Question(s) from Search: IIT

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)

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)

Let f(x) be a continuous function given by
 

Find the area of the region in the third quadrant bounded by the curve x = − 2y² and y = f(x) lying on the left of the line 8x + 1 = 0.

a) 192

b) 320

c) 761/192

d) 320/761

Let d be the perpendicular distance from the centre of the ellipse  to the tangent at a point P on the ellipse. Let F₁ and F₂ be the two focii of the ellipse, then show that

Find the area of the region bounded by the curves y = x², y = |2 – x²| and y = 2 which lies to the right of the line x = 1.

a)

b)

c)

d)

Prove that in an ellipse the perpendicular from a focus upon a tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix.

A curve passing through the point  has the property that the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of P from the X-axis. Determine the equation of the curve.

Let f : ℝ → ℝ be any function. Define g : ℝ → ℝ by g(x) = |f(x)| for all x. Then g is

a) Onto if f is onto

b) One–one if f is one–one

c) Continuous if f is continuous

d) Differentiable if f is differentiable

f(x) is a differentiable function and g(x) is a double differentiable function such that  
If  prove that there exists some c ε (−3, 3) such that .