Question(s) from Search: IIT

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)

A plane passes through (1, −2, 1) and is perpendicular to the two planes  and  The distance of the plane from the point (1, 2, 2) is.

What normal to the curve y = x² forms the shortest normal?

a)

b)

c)

d) y = x + 1

(Multiple choices)
The value of θ lying between θ = 0 and θ =  and satisfying the equation
 = 0 are

a)

b)

c)

d)

Let a complex number α, α ≠ 1, be root of the equation  where p and q are distinct primes. Show that either  or , but not together.

The circle x² + y² = 1 cuts the X–axis at P and Q. Another circle with centre at Q and variable radius intersects the first circle at R above the X–axis and the line segment PQ at S. Find the maximum area of ΔQRS.

a)

b)

c)

d)

From a point A common tangents are drawn to the circle  and the parabola . Find the area of the quadrilateral formed by the common tangents drawn from A and the chords of contact of the circle and the parabola.

True/False
For the complex numbers  and  we write  and  then for all complex numbers z with  we have  . 

a) True

b) False

Let
where a is a positive constant. Find the interval in which  is increasing.

a)

b)

c)

d)

Let S be a square of unit area. Consider any quadrilateral which has one vertex on each side of S. If a, b, c and d denote the lengths of the sides of the quadrilateral; prove that
2 ≤ a² + b² + c² + d² ≤ 4

The number of ordered pairs satisfying the equations
 is

a) 4

b) 2

c) 0

d) 1

Let O (0, 0), A(2, 0) and  be the vertices of a triangle. Let R be the region consisting of all those points P inside ΔOAB which satisfies d(P, OA) ≤ d(P, OB) . d(P, AB), where d denotes the distance from the point to the corresponding line. Sketch the region R and find its area.

a)

b)

c)

d)

The function f(x) = |px – q|+ r|x|, x  when p > 0, q > 0, r > 0 assumes minimum value only on one point if

a)  p ≠ q

b)  r ≠ q

c)  r ≠ p

d)  p = q = r

Let b ≠ 0 and j = 0, 1, 2, .  .  . , n. Let S_j be the area of the region bounded by Y–axis and the curve
.

Show that S₀, S₁, S₂, .  .  .  , S_n are in geometric progression. Also find the sum for a = − 1 and b = π.

a)

b)

c)

d)

A tangent to the ellipse x² + 4y² = 4 meets the ellipse x² + 2y² = 6 at P and Q. Prove that tangents at P and Q of the ellipse x² + 2y² = 6 are at right angles.

Let f(θ) = sinθ (sinθ + sin3θ) then f(θ)

a) ≥ 0 only when θ ≥ 0

b)  ≤ 0 for all real θ

c)  ≥ 0 for all real θ

d) ≤ θ only when θ ≤ 0

Let y = f(x) is a cubic polynomial having maximum at x = − 1 and  has a minimum at x = 1 and f(−1) = 10, f(1) = − 6. Find the cubic polynomial and also find the distance between the points which are maxima or minima.

a)

b)

c)

d)

Each of the following four inequalities given below define a region in the XY–plane. One of these four regions does not have the following property: For any two points (x₁, y₁) and (x₂, y₂) in the region, point  is also in the region. The inequality defining the region that does not have this property is

a) x² + 2y² ≤ 1

b) max (|x|, |y|) ≤ 1

c) x² – y² ≥ 1

d) y² – x ≤ 0

The domain of definition of the function           is

a)  

b)  

c)  

d)  

The set of values of x which ln(1 + x) ≤ x is equal to .  .  .  .

a) (−∞, −1)

b) (−1, 0)

c) (0, 1)

d) (1, ∞)

For any positive integers m, n (with n ≥ m), we are given that
  
Deduce that