Question(s) from Search: IIT

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

If  then f(x) is

a) Increasing on

b) Decreasing on ℝ

c) Increasing on ℝ

d) Decreasing on

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

Let  be the equation of pair of tangents from the origin O to a circle of radius 3 with centre in the first quadrant. If A is a point of contact, find the length of OA.

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)

Let I_n represents area of n sided regular polygon inscribed in a unit circle and O_n the area of n–sided regular polygon circumscribing it. Prove that

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

Minimum area of the triangle formed by the tangent to the ellipse

 with co-ordinate axes is

a)

b)

c)

d) ab

If A and B are points in the plane such that (constant) for all P on a given circle then the value of k cannot be equal to - -  - - -.

If a, b, c are in Geometric Progression then the equations  
have a common root if  are in

a) Arithmetic Progression

b) Geometric Progression

c) Harmonic Progression

d) None of these

Multiple choice

Let h(x) = f(x) – (f(x))² + (f(x))³ for every real number x, then

a) h increases whenever f is increasing

b) h is increasing whenever f is decreasing

c) h is decreasing whenever f is decreasing

d) nothing can be said in general

From the origin chords are drawn to the circle . The equation of the locus of the mid points of these chords is . . . . .

Solve for x in the following equation

Let T_r be the r^th term of an Arithmetic Progression for  If for some positive integers m, n we have  and  then T_mn equals

a)

b)

c)

d)

The area of the triangle formed by the tangents from the point (4, 3) to the circle  and the line joining their point of contact is .

Consider an infinite geometric series with first term a and common ratio r. If its sum is four and the second term is  then

a)

b)

c)

d)

The area of triangle formed by the positive X–axis and the normal and tangent to the circle  at  is . . . . . .

For every positive integer n, prove that
 
Hence or otherwise prove that
 
Where [ ] denotes greatest integer not exceeding x.

If  are positive real numbers whose product is a fixed number c then the minimum value of
  is

a)

b)

c)

d)

Intercepts on the line y = x by the circle  is AB. Equation of the circle with AB as diameter is  . . . . .

Let a and b the roots of the equation  and those of  are c and d, then find the value of a + b + c + d when a ≠ b ≠ c ≠ d.

Find the coordinates of the point on the curve  where the tangent to the curve has the greatest slope.

a) (0, 0)

b)

c)

d)

Fill in the blanks

If  is a root of the equation  where p and q are real then (p, q)  …………