Question(s) from Search: IIT

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

If two functions f and g satisfy the given conditions  x, y ε ℝ, f(x – y) = f(x)g(y) – f(y)g(x) and g(x – y) = g(x) . g(y) + f(x) . f(y).

If the RHD at x = 0 exists for f(x) then find the derivative of g(x) at x = 0.

Let

be a real valued function. The set of points where f(x) is not differentiable are

a) {0}

b) {1}

c) {0, 1}

d) {∅}

Multiple choice

Let  and

Then g(x) has

a) Local maximum at x = 1 + ln2 and local minima at x = e

b) Local maximum at x = 1 and local minima at x = 2

c) No local maximas

d) No local minimas

For all x in [0, 1], let the second derivative  of a function f(x) exists and satisfies . If f(0) = f(1) then for all x ε [0, 1]

a)  

b)  

c) None of these

Match the following

Let the function defined in column 1 have domain  and range ()

Let f(x) = [x] where [.] denotes the greatest integer function. Then the domain of f is .  .  .  ., points of discontinuity of f are .  .  .  .

a) ∀ x ε I

b) ∀ x ε I − {0}

c) ∀ x ε I – {0, 1}

d) ∀ x ε I – {0, 1, 2}

PQ and PR are two infinite rays, QAR is an arc.

		U

Points lying in the shaded region excluding the boundary satisfies

a)   |z + 1| > 2; |arg(z + 1)| <

b)   |z + 1| < 2; |arg(z + 1)| <

c)  

d)  

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

Find three dimensional vectors u₁, u₂, u₃ satisfying
u₁.u₁ = 4; u₁.u₂ = −2; u₁.u₃ = 6; u₂.u₂  = 2; u₂.u₃ = −5; u₃.u₃ = 29

If a continuous function f defined on the real line ℝ, assumes positive and negative values in ℝ then the equation f(x) = 0 has a root in ℝ. For example, it is known that if a continuous function f on ℝ is positive at some points and its minimum value is negative then the equation f(x) = 0 has a root in ℝ. Consider the function f(x) =  for all real x where k is a real constant.

For k > 0, the set of all values of k for which  has two distinct roots is

a)

b)

c)

d) (0, 1)

Let f(x) = x³ – x² + x + 1 and
 
Discuss the continuity and differentiability of f(x) in the interval (0, 2)

a) Continuous and differentiable in (0, 2)

b) Continuous and differentiable in (0, 2)except x = 1

c) Continuous in (0, 2). Differentiable in (0, 2) except x = 1

d) None of the above

A relation R on the set of complex numbers is defined by iff  is real. Show that R is an equivalence relation.

Find the point on the curve 4x² + a²y² = 4a², 4 < a² < 8 that is farthest from the point (0, −2).

a) (a, 0)

b)

c)

d) (0, - 2)

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y² = 4ax is another parabola with directrix

a) x = −a

b)

c)

d)

Complex numbers  are the vertices A, B, C respectively of an isosceles right angled triangle with right angle at B. Show that

Find all maximum and minimum of the curve y = x(x – 1)², 0 ≤ x ≤ 2. Also find the area bounded by the curve y = x(x – 2)², the Y–axis and the line y = 2.

a) Local minimum at x = 1, Local maximum at x = , Area =

b) Local minimum at x = , Local maximum at x =1, Area =

c) Local minimum at x = 2, Local maximum at x = , Area =

d) Local minimum at x = , Local maximum at x =2, Area =

A line is perpendicular to  and passes through (0, 1, 0). Then the perpendicular distance of this line from the origin is  . . .