Question(s) from Search: IIT

Let  be the vertices of the triangle. A parallelogram AFDE is drawn with the vertices D, E and F on the line segments BC, CA and AB respectively. Using calculus find the area of the parallelogram.

a)  

b)  

c)  

d)  

Two rays in the first quadrant x + y = |a| and ax – y = 1 intersect each other in the interval a ε (a₀, ∞). The value of a₀ is

Find the area of the region bounded by the curve C: y = tanx, tangent drawn to C at  and the X–axis.

a) ln2 – 1

b)

c)

d)

then tan t =

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

A curve y = f(x) passes through the point P:(1, 1). The equation to the normal at (1, 1) to the curve y = f(x) is (x – 1) + a(y – 1)  = 0 and the slope of the tangent at any point on the curve is proportional to the ordinate of the point. Determine the equation of the curve. Also obtain the area bounded by the Y–axis, the curve and the normal at P.

a)

b) y = ;

c)  ;

d)

Consider the circle x² + y² = 9 and the parabola y² = 8x. They intersect P and Q in the first and fourth quadrants respectively. Tangents to the circle at P and Q intersect the X–axis at R and tangents to the parabola at P and Q intersect the X- axis at S. The ratio of areas of the triangle PQS and PQR is

a)

b) 1:2

c)

d) 1:8

Let a + b = 4 where a < 2 and let g(x) be a differentiable function. If  for all x, prove that  increases as (b – a) increases.

A and B are two separate reservoirs of water. Capacity of reservoir A is double the capacity of reservoir B. Both the reservoirs are filled completely with water, their inlets are closed and then water is released simultaneously from both the reservoirs. The rate of flow of water out of each reservoir at any instant of time is proportionate to the quantity of water in the reservoir at the time. One hour after the water is released the quantity of water in reservoir A is   times the quantity of water in reservoir B. After how many hours do both the reservoirs have the same quantity of water?

a)

b)

c) ln2

d)  

The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse  is

a)  square units

b)

c)  square units

d) 27 square units

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)