Question(s) from Search: IIT

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)

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)

Sketch the curves and identify the region bounded by
 

Consider the following linear equations
ax + by + cz = 0
bx + cy + az = 0
cx + ay + bz = 0
Match the statements/expressions in column 1 with column 2

The domain of the function y(x) given by the equation  is

a) 0 < x ≤ 1

b) 0 ≤ x ≤ 1

c)  < x ≤ 0

d)  < x < 1

If A = , 6A^-1 = A² + cA + dI

then (c, d ) is

a) (−11, 6)

b) (−6, 11)

c)  (6, 11 )

d)  (11, 6 )

Prove that

Tangent at a point P₁ (other than (10, 0)) on the curve y = x³ meets the curve again at P₂. The tangent at P₂ meets the curve at P₃ and so on. Show that the abscissae of P₁, P₂, P₃, .  .  . , P_n form a Geometric Progression. Also find the ratio .

a) 32

b) 16

c)

d)

In what ratio does the X–axis divide the area of the region bounded by the parabolas y = 4x – x² and y = x² – x

a) 1:4

b) 21:1

c) 21:4

d) 3:4

Let C₁ and C_2, be respectively, the parabolas  and  . Let P be any point on C₁ and Q be any point on C₂. Let P₁ and Q₁ be the reflections of P and Q respectively with respect to y = x . Prove that P₁ lies on C₂ and Q₁ lies on C₁ and  . Hence or otherwise determine points P₂ and Q₂ on the parabolas C₁ and C₂ respectively such that  for all points (P, Q) with P on C₁ and Q on C₂ .

Suppose , , are the vertices of an equilateral triangle inscribed in the circle  = 2. If = 1 + i, then find  and .

a)

b)

c)

d) None of the above