Question(s) from Search: IIT

Without expanding a determinant at any stage show that
 = Ax + B

where A, B are non-zero constants

True/False
If the complex numbers  represent the vertices of an equilateral triangle with  then .

a) True

b) False

The order of the differential equation whose general solution is given by  is

a) 5

b) 4

c) 3

d) 2

If f (x) =

a) f (x) is a strictly increasing function

b) f (x) has a local maxima

c) f (x) is a strictly decreasing function

d) f (x) is bounded

Let Δa =
Then show that  = c, a constant.

For any two complex numbers  and any real numbers  is equal to .  .  .  .

a)  

b)  

c)  

d)  

The locus of a variable point whose distance from  is  times its distance from the line  is

a) Ellipse

b) Parabola

c) Hyperbola

d) None of these

If  and  then  equals

a)

b)

c)

d) 1

The second degree polynomial satisfying f (0) = 0, f (1) = 1,  for all x ε [0, 1] is

a)

b) No such polynomial

c)

d)

For a > 0, d > 0, find the value of the determinant
 

a) 0

b) 1

c)

d)

Multiple choices

For real x, the function  will assume all real values provided

a)

b)

c)

d)

If the matrix A is equal to where a, b, c are real positive numbers, abc = 1 and A^TA = I then find the value of a³ + b³ + c³.

a) 1

b) 2

c) 3

d) 4

Prove if α, β are roots of the equation  and γ, δ are roots of  then show that
 

If the function f: [0, 4] → ℝ is differentiable, then for a, b ε [0, 4]

a) 8 f (a) f (b)

b) 8 f (a) f '(b)

c) 8 f '(a) f (b)

d) 8 f '(a) f '(b)

Find the equation of the common tangent in the first quadrant to the circle  and the ellipse   . Also find the length of the intercept of the tangent between the coordinate axis.

A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the value of the determinant chosen is positive is

a)

b)

c)

d)

If one root of  is equal to the power of the other then show that
 

For 0 < a < x the minimum value of the function log_ax + log_xa is 2.

a) True

b) False

The curve described parametrically by   ,  represents

a) A pair of straight lines

b) An ellipse

c) A parabola

d) A hyperbola

For the hyperbola  which of the following remains constant with change in α

a) Abscissae of vertices

b) Abscissae of focii

c) Eccentricity

d) Directrix

The value of  is

a) 0

b) 1

c) 2

d) 4

For a ≤ 0, determine all real roots of the equation
 

If a, b, c, d and p are distinct real numbers such that
 
then a, b, c, d

a) Are in Arithmetic Progression

b) Are in Geometric Progression

c) Are in Harmonic Progression

d) Satisfy ab = cd

e) Satisfy none of these

Suppose f(x) is a function satisfying the following conditions

i) f(0) = 2, f(1) = 1

ii) f has a minimum value at x = 5/2 and

iii) for all x

 
where a, b are constants. Determine the constants a and b, and the function f(x).

a)

b)

c)

d)

Solve