and tanD is (A) 39x2 – 16x – 48 = 0 (B) 39x2 + 88x + 48 = 0 (C) 39x2 – 88x + 48 = 0 (D) 39x2 – 16x + 48 = 0

35.

If in a triangle whose circumcentre is origin, a  sinA, then for any point (a, b) lying inside the circumcircle of ABC, (A) |ab| < 1/8 (B) 1/8 < |ab| < 1/2 (C) |ab| > 1/2 (D) Nothing can be said Page # 5

36.

ABC is isosceles with AB= AC and CAB = 106°. Point M is an interior point such that MBA = 7° and MAB = 23°. Then AMC = (A)87° (B) 67°

37.

(C) 74°

x2 y2 + = 1 and P in any point on it, then the range of values of SP.S .SP is 25 16

If S & S are the foci of the ellipse (A) [9, 16]

(D) 83°

(B) [9, 25]

(C) [16, 25]

(D) [1, 16]

38.

The sides of a triangle are a = x2 + x + 1, b = 2x + 1, c = x2 – 1. Then the greatest angle of the triangle is (A) 90° (B) 120° (C) 60° (D) 30°

39.

Let a  b  c be the sides of a triangle. If a2 + b2 < c2, then the triangle is (A) acute angled (C) obtuse angled

40.

Consider the ellipse

(B) right angled (D) nothing can be said

x2 f (k 2  2k  5)



y2 = 1, where f(x) is a decreasing function, then f (k  11)

(A) the set of values of k for which the major axis of the ellipse is x-axis is (–3, 2) (B) the set of values of k for which the major axis of the ellipse is y-axis is (–, 2) (C) the set of values of k for which the major axis of the ellipse is y-axis is (–, –3)  (2, ) (D) the set of values of k for which the major axis of the ellipse is x-axis is (–3, ) 41.

Two concentric ellipses are such that the foci of one lie on the other and the length of their major-axes are equal. If e1 & e2 be their eccentricities, then (A) the quadrilateral formed by joining their foci is a parallelogram 1

(B) the angle between their axes is given by cos  =

e12



1 e 22

–

1 e12 e 22

(C) their axes are perpendicular if e1 = 1 – e 22 (D) None of these 42.

If the tangent at the point P() to the ellipse 16x2 + 11y2 = 256 is also a tangent to the circle x2 + y2 – 2x = 15 then  = (A)

43.

2 3

(B)

4 3

A tangent drawn to the hyperbola

(C)

x2 a

2

–

5 3

(D)

 3

y2

  = 1 at P   forms a triangle of area 3a2 sq. units. with co-ordinate 6 b 2

axes. If the eccenticity of the hyperbola is 'e', then the value of e2 – 9 is.

Page # 6

44.

Tangents drawn from a point P to the ellispe

x2 a2



y2 b2

= 1 (a > b) make angles  and  with the major axis of

the ellipse. Then find the locus of P if Column- I

45.

(A)

+=

(B)

n (n  N) 2

Column- II (p)

circle

tan  tan  =  (  R)

(q)

ellipse

(C)

tan  + tan  =  (  R)

(r)

hyperbola

(D)

cot  + cot  = c (c  R)

(s)

pair of straight lines

If e1 and e2 are the roots of the equation x2 – ax + 2 = 0, then match the following Column-I (A) If e1 & e2 are the eccentricities of an ellipse and a hyperbola, then a

Column-II (p)

6

(B) e1 & e2 are the eccentricities of two hyperbolas, then a can take values

(q)

57/20

(C) e1 & e2 are the eccentricities of a hyperbola and its conjugate, then a

(r)

2 2

(s)

5

can take values

can take values (D) e1 is eccentricity of a hyperbola for which there exist infinite number of points from which perpendicular tangents can be drawn & e2 the eccentricity of a hyperbola for which no such points exist, then a can take values

Page # 7