Question(s) from Search: IIT

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)

True/False

If  for some non zero vector X then  

a) True

b) False

If  then  

a) True

b) False

Let  and  where O, A and B are non-collinear points. Let p denote the area of the quadrilateral OABC and let q denote the area of the quadrilateral with OA and OC as adjacent sides. If p = kq then k = .  .  .  .  .

Prove that  = 2[cosx + cos3x + cos5x + … + cos(2k−1)x] for any positive integer k. Hence prove that  =

The function
f(x) =|px – q| + r |x|, x ε (−, )
where p > 0, q > 0, r > 0 assumes minimum value on one point if

a) p ≠ q

b) r = q

c) r ≠ p

d) r = p = q

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)

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

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.

A ladder rests against a wall at an angle α to the horizontal. If its foot is pulled away from the wall through a distance a, so that it slides a distance b down the wall making an angle β with the horizontal, then .

a) True

b) False

Let be the vertices of an n sided regular polygon such that   . Then find n.

a) 5

b) 6

c) 7

d) 8

A variable plane at a distance of one unit from the origin cuts the coordinate axes at A, B and C. If the centroid D(x, y, z) of triangle ABC satisfies the relation  then the value of k is

a) 9

b)

c) 1

d) 3

Find the equation of the plane passing through the points (2, 1, 0), (4, 1, 1), (5, 0, 1). Find the point Q such that its distance from the plane is equal to the point P(2, 1, 6) from the plane and the line joining P and Q is perpendicular to the plane.