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1101 |
Multiple choices
If f(x) =  where [x] stands for the greatest integer function then a)  b)  c)  d) 
Multiple choices
If f(x) = where [x] stands for the greatest integer function then a) b) c) d)
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IIT 1991 |
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1102 |
 = 
where t2 = cot2x – 1 a) True b) False
= where t2 = cot2x – 1 a) True b) False
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IIT 1987 |
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1103 |
The function  is not one to one a) True b) False
The function is not one to one a) True b) False
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IIT 1983 |
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1104 |
Let A be a set of n distinct elements. Then find the total number of distinct functions from A to A is and out of these onto functions are . . .
Let A be a set of n distinct elements. Then find the total number of distinct functions from A to A is and out of these onto functions are . . .
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IIT 1985 |
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1105 |
Match the following
Let the function defined in column 1 have domain  and range (−∞ ∞)
| Column1 | Column2 | | i) 1+2x | A) Onto but not one – one | | ii) tanx | B) One to one but not onto | | | C) One to one and onto | | | D) Neither one to one nor onto |
Match the following
Let the function defined in column 1 have domain and range (−∞ ∞)
| Column1 | Column2 | | i) 1+2x | A) Onto but not one – one | | ii) tanx | B) One to one but not onto | | | C) One to one and onto | | | D) Neither one to one nor onto |
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IIT 1992 |
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1106 |
Let a, b, c be real numbers such that
 
Then ax2 + bx + c = 0 has a) No root in (0, 2) b) At least one root in (0, 2) c) A double root in (0, 2) d) Two imaginary roots
Let a, b, c be real numbers such that
Then ax2 + bx + c = 0 has a) No root in (0, 2) b) At least one root in (0, 2) c) A double root in (0, 2) d) Two imaginary roots
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IIT 1981 |
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1107 |
The total number of local maximum and minimum of the function 
is a) 0 b) 1 c) 2 d) 3
The total number of local maximum and minimum of the function
is a) 0 b) 1 c) 2 d) 3
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IIT 2008 |
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1108 |
Let S be a square of unit area. Consider any quadrilateral which has one vertex on each side of S. If a, b, c and d denote the lengths of the sides of the quadrilateral; prove that
2 ≤ a2 + b2 + c2 + d2 ≤ 4
Let S be a square of unit area. Consider any quadrilateral which has one vertex on each side of S. If a, b, c and d denote the lengths of the sides of the quadrilateral; prove that
2 ≤ a2 + b2 + c2 + d2 ≤ 4
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IIT 1997 |
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1109 |
If Cr stands for  then the sum of the series
where n is a positive integer, is equal to a) 0 b) (−)n/2(n + 1) c) (−)n/2 (n + 2) d) None of these
If Cr stands for then the sum of the series
where n is a positive integer, is equal to a) 0 b) (−)n/2(n + 1) c) (−)n/2 (n + 2) d) None of these
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IIT 1986 |
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1110 |
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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1111 |
The sum  if p > q is maximum when m is a) 5 b) 10 c) 15 d) 20
The sum if p > q is maximum when m is a) 5 b) 10 c) 15 d) 20
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IIT 2002 |
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1112 |
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 |
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1113 |
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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1114 |
Prove that 
Prove that
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IIT 1979 |
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1115 |
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 |
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1116 |
Use mathematical induction to prove: If n is an odd positive integer
then  is divisible by 24.
Use mathematical induction to prove: If n is an odd positive integer
then is divisible by 24.
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IIT 1983 |
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1117 |
Using mathematical induction, prove that
m, n, k are positive integers and  for p < q
Using mathematical induction, prove that
m, n, k are positive integers and for p < q
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IIT 1989 |
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1118 |
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 |
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1119 |
If  for all k ≥ n then show that 
If for all k ≥ n then show that
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IIT 1992 |
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1120 |
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 |
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1121 |
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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1122 |
Let A, B , C be three mutually independent events. Consider the two statements S1 and S2 S1 : A and B ∪ C are 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 |
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1123 |
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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1124 |
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 |
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1125 |
(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 |
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