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1251 |
Use the function  , x > 0 to determine the bigger of the numbers eπ and πe. a) eπ b) πe
Use the function , x > 0 to determine the bigger of the numbers eπ and πe. a) eπ b) πe
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IIT 1981 |
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1252 |
In a triangle ABC, D and E are points on  and  respectively such that  and  . Let P be the point of intersection of  and  . Find  using vector method. a)  b)  c)  d) 2
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IIT 1993 |
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1253 |
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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1254 |
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 |
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1255 |
Find the area of the region bounded by the X–axis and the curve defined by
 a) ln2 b) 2ln2 c)  d) 
Find the area of the region bounded by the X–axis and the curve defined by
a) ln2 b) 2ln2 c) d)
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IIT 1984 |
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1256 |
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 circle touching the line L and the circle C1 externally such that both the circles are on the same side of the line, then the locus of the centre of circle is a) Ellipse b) Hyperbola c) Parabola d) Pair of straight lines
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 circle touching the line L and the circle C1 externally such that both the circles are on the same side of the line, then the locus of the centre of circle is a) Ellipse b) Hyperbola c) Parabola d) Pair of straight lines
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IIT 2006 |
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1257 |
Find three dimensional vectors u1, u2, u3 satisfying
u1.u1 = 4; u1.u2 = −2; u1.u3 = 6; u2.u2 = 2; u2.u3 = −5; u3.u3 = 29
Find three dimensional vectors u1, u2, u3 satisfying
u1.u1 = 4; u1.u2 = −2; u1.u3 = 6; u2.u2 = 2; u2.u3 = −5; u3.u3 = 29
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IIT 2001 |
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1258 |
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 |
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1259 |
Let f(x) = x3 – x2 + x + 1 and
Discuss the continuity and differentiability of f(x) in the interval (0, 2) a) Continuous and differentiable in (0, 2) b) Continuous and differentiable in (0, 2)except x = 1 c) Continuous in (0, 2). Differentiable in (0, 2) except x = 1 d) None of the above
Let f(x) = x3 – x2 + x + 1 and
Discuss the continuity and differentiability of f(x) in the interval (0, 2) a) Continuous and differentiable in (0, 2) b) Continuous and differentiable in (0, 2)except x = 1 c) Continuous in (0, 2). Differentiable in (0, 2) except x = 1 d) None of the above
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IIT 1985 |
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1260 |
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 |
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1261 |
Find the point on the curve 4x2 + a2y2 = 4a2, 4 < a2 < 8 that is farthest from the point (0, −2). a) (a, 0) b)  c)  d) (0, - 2)
Find the point on the curve 4x2 + a2y2 = 4a2, 4 < a2 < 8 that is farthest from the point (0, −2). a) (a, 0) b) c) d) (0, - 2)
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IIT 1987 |
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1262 |
The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y2 = 4ax is another parabola with directrix a) x = −a b)  c)  d) 
The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y2 = 4ax is another parabola with directrix a) x = −a b) c) d)
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IIT 2002 |
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1263 |
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IIT 2006 |
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1264 |
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 |
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1265 |
Find all maximum and minimum of the curve y = x(x – 1)2, 0 ≤ x ≤ 2. Also find the area bounded by the curve y = x(x – 2)2, the Y–axis and the line y = 2. a) Local minimum at x = 1, Local maximum at x =  , Area =  b) Local minimum at x = , Local maximum at x =1, Area = c) Local minimum at x = 2, Local maximum at x =  , Area =  d) Local minimum at x = , Local maximum at x =2, Area =
Find all maximum and minimum of the curve y = x(x – 1)2, 0 ≤ x ≤ 2. Also find the area bounded by the curve y = x(x – 2)2, the Y–axis and the line y = 2. a) Local minimum at x = 1, Local maximum at x = , Area = b) Local minimum at x = , Local maximum at x =1, Area = c) Local minimum at x = 2, Local maximum at x = , Area = d) Local minimum at x = , Local maximum at x =2, Area =
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IIT 1989 |
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1266 |
A line is perpendicular to  and passes through (0, 1, 0). Then the perpendicular distance of this line from the origin is . . .
A line is perpendicular to and passes through (0, 1, 0). Then the perpendicular distance of this line from the origin is . . .
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IIT 2006 |
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|
1267 |
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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1268 |
The curve y = ax3 + bx2 + cx + 5 touches the X – axis at (− 2, 0) and cuts the Y–axis at a point Q where the gradient is 3. Find a, b, c. a)  b)  c)  d) 
The curve y = ax3 + bx2 + cx + 5 touches the X – axis at (− 2, 0) and cuts the Y–axis at a point Q where the gradient is 3. Find a, b, c. a) b) c) d)
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IIT 1994 |
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1269 |
Points A, B, C lie on the parabola  . The tangents to the parabola at A, B, C taken in pair intersect at the points P, Q, R. Determine the ratios of the areas of ΔABC and ΔPQR.
Points A, B, C lie on the parabola . The tangents to the parabola at A, B, C taken in pair intersect at the points P, Q, R. Determine the ratios of the areas of ΔABC and ΔPQR.
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IIT 1996 |
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1270 |
Consider the lines given by L1 : x + 3y – 5 = 0; L2 = 3x – ky – 1 = 0; L3 = 5x + 2y −12 = 0. Match the statement/expressions in column 1 with 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, if | 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 given by L1 : x + 3y – 5 = 0; L2 = 3x – ky – 1 = 0; L3 = 5x + 2y −12 = 0. Match the statement/expressions in column 1 with 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, if | 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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|
1271 |
 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 |
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1272 |
Let a line passing through the fixed point (h, k) cut the X–axis at P and Y–axis at Q. Then find the minimum area of ΔOPQ. a) hk b) h2/k c) k2/h d) 2hk
Let a line passing through the fixed point (h, k) cut the X–axis at P and Y–axis at Q. Then find the minimum area of ΔOPQ. a) hk b) h2/k c) k2/h d) 2hk
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IIT 1995 |
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1273 |
Match the following | Column 1 | Column 2 | | i) Re z = 0 | A) Re  = 0 | | ii) Arg z = π/4 | B) Im = 0 | | | C) Re = Im |
Match the following | Column 1 | Column 2 | | i) Re z = 0 | A) Re = 0 | | ii) Arg z = π/4 | B) Im = 0 | | | C) Re = Im |
|
IIT 1992 |
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1274 |
Let An be the area bounded by the curve y = (tanx)n and the line
x = 0, y = 0 and  . Prove that for  . Hence deduce that
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IIT 1996 |
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1275 |
Consider the circle x2 + y2 = 9 and the parabola y2 = 8x. They intersect P and Q in the first and fourth quadrants respectively. Tangents to the circle at P and Q intersect the X–axis at R and tangents to the parabola at P and Q intersect the X- axis at S. The radius of the incircle of △PQR is a) 4 b) 3 c)  d) 2
Consider the circle x2 + y2 = 9 and the parabola y2 = 8x. They intersect P and Q in the first and fourth quadrants respectively. Tangents to the circle at P and Q intersect the X–axis at R and tangents to the parabola at P and Q intersect the X- axis at S. The radius of the incircle of △PQR is a) 4 b) 3 c) d) 2
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IIT 2007 |
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