Question(s) from Search: IIT

Using permutation or otherwise prove that    is an integer, where n is a positive integer.

Three circles of radii 3, 4 and 5 units touch each other externally and tangents drawn at the points of contact intersect at P. Find the distance between P and the point of contact.

In ΔABC, D is the midpoint of BC. If AD is perpendicular to AC then .

a) True

b) False

A function f : R  R where R is the set of real numbers is defined by f (x) = . Find the interval of values of α for which f is onto. Is the function one to one for α = 3? Justify your answer.

a) 2 ≤ α ≤ 14

b) α ≥ 2

c) α ≤ 14

d) none of the above

If f₁ ( x ) and f₂ ( x ) are defined by domains D₁ and D₂ respectively, then f₁ ( x ) + f₂ ( x ) is defined as on D₁ D₂.

a) True

b) False

a) ln2

b) ln3

c) ln6

d) ln2 ln3

For all complex numbers satisfying  = 5, the minimum value of

a) 0

b) 2

c) 7

d) 17

If f(x) and g(x) are differentiable functions for 0 ≤ x ≤ 1 such that f(0) = 2, g(0) = 0, f(1) = 6, g(1) = 2 then show that there exists c satisfying 0 < c < 1 and .

a) 0 < c < 1 and

b) 0 < c < 1 and

c) 0 < c < 1 and

d) 0 < c < 1 and

Let a > 0, b > 0, c > 0 then both the roots of the equation  

a) are real and positive

b) have negative real parts

c) have positive real parts

d) none of these

If f(x) is a continuous function defined for 1 ≤ x ≤ 3. If f(x) takes rational values for all x and f(2) = 10 then f(1.5) = .  .  .  .

a) 2

b) 5

c) 10

d) 20

If x, y, z are real and distinct then  is always

a) Non – negative

b) Non – positive

c) Zero

d) None of these

Match the following
Let [x] denote the greatest integer less than or equal to x

a) (i)→ A, B, C, (ii)→ A, D, (iii)→ C, D, (iv)→ A, B

b) (i)→ A, (ii)→ A, (iii)→ C, (iv)→ B

c) (i)→ B, (ii)→ D, (iii)→ C, (iv)→ A

d) (i)→ A, B, (ii)→ A, D, (iii)→ C, D, (iv)→ B

(One or more than one correct answer)
If  are complex numbers such that  and  then the pair of complex numbers  and  satisfy

a)

b)

c)

d) None of these

Let ABCD be a square with side of length 2 units. C₂ is the circle through the vertices A, B, C, D and C₁ is the circle touching all the sides of the square ABCD. L is a line through A.

A line M is drawn through A parallel to BD. Point S moves such that the distance from the line BD and the vertex A are equal. If the locus of S cuts M at T₂ and T₃ and AC at T₁, then find the area of △T₁T₂T₃.

Express  in the form A + iB

a)

b)

c)

d)

Find the area bounded by the curves
 

a) 1/6

b) 1/3

c) π

d)

If the line x – 1 = 0 is the directrix of the parabola y² – kx + 8 = 0, then one of the values of k is

a)

b) 8

c) 4

d)

Find the area bounded by the curves x² + y² = 25, 4y = |4 – x²| and x = 0 above the X–axis.

a)

b)

c)

d)

If sinA sinB sinC + cosA cosB = 1then the value of sinC is

Let = 10 + 6i and  . If z is a complex number such that argument of  is  then prove that  .

Compute the area of the region bounded by the curves
y = exlnx and

a)

b)

c)

d)

A plane passes through (1, −2, 1) and is perpendicular to the two planes  and  The distance of the plane from the point (1, 2, 2) is.

What normal to the curve y = x² forms the shortest normal?

a)

b)

c)

d) y = x + 1

(Multiple choices)
The value of θ lying between θ = 0 and θ =  and satisfying the equation
 = 0 are

a)

b)

c)

d)

Let a complex number α, α ≠ 1, be root of the equation  where p and q are distinct primes. Show that either  or , but not together.