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1226

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
1227

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
1228

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
1229

Let  then

a)

b)

c)

d)

Let  then

a)

b)

c)

d)

IIT 1987
1230

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
1231

Find all solutions of

a)

b)

c)

d)

Find all solutions of

a)

b)

c)

d)

IIT 1983
1232

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
1233

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
1234

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
1235

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
1236

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
1237

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
1238

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
1239

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
1240

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
1241

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
1242

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
1243

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
1244

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
1245

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
1246

                      

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
1247

The product of r consecutive natural numbers is divisible by r!

a) True

b) False

The product of r consecutive natural numbers is divisible by r!

a) True

b) False

IIT 1985
1248

The area bounded by the curve y = f(x), the X–axis and the ordinates x = 1, x = b is (b – 1) sin (3b + 4). Then f(x) is

a) (x – 1) cos (3x + b)

b) sin (3x + 4)

c) sin (3x + 4) + 3 (x – 1) cos (3x + 4)

d) none of these

The area bounded by the curve y = f(x), the X–axis and the ordinates x = 1, x = b is (b – 1) sin (3b + 4). Then f(x) is

a) (x – 1) cos (3x + b)

b) sin (3x + 4)

c) sin (3x + 4) + 3 (x – 1) cos (3x + 4)

d) none of these

IIT 2005
1249

The sum  where  equals

a) i

b) i – 1

c) – i

d) 0

The sum  where  equals

a) i

b) i – 1

c) – i

d) 0

IIT 1998
1250

Fill in the blank

The value of f (x) =  lies in the interval …………….

a)

b)

c)

d)

Fill in the blank

The value of f (x) =  lies in the interval …………….

a)

b)

c)

d)

IIT 1983

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