1001 |
Let and f = R – [R] where [ ] denotes the greatest integer function. Prove that Rf = 42n + 4
Let and f = R – [R] where [ ] denotes the greatest integer function. Prove that Rf = 42n + 4
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IIT 1988 |
|
1002 |
Using induction or otherwise, prove that for any non-negative integers m, n, r and k
Using induction or otherwise, prove that for any non-negative integers m, n, r and k
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IIT 1991 |
|
1003 |
For any real t, , is a point on the hyperbola x2 – y2 = 1. Find the area bounded by the hyperbola and the line joining the centre to the points corresponding to t1 and –t1.
For any real t, , is a point on the hyperbola x2 – y2 = 1. Find the area bounded by the hyperbola and the line joining the centre to the points corresponding to t1 and –t1.
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IIT 1982 |
|
1004 |
Let a and b are non-zero real numbers. Then the equation (ax2 + by2 + c) (x2 – 5xy + 6y2) = 0 represents a) Four straight lines when c = 0 and a, b are of the same sign b) Two straight lines and a circle when a = b and c is of sign opposite to that of a. c) Two straight lines and a hyperbola when a and b are of the same sign d) A circle and an ellipse when a and b are of the same sign and c is of sign opposite to that of a.
Let a and b are non-zero real numbers. Then the equation (ax2 + by2 + c) (x2 – 5xy + 6y2) = 0 represents a) Four straight lines when c = 0 and a, b are of the same sign b) Two straight lines and a circle when a = b and c is of sign opposite to that of a. c) Two straight lines and a hyperbola when a and b are of the same sign d) A circle and an ellipse when a and b are of the same sign and c is of sign opposite to that of a.
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IIT 2008 |
|
1005 |
Let 0 < Ai < π for i = 1, 2, . . . n. Use mathematical induction to prove that where n ≥ 1 is a natural number.
Let 0 < Ai < π for i = 1, 2, . . . n. Use mathematical induction to prove that where n ≥ 1 is a natural number.
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IIT 1997 |
|
1006 |
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. Points E and F are given by 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. Points E and F are given by a) b) c) d)
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IIT 2008 |
|
1007 |
Solve
Solve
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IIT 1978 |
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1008 |
ConsiderL1: 2x + 3y + p – 3 = 0; L2: 2x + 3y + p + 3 = 0 where p is a real number and C : x2 + y2 + 6x – 10y + 30 = 0 Statement 1 – If the line L1 is a chord of the circle C then L2 is not always a diameter of C. Statement 2 - If the line L1 is a diameter of the circle C then L2 is not a chord of the circle. Which of the following four statements is true? a) Statement 1 and 2 are true. Statement 2 is a correct explanation for statement 1. b) Statement 1 and 2 are true. Statement 2 is not a correct explanation for statement 1. c) Statement 1 is true. Statement 2 is false. d) Statement 1 is false. Statement 2 is true
ConsiderL1: 2x + 3y + p – 3 = 0; L2: 2x + 3y + p + 3 = 0 where p is a real number and C : x2 + y2 + 6x – 10y + 30 = 0 Statement 1 – If the line L1 is a chord of the circle C then L2 is not always a diameter of C. Statement 2 - If the line L1 is a diameter of the circle C then L2 is not a chord of the circle. Which of the following four statements is true? a) Statement 1 and 2 are true. Statement 2 is a correct explanation for statement 1. b) Statement 1 and 2 are true. Statement 2 is not a correct explanation for statement 1. c) Statement 1 is true. Statement 2 is false. d) Statement 1 is false. Statement 2 is true
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IIT 2008 |
|
1009 |
If E and F are events with P (E) ≤ P (F) and P (E ∩ F) > 0 then a) occurrence of E ⇒ occurrence of F b) occurrence of F ⇒ occurrence of E c) non-occurrence of E ⇒ non-occurrence of F d) none of the above occurrences hold
If E and F are events with P (E) ≤ P (F) and P (E ∩ F) > 0 then a) occurrence of E ⇒ occurrence of F b) occurrence of F ⇒ occurrence of E c) non-occurrence of E ⇒ non-occurrence of F d) none of the above occurrences hold
|
IIT 1998 |
|
1010 |
Let denotes the complement of an event E. Let E, F, G are pair wise independent events with P (G) > 0 and P (E ∩ F ∩ G) = 0 then equals a) b) c) d)
Let denotes the complement of an event E. Let E, F, G are pair wise independent events with P (G) > 0 and P (E ∩ F ∩ G) = 0 then equals a) b) c) d)
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IIT 2007 |
|
1011 |
(One or more correct answers) For any two events in the sample space a) is always true b) does not hold c) if A and B are independent d) if A and B are disjoint
(One or more correct answers) For any two events in the sample space a) is always true b) does not hold c) if A and B are independent d) if A and B are disjoint
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IIT 1991 |
|
1012 |
If are unit coplanar vectors then the scalar triple product a) 0 b) 1 c) d)
If are unit coplanar vectors then the scalar triple product a) 0 b) 1 c) d)
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IIT 2000 |
|
1013 |
Domain of definition of the function f (x) = for real valued x is a) b) c) d)
Domain of definition of the function f (x) = for real valued x is a) b) c) d)
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IIT 2003 |
|
1014 |
If a, b, c; u, v, w are complex numbers representing the vertices of two triangles such that c = (1 − r)a + rb, w = (1 − r)u + rv where r is a complex number. The two triangles a) have the same area b) are similar c) are congruent d) none of these
If a, b, c; u, v, w are complex numbers representing the vertices of two triangles such that c = (1 − r)a + rb, w = (1 − r)u + rv where r is a complex number. The two triangles a) have the same area b) are similar c) are congruent d) none of these
|
IIT 1985 |
|
1015 |
Match the following Let the function defined in column 1 have domain and range () Column 1 | Column 2 | i) 1 + 2x | A) Onto but not one-one | ii) tan x | B) One-one but not onto | | C) One-one and onto | | D) Neither one |
Match the following Let the function defined in column 1 have domain and range () Column 1 | Column 2 | i) 1 + 2x | A) Onto but not one-one | ii) tan x | B) One-one but not onto | | C) One-one and onto | | D) Neither one |
|
IIT 1992 |
|
1016 |
If are three non-coplanar unit vectors and α, β, γ are the angles between , v and w, w and u respectively and x, y and z are unit vectors along the bisector of the angles α, β, γ respectively. Prove that
|
IIT 2003 |
|
1017 |
Let be a regular hexagon in a circle of unit radius. Then the product of the length of the segments , and is a) b) c) 3 d)
Let be a regular hexagon in a circle of unit radius. Then the product of the length of the segments , and is a) b) c) 3 d)
|
IIT 1998 |
|
1018 |
Find the equation of the plane at a distance from the point and containing the line .
|
IIT 2005 |
|
1019 |
Let AB be a chord of the circle subtending a right angle at the centre then the locus of the centroid of the triangle PAB as P moves on the circle is a) A parabola b) A circle c) An ellipse d) A pairing straight line
Let AB be a chord of the circle subtending a right angle at the centre then the locus of the centroid of the triangle PAB as P moves on the circle is a) A parabola b) A circle c) An ellipse d) A pairing straight line
|
IIT 2000 |
|
1020 |
The point (α, β, γ) lies on the plane . Let a = . . . . .
The point (α, β, γ) lies on the plane . Let a = . . . . .
|
IIT 2006 |
|
1021 |
Sides a, b, c of a triangle ABC are in arithmetic progression and then
Sides a, b, c of a triangle ABC are in arithmetic progression and then
|
IIT 2006 |
|
1022 |
Let ABCD be a quadrilateral with area 18 with side AB parallel to CD and AB = 2CD. Let AD be perpendicular to AB and CD. A circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is a) 3 b) 2 c) d) 1
Let ABCD be a quadrilateral with area 18 with side AB parallel to CD and AB = 2CD. Let AD be perpendicular to AB and CD. A circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is a) 3 b) 2 c) d) 1
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IIT 2007 |
|
1023 |
Find the equation of the circle passing through ( 4, 3) and touching the lines x + y = 4 and .
Find the equation of the circle passing through ( 4, 3) and touching the lines x + y = 4 and .
|
IIT 1982 |
|
1024 |
A circle touches the line y = x at a point P such that , where O is the origin. The circle contains the point in its interior and the length of its chord on the line is . Determine its equation.
|
IIT 1990 |
|
1025 |
equals a) b) c) d)
|
IIT 1997 |
|