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1001

A tangent PT is drawn to the circle x2 + y2 = 4 at the point P(3,1)

. A straight line L, perpendicular to PT is tangent to the circle (x – 3)2 + y2 = 1A common tangent to the circles is

a) x = 4

b) y = 2

c) x+3y=4

d) x+22y=6

A tangent PT is drawn to the circle x2 + y2 = 4 at the point P(3,1)

. A straight line L, perpendicular to PT is tangent to the circle (x – 3)2 + y2 = 1A common tangent to the circles is

a) x = 4

b) y = 2

c) x+3y=4

d) x+22y=6

IIT 2012
1002

The integer n, for which  is a finite

non–zero number is

a) 1

b) 2

c) 3

d) 4

The integer n, for which  is a finite

non–zero number is

a) 1

b) 2

c) 3

d) 4

IIT 2002
1003

The locus of the middle points of the chord of tangents drawn from points lying on the straight line 4x – 5y = 20 to the circle x2 + y2 = 9 is

a) 20(x2 + y2) – 36x + 45y = 0

b) 20(x2 + y2) + 36x − 45y = 0

c) 36(x2 + y2) – 20x + 45y = 0

d) 36(x2 + y2) + 20x − 45y = 0

The locus of the middle points of the chord of tangents drawn from points lying on the straight line 4x – 5y = 20 to the circle x2 + y2 = 9 is

a) 20(x2 + y2) – 36x + 45y = 0

b) 20(x2 + y2) + 36x − 45y = 0

c) 36(x2 + y2) – 20x + 45y = 0

d) 36(x2 + y2) + 20x − 45y = 0

IIT 2012
1004

Let  be a regular hexagon in a circle of unit radius. Then the product of the length of the segments  ,  and  is

a)

b)

c) 3

d)

Let  be a regular hexagon in a circle of unit radius. Then the product of the length of the segments  ,  and  is

a)

b)

c) 3

d)

IIT 1998
1005

f(x) is twice differentiable polynomial function such that f (1) = 1, f (2) = 4, f (3) = 9, then

a) there exists at least one x  (1, 2) such that

b) there exists at least one x  (2, 3) such that

  

c)

d) there exists at least one x  (1, 3) such that

f(x) is twice differentiable polynomial function such that f (1) = 1, f (2) = 4, f (3) = 9, then

a) there exists at least one x  (1, 2) such that

b) there exists at least one x  (2, 3) such that

  

c)

d) there exists at least one x  (1, 3) such that

IIT 2005
1006

The radius of a circle having minimum area which touches the curve y = 4 – x2 and the line y = |x| is

a) 22

b) 2(21)

c) 4(21)

d) 4(2+1)

The radius of a circle having minimum area which touches the curve y = 4 – x2 and the line y = |x| is

a) 22

b) 2(21)

c) 4(21)

d) 4(2+1)

IIT 2017
1007

Let AB be a chord of the circle subtending a right angle at the centre then the locus of the centroid of the triangle PAB as P moves on the circle is

a) A parabola

b) A circle

c) An ellipse

d) A pairing straight line

Let AB be a chord of the circle subtending a right angle at the centre then the locus of the centroid of the triangle PAB as P moves on the circle is

a) A parabola

b) A circle

c) An ellipse

d) A pairing straight line

IIT 2000
1008

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
1009

Let  then

a)

b)

c)

d)

Let  then

a)

b)

c)

d)

IIT 1987
1010

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
1011

Find all solutions of

a)

b)

c)

d)

Find all solutions of

a)

b)

c)

d)

IIT 1983
1012

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
1013

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
1014

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
1015

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
1016

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
1017

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
1018

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
1019

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
1020

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
1021

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
1022

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
1023

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
1024

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
1025

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

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