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1051

The circles  and  intersect each other in distinct points if

a) r < 2

b) r > 8

c) 2 < r < 8

d) 2 ≤ r ≤ 8

The circles  and  intersect each other in distinct points if

a) r < 2

b) r > 8

c) 2 < r < 8

d) 2 ≤ r ≤ 8

IIT 1994
1052

Prove by induction that
Pn = Aαn + Bβn for all n ≥ 1
Where α and β are roots of the quadratic equation
x2 – (1 – P) x – P (1 – P) = 0,
P1 = 1, P2 = 1 – P2, .  .  .,
Pn = (1 – P) Pn – 1 + P (1 – P) Pn – 2  n ≥ 3,
and ,

Prove by induction that
Pn = Aαn + Bβn for all n ≥ 1
Where α and β are roots of the quadratic equation
x2 – (1 – P) x – P (1 – P) = 0,
P1 = 1, P2 = 1 – P2, .  .  .,
Pn = (1 – P) Pn – 1 + P (1 – P) Pn – 2  n ≥ 3,
and ,

IIT 2000
1053

Let P be a point on the parabola y2 = 8x which is at a minimum distance from the centre C of the circle x2 + (y + 6)2 = 1. Then the equation of the circle passing through C and having its centre at P is

a) x2 + y2 – 4x + 8y + 12 = 0

b) x2 + y2 –x + 4y − 12 = 0

c) x2 + y2 –x + 2y − 24 = 0

d) x2 + y2 – 4x + 9y + 18 = 0

Let P be a point on the parabola y2 = 8x which is at a minimum distance from the centre C of the circle x2 + (y + 6)2 = 1. Then the equation of the circle passing through C and having its centre at P is

a) x2 + y2 – 4x + 8y + 12 = 0

b) x2 + y2 –x + 4y − 12 = 0

c) x2 + y2 –x + 2y − 24 = 0

d) x2 + y2 – 4x + 9y + 18 = 0

IIT 2016
1054

Let  then points where f (x) is not differentiable is (are)

a) 0

b) 1

c) ± 1

d) 0, ± 1

Let  then points where f (x) is not differentiable is (are)

a) 0

b) 1

c) ± 1

d) 0, ± 1

IIT 2005
1055

The slope of the line touching both parabolas y2 = 4x and x2 = −32y is

a) 12

b) 32

c) 18

d) 23

The slope of the line touching both parabolas y2 = 4x and x2 = −32y is

a) 12

b) 32

c) 18

d) 23

IIT 2014
1056

Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and QR intersect at a point x on the circumference of the circle, then 2r equals

a)

b)

c)

d)

Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and QR intersect at a point x on the circumference of the circle, then 2r equals

a)

b)

c)

d)

IIT 2001
1057

Multiple choices

Let [x] denote the greatest integer less than or equal to x. If

f (x) = [xsinπx] then f(x) is

a) Continuous at x = 0

b) Continuous in  

c) f (x) is differentiable at x = 1

d) differentiable in

e) None of these

Multiple choices

Let [x] denote the greatest integer less than or equal to x. If

f (x) = [xsinπx] then f(x) is

a) Continuous at x = 0

b) Continuous in  

c) f (x) is differentiable at x = 1

d) differentiable in

e) None of these

IIT 1986
1058

Let  then

a)

b)

c)

d)

Let  then

a)

b)

c)

d)

IIT 1987
1059

Let a, r, s, t be non-zero real numbers. Let P(at2, 2at), Q, R(ar2, 2ar and S(as2, 2as) be distinct points on the parabola y2 = 4ax. Suppose PQ is the focal chord and QR and PK are parallel, where K is point (2a, 0)

The value of r is

a) 1t

b) t2+1t

c) 1t

d) t21t

Let a, r, s, t be non-zero real numbers. Let P(at2, 2at), Q, R(ar2, 2ar and S(as2, 2as) be distinct points on the parabola y2 = 4ax. Suppose PQ is the focal chord and QR and PK are parallel, where K is point (2a, 0)

The value of r is

a) 1t

b) t2+1t

c) 1t

d) t21t

IIT 2014
1060

Find all solutions of

a)

b)

c)

d)

Find all solutions of

a)

b)

c)

d)

IIT 1983
1061

Multiple choices

Which of the following functions are continuous on (0, π)

a) tanx

b)

c)

d)

Multiple choices

Which of the following functions are continuous on (0, π)

a) tanx

b)

c)

d)

IIT 1991
1062

One or more than one correct option

If the normals of the parabola y2 = 4x drawn at the end points of the latus rectum are tangents to the circle (x − 3)2 + (y + 2)2 = r2 then the value of r2 is

a) 4

b) 1

c) 2

d) 0

One or more than one correct option

If the normals of the parabola y2 = 4x drawn at the end points of the latus rectum are tangents to the circle (x − 3)2 + (y + 2)2 = r2 then the value of r2 is

a) 4

b) 1

c) 2

d) 0

IIT 2015
1063

Multiple choices

Let  for every real number x then

a) h (x) is continuous for all x

b) h is differentiable for all x

c)  for all x > 1

d) h is not differentiable for two values of x

Multiple choices

Let  for every real number x then

a) h (x) is continuous for all x

b) h is differentiable for all x

c)  for all x > 1

d) h is not differentiable for two values of x

IIT 1998
1064

Number of divisors of the form 4n + 2(n ≥ 0) of integer 240 is

a) 4

b) 8

c) 10

d) 3

Number of divisors of the form 4n + 2(n ≥ 0) of integer 240 is

a) 4

b) 8

c) 10

d) 3

IIT 1998
1065

The smallest positive root of the equation tan x – x = 0 lies in

a)

b)

c)

d)

e) None of these

The smallest positive root of the equation tan x – x = 0 lies in

a)

b)

c)

d)

e) None of these

IIT 1987
1066

Let f (x) be defined on the interval  such that

 

g (x) = f (|x|) + |f(x)|

Test the differentiability of g (x) in

a) g(x) is differentiable at all x  ℝ

b) g(x) is differentiable at all x  ℝ except at x = 1

c) g(x) is differentiable at all x  ℝ except at x = 0, 1

d) g(x) is differentiable at all x  ℝ except at x = 0, 1, 2

Let f (x) be defined on the interval  such that

 

g (x) = f (|x|) + |f(x)|

Test the differentiability of g (x) in

a) g(x) is differentiable at all x  ℝ

b) g(x) is differentiable at all x  ℝ except at x = 1

c) g(x) is differentiable at all x  ℝ except at x = 0, 1

d) g(x) is differentiable at all x  ℝ except at x = 0, 1, 2

IIT 1986
1067

If the LCM of p, q is  where r, s, t are prime numbers and p, q are positive integers then the number of ordered pairs (p, q) is

a) 252

b) 254

c) 225

d) 224

If the LCM of p, q is  where r, s, t are prime numbers and p, q are positive integers then the number of ordered pairs (p, q) is

a) 252

b) 254

c) 225

d) 224

IIT 2006
1068

Consider a family of circles passing through two fixed points A (3, 7) and B (6, 5). Show that the chords in which the circle  cuts the members of the family are concurrent at a point. Find the coordinates of this point.

Consider a family of circles passing through two fixed points A (3, 7) and B (6, 5). Show that the chords in which the circle  cuts the members of the family are concurrent at a point. Find the coordinates of this point.

IIT 1993
1069

In how many ways can a pack of 52 cards be divided into four groups of 13 cards each.

In how many ways can a pack of 52 cards be divided into four groups of 13 cards each.

IIT 1979
1070

In a triangle ABC, let ∠ C = . If r is the inradius and R is the circumradius of the triangle then 2(r+R) = ………….

a) a+b

b) b+c

c) c+a

d) a+b+c

In a triangle ABC, let ∠ C = . If r is the inradius and R is the circumradius of the triangle then 2(r+R) = ………….

a) a+b

b) b+c

c) c+a

d) a+b+c

IIT 2000
1071

Determine the values of x for which the following function fails to be continuous or differentiable.

 

Justify your answer.

a) f(x) is continuous and differentiable

b) f(x) is continuous everywhere but not differentiable at
x = 1, 2

c) f(x) is continuous everywhere but not differentiable at x = 2

d) f(x) is neither continuous nor differentiable at x = 1, 2

Determine the values of x for which the following function fails to be continuous or differentiable.

 

Justify your answer.

a) f(x) is continuous and differentiable

b) f(x) is continuous everywhere but not differentiable at
x = 1, 2

c) f(x) is continuous everywhere but not differentiable at x = 2

d) f(x) is neither continuous nor differentiable at x = 1, 2

IIT 1997
1072

Let  

And

where a and b are non-negative real numbers. Determine the composite function gof. If (gof)(x) is continuous for all real x, determine the values of a and b. Is gof differentiable at x = 0?

a) a = b = 0

b) a = 0, b = 1

c) a = 1, b = 0

d) a = b = 1

Let  

And

where a and b are non-negative real numbers. Determine the composite function gof. If (gof)(x) is continuous for all real x, determine the values of a and b. Is gof differentiable at x = 0?

a) a = b = 0

b) a = 0, b = 1

c) a = 1, b = 0

d) a = b = 1

IIT 2002
1073

Find the equation of the circle touching the line 2x + 3y + 1 = 0 at the point (1, −1) and is orthogonal to the circle which has the line segment having end points (0, −1) and (−2, 3) as diameter.

Find the equation of the circle touching the line 2x + 3y + 1 = 0 at the point (1, −1) and is orthogonal to the circle which has the line segment having end points (0, −1) and (−2, 3) as diameter.

IIT 2004
1074

Show that the value of  wherever defined

a) always lies between  and 3

b) never lies between  and 3

c) depends on the value of x

Show that the value of  wherever defined

a) always lies between  and 3

b) never lies between  and 3

c) depends on the value of x

IIT 1992
1075

                      

Show that f(x) is differentiable at the value of α = 1. Also,

a) b2 +c2 = 4

b) 4 b2  = 4 − c2  

c) 64 b2 = 4 − c2

d) 64 b2 = 4 + c2

                      

Show that f(x) is differentiable at the value of α = 1. Also,

a) b2 +c2 = 4

b) 4 b2  = 4 − c2  

c) 64 b2 = 4 − c2

d) 64 b2 = 4 + c2

IIT 2004

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