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1226

Let f and g be increasing and decreasing functions, respectively from [0, ∞) to [0, ∞). Let h(x) =f(g(x)). If h(0) = 0 then h(x) – h(t) is

a) Always zero

b) Always negative

c) Always positive

d) Strictly increasing

e) None of these

Let f and g be increasing and decreasing functions, respectively from [0, ∞) to [0, ∞). Let h(x) =f(g(x)). If h(0) = 0 then h(x) – h(t) is

a) Always zero

b) Always negative

c) Always positive

d) Strictly increasing

e) None of these

IIT 1988
1227

A rectangle with sides (2m – 1) and (2n – 1) is divided into squares of unit length by drawing parallel lines. Then the number of rectangles possible with odd side lengths is

a) mn (m + 1)(n + 1)

b)

c)

d)

A rectangle with sides (2m – 1) and (2n – 1) is divided into squares of unit length by drawing parallel lines. Then the number of rectangles possible with odd side lengths is

a) mn (m + 1)(n + 1)

b)

c)

d)

IIT 2005
1228

If the normal to the curve y = f(x) at the point (3, 4) makes an angle  with the positive X–axis then

a) – 1

b)

c)

d) 1

If the normal to the curve y = f(x) at the point (3, 4) makes an angle  with the positive X–axis then

a) – 1

b)

c)

d) 1

IIT 2000
1229

A circle passes through points A, B and C with the line segment AC as its diameter. A line passing through A intersects the chord BC at D inside the circle. If ∠DAB and ∠CAB are α and β respectively and the distance between the point A and the midpoint of the line segment DC is d, prove that the area of the circle is
 

A circle passes through points A, B and C with the line segment AC as its diameter. A line passing through A intersects the chord BC at D inside the circle. If ∠DAB and ∠CAB are α and β respectively and the distance between the point A and the midpoint of the line segment DC is d, prove that the area of the circle is
 

IIT 1996
1230

Domain of definition of the function f (x) =  for real valued x is

a)

b)

c)

d)

Domain of definition of the function f (x) =  for real valued x is

a)

b)

c)

d)

IIT 2003
1231

Find the values of a and b, so that the functions

 

Is continuous for 0 ≤ x ≤ π

a)

b)

c)

d)

Find the values of a and b, so that the functions

 

Is continuous for 0 ≤ x ≤ π

a)

b)

c)

d)

IIT 1989
1232

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
1233

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
1234

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
1235

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
1236

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
1237

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
1238

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
1239

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
1240

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
1241

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
1242

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
1243

If  for all positive x where a > 0 and b > 0 then

a) 9ab2 ≥ 4c3

b) 27ab2 ≥ 4c3

c) 9ab2 ≤ 4c3

d) 27ab2 ≤ 4c3

If  for all positive x where a > 0 and b > 0 then

a) 9ab2 ≥ 4c3

b) 27ab2 ≥ 4c3

c) 9ab2 ≤ 4c3

d) 27ab2 ≤ 4c3

IIT 1989
1244

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.

If P is a point on C1 and Q is another point on C2, then  is equal to

a) 0.75

b) 1.25

c) 1

d) 0.5

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.

If P is a point on C1 and Q is another point on C2, then  is equal to

a) 0.75

b) 1.25

c) 1

d) 0.5

IIT 2006
1245

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.

The positive value of k for which  has only one root is

a)

b) 1

c) e

d) ln2

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.

The positive value of k for which  has only one root is

a)

b) 1

c) e

d) ln2

IIT 2007
1246

Let . Find the intervals in which λ should lie in order that f(x) has exactly one minimum and exactly one maximum.

a)

b)

c)

d)

Let . Find the intervals in which λ should lie in order that f(x) has exactly one minimum and exactly one maximum.

a)

b)

c)

d)

IIT 1985
1247

Consider a circle with centre lying on the focus of the parabola  such that it touches the directrix of the parabola. Then a point of intersection of the circle and parabola is

a) or

b)

c)

d)

Consider a circle with centre lying on the focus of the parabola  such that it touches the directrix of the parabola. Then a point of intersection of the circle and parabola is

a) or

b)

c)

d)

IIT 1995
1248

Find the equation of the plane at a distance  from the point  and containing the line
 .

Find the equation of the plane at a distance  from the point  and containing the line
 .

IIT 2005
1249

Let the complex numbers  are vertices of an equilateral triangle. If  be the circumcentre of the triangle, then prove that

Let the complex numbers  are vertices of an equilateral triangle. If  be the circumcentre of the triangle, then prove that

IIT 1981
1250

A two metre long object is fired vertically upwards from the mid-point of two locations A and B, 8 metres apart. The speed of the object after t seconds is given by  metres per second. Let α and β be the angles subtended by the objects A and B respectively after one and two seconds. Find the value of cos(α − β).

a)

b)

c)

d)

A two metre long object is fired vertically upwards from the mid-point of two locations A and B, 8 metres apart. The speed of the object after t seconds is given by  metres per second. Let α and β be the angles subtended by the objects A and B respectively after one and two seconds. Find the value of cos(α − β).

a)

b)

c)

d)

IIT 1989

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