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1201 |
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 line M is drawn through A parallel to BD. Point S moves such that the distance from the line BD and the vertex A are equal. If the locus of S cuts M at T2 and T3 and AC at T1, then find the area of △T1T2T3.
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 line M is drawn through A parallel to BD. Point S moves such that the distance from the line BD and the vertex A are equal. If the locus of S cuts M at T2 and T3 and AC at T1, then find the area of △T1T2T3.
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IIT 2006 |
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1202 |
Express in the form A + iB a)  b)  c)  d) 
Express in the form A + iB a)  b)  c)  d) 
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IIT 1979 |
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1203 |
Find the area bounded by the curves a) 1/6 b) 1/3 c) π d) 
Find the area bounded by the curves a) 1/6 b) 1/3 c) π d) 
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IIT 1986 |
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1204 |
If the line x – 1 = 0 is the directrix of the parabola y2 – kx + 8 = 0, then one of the values of k is a)  b) 8 c) 4 d) 
If the line x – 1 = 0 is the directrix of the parabola y2 – kx + 8 = 0, then one of the values of k is a)  b) 8 c) 4 d) 
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IIT 2000 |
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1205 |
Find the area bounded by the curves x2 + y2 = 25, 4y = |4 – x2| and x = 0 above the X–axis. a)  b)  c)  d) 
Find the area bounded by the curves x2 + y2 = 25, 4y = |4 – x2| and x = 0 above the X–axis. a)  b)  c)  d) 
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IIT 1987 |
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1206 |
If sinA sinB sinC + cosA cosB = 1then the value of sinC is
If sinA sinB sinC + cosA cosB = 1then the value of sinC is
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IIT 2006 |
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1207 |
Let = 10 + 6i and . If z is a complex number such that argument of is then prove that .
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IIT 1990 |
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1208 |
Compute the area of the region bounded by the curves y = exlnx and  a)  b)  c)  d) 
Compute the area of the region bounded by the curves y = exlnx and  a)  b)  c)  d) 
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IIT 1990 |
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1209 |
A plane passes through (1, −2, 1) and is perpendicular to the two planes and The distance of the plane from the point (1, 2, 2) is.
A plane passes through (1, −2, 1) and is perpendicular to the two planes and The distance of the plane from the point (1, 2, 2) is.
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IIT 2006 |
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1210 |
What normal to the curve y = x2 forms the shortest normal? a)  b)  c)  d) y = x + 1
What normal to the curve y = x2 forms the shortest normal? a)  b)  c)  d) y = x + 1
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IIT 1992 |
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1211 |
(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) 
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IIT 1988 |
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1212 |
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.
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IIT 2002 |
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1213 |
The circle x2 + y2 = 1 cuts the X–axis at P and Q. Another circle with centre at Q and variable radius intersects the first circle at R above the X–axis and the line segment PQ at S. Find the maximum area of ΔQRS. a)  b)  c)  d) 
The circle x2 + y2 = 1 cuts the X–axis at P and Q. Another circle with centre at Q and variable radius intersects the first circle at R above the X–axis and the line segment PQ at S. Find the maximum area of ΔQRS. a)  b)  c)  d) 
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IIT 1994 |
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1214 |
From a point A common tangents are drawn to the circle and the parabola . Find the area of the quadrilateral formed by the common tangents drawn from A and the chords of contact of the circle and the parabola.
From a point A common tangents are drawn to the circle and the parabola . Find the area of the quadrilateral formed by the common tangents drawn from A and the chords of contact of the circle and the parabola.
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IIT 1996 |
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1215 |
True/False For the complex numbers and we write and then for all complex numbers z with we have . a) True b) False
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IIT 1981 |
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1216 |
Let  where a is a positive constant. Find the interval in which is increasing. a)  b)  c)  d) 
Let  where a is a positive constant. Find the interval in which is increasing. a)  b)  c)  d) 
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IIT 1996 |
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1217 |
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.
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IIT 1997 |
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1218 |
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)
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IIT 1997 |
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1219 |
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
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IIT 2003 |
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1220 |
The function f(x) = |px – q|+ r|x|, x when p > 0, q > 0, r > 0 assumes minimum value only on one point if a) p ≠ q b) r ≠ q c) r ≠ p d) p = q = r
The function f(x) = |px – q|+ r|x|, x when p > 0, q > 0, r > 0 assumes minimum value only on one point if a) p ≠ q b) r ≠ q c) r ≠ p d) p = q = r
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IIT 1995 |
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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) 
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IIT 2001 |
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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.
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IIT 1997 |
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1223 |
Let f(θ) = sinθ (sinθ + sin3θ) then f(θ) a) ≥ 0 only when θ ≥ 0 b) ≤ 0 for all real θ c) ≥ 0 for all real θ d) ≤ θ only when θ ≤ 0
Let f(θ) = sinθ (sinθ + sin3θ) then f(θ) a) ≥ 0 only when θ ≥ 0 b) ≤ 0 for all real θ c) ≥ 0 for all real θ d) ≤ θ only when θ ≤ 0
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IIT 2000 |
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1224 |
Let y = f(x) is a cubic polynomial having maximum at x = − 1 and has a minimum at x = 1 and f(−1) = 10, f(1) = − 6. Find the cubic polynomial and also find the distance between the points which are maxima or minima. a)  b)  c)  d) 
Let y = f(x) is a cubic polynomial having maximum at x = − 1 and has a minimum at x = 1 and f(−1) = 10, f(1) = − 6. Find the cubic polynomial and also find the distance between the points which are maxima or minima. a)  b)  c)  d) 
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IIT 2005 |
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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
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IIT 1981 |
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