Question(s) from Search: IIT

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 –1 ≤ p ≤ 1. Show that the equation 4x³ – 3x – p = 0 has a unique root in the interval  and identify it.

a) p

b) p/3

c)

d)

Find the coordinates of all points P on the ellipse , for which the area of △PON is maximum where O denotes the origin and N the feet of perpendicular from O to the tangent at P.

Determine the equation of the curve passing through origin in the form  which satisfies the differential equation

If α, β are roots of  and γ, δ are roots of  then evaluate  in terms of p, q, r, s.

If p(x) = 51x¹⁰¹ – 2323x¹⁰⁰ – 45x + 1035, using Rolle’s theorem prove that at least one root lies between .

For what values of m does the system of equations 3x + my = m, 2x – 5y = 20 have solutions satisfying x > 0, y > 0?

a) m ε (

b) m ε (

c) m ε ( ∪ (

d) m ε (

Given

 
and f(x) is a quadratic polynomial. V is a point of maximum of f(x) and ‘A’ is the point where f(x) cuts the X–axis. ‘B’ is a point such that AB subtends a right angle at V. Find the area between chord AB and f(x).

a) 125

b) 125/2

c) 125/3

d) 125/6

The area enclosed within the curve |x| + |y| = 1 is .  .  .

a) 1

b)

c)

d) 2

Let a hyperbola pass through the foci of the ellipse  . The transverse and conjugate axes of the hyperbola coincide with the major and minor axes of the given ellipse. Also the product of the eccentricity of the given ellipse and hyperbola is 1 then

a) Equation of the hyperbola is

b) Equation of the hyperbola is

c) Focus of the hyperbola is (5, 0)

d) Vertex of the hyperbola is

The integral $\underset{2}{\overset{4}{\int }}\frac{{\mathrm{logx}}^2}{{\mathrm{logx}}^2+\log \left(x^2-12x+36\right)}\mathrm{dx}$

is equal to

a) 2

b) 4

c) 1

d) 6

Fifteen coupons are numbered 1, 2, 3, .  .  .   ., 15 respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is 9 is

a)

b)

c)

d) None of these

Match the statement of column 1 and the properties of column 2

The value of the integral $\underset{\sqrt{\log 2}}{\overset{\sqrt{\log 3}}{\int }}\frac{x\sin x^2}{\sin x^2+\sin \left(\log 6-x^2\right)}\mathrm{dx}$

is equal to

a) $\frac{1}{4}\log \frac{3}{2}$

b) $\frac{1}{2}\log \frac{3}{2}$

c) $\log \frac{3}{2}$

d) $\frac{1}{6}\log \frac{3}{2}$

Let g(x) be a function of x defined on (−1, 1). If the area of the equilateral triangle with two of its vertices as (0, 0) and [x, g(x)] is , then the function g(x) is

a)

b)

c)

d) None of the above

Show that the integral of   is