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1001

The sum if p > q is maximum when m is

a) 5

b) 10

c) 15

d) 20

The sum if p > q is maximum when m is

a) 5

b) 10

c) 15

d) 20

IIT 2002
1002

At present a firm is manufacturing 2000 items. It is estimated that the rate of change of production P with respect to additional number of workers x is given by dPdx=10012x

. If the firm employs 25 more workers then the new level of production of items is

a) 2500

b) 3000

c) 3500

d) 4500

At present a firm is manufacturing 2000 items. It is estimated that the rate of change of production P with respect to additional number of workers x is given by dPdx=10012x

. If the firm employs 25 more workers then the new level of production of items is

a) 2500

b) 3000

c) 3500

d) 4500

IIT 2013
1003

If a, b, c; u, v, w are complex numbers representing the vertices of two triangles such that c = (1 − r)a + rb, w = (1 − r)u + rv where r is a complex number. The two triangles

a) have the same area

b) are similar

c) are congruent

d) none of these

If a, b, c; u, v, w are complex numbers representing the vertices of two triangles such that c = (1 − r)a + rb, w = (1 − r)u + rv where r is a complex number. The two triangles

a) have the same area

b) are similar

c) are congruent

d) none of these

IIT 1985
1004

Prove that

 

Prove that

 

IIT 1979
1005

The question contains Statement – 1(assertion) and Statement – 2 (reason). Let f (x) = 2 + cosx for all real x.

Statement 1: For each real t, there exists a point c in [t, t + π] such that  because

Statement 2: f (t) = f[t, t + 2π] for each real t

a) Statement 1 and 2 are true. Statement 2 is a correct explanation of Statement 1.

b) Statement 1 and 2 are true. Statement 2 is not a correct explanation of Statement 1.

c) Statement 1 is true and Statement 2 is false.

d) Statement 1 is false. Statement 2 is true.

The question contains Statement – 1(assertion) and Statement – 2 (reason). Let f (x) = 2 + cosx for all real x.

Statement 1: For each real t, there exists a point c in [t, t + π] such that  because

Statement 2: f (t) = f[t, t + 2π] for each real t

a) Statement 1 and 2 are true. Statement 2 is a correct explanation of Statement 1.

b) Statement 1 and 2 are true. Statement 2 is not a correct explanation of Statement 1.

c) Statement 1 is true and Statement 2 is false.

d) Statement 1 is false. Statement 2 is true.

IIT 2007
1006

Let f(x) = (1 – x)2 sin2x + x2 and g(x)=1x(2(t1)t+1lnt)f(t)dt

Which of the following is true?

a) g is increasing on (1, ∞)

b) g is decreasing on (1, ∞)

c) g is increasing on (1, 2) and decreasing on (2, ∞)

d) g is decreasing on (1, 2) and increasing on (2, ∞)

Let f(x) = (1 – x)2 sin2x + x2 and g(x)=1x(2(t1)t+1lnt)f(t)dt

Which of the following is true?

a) g is increasing on (1, ∞)

b) g is decreasing on (1, ∞)

c) g is increasing on (1, 2) and decreasing on (2, ∞)

d) g is decreasing on (1, 2) and increasing on (2, ∞)

IIT 2013
1007

Use mathematical induction to prove: If n is an odd positive integer
then  is divisible by 24.

Use mathematical induction to prove: If n is an odd positive integer
then  is divisible by 24.

IIT 1983
1008

Given
 
 
Prove that
 

Given
 
 
Prove that
 

IIT 1984
1009

The coordinates of the in centre of the triangle that has the co ordinates of the mid points of its sides as (0, 1), (1, 1) and (1, 0) is

a) 2+2

b) 22

c) 1+2

d) 12

The coordinates of the in centre of the triangle that has the co ordinates of the mid points of its sides as (0, 1), (1, 1) and (1, 0) is

a) 2+2

b) 22

c) 1+2

d) 12

IIT 2013
1010

Using mathematical induction, prove that

 for n > 1

Using mathematical induction, prove that

 for n > 1

IIT 1986
1011

If f(x) =  then on the interval [0, π]

a) tan  and  are both continuous

b) tan  and  are both discontinuous

c) tan  and  are both continuous

d) tan  is continuous but  is not

If f(x) =  then on the interval [0, π]

a) tan  and  are both continuous

b) tan  and  are both discontinuous

c) tan  and  are both continuous

d) tan  is continuous but  is not

IIT 1989
1012

One or more than one correct option

A ray of light along x+3y=3

gets reflected upon reaching X- axis, the equation of the reflected ray is

a) y=x+3

b) 3y=x3

c) y=3x3

d) 3y=x1

One or more than one correct option

A ray of light along x+3y=3

gets reflected upon reaching X- axis, the equation of the reflected ray is

a) y=x+3

b) 3y=x3

c) y=3x3

d) 3y=x1

IIT 2013
1013

If  and  where 0 < x ≤1, then in this interval

a) Both f (x) and g (x) are increasing functions

b) Both f (x) and g (x) are decreasing functions

c) f (x) is an increasing function

d) g (x) is an increasing function

If  and  where 0 < x ≤1, then in this interval

a) Both f (x) and g (x) are increasing functions

b) Both f (x) and g (x) are decreasing functions

c) f (x) is an increasing function

d) g (x) is an increasing function

IIT 1997
1014

The number of common tangents to the circles x2 + y2 – 4x − 6y – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is

a) 1

b) 2

c) 3

d) 4

The number of common tangents to the circles x2 + y2 – 4x − 6y – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is

a) 1

b) 2

c) 3

d) 4

IIT 2015
1015

Let p ≥ 3 be an integer and α, β be the roots of x2 – (p + 1) x + 1 = 0. Using mathematical induction show that αn + βn
i) is an integer
ii) and is not divisible by p.

Let p ≥ 3 be an integer and α, β be the roots of x2 – (p + 1) x + 1 = 0. Using mathematical induction show that αn + βn
i) is an integer
ii) and is not divisible by p.

IIT 1992
1016

The function  is not differentiable at

a) – 1

b) 0

c) 1

d) 2

The function  is not differentiable at

a) – 1

b) 0

c) 1

d) 2

IIT 1999
1017

One or more than one correct option

Let RS be a diameter of the circle x2 + y2 = 1 where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and the tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersect a line drawn through Q parallel to RS at a point E. Then the locus of E passes through the point(s)

a) (13,13)

b) (14,12)

c) (13,13)

d) (14,12)

One or more than one correct option

Let RS be a diameter of the circle x2 + y2 = 1 where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and the tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersect a line drawn through Q parallel to RS at a point E. Then the locus of E passes through the point(s)

a) (13,13)

b) (14,12)

c) (13,13)

d) (14,12)

IIT 2016
1018

If x is not an integral multiple of 2π use mathematical induction to prove that
 

If x is not an integral multiple of 2π use mathematical induction to prove that
 

IIT 1994
1019

A circle passing through (1, −2) and touching the axis of X at (3, 0) also passes through the point

a) (−5, 2)

b) (2, −5)

c) (5, −2)

d) (−2, 5)

A circle passing through (1, −2) and touching the axis of X at (3, 0) also passes through the point

a) (−5, 2)

b) (2, −5)

c) (5, −2)

d) (−2, 5)

IIT 2013
1020

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
1021

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
1022

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
1023

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
1024

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
1025

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

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