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1226 |
Let ABC be an equilateral triangle inscribed in the circle x2 + y2 = a2. Suppose perpendiculars from A, B, C to the major axis of the ellipse  (a > b) meet the ellipse respectively at P, Q, R so that P, Q, R are on the same side of the major axis. Prove that the normals drawn at the points P, Q and R are concurrent.
Let ABC be an equilateral triangle inscribed in the circle x2 + y2 = a2. Suppose perpendiculars from A, B, C to the major axis of the ellipse (a > b) meet the ellipse respectively at P, Q, R so that P, Q, R are on the same side of the major axis. Prove that the normals drawn at the points P, Q and R are concurrent.
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IIT 2000 |
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1227 |
Which of the following pieces of data does not uniquely determine an acute angled triangle ABC (R being the radius of the circumcircle). a) a, sinA, sinB b) a, b , c c) a, sinB, R d) a, sinA, R
Which of the following pieces of data does not uniquely determine an acute angled triangle ABC (R being the radius of the circumcircle). a) a, sinA, sinB b) a, b , c c) a, sinB, R d) a, sinA, R
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IIT 2002 |
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1228 |
Let f(x), x ≥ 0 be a non-negative function and let F(x) =  . For some c > 0, f(x) ≤ cF(x) for all x ≥ 0. Then for all x ≥ 0, f(x) = a) 0 b) 1 c) 2 d) 4
Let f(x), x ≥ 0 be a non-negative function and let F(x) = . For some c > 0, f(x) ≤ cF(x) for all x ≥ 0. Then for all x ≥ 0, f(x) = a) 0 b) 1 c) 2 d) 4
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IIT 2001 |
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1229 |
Tangents are drawn from P (6, 8) to the circle  . Find the radius of the circle such that the area of the triangle formed by tangents and chord of contact is maximum.
Tangents are drawn from P (6, 8) to the circle . Find the radius of the circle such that the area of the triangle formed by tangents and chord of contact is maximum.
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IIT 2003 |
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1230 |
Find the natural number a for which
where the function f satisfies the relation f (x + y) = f (x) . f (y)
for all natural numbers x and y and further f (1) = 2 a) 1 b) 2 c) 3 d) 4
Find the natural number a for which
where the function f satisfies the relation f (x + y) = f (x) . f (y)
for all natural numbers x and y and further f (1) = 2 a) 1 b) 2 c) 3 d) 4
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IIT 1992 |
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1231 |
In a certain test  students gave wrong answers to at least i questions where i = 1, 2, …, k. No student gave more than k correct answers. Total number of wrong answers given is . . .
In a certain test students gave wrong answers to at least i questions where i = 1, 2, …, k. No student gave more than k correct answers. Total number of wrong answers given is . . .
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IIT 1982 |
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1232 |
Multiple choice If  a) f(x) is increasing on [– 1, 2] b) f(x) is continuous on [– 1, 3] c)  does not exist d) f(x) has maximum value at x = 2
Multiple choice If a) f(x) is increasing on [– 1, 2] b) f(x) is continuous on [– 1, 3] c) does not exist d) f(x) has maximum value at x = 2
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IIT 1993 |
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1233 |
If arg(z) < 0 then arg(−z) – arg(z) is equal to a) π b) –π c) – π/2 d) π/2
If arg(z) < 0 then arg(−z) – arg(z) is equal to a) π b) –π c) – π/2 d) π/2
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IIT 2000 |
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|
1234 |
Multiple choice f(x) is a cubic polynomial with f(2) = 18 and f(1) = − 1. Also f(x) has a local maxima at x = − 1 and  has a local minima at x = 0 then a) The distance between (− 1, 2) and (a, f(a)), where x = a is the point of local minimum, is  b) f(x) is increasing for  c) f(x) has a local minima at x = 1 d) The value of f(0) = 15
Multiple choice f(x) is a cubic polynomial with f(2) = 18 and f(1) = − 1. Also f(x) has a local maxima at x = − 1 and has a local minima at x = 0 then a) The distance between (− 1, 2) and (a, f(a)), where x = a is the point of local minimum, is b) f(x) is increasing for c) f(x) has a local minima at x = 1 d) The value of f(0) = 15
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IIT 2006 |
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1235 |
From the point A (0, 3) on the circle  , a chord AB is drawn and extended to a point M such that AˆM = 2AˆB. The equation of locus of M is . . . . .
From the point A (0, 3) on the circle , a chord AB is drawn and extended to a point M such that AˆM = 2AˆB. The equation of locus of M is . . . . .
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IIT 1986 |
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1236 |
In Δ ABC the median to the side BC is of length  and divides ∠A into 30° and 45°. Then find the length of side BC. a) 1 b) 2 c)  d) 
In Δ ABC the median to the side BC is of length and divides ∠A into 30° and 45°. Then find the length of side BC. a) 1 b) 2 c) d)
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IIT 1985 |
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1237 |
If f is an even function defined on (−5, 5) then the real values of x satisfying the equation f (x) =  are …………… a)  b)  c)  d) 
If f is an even function defined on (−5, 5) then the real values of x satisfying the equation f (x) = are …………… a) b) c) d)
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IIT 1996 |
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1238 |
If ω(≠1) be a cube root of unity and  then the least positive value of n is a) 2 b) 3 c) 5 d) 6
If ω(≠1) be a cube root of unity and then the least positive value of n is a) 2 b) 3 c) 5 d) 6
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IIT 2004 |
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1239 |
A circle is inscribed in an equilateral triangle of side a. The area of any square inscribed in the circle is . . . . .
A circle is inscribed in an equilateral triangle of side a. The area of any square inscribed in the circle is . . . . .
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IIT 1994 |
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1240 |
The number of all possible triplets  such that
 for all x is a) Zero b) One c) Three d) Infinite e) None
The number of all possible triplets such that
for all x is a) Zero b) One c) Three d) Infinite e) None
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IIT 1987 |
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1241 |
A swimmer S is in the sea at a distance d km. from the closest point A on a straight shore. The house of the swimmer is on the shore at a distance L km. from A. He can swim at a speed of
u km/hour and walk at a speed of v km/hr (v > u). At what point on the shore should he land so that he reaches his house in the shortest possible time. a)  b)  c)  d) 
A swimmer S is in the sea at a distance d km. from the closest point A on a straight shore. The house of the swimmer is on the shore at a distance L km. from A. He can swim at a speed of
u km/hour and walk at a speed of v km/hr (v > u). At what point on the shore should he land so that he reaches his house in the shortest possible time. a) b) c) d)
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IIT 1983 |
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1242 |
Sketch the region bounded by the curves
 and y = |x – 1|
and find its area. a)  b)  c)  d) 5π + 2
Sketch the region bounded by the curves
and y = |x – 1|
and find its area. a) b) c) d) 5π + 2
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IIT 1985 |
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1243 |
Tangents are drawn from the point (17, 7) to the circle  .
Statement 1 – The tangents are mutually perpendicular, because Statement 2 – The locus of points from which mutually perpendicular tangents are drawn to the given circle is  . The question contains statement – 1 (assertion) and statement 2 (reason). Of these statements mark correct choice if 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
Tangents are drawn from the point (17, 7) to the circle .
Statement 1 – The tangents are mutually perpendicular, because Statement 2 – The locus of points from which mutually perpendicular tangents are drawn to the given circle is . The question contains statement – 1 (assertion) and statement 2 (reason). Of these statements mark correct choice if 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 2007 |
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1244 |
Let  be the vertices of the triangle. A parallelogram AFDE is drawn with the vertices D, E and F on the line segments BC, CA and AB respectively. Using calculus find the area of the parallelogram. a)  b)  c)  d) 
Let be the vertices of the triangle. A parallelogram AFDE is drawn with the vertices D, E and F on the line segments BC, CA and AB respectively. Using calculus find the area of the parallelogram. a) b) c) d)
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IIT 1986 |
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1245 |
Two rays in the first quadrant x + y = |a| and ax – y = 1 intersect each other in the interval a ε (a0, ∞). The value of a0 is
Two rays in the first quadrant x + y = |a| and ax – y = 1 intersect each other in the interval a ε (a0, ∞). The value of a0 is
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IIT 2006 |
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1246 |
Find the area of the region bounded by the curve C: y = tanx, tangent drawn to C at  and the X–axis. a) ln2 – 1 b)  c)  d) 
Find the area of the region bounded by the curve C: y = tanx, tangent drawn to C at and the X–axis. a) ln2 – 1 b) c) d)
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IIT 1988 |
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|
1247 |
 then tan t =
then tan t =
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IIT 2006 |
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|
1248 |
Sketch the curves and identify the region bounded by
Sketch the curves and identify the region bounded by
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IIT 1991 |
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|
1249 |
Consider the following linear equations
ax + by + cz = 0
bx + cy + az = 0
cx + ay + bz = 0
Match the statements/expressions in column 1 with column 2
| Column 1 | Column2 | | i. a + b + c ≠ 0 and a2 + b2 + c2 = ab + bc + ca | A. Equations represent planes meeting at only one single point | | ii. a + b + c = 0 and a2 + b2 + c2 ≠ ab + bc + ca | B. The equations represent the line x = y = z | | iii. a + b + c ≠ 0 and a2 + b2 + c2 ≠ ab + bc + ca | C. The equations represent identical planes | | iv. a + b + c = 0 and a2 + b2 + c2 = ab + bc + ca | D.The equations represent the whole of the three dimensional space |
Consider the following linear equations
ax + by + cz = 0
bx + cy + az = 0
cx + ay + bz = 0
Match the statements/expressions in column 1 with column 2
| Column 1 | Column2 | | i. a + b + c ≠ 0 and a2 + b2 + c2 = ab + bc + ca | A. Equations represent planes meeting at only one single point | | ii. a + b + c = 0 and a2 + b2 + c2 ≠ ab + bc + ca | B. The equations represent the line x = y = z | | iii. a + b + c ≠ 0 and a2 + b2 + c2 ≠ ab + bc + ca | C. The equations represent identical planes | | iv. a + b + c = 0 and a2 + b2 + c2 = ab + bc + ca | D.The equations represent the whole of the three dimensional space |
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IIT 2007 |
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|
1250 |
The domain of the function y(x) given by the equation  is a) 0 < x ≤ 1 b) 0 ≤ x ≤ 1 c)  < x ≤ 0 d) < x < 1
The domain of the function y(x) given by the equation is a) 0 < x ≤ 1 b) 0 ≤ x ≤ 1 c) < x ≤ 0 d) < x < 1
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IIT 2000 |
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