Question(s) from Search: IIT

Let f be a one–one function with domain {x, y, z} and range {1, 2, 3}. It is given that exactly one of the following statements is true and remaining statements are false f (1) = 1, f (y) ≠ 1, f (z) ≠ 2. Determine  

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.

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

If S₁, S₂, .  .  .  .,S_n 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  are three non–coplanar vectors, then

  equals

a) 0

b)

c)

d)

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
 

a) True

b) False

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)

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)

Find the range of values of t for which  

a) (−, −)

b) ( ,  )

c) (− , −  ) U ( ,  )

d) (−,  )

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.

The value of  is equal to

a)

b)

c)

d)

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.

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.

For any two vectors u and v prove that

i)

ii)

The points with position vectors  and  are collinear for all real values of k.

a) True

b) False

Multiple choices
Let and  (x is measured in radians) then x lies in the interval

a)

b)

c)

d)

If  

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

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

 lies between –4 and 10.

a) True

b) False

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

a) 30°

b) 50°

c) 55°

d) 60°

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

a) x = 0

b) x = ±  

c) x = 0 , x = ±  

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

a) R  { 0 }

b) R

c) R

d) R

The greater of the two angles
 and  is

a) A

b) B

c) Both are equal

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

a)

b)

c)

d)