Question(s) from Search: IIT

Domain of definition of the function   for real values of x is

a)

b)

c)

d)

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.

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.

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

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

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.

The function  is not differentiable at

a) – 1

b) 0

c) 1

d) 2

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) $\left(\frac{1}{3},\frac{1}{\sqrt{3}}\right)$

b) $\left({\frac{1}{4},}^{}\frac{1}{2}\right)$

c) $\left(\frac{1}{3},-\frac{1}{3}\right)$

d) $\left(\frac{1}{4},-\frac{1}{2}\right)$

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

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)

The circles  and  intersect each other in distinct points if

a) r < 2

b) r > 8

c) 2 < r < 8

d) 2 ≤ r ≤ 8

Prove by induction that
P_n = 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, .  .  .,
P_n = (1 – P) P_{n – 1} + P (1 – P) P_{n – 2}  n ≥ 3,
and ,