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101 |
Let f (x + y) = f (x) f (y) for all x, y. Suppose that f (5) = 2 and  (0) = 3. Find f (5). a) 1 b) 2 c) 3 d) 6
Let f (x + y) = f (x) f (y) for all x, y. Suppose that f (5) = 2 and (0) = 3. Find f (5). a) 1 b) 2 c) 3 d) 6
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IIT 1981 |
03:33 min
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|
102 |
If a function f :  is an odd function such that  for x ε [a, 2a] and the left hand derivative at x = a is 0 then find the left hand derivative at x =  a) 0 b) 1 c) a d) 2a
If a function f : is an odd function such that for x ε [a, 2a] and the left hand derivative at x = a is 0 then find the left hand derivative at x = a) 0 b) 1 c) a d) 2a
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IIT 2003 |
03:55 min
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103 |
A country produces 90% of its food diet. The population grows continuously at a rate of 3% per year. Its annual food production every year is 4% more than that of last year. Assuming that the average food requirement per person remains constant, prove that the country will become self sufficient in food after n years, where n is the smallest integer bigger than or equal to 
A country produces 90% of its food diet. The population grows continuously at a rate of 3% per year. Its annual food production every year is 4% more than that of last year. Assuming that the average food requirement per person remains constant, prove that the country will become self sufficient in food after n years, where n is the smallest integer bigger than or equal to
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IIT 2000 |
04:17 min
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104 |
If f(x) is a polynomial of degree less than or equal to 2 and S be the set of all such polynomials so that P(0) = 0 P(1) = 1, and 
Then a) S = ɸ b) S = ax + (1 – a) x2 ⩝ a ε (0, 2) c) S = ax + (1 – a) x2 ⩝ a ε (0, ∞) d) S = ax + (1 – a) x2 ⩝ a ε (0, 1)
If f(x) is a polynomial of degree less than or equal to 2 and S be the set of all such polynomials so that P(0) = 0 P(1) = 1, and Then a) S = ɸ b) S = ax + (1 – a) x2 ⩝ a ε (0, 2) c) S = ax + (1 – a) x2 ⩝ a ε (0, ∞) d) S = ax + (1 – a) x2 ⩝ a ε (0, 1)
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IIT 2005 |
02:32 min
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|
105 |
The line  is a diameter of the circle  a) True b) False
The line is a diameter of the circle a) True b) False
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IIT 1989 |
01:39 min
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|
106 |
One or more correct answers
In a triangle PQR, sin P, sin Q, sin R are in arithmetic progression then a) Altitudes are in arithmetic progression b) Altitudes are in harmonic progression c) Medians are in geometric progression d) Medians are in arithmetic progression
One or more correct answers
In a triangle PQR, sin P, sin Q, sin R are in arithmetic progression then a) Altitudes are in arithmetic progression b) Altitudes are in harmonic progression c) Medians are in geometric progression d) Medians are in arithmetic progression
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IIT 1998 |
03:36 min
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107 |
f(x) is a function such that  and the tangent at any point passes through (1, 2). Find the equation of the tangent. a) x = 2 b) y = 2 c) x + y = 2 d) x – y = 2
f(x) is a function such that and the tangent at any point passes through (1, 2). Find the equation of the tangent. a) x = 2 b) y = 2 c) x + y = 2 d) x – y = 2
|
IIT 2005 |
03:06 min
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|
108 |
The lines  and  are tangents to the same circle. The radius of this circle is . . . . .
The lines and are tangents to the same circle. The radius of this circle is . . . . .
|
IIT 1984 |
02:30 min
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|
109 |
The external radii  of ΔABC are in harmonic progression then prove that a, b, c are in arithmetic progression a) True b) False
The external radii of ΔABC are in harmonic progression then prove that a, b, c are in arithmetic progression a) True b) False
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IIT 1983 |
01:51 min
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|
110 |
True / False If f (x) = ( a – xn )1/n where a > 0 and n is a positive integer then f ( f ( x ) ) = x. a) True b) False
True / False If f (x) = ( a – xn )1/n where a > 0 and n is a positive integer then f ( f ( x ) ) = x. a) True b) False
|
IIT 1983 |
01:23 min
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|
111 |
Let f(x) =  If f is continuous for all x, then k is equal to a) 3 b) 5 c) 7 d) 9
Let f(x) = If f is continuous for all x, then k is equal to a) 3 b) 5 c) 7 d) 9
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IIT 1981 |
03:32 min
|
|
112 |
Fill in the blank The domain of the function f (x) =  is a) [− 2, − 1] b) [1, 2] c) [− 2, − 1] ⋃ [1, 2] d) None of the above
Fill in the blank The domain of the function f (x) = is a) [− 2, − 1] b) [1, 2] c) [− 2, − 1] ⋃ [1, 2] d) None of the above
|
IIT 1984 |
02:48 min
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|
113 |
Then 
a) 0 b) 1 c) 2 d) 4
|
IIT 1981 |
01:26 min
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|
114 |
The complex numbers  satisfying  are the vertices of the triangle which is a) of zero area b) right angle isosceles c) equilateral d) obtuse angled isosceles
The complex numbers satisfying are the vertices of the triangle which is a) of zero area b) right angle isosceles c) equilateral d) obtuse angled isosceles
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IIT 2001 |
05:10 min
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|
115 |
Let x and y be two real variables such that x > 0 and xy = 1. Find the minimum value of x + y. a) 1 b) 2 c) 3 d) 4
Let x and y be two real variables such that x > 0 and xy = 1. Find the minimum value of x + y. a) 1 b) 2 c) 3 d) 4
|
IIT 1981 |
01:44 min
|
|
116 |
ABC is an isosceles triangle in a circle of radius r. If AB = AC and h is the altitude from A to BC then the triangle ABC has perimeter  , area A = . . . . . Also  . . . . .
ABC is an isosceles triangle in a circle of radius r. If AB = AC and h is the altitude from A to BC then the triangle ABC has perimeter , area A = . . . . . Also . . . . .
|
IIT 1989 |
07:12 min
|
|
117 |
Let f(x) = x|x|. The set of points where f(x) is twice differentiable is . . . . a) ℝ b) 0 c) ℝ − {0, 1}
Let f(x) = x|x|. The set of points where f(x) is twice differentiable is . . . . a) ℝ b) 0 c) ℝ − {0, 1}
|
IIT 1992 |
02:00 min
|
|
118 |
Find the shortest distance of the point (0, c) from the parabola
y = x2, where 0 ≤ c ≤ 5. a)  b)  c)  d) 
Find the shortest distance of the point (0, c) from the parabola
y = x2, where 0 ≤ c ≤ 5. a) b) c) d)
|
IIT 1982 |
03:58 min
|
|
119 |
Both roots of the equation ( x – b) ( x – c) + (x – c) ( x – a) + (x – a) (x – b) = 0 are always a) positive b) negative c) real d) none of these
Both roots of the equation ( x – b) ( x – c) + (x – c) ( x – a) + (x – a) (x – b) = 0 are always a) positive b) negative c) real d) none of these
|
IIT 1980 |
02:52 min
|
|
120 |
a) – 1 b) 0 c) 1 d) 2
a) – 1 b) 0 c) 1 d) 2
|
IIT 1997 |
02:51 min
|
|
121 |
If  is purely real where ω = α + iβ, β ≠ 0 and z ≠ 1 then the set of real values of z is a)  b)  c)  d) 
If is purely real where ω = α + iβ, β ≠ 0 and z ≠ 1 then the set of real values of z is a) b) c) d)
|
IIT 2006 |
05:43 min
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|
122 |
Two vertices of an equilateral triangle are (- 1, 0) and (1, 0) and its third vertex lies above the X–axis, the equation of circumcircle is . . .
Two vertices of an equilateral triangle are (- 1, 0) and (1, 0) and its third vertex lies above the X–axis, the equation of circumcircle is . . .
|
IIT 1997 |
04:55 min
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|
123 |
Two towns A and B are 60 meters apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by all the 200 students is to be as small as possible then the school should be built at a) Town B b) 45 km from town A c) Town A d) 45 km from town B
Two towns A and B are 60 meters apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by all the 200 students is to be as small as possible then the school should be built at a) Town B b) 45 km from town A c) Town A d) 45 km from town B
|
IIT 1982 |
01:37 min
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|
124 |
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 line y = x meets y =  for k ≤ 0 at a) No point b) One point c) Two points d) More than two points
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 line y = x meets y = for k ≤ 0 at a) No point b) One point c) Two points d) More than two points
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IIT 2007 |
02:08 min
|
|
125 |
(One or more than one correct answer)
Let  and  be complex numbers such that  and  . If  has positive real part and has negative imaginary part, then  may be a) Zero b) Real and positive c) Real and negative d) None of these
(One or more than one correct answer)
Let and be complex numbers such that and . If has positive real part and has negative imaginary part, then may be a) Zero b) Real and positive c) Real and negative d) None of these
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IIT 1986 |
05:31 min
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