Question(s) from Search: IIT

Solve

 for every 0 < α, β < 2.

Let (x, y) be any point on the parabola y² = 4x. Let P be the point that divides the line segment from (0, 0) to (x, y) in the ratio of 1 : 3. Then the locus of P is

a) x² = y

b) y² = 2x

c) y² = x

d) x² = 2y

The value of  where x > 0 is

a) 0

b) – 1

c) 1

d) 2

The value of

a) 5050

b) 5051

c) 100

d) 101

Let the curve C be the mirror image of the parabola y² = 4x with respect to the line x + y + 4 = 0. If A and B are points of intersection of C with the line y = −5 then the distance between A and B is . . .?

Consider the parabola y² = 8x. Let △₁ be the area of the triangle formed by the end points of its latus rectum and the point $P\left(\frac{1}{2},2\right)$

on the parabola and △₂ be the area of the triangle formed by drawing tangent at P and the end points of the latus rectum. Then $\frac{{△}_1}{{△}_2}$ is

Multiple choices

Let g (x) = x f (x), where   at x = 0

a) g is  but  is not continuous

b) g is  while f is not

c) f and g are both differentiable

d) g is  and  is continuous

A five digit number divisible by 3 is formed using the numerals 0, 1, 2, 3, 4, and 5 without repetition. Total number of ways this can be done is

a) At least 30

b) At most 20

c) Exactly 25

d) None of these

A rectangle with sides (2m – 1) and (2n – 1) is divided into squares of unit length by drawing parallel lines. Then the number of rectangles possible with odd side lengths is

a) mn (m + 1)(n + 1)

b)

c)

d)

If the normal to the curve y = f(x) at the point (3, 4) makes an angle  with the positive X–axis then

a) – 1

b)

c)

d) 1

Find the interval in which ‘a’ lies for which the line y + x = 0 bisects the chord drawn from the point  to the circle

The points on the curve   where the tangent is vertical, is (are)

a)

b)

c)

d)

Let T₁, T₂ be two tangents drawn from (−2, 0) onto the circle C: x² + y² = 1. Determine the circle touching C and having T₁, T₂ as their pair of tangents. Further find the equation of all possible common tangents to the circles, when taken two at a time.

Let  for all real x and y. If   exists and  then find f(2)

a) – 1

b) 0

c) 1

d) 2

Let  and  where  are continuous functions. If A(t) and B(t) are non-zero vectors for all t and

A(0) =

then A(t) and b(t) are parallel for some t.

a) True

b) False

Let n be any positive integer. Prove that
For each non negative integer m ≤ n

Find the centre and radius of the circle formed by all the points represented by  satisfying the relation  where α and β are complex numbers given by
 

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