Question(s) from Search: IIT

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.

The unit vector perpendicular to the plane determined by
 is.

Consider the lines

 ;

 
The shortest distance between L₁ and L₂ is

a) 0

b)

c)

d)

Let ABCD is the base of parallelopiped T and Aʹ.BʹCʹDʹ be the upper face. The parallelopiped is compressed so that the vertex Aʹ shifts to Aʹʹ on a parallelepiped S. If the volume of the new parallelopiped is 90% of the parallelopiped T, prove that the locus of Aʹʹ is a plane.

Show that  =

For all A, B, C, P, Q, R show that
 = 0

Let f(x) = |x – 1|, then

a)

b)

c)

d) None of these

The differential equation representing the family of curves  where c is a positive parameter, is of

a) Order 1

b) Order 2

c) Degree 3

d) Degree 4

Let a, b, c be real numbers with a² + b² + c² = 1. Show that the equation represents a straight line
 = 0

Let , then the set  is

a)  

b)  

c)  

d)  ϕ

A normal is drawn at a point  of a curve meeting X-axis at Q. If PQ is of constant length k, then show that the differential equation of the curve is  

If f(x) = 3x – 5 then  

a) is given by

b) is given by

c) does not exist because f is not one-one

d) does not exist because f is not onto

Find the integral solutions of the following system of inequality
 

a) x = 1

b) x = 2

c) x = 3

d) x = 4

Let f(θ) = sinθ (sinθ + sin3θ) then f(θ)

a) ≥ 0 only when θ ≥ 0

b)  ≤ 0 for all real θ

c)  ≥ 0 for all real θ

d) ≤ θ only when θ ≤ 0

Let y = f(x) is a cubic polynomial having maximum at x = − 1 and  has a minimum at x = 1 and f(−1) = 10, f(1) = − 6. Find the cubic polynomial and also find the distance between the points which are maxima or minima.

a)

b)

c)

d)

Each of the following four inequalities given below define a region in the XY–plane. One of these four regions does not have the following property: For any two points (x₁, y₁) and (x₂, y₂) in the region, point  is also in the region. The inequality defining the region that does not have this property is

a) x² + 2y² ≤ 1

b) max (|x|, |y|) ≤ 1

c) x² – y² ≥ 1

d) y² – x ≤ 0

The domain of definition of the function           is

a)  

b)  

c)  

d)  

The set of values of x which ln(1 + x) ≤ x is equal to .  .  .  .

a) (−∞, −1)

b) (−1, 0)

c) (0, 1)

d) (1, ∞)

For any positive integers m, n (with n ≥ m), we are given that
  
Deduce that
  

If A and B are two independent events such that P (A) > 0 and P (B) ≠ 1 then  is equal to

a)

b)

c)

d)

If,  then g(f(x)) is invertible in the domain

a)

b)

c)

d)

Evaluate

a)

b)

c)

d)