Question(s) from Search: IIT

The value of  is equal to

a)

b)

c)

d)

If  then  

a) True

b) False

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

Let p, q, r be three mutually perpendicular vectors of the same magnitude. If x satisfies the equation p  ((x – q)  p) + q ((x – r)  q) + r  ((x – p)  r) = 0 then x is given by

a)

b)

c)

d)

Let f : R → R be any function defined g : R → R by g (x) = |f (x)| for all x. Then g is

a) onto if f is onto

b) one to one if f is one to one

c) continuous if f is continuous

d) differentiable if f is differentiable

Let and a unit vector c be coplanar. If c is perpendicular to a then c is equal to

a)

b)

c)

d)

If f : [ 1,  → [ 2, ] is given by f (x) = x +  then ( x ) is given by

a)

b)

c)

d) 1 +

The function of f : R → R be defined by f (x) = 2x + sinx for x ε R . Then f is

a) one-one and onto

b) one-one but not onto

c) onto but not one-one

d) neither one-one nor onto

Multiple choice

The vector  is

a) A unit vector

b) Makes an angle  with the vector

c) Parallel to vector

d) Perpendicular to the vector

A₁, A₂, …… , A_n are the vertices of  a regular polygon with n sides and O is the centre. Show that
 

Multiple choice

There exists a triangle ABC satisfying the conditions

a) bsinA = a, A <

b) bsinA > a, A >

c) bsinA > a, A <

d) bsinA < a, A <, b > a

e) bsinA < a, A >, b = a

With usual notation if in a triangle ABC,  then

 .

a) True

b) False

If A, B, C are such that |B| = |C|. Prove that

If in a triangle ABC, cosA cosB + sinA sinB sin C = 1 then show that  a : b : c = 1 : 1 :

a) True

b) False

Let u and v be unit vectors. If w is a vector such that , then prove that  and that equality holds if and only if  is perpendicular to

If the lines  and  intersect then the value of k is

a)

b)

c)

d)

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

a) True

b) False

If  

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

The area of a triangle whose vertices are
 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

If b > a then the equation ( x – a ) ( x – b )1 = 0 has

a) Both roots in [ a, b ]

b) Both roots in ( , a )

c) Both roots in (  )

d) One root in ( , a ) and other in ( )

Prove that for all values of θ
 = 0

A = , B = , U = , V =

If AX = U has infinitely many solutions, prove that BX = V has no unique solution. Also prove that if afd ≠ 0 then BX = V has no solution. X is a vector.

Let λ and α be real. Find the set of all values of λ for which the system of linear equations
 
 
 
has a non-trivial solution. For λ = 1 find the value of α.