1076 |
f(x) is twice differentiable polynomial function such that f (1) = 1, f (2) = 4, f (3) = 9, then a) there exists at least one x (1, 2) such that b) there exists at least one x (2, 3) such that c) d) there exists at least one x (1, 3) such that
f(x) is twice differentiable polynomial function such that f (1) = 1, f (2) = 4, f (3) = 9, then a) there exists at least one x (1, 2) such that b) there exists at least one x (2, 3) such that c) d) there exists at least one x (1, 3) such that
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IIT 2005 |
|
1077 |
Multiple choices The function f (x) = max is a) continuous at all points b) differentiable at all points c) differentiable at all points except x = 1 and x = d) continuous at all points except at x=1 and x=-1 where it is discontinuous
Multiple choices The function f (x) = max is a) continuous at all points b) differentiable at all points c) differentiable at all points except x = 1 and x = d) continuous at all points except at x=1 and x=-1 where it is discontinuous
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IIT 1995 |
|
1078 |
a) b) c) d)
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IIT 2005 |
|
1079 |
Let g (x) be a polynomial of degree one and f (x) be defined by Find the continuous function f (x) satisfying a) b) c) d) None of the above
Let g (x) be a polynomial of degree one and f (x) be defined by Find the continuous function f (x) satisfying a) b) c) d) None of the above
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IIT 1987 |
|
1080 |
In how many ways can a pack of 52 cards be divided equally amongst 4 players in order?
In how many ways can a pack of 52 cards be divided equally amongst 4 players in order?
|
IIT 1979 |
|
1081 |
Let for all real x and y. If exists and then find f(2) a) – 1 b) 0 c) 1 d) 2
Let for all real x and y. If exists and then find f(2) a) – 1 b) 0 c) 1 d) 2
|
IIT 1995 |
|
1082 |
Let and where are continuous functions. If A(t) and B(t) are non-zero vectors for all t and A(0) = then A(t) and b(t) are parallel for some t. a) True b) False
Let and where are continuous functions. If A(t) and B(t) are non-zero vectors for all t and A(0) = then A(t) and b(t) are parallel for some t. a) True b) False
|
IIT 2001 |
|
1083 |
Let n be any positive integer. Prove that For each non negative integer m ≤ n
Let n be any positive integer. Prove that For each non negative integer m ≤ n
|
IIT 1999 |
|
1084 |
Using permutation or otherwise prove that is an integer, where n is a positive integer.
Using permutation or otherwise prove that is an integer, where n is a positive integer.
|
IIT 2004 |
|
1085 |
For all complex numbers satisfying = 5, the minimum value of a) 0 b) 2 c) 7 d) 17
For all complex numbers satisfying = 5, the minimum value of a) 0 b) 2 c) 7 d) 17
|
IIT 2002 |
|
1086 |
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)
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IIT 2005 |
|
1087 |
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
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IIT 1992 |
|
1088 |
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)
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IIT 2007 |
|
1089 |
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.
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IIT 1982 |
|
1090 |
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
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IIT 1986 |
|
1091 |
Prove that for complex numbers z and ω, iff z = ω or .
Prove that for complex numbers z and ω, iff z = ω or .
|
IIT 1999 |
|
1092 |
is a circle inscribed in a square whose one vertex is . Find the remaining vertices. a) b) c) d)
is a circle inscribed in a square whose one vertex is . Find the remaining vertices. a) b) c) d)
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IIT 2005 |
|
1093 |
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)
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)
|
IIT 1993 |
|
1094 |
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 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)
|
IIT 1994 |
|
1095 |
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)
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)
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IIT 2001 |
|
1096 |
Find the area bounded by the curves x2 = y, x2 = − y and y2 = 4x – 3 a) 1 b) 3 c) 1/3 d) 1/9
Find the area bounded by the curves x2 = y, x2 = − y and y2 = 4x – 3 a) 1 b) 3 c) 1/3 d) 1/9
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IIT 2005 |
|
1097 |
Let E = {1, 2, 3, 4} and F = {1, 2}, then the number of onto functions from E to F is a) 14 b) 16 c) 12 d) 8
Let E = {1, 2, 3, 4} and F = {1, 2}, then the number of onto functions from E to F is a) 14 b) 16 c) 12 d) 8
|
IIT 2001 |
|
1098 |
For a twice differentiable function f(x), g(x) is defined as If for a < b < c < d < e, f(a) = 0, f(b) = 2, f(c) = − 1, f(d) = 2, f(e) = 0 then find the maximum number of zeros of g(x). a) 1 b) 2 c) 3 d) 6
For a twice differentiable function f(x), g(x) is defined as If for a < b < c < d < e, f(a) = 0, f(b) = 2, f(c) = − 1, f(d) = 2, f(e) = 0 then find the maximum number of zeros of g(x). a) 1 b) 2 c) 3 d) 6
|
IIT 2006 |
|
1099 |
The larger of cos (lnθ) and ln (cosθ) if is a) cos(lnθ) b) ln(cosθ)
The larger of cos (lnθ) and ln (cosθ) if is a) cos(lnθ) b) ln(cosθ)
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IIT 1983 |
|
1100 |
X and Y are two sets and f : X → Y. If then the true statement is a) b) c) , d)
X and Y are two sets and f : X → Y. If then the true statement is a) b) c) , d)
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IIT 2005 |
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