Question(s) from Search: IIT

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 circle touching the line L and the circle C₁ externally such that both the circles are on the same side of the line, then the locus of the centre of circle is

a) Ellipse

b) Hyperbola

c) Parabola

d) Pair of straight lines

Find three dimensional vectors u₁, u₂, u₃ satisfying
u₁.u₁ = 4; u₁.u₂ = −2; u₁.u₃ = 6; u₂.u₂  = 2; u₂.u₃ = −5; u₃.u₃ = 29

If a continuous function f defined on the real line ℝ, assumes positive and negative values in ℝ then the equation f(x) = 0 has a root in ℝ. For example, it is known that if a continuous function f on ℝ is positive at some points and its minimum value is negative then the equation f(x) = 0 has a root in ℝ. Consider the function f(x) =  for all real x where k is a real constant.

For k > 0, the set of all values of k for which  has two distinct roots is

a)

b)

c)

d) (0, 1)

Let f(x) = x³ – x² + x + 1 and
 
Discuss the continuity and differentiability of f(x) in the interval (0, 2)

a) Continuous and differentiable in (0, 2)

b) Continuous and differentiable in (0, 2)except x = 1

c) Continuous in (0, 2). Differentiable in (0, 2) except x = 1

d) None of the above

A relation R on the set of complex numbers is defined by iff  is real. Show that R is an equivalence relation.

Find the point on the curve 4x² + a²y² = 4a², 4 < a² < 8 that is farthest from the point (0, −2).

a) (a, 0)

b)

c)

d) (0, - 2)

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y² = 4ax is another parabola with directrix

a) x = −a

b)

c)

d)

Complex numbers  are the vertices A, B, C respectively of an isosceles right angled triangle with right angle at B. Show that

Find all maximum and minimum of the curve y = x(x – 1)², 0 ≤ x ≤ 2. Also find the area bounded by the curve y = x(x – 2)², the Y–axis and the line y = 2.

a) Local minimum at x = 1, Local maximum at x = , Area =

b) Local minimum at x = , Local maximum at x =1, Area =

c) Local minimum at x = 2, Local maximum at x = , Area =

d) Local minimum at x = , Local maximum at x =2, Area =

A line is perpendicular to  and passes through (0, 1, 0). Then the perpendicular distance of this line from the origin is  . . .

Prove that for complex numbers z and ω,   iff z = ω or .

The curve y = ax³ + bx² + cx + 5 touches the X – axis at (− 2, 0) and cuts the Y–axis at a point Q where the gradient is 3. Find a, b, c.

a)

b)

c)

d)

Points A, B, C lie on the parabola . The tangents to the parabola at A, B, C taken in pair intersect at the points P, Q, R. Determine the ratios of the areas of ΔABC and ΔPQR.

Consider the lines given by L₁ : x + 3y – 5 = 0; L₂ = 3x – ky – 1 = 0; L₃ = 5x + 2y −12 = 0. Match the statement/expressions in column 1 with column 2.

 is a circle inscribed in a square whose one vertex is . Find the remaining vertices.

a)

b)

c)

d)

Let a line passing through the fixed point (h, k) cut the X–axis at P and Y–axis at Q. Then find the minimum area of ΔOPQ.

a) hk

b) h²/k

c) k²/h

d) 2hk

Match the following

Let A_n be the area bounded by the curve y = (tanx)ⁿ and the line
x = 0, y = 0 and . Prove that for  . Hence deduce that
 

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 radius of the incircle of △PQR is

a) 4

b) 3

c)

d) 2

ABCD is a rhombus. The diagonals AC and BD intersect at the point M and satisfy BD = 2AC. If the points D and M represent the complex numbers 1 + i and (2 – i) respectively then find the complex number x + iy represented by A.

a)  

b)  

c)  

d)  

Find all possible values of b > 0, so that the area of the bounded region enclosed between the parabolas  and  is maximum.

a) b = 1

b) b ≥ 1

c) b ≤ 1

d) 0 < b < 1

Let f(x) = sinx and g(x) = ln|x|. If the ranges of the composition function fog and gof are R₁ and R₂ respectively then

a)

b) ,

c)

d)

Let C₁ and C₂ be the graph of the function y = x² and y = 2x respectively. Let C₃ be the graph of the function
y = f (x), 0 ≤ x ≤ 1, f (0) = 0. Consider a point P on C₁. Let the lines through P, parallel to the axes meet C₂ and C₃ at Q and R respectively (see figure). If for every position of P (on C₁) the area of the shaded regions OPQ and OPR are equal, determine the function f(x).

a) x² – 1

b) x³ – 1

c) x³ – x²

d) 1 + x² + x³

A hemispherical tank of radius 2 meters is initially full of water and has an outlet of 12cm² cross section area at the bottom. The outlet is opened at some instant. The flow through the outlet is according to the law  where g(t) and h(t) are respectively the velocity of the flow through the outlet and the height of the water level above the outlet at the time t, and g is the acceleration due to gravity. Find the time it takes to empty the tank. (Hint: Form a differential equation by relating the decrease of water level to the outflow).

a)

b)

c)

d)