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1201

C1 and C2 are two concentric circles, the radius of C2 being twice of C1 . From a point on C2 tangents PA and PB are drawn to C1. Prove that the centroid of ΔPAB lies on C1.

C1 and C2 are two concentric circles, the radius of C2 being twice of C1 . From a point on C2 tangents PA and PB are drawn to C1. Prove that the centroid of ΔPAB lies on C1.

IIT 1998
1202

In [0, 1], Lagrange’s Mean Value theorem is not applicable to

a)

b)

c)

d)

In [0, 1], Lagrange’s Mean Value theorem is not applicable to

a)

b)

c)

d)

IIT 2003
1203

Let α ε ℝ, then a function f : ℝ → ℝ is differentiable at α if and only if there is a function g : ℝ → ℝ which is continuous at α and satisfies f(x) – f(α) = g(x) (x – α) for all x ε ℝ.

a) True

b) False

Let α ε ℝ, then a function f : ℝ → ℝ is differentiable at α if and only if there is a function g : ℝ → ℝ which is continuous at α and satisfies f(x) – f(α) = g(x) (x – α) for all x ε ℝ.

a) True

b) False

IIT 2001
1204

The area bounded by the angle bisectors of the lines

x2 – y2 + 2y = 1 and the line x + y = 3 is

a) 2

b) 3

c) 4

d) 6

The area bounded by the angle bisectors of the lines

x2 – y2 + 2y = 1 and the line x + y = 3 is

a) 2

b) 3

c) 4

d) 6

IIT 2004
1205

If two functions f and g satisfy the given conditions  x, y ε ℝ, f(x – y) = f(x)g(y) – f(y)g(x) and g(x – y) = g(x) . g(y) + f(x) . f(y).

If the RHD at x = 0 exists for f(x) then find the derivative of g(x) at x = 0.

If two functions f and g satisfy the given conditions  x, y ε ℝ, f(x – y) = f(x)g(y) – f(y)g(x) and g(x – y) = g(x) . g(y) + f(x) . f(y).

If the RHD at x = 0 exists for f(x) then find the derivative of g(x) at x = 0.

IIT 2005
1206

Let

be a real valued function. The set of points where f(x) is not differentiable are

a) {0}

b) {1}

c) {0, 1}

d) {∅}

Let

be a real valued function. The set of points where f(x) is not differentiable are

a) {0}

b) {1}

c) {0, 1}

d) {∅}

IIT 1981
1207

Multiple choice

Let  and

 

Then g(x) has

a) Local maximum at x = 1 + ln2 and local minima at x = e

b) Local maximum at x = 1 and local minima at x = 2

c) No local maximas

d) No local minimas

Multiple choice

Let  and

 

Then g(x) has

a) Local maximum at x = 1 + ln2 and local minima at x = e

b) Local maximum at x = 1 and local minima at x = 2

c) No local maximas

d) No local minimas

IIT 2006
1208

For all x in [0, 1], let the second derivative  of a function f(x) exists and satisfies . If f(0) = f(1) then for all x ε [0, 1]

a)  

b)  

c) None of these

For all x in [0, 1], let the second derivative  of a function f(x) exists and satisfies . If f(0) = f(1) then for all x ε [0, 1]

a)  

b)  

c) None of these

IIT 1981
1209

Match the following

Let the function defined in column 1 have domain  and range ()

Column 1

Column 2

i) 1 + 2x

A) Onto but not one-one

ii) tan x

B) One-one but not onto

C) One-one and onto

D) Neither one

Match the following

Let the function defined in column 1 have domain  and range ()

Column 1

Column 2

i) 1 + 2x

A) Onto but not one-one

ii) tan x

B) One-one but not onto

C) One-one and onto

D) Neither one

IIT 1992
1210

Let f(x) = [x] where [.] denotes the greatest integer function. Then the domain of f is .  .  .  ., points of discontinuity of f are .  .  .  .

a) ∀ x ε I

b) ∀ x ε I − {0}

c) ∀ x ε I – {0, 1}

d) ∀ x ε I – {0, 1, 2}

Let f(x) = [x] where [.] denotes the greatest integer function. Then the domain of f is .  .  .  ., points of discontinuity of f are .  .  .  .

a) ∀ x ε I

b) ∀ x ε I − {0}

c) ∀ x ε I – {0, 1}

d) ∀ x ε I – {0, 1, 2}

IIT 1996
1211

PQ and PR are two infinite rays, QAR is an arc.

U


Points lying in the shaded region excluding the boundary satisfies

a)   |z + 1| > 2; |arg(z + 1)| <

b)   |z + 1| < 2; |arg(z + 1)| <

c)  

d)  

PQ and PR are two infinite rays, QAR is an arc.

U


Points lying in the shaded region excluding the boundary satisfies

a)   |z + 1| > 2; |arg(z + 1)| <

b)   |z + 1| < 2; |arg(z + 1)| <

c)  

d)  

IIT 2005
1212

The minimum value of  where a, b c are all not equal integers and ω(≠1) a cube root of unity is

a) 1

b) 0

c)

d)

The minimum value of  where a, b c are all not equal integers and ω(≠1) a cube root of unity is

a) 1

b) 0

c)

d)

IIT 2005
1213

Match the following
Let the functions defined in column 1 have domain

Column 1

Column 2

    i) sin(π[x])

    A) differentiable everywhere

    ii) sinπ(x-[x])

    B) nowhere differentiable

    C) not differentiable at 1, 1

a) i) → A, ii) → B

b) i) → A, ii) → C

c) i) → C, ii) → A

d) i) → B, ii) → C

Match the following
Let the functions defined in column 1 have domain

Column 1

Column 2

    i) sin(π[x])

    A) differentiable everywhere

    ii) sinπ(x-[x])

    B) nowhere differentiable

    C) not differentiable at 1, 1

a) i) → A, ii) → B

b) i) → A, ii) → C

c) i) → C, ii) → A

d) i) → B, ii) → C

IIT 1992
1214

Find the area of the region bounded by the X–axis and the curve defined by
 
 

a) ln2

b) 2ln2

c)

d)

Find the area of the region bounded by the X–axis and the curve defined by
 
 

a) ln2

b) 2ln2

c)

d)

IIT 1984
1215

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

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

IIT 2006
1216

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

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

IIT 2001
1217

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)

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)

IIT 2007
1218

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

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

IIT 1985
1219

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

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

IIT 1982
1220

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)

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)

IIT 1987
1221

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)

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)

IIT 2002
1222

  

  

IIT 2006
1223

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

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

IIT 1986
1224

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 =

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 =

IIT 1989
1225

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

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

IIT 2006

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