Question(s) from Search: IIT

Let f (x) be defined on the interval  such that

g (x) = f (|x|) + |f(x)|

Test the differentiability of g (x) in

a) g(x) is differentiable at all x  ℝ

b) g(x) is differentiable at all x  ℝ except at x = 1

c) g(x) is differentiable at all x  ℝ except at x = 0, 1

d) g(x) is differentiable at all x  ℝ except at x = 0, 1, 2

If the LCM of p, q is  where r, s, t are prime numbers and p, q are positive integers then the number of ordered pairs (p, q) is

a) 252

b) 254

c) 225

d) 224

Consider a family of circles passing through two fixed points A (3, 7) and B (6, 5). Show that the chords in which the circle  cuts the members of the family are concurrent at a point. Find the coordinates of this point.

In how many ways can a pack of 52 cards be divided into four groups of 13 cards each.

In a triangle ABC, let ∠ C = . If r is the inradius and R is the circumradius of the triangle then 2(r+R) = ………….

a) a+b

b) b+c

c) c+a

d) a+b+c

Determine the values of x for which the following function fails to be continuous or differentiable.

Justify your answer.

a) f(x) is continuous and differentiable

b) f(x) is continuous everywhere but not differentiable at
x = 1, 2

c) f(x) is continuous everywhere but not differentiable at x = 2

d) f(x) is neither continuous nor differentiable at x = 1, 2

Let  

And

where a and b are non-negative real numbers. Determine the composite function gof. If (gof)(x) is continuous for all real x, determine the values of a and b. Is gof differentiable at x = 0?

a) a = b = 0

b) a = 0, b = 1

c) a = 1, b = 0

d) a = b = 1

Find the equation of the circle touching the line 2x + 3y + 1 = 0 at the point (1, −1) and is orthogonal to the circle which has the line segment having end points (0, −1) and (−2, 3) as diameter.

Show that the value of  wherever defined

a) always lies between  and 3

b) never lies between  and 3

c) depends on the value of x

Show that f(x) is differentiable at the value of α = 1. Also,

a) b² +c² = 4

b) 4 b²  = 4 − c²  

c) 64 b² = 4 − c²

d) 64 b² = 4 + c²

The product of r consecutive natural numbers is divisible by r!

a) True

b) False

The area bounded by the curve y = f(x), the X–axis and the ordinates x = 1, x = b is (b – 1) sin (3b + 4). Then f(x) is

a) (x – 1) cos (3x + b)

b) sin (3x + 4)

c) sin (3x + 4) + 3 (x – 1) cos (3x + 4)

d) none of these

The sum  where  equals

a) i

b) i – 1

c) – i

d) 0

Fill in the blank

The value of f (x) =  lies in the interval …………….

a)

b)

c)

d)

Find the area bounded by the curve x² = 4y and the straight line
x = 4y – 2.

a) 3/2

b) 3/4

c) 9/4

d) 9/8

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)