1201 |
The sum where equals a) i b) i – 1 c) – i d) 0
The sum where equals a) i b) i – 1 c) – i d) 0
|
IIT 1998 |
|
1202 |
If f(x) is a continuous function defined for 1 ≤ x ≤ 3. If f(x) takes rational values for all x and f(2) = 10 then f(1.5) = . . . . a) 2 b) 5 c) 10 d) 20
If f(x) is a continuous function defined for 1 ≤ x ≤ 3. If f(x) takes rational values for all x and f(2) = 10 then f(1.5) = . . . . a) 2 b) 5 c) 10 d) 20
|
IIT 1997 |
|
1203 |
If x, y, z are real and distinct then is always a) Non – negative b) Non – positive c) Zero d) None of these
If x, y, z are real and distinct then is always a) Non – negative b) Non – positive c) Zero d) None of these
|
IIT 2005 |
|
1204 |
Match the following Let [x] denote the greatest integer less than or equal to x Column 1 | Column 2 | i) x|x| | A)continuous in | ii) | B)Differentiable in | iii) x + [x] | C)Steadily increasing in | iv) |x – 1| + |x + 1| | D) Not differentiable at least at one point in | a) (i)→ A, B, C, (ii)→ A, D, (iii)→ C, D, (iv)→ A, B b) (i)→ A, (ii)→ A, (iii)→ C, (iv)→ B c) (i)→ B, (ii)→ D, (iii)→ C, (iv)→ A d) (i)→ A, B, (ii)→ A, D, (iii)→ C, D, (iv)→ B
Match the following Let [x] denote the greatest integer less than or equal to x Column 1 | Column 2 | i) x|x| | A)continuous in | ii) | B)Differentiable in | iii) x + [x] | C)Steadily increasing in | iv) |x – 1| + |x + 1| | D) Not differentiable at least at one point in | a) (i)→ A, B, C, (ii)→ A, D, (iii)→ C, D, (iv)→ A, B b) (i)→ A, (ii)→ A, (iii)→ C, (iv)→ B c) (i)→ B, (ii)→ D, (iii)→ C, (iv)→ A d) (i)→ A, B, (ii)→ A, D, (iii)→ C, D, (iv)→ B
|
IIT 2007 |
|
1205 |
(One or more than one correct answer) If are complex numbers such that and then the pair of complex numbers and satisfy a) b) c) d) None of these
|
IIT 1985 |
|
1206 |
Express in the form A + iB a) b) c) d)
Express in the form A + iB a) b) c) d)
|
IIT 1979 |
|
1207 |
Let = 10 + 6i and . If z is a complex number such that argument of is then prove that .
|
IIT 1990 |
|
1208 |
(Multiple choices) The value of θ lying between θ = 0 and θ = and satisfying the equation = 0 are a) b) c) d)
(Multiple choices) The value of θ lying between θ = 0 and θ = and satisfying the equation = 0 are a) b) c) d)
|
IIT 1988 |
|
1209 |
Let a complex number α, α ≠ 1, be root of the equation where p and q are distinct primes. Show that either or , but not together.
|
IIT 2002 |
|
1210 |
True/False For the complex numbers and we write and then for all complex numbers z with we have . a) True b) False
|
IIT 1981 |
|
1211 |
Find the area of the region bounded by the curves y = x2, y = |2 – x2| and y = 2 which lies to the right of the line x = 1. a) b) c) d)
Find the area of the region bounded by the curves y = x2, y = |2 – x2| and y = 2 which lies to the right of the line x = 1. a) b) c) d)
|
IIT 2002 |
|
1212 |
Let f : ℝ → ℝ be any function. Define g : ℝ → ℝ by g(x) = |f(x)| for all x. Then g is a) Onto if f is onto b) One–one if f is one–one c) Continuous if f is continuous d) Differentiable if f is differentiable
Let f : ℝ → ℝ be any function. Define g : ℝ → ℝ by g(x) = |f(x)| for all x. Then g is a) Onto if f is onto b) One–one if f is one–one c) Continuous if f is continuous d) Differentiable if f is differentiable
|
IIT 2000 |
|
1213 |
f(x) is a differentiable function and g(x) is a double differentiable function such that If prove that there exists some c ε (−3, 3) such that .
|
IIT 2005 |
|
1214 |
If (x – r) is a factor of the polynomial f(x) = anxn + . . . + a0, repeated m times (1 < m ≤ n) then r is a root of repeated m times. a) True b) False
If (x – r) is a factor of the polynomial f(x) = anxn + . . . + a0, repeated m times (1 < m ≤ n) then r is a root of repeated m times. a) True b) False
|
IIT 1983 |
|
1215 |
A function f : ℝ → ℝ, where ℝ is the set of real numbers, is defined by . Find the interval of values of α for which f is onto. Is the function one to one for α= 3? Justify your answer.
A function f : ℝ → ℝ, where ℝ is the set of real numbers, is defined by . Find the interval of values of α for which f is onto. Is the function one to one for α= 3? Justify your answer.
|
IIT 1996 |
|
1216 |
Let f and g be real valued functions on (−1, 1) such that g’(x) is continuous, g(0) ≠ 0, g’(0) = 0, g’’(0) ≠ 0 and f(x) = g(x)sinx Statement 1 - Statement 2 – f’(0) = g(0) a) Statement 1 is true. Statement 2 is true. Statement 2 is a correct explanation of statement 1 b) Statement 1 is true. Statement 2 is true. Statement 2 is not a correct explanation of statement 1 c) Statement 1 is true. Statement 2 is false. d) Statement 1 is false. Statement 2 is true.
Let f and g be real valued functions on (−1, 1) such that g’(x) is continuous, g(0) ≠ 0, g’(0) = 0, g’’(0) ≠ 0 and f(x) = g(x)sinx Statement 1 - Statement 2 – f’(0) = g(0) a) Statement 1 is true. Statement 2 is true. Statement 2 is a correct explanation of statement 1 b) Statement 1 is true. Statement 2 is true. Statement 2 is not a correct explanation of statement 1 c) Statement 1 is true. Statement 2 is false. d) Statement 1 is false. Statement 2 is true.
|
IIT 2008 |
|
1217 |
Differentiate from first principles (or ab initio) a) 2xcos(x2 + 1) b) xcos(x2 + 1) c) 2cosx(x2 + 1) d) 2xcosx(x2 + 1) + sin(x2 + 1)
Differentiate from first principles (or ab initio) a) 2xcos(x2 + 1) b) xcos(x2 + 1) c) 2cosx(x2 + 1) d) 2xcosx(x2 + 1) + sin(x2 + 1)
|
IIT 1978 |
|
1218 |
Let a + b = 4 where a < 2 and let g(x) be a differentiable function. If for all x, prove that increases as (b – a) increases.
Let a + b = 4 where a < 2 and let g(x) be a differentiable function. If for all x, prove that increases as (b – a) increases.
|
IIT 1997 |
|
1219 |
A and B are two separate reservoirs of water. Capacity of reservoir A is double the capacity of reservoir B. Both the reservoirs are filled completely with water, their inlets are closed and then water is released simultaneously from both the reservoirs. The rate of flow of water out of each reservoir at any instant of time is proportionate to the quantity of water in the reservoir at the time. One hour after the water is released the quantity of water in reservoir A is times the quantity of water in reservoir B. After how many hours do both the reservoirs have the same quantity of water? a) b) c) ln2 d)
A and B are two separate reservoirs of water. Capacity of reservoir A is double the capacity of reservoir B. Both the reservoirs are filled completely with water, their inlets are closed and then water is released simultaneously from both the reservoirs. The rate of flow of water out of each reservoir at any instant of time is proportionate to the quantity of water in the reservoir at the time. One hour after the water is released the quantity of water in reservoir A is times the quantity of water in reservoir B. After how many hours do both the reservoirs have the same quantity of water? a) b) c) ln2 d)
|
IIT 1997 |
|
1220 |
The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse is a) square units b) c) square units d) 27 square units
The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse is a) square units b) c) square units d) 27 square units
|
IIT 2003 |
|
1221 |
Let b ≠ 0 and j = 0, 1, 2, . . . , n. Let Sj be the area of the region bounded by Y–axis and the curve . Show that S0, S1, S2, . . . , Sn are in geometric progression. Also find the sum for a = − 1 and b = π. a) b) c) d)
Let b ≠ 0 and j = 0, 1, 2, . . . , n. Let Sj be the area of the region bounded by Y–axis and the curve . Show that S0, S1, S2, . . . , Sn are in geometric progression. Also find the sum for a = − 1 and b = π. a) b) c) d)
|
IIT 2001 |
|
1222 |
A tangent to the ellipse x2 + 4y2 = 4 meets the ellipse x2 + 2y2 = 6 at P and Q. Prove that tangents at P and Q of the ellipse x2 + 2y2 = 6 are at right angles.
A tangent to the ellipse x2 + 4y2 = 4 meets the ellipse x2 + 2y2 = 6 at P and Q. Prove that tangents at P and Q of the ellipse x2 + 2y2 = 6 are at right angles.
|
IIT 1997 |
|
1223 |
Let U1 = 1, U2 = 1, Un + 2 = Un + 1 + Un, n > 1. Use mathematical induction to show that for all integers n > 1
Let U1 = 1, U2 = 1, Un + 2 = Un + 1 + Un, n > 1. Use mathematical induction to show that for all integers n > 1
|
IIT 1981 |
|
1224 |
Given Prove that
|
IIT 1984 |
|
1225 |
Each of the following four inequalities given below define a region in the XY–plane. One of these four regions does not have the following property: For any two points (x1, y1) and (x2, y2) in the region, point is also in the region. The inequality defining the region that does not have this property is a) x2 + 2y2 ≤ 1 b) max (|x|, |y|) ≤ 1 c) x2 – y2 ≥ 1 d) y2 – x ≤ 0
Each of the following four inequalities given below define a region in the XY–plane. One of these four regions does not have the following property: For any two points (x1, y1) and (x2, y2) in the region, point is also in the region. The inequality defining the region that does not have this property is a) x2 + 2y2 ≤ 1 b) max (|x|, |y|) ≤ 1 c) x2 – y2 ≥ 1 d) y2 – x ≤ 0
|
IIT 1981 |
|