1101 |
Consider the parabola y2 = 8x. Let △1 be the area of the triangle formed by the end points of its latus rectum and the point on the parabola and △2 be the area of the triangle formed by drawing tangent at P and the end points of the latus rectum. Then is
Consider the parabola y2 = 8x. Let △1 be the area of the triangle formed by the end points of its latus rectum and the point on the parabola and △2 be the area of the triangle formed by drawing tangent at P and the end points of the latus rectum. Then is
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IIT 2011 |
|
1102 |
Let O (0, 0), A(2, 0) and be the vertices of a triangle. Let R be the region consisting of all those points P inside ΔOAB which satisfies d(P, OA) ≤ d(P, OB) . d(P, AB), where d denotes the distance from the point to the corresponding line. Sketch the region R and find its area. a) b) c) d)
Let O (0, 0), A(2, 0) and be the vertices of a triangle. Let R be the region consisting of all those points P inside ΔOAB which satisfies d(P, OA) ≤ d(P, OB) . d(P, AB), where d denotes the distance from the point to the corresponding line. Sketch the region R and find its area. a) b) c) d)
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IIT 1997 |
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1103 |
Let f(x) be a continuous function given by Find the area of the region in the third quadrant bounded by the curve x = − 2y2 and y = f(x) lying on the left of the line 8x + 1 = 0. a) 192 b) 320 c) 761/192 d) 320/761
Let f(x) be a continuous function given by Find the area of the region in the third quadrant bounded by the curve x = − 2y2 and y = f(x) lying on the left of the line 8x + 1 = 0. a) 192 b) 320 c) 761/192 d) 320/761
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IIT 1999 |
|
1104 |
Let d be the perpendicular distance from the centre of the ellipse to the tangent at a point P on the ellipse. Let F1 and F2 be the two focii of the ellipse, then show that
Let d be the perpendicular distance from the centre of the ellipse to the tangent at a point P on the ellipse. Let F1 and F2 be the two focii of the ellipse, then show that
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IIT 1995 |
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1105 |
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
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IIT 2002 |
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1106 |
Prove that in an ellipse the perpendicular from a focus upon a tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix.
Prove that in an ellipse the perpendicular from a focus upon a tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix.
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IIT 2002 |
|
1107 |
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 |
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1108 |
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 |
|
1109 |
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 |
|
1110 |
A hyperbola having the transverse axis of length 2sinθ is confocal with the ellipse . Then its equation is a) b) c) d)
A hyperbola having the transverse axis of length 2sinθ is confocal with the ellipse . Then its equation is a) b) c) d)
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IIT 2007 |
|
1111 |
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 |
|
1112 |
The angle between the pair of tangents from a point P to the parabola y2 = 4ax is 45°. Show that the locus of the point P is a hyperbola.
The angle between the pair of tangents from a point P to the parabola y2 = 4ax is 45°. Show that the locus of the point P is a hyperbola.
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IIT 1998 |
|
1113 |
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 |
|
1114 |
A box contains 24 identical balls of which 12 are white and 12 are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the fourth time on the seventh draw is a) b) c) d)
A box contains 24 identical balls of which 12 are white and 12 are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the fourth time on the seventh draw is a) b) c) d)
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IIT 1984 |
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1115 |
Let A, B , C be three mutually independent events. Consider the two statements S1 and S2 S1 : A and B ∪ Care independent S2 : A and B ∩ C are independent. Then a) Both S1 and S2 are true b) Only S1 is true c) Only S2 is true d) Neither S1 nor S2 is true
Let A, B , C be three mutually independent events. Consider the two statements S1 and S2 S1 : A and B ∪ Care independent S2 : A and B ∩ C are independent. Then a) Both S1 and S2 are true b) Only S1 is true c) Only S2 is true d) Neither S1 nor S2 is true
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IIT 1994 |
|
1116 |
A circle C of radius 1 is inscribed in an equilateral triangle PQR. The point of contacts of C with its sides PQ, QR and RP are D, E, F respectively. The line PQ is given by and the point D is . Further, it is given that the origin and the centre of C are on the same side of the line PQ. Equations of lines QR and RP are a) b) c) d)
A circle C of radius 1 is inscribed in an equilateral triangle PQR. The point of contacts of C with its sides PQ, QR and RP are D, E, F respectively. The line PQ is given by and the point D is . Further, it is given that the origin and the centre of C are on the same side of the line PQ. Equations of lines QR and RP are a) b) c) d)
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IIT 2008 |
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1117 |
Prove that for complex numbers z and ω, iff z = ω or .
Prove that for complex numbers z and ω, iff z = ω or .
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IIT 1999 |
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1118 |
Consider the lines L1: x + 3y – 5 = 0, L2: 3x – ky – 1 = 0, L3: 5x + 2y – 12 = 0. Match the statement/expressions in column 1 with the statement/expression in column 2. Column 1 | Column 2 | A) L1, L2, L3 are concurrent if | p) k = − 9 | B) One of L1, L2, L3 is parallel to at least one of the other two | q) | C) L1, L2, L3 form a triangle if | r) | D) L1, L2, L3 do not form a triangle if | s) k = 5 |
Consider the lines L1: x + 3y – 5 = 0, L2: 3x – ky – 1 = 0, L3: 5x + 2y – 12 = 0. Match the statement/expressions in column 1 with the statement/expression in column 2. Column 1 | Column 2 | A) L1, L2, L3 are concurrent if | p) k = − 9 | B) One of L1, L2, L3 is parallel to at least one of the other two | q) | C) L1, L2, L3 form a triangle if | r) | D) L1, L2, L3 do not form a triangle if | s) k = 5 |
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IIT 2008 |
|
1119 |
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 |
|
1120 |
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)
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IIT 1993 |
|
1121 |
(Multiple correct answers) Let M and N are two events, the probability that exactly one of them occurs is a) P (M) + P (N) − 2P (M ∩ N) b) P (M) + P (N) − P () c) d)
(Multiple correct answers) Let M and N are two events, the probability that exactly one of them occurs is a) P (M) + P (N) − 2P (M ∩ N) b) P (M) + P (N) − P () c) d)
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IIT 1984 |
|
1122 |
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)
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IIT 1994 |
|
1123 |
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 |
|
1124 |
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 |
|
1125 |
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
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IIT 2001 |
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