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Question(s) from Search: IIT

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776

The value of . Given that a, x, y, z, b are in Arithmetic Progression while the value of . If a, x, y, z, b are in Harmonic Progression then find a and b.

The value of . Given that a, x, y, z, b are in Arithmetic Progression while the value of . If a, x, y, z, b are in Harmonic Progression then find a and b.

IIT 1978
777

Let {x} and [x] denote the fractional and integral part of a real number x respectively. Solve 4{x} = x + [x]

Let {x} and [x] denote the fractional and integral part of a real number x respectively. Solve 4{x} = x + [x]

IIT 1994
778

If S1, S2, .  .  .  .,Sn are the sums of infinite geometric series whose first terms are 1, 2, 3,   .  .  ., n and whose common ratios are  respectively, then find the value of

If S1, S2, .  .  .  .,Sn are the sums of infinite geometric series whose first terms are 1, 2, 3,   .  .  ., n and whose common ratios are  respectively, then find the value of

IIT 1991
779

If  are three non–coplanar vectors, then

  equals

a) 0

b)

c)

d)

If  are three non–coplanar vectors, then

  equals

a) 0

b)

c)

d)

IIT 1995
780

Let a, b are real positive numbers. If a, A1, A2, b are in Arithmetic Progression, a, G1, G2, b are in Geometric Progression and a, H1, H2, b are in Harmonic Progression show that
 

Let a, b are real positive numbers. If a, A1, A2, b are in Arithmetic Progression, a, G1, G2, b are in Geometric Progression and a, H1, H2, b are in Harmonic Progression show that
 

IIT 2002
781

  

a) True

b) False

  

a) True

b) False

IIT 1978
782

Multiple choice

Let  be three vectors. A vector in the plane of b and c whose projection on a is of magnitude  is

a)

b)

c)

d)

Multiple choice

Let  be three vectors. A vector in the plane of b and c whose projection on a is of magnitude  is

a)

b)

c)

d)

IIT 1993
783

Let A be vector parallel to the line of intersection of planes P1 and P2. Plane P1 is parallel to the vectors   and  and that P2 is parallel to  and , then the angle between vector A and a given vector  is

a)

b)

c)

d)

Let A be vector parallel to the line of intersection of planes P1 and P2. Plane P1 is parallel to the vectors   and  and that P2 is parallel to  and , then the angle between vector A and a given vector  is

a)

b)

c)

d)

IIT 2006
784

Find the range of values of t for which  

a) (−, −)

b) ( ,  )

c) (− , −  ) U ( ,  )

d) (−,  )

Find the range of values of t for which  

a) (−, −)

b) ( ,  )

c) (− , −  ) U ( ,  )

d) (−,  )

IIT 2005
785

A vector A has components A1, A2, A3 in a right handed rectangular cartesian coordinate system OXYZ. The coordinate system is rotated about the X–axis through an angle . Find the components of A in the new co-ordinate system in terms of A1, A2, A3.

A vector A has components A1, A2, A3 in a right handed rectangular cartesian coordinate system OXYZ. The coordinate system is rotated about the X–axis through an angle . Find the components of A in the new co-ordinate system in terms of A1, A2, A3.

IIT 1983
786

The value of  is equal to

a)

b)

c)

d)

The value of  is equal to

a)

b)

c)

d)

IIT 1991
787

In a triangle OAB, E is the midpoint of BO and D is a point on AB such that AD : DB = 2 : 1. If OD and AE intercept at P determine the ratio OP : PD using vector methods.

In a triangle OAB, E is the midpoint of BO and D is a point on AB such that AD : DB = 2 : 1. If OD and AE intercept at P determine the ratio OP : PD using vector methods.

IIT 1989
788

The position vectors of the vertices A, B, C of a tetrahedron are  respectively. The altitude from the vertex D to the opposite face ABC meets the median line through A of the triangle ABC at E. If the length of the side AD is 4 and the volume of the tetrahedron is . Find the position vector of E or all possible positions.

The position vectors of the vertices A, B, C of a tetrahedron are  respectively. The altitude from the vertex D to the opposite face ABC meets the median line through A of the triangle ABC at E. If the length of the side AD is 4 and the volume of the tetrahedron is . Find the position vector of E or all possible positions.

IIT 1996
789

For any two vectors u and v prove that

i)

ii)

For any two vectors u and v prove that

i)

ii)

IIT 1998
790

True/False

If  for some non zero vector X then  

a) True

b) False

True/False

If  for some non zero vector X then  

a) True

b) False

IIT 1983
791

If  

and the vectors (1, a, a2), (1, b, b2), (1, c, c2) are non-coplanar then the product abc is

If  

and the vectors (1, a, a2), (1, b, b2), (1, c, c2) are non-coplanar then the product abc is

IIT 1985
792

Let  and c be two vectors perpendicular to each other in the XY–plane. All vectors in the same plane having projections 1 and 2 along b and c respectively, are given by

Let  and c be two vectors perpendicular to each other in the XY–plane. All vectors in the same plane having projections 1 and 2 along b and c respectively, are given by

IIT 1987
793

 lies between –4 and 10.

a) True

b) False

 lies between –4 and 10.

a) True

b) False

IIT 1979
794

Determine the smallest positive value of x (in degrees) for which  

a) 30°

b) 50°

c) 55°

d) 60°

Determine the smallest positive value of x (in degrees) for which  

a) 30°

b) 50°

c) 55°

d) 60°

IIT 1993
795

The real roots of the equation x +  = 1 in the interval (−π, π) are …...........

a) x = 0

b) x = ±  

c) x = 0 , x = ±  

The real roots of the equation x +  = 1 in the interval (−π, π) are …...........

a) x = 0

b) x = ±  

c) x = 0 , x = ±  

IIT 1997
796

The domain of the derivative of the function
f (x) =

a) R  { 0 }

b) R

c) R

d) R

The domain of the derivative of the function
f (x) =

a) R  { 0 }

b) R

c) R

d) R

IIT 2002
797

The greater of the two angles
 and  is

a) A

b) B

c) Both are equal

The greater of the two angles
 and  is

a) A

b) B

c) Both are equal

IIT 1989
798

If f (x) = sinx + cosx, g (x) = x2 – 1 then g ( f (x)) is invertible in the domain

a)

b)

c)

d)

If f (x) = sinx + cosx, g (x) = x2 – 1 then g ( f (x)) is invertible in the domain

a)

b)

c)

d)

IIT 2004
799

One or more correct answers
In a triangle the length of the two larger sides are 10 and 9 respectively. If the angles are in arithmetic progression then the length of the third side can be

a)

b)

c) 5

d)

e) None of these

One or more correct answers
In a triangle the length of the two larger sides are 10 and 9 respectively. If the angles are in arithmetic progression then the length of the third side can be

a)

b)

c) 5

d)

e) None of these

IIT 1987
800

Let f (x) = Ax2 + Bx + C where A, B , C are real numbers. Prove that if f (x) is an integer then the numbers 2A, A + B and C are all integers. Conversely prove that if the numbers 2A, A + B and C are all integers then f ( x ) is an integer whenever x is an integer.

Let f (x) = Ax2 + Bx + C where A, B , C are real numbers. Prove that if f (x) is an integer then the numbers 2A, A + B and C are all integers. Conversely prove that if the numbers 2A, A + B and C are all integers then f ( x ) is an integer whenever x is an integer.

IIT 1998

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