Question(s) from Search: IIT

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

A circle passes through points A, B and C with the line segment AC as its diameter. A line passing through A intersects the chord BC at D inside the circle. If ∠DAB and ∠CAB are α and β respectively and the distance between the point A and the midpoint of the line segment DC is d, prove that the area of the circle is
 

Domain of definition of the function f (x) =  for real valued x is

a)

b)

c)

d)

Find the values of a and b, so that the functions

Is continuous for 0 ≤ x ≤ π

a)

b)

c)

d)

C₁ and C₂ are two concentric circles, the radius of C₂ being twice of C₁ . From a point on C₂ tangents PA and PB are drawn to C₁. Prove that the centroid of ΔPAB lies on C₁.

In [0, 1], Lagrange’s Mean Value theorem is not applicable to

a)

b)

c)

d)

Let α ε ℝ, then a function f : ℝ → ℝ is differentiable at α if and only if there is a function g : ℝ → ℝ which is continuous at α and satisfies f(x) – f(α) = g(x) (x – α) for all x ε ℝ.

a) True

b) False

The area bounded by the angle bisectors of the lines

x² – y² + 2y = 1 and the line x + y = 3 is

a) 2

b) 3

c) 4

d) 6

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

Use the function  , x > 0 to determine the bigger of the numbers e^π and π^e.

a) e^π

b) π^e

In a triangle ABC, D and E are points on  and  respectively such that  and . Let P be the point of intersection of  and . Find  using vector method.

a)

b)

c)

d) 2

The minimum value of  where a, b c are all not equal integers and ω(≠1) a cube root of unity is

a) 1

b) 0

c)

d)

Match the following
Let the functions defined in column 1 have domain

a) i) → A, ii) → B

b) i) → A, ii) → C

c) i) → C, ii) → A

d) i) → B, ii) → C

Find the area of the region bounded by the X–axis and the curve defined by
 
 

a) ln2

b) 2ln2

c)

d)

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 circle touching the line L and the circle C₁ externally such that both the circles are on the same side of the line, then the locus of the centre of circle is

a) Ellipse

b) Hyperbola

c) Parabola

d) Pair of straight lines