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1101 |
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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1102 |
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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1103 |
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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1104 |
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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1105 |
 then tan t =
then tan t =
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IIT 2006 |
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1106 |
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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1107 |
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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1108 |
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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1109 |
If A = , 6A-1 = A2 + cA + dI then (c, d ) is a) (−11, 6) b) (−6, 11) c) (6, 11 ) d) (11, 6 )
If A = , 6A-1 = A2 + cA + dI then (c, d ) is a) (−11, 6) b) (−6, 11) c) (6, 11 ) d) (11, 6 )
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IIT 2005 |
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1110 |
Prove that 
Prove that
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IIT 1997 |
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1111 |
Let  be a line in the complex plane where  is the complex conjugate of b. If a point  is the deflection of a point  through the line, show that  .
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IIT 1997 |
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1112 |
Let  Find all possible values of b such that f(x) has the smallest value at x = 1. a) (−2, ∞) b) (−2, −1) c) (1, ∞) d) (−2, −1) ∪ (1, ∞)
Let Find all possible values of b such that f(x) has the smallest value at x = 1. a) (−2, ∞) b) (−2, −1) c) (1, ∞) d) (−2, −1) ∪ (1, ∞)
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IIT 1993 |
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1113 |
Use mathematical induction for
to prove that
Im = mπ, m = 0, 1, 2 . . . .
Use mathematical induction for
to prove that
Im = mπ, m = 0, 1, 2 . . . .
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IIT 1995 |
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1114 |
Determine the points of maxima and minima of the function
 where b ≥ 0 is a constant. a) Minima at x = x1, maxima at x = x2 b) Minima at x = x2, maxima at x = x1 c) Minima at x = x1, x2, no maxima d) Maxima at x =x1, x2, no minima where x1 =  and x2 = 
Determine the points of maxima and minima of the function
where b ≥ 0 is a constant. a) Minima at x = x1, maxima at x = x2 b) Minima at x = x2, maxima at x = x1 c) Minima at x = x1, x2, no maxima d) Maxima at x =x1, x2, no minima where x1 = and x2 =
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IIT 1996 |
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1115 |
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 circum circle of △PRS is a) 5 b)  c) 3 d) 
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 circum circle of △PRS is a) 5 b) c) 3 d)
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IIT 2007 |
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1116 |
Let  where 0 ≤ x ≤ 1. Determine the area bounded by y = f (x), X–axis, x = 0 and x = 1. a)  b)  c)  d) 
Let where 0 ≤ x ≤ 1. Determine the area bounded by y = f (x), X–axis, x = 0 and x = 1. a) b) c) d)
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IIT 1997 |
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1117 |
Which of the following function is periodic? a) f(x) = x – [x] where [x] denotes the greatest integer less than equal to the real number x b)  c) f(x) = x cos(x) d) None of these
Which of the following function is periodic? a) f(x) = x – [x] where [x] denotes the greatest integer less than equal to the real number x b) c) f(x) = x cos(x) d) None of these
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IIT 1983 |
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1118 |
A curve C has the property that the tangent drawn at any point P on C meets the co-ordinate axes at A and B, and P is the mid-point of AB. The curve passes through the point (1, 1). Determine the equation of the curve. a) x2y = 1 b) x = y c) xy = 1 d) x2 = y
A curve C has the property that the tangent drawn at any point P on C meets the co-ordinate axes at A and B, and P is the mid-point of AB. The curve passes through the point (1, 1). Determine the equation of the curve. a) x2y = 1 b) x = y c) xy = 1 d) x2 = y
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IIT 1998 |
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1119 |
Let –1 ≤ p ≤ 1. Show that the equation 4x3 – 3x – p = 0 has a unique root in the interval  and identify it. a) p b) p/3 c)  d) 
Let –1 ≤ p ≤ 1. Show that the equation 4x3 – 3x – p = 0 has a unique root in the interval and identify it. a) p b) p/3 c) d)
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IIT 2001 |
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1120 |
Find the coordinates of all points P on the ellipse  , for which the area of △PON is maximum where O denotes the origin and N the feet of perpendicular from O to the tangent at P.
Find the coordinates of all points P on the ellipse , for which the area of △PON is maximum where O denotes the origin and N the feet of perpendicular from O to the tangent at P.
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IIT 1999 |
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1121 |
Determine the equation of the curve passing through origin in the form  which satisfies the differential equation 
Determine the equation of the curve passing through origin in the form which satisfies the differential equation
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IIT 1996 |
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1122 |
If α, β are roots of  and γ, δ are roots of  then evaluate  in terms of p, q, r, s.
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IIT 1979 |
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1123 |
If p(x) = 51x101 – 2323x100 – 45x + 1035, using Rolle’s theorem prove that at least one root lies between  .
If p(x) = 51x101 – 2323x100 – 45x + 1035, using Rolle’s theorem prove that at least one root lies between .
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IIT 2004 |
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1124 |
For what values of m does the system of equations 3x + my = m, 2x – 5y = 20 have solutions satisfying x > 0, y > 0? a) m ε ( b) m ε ( c) m ε ( ∪ ( d) m ε (
For what values of m does the system of equations 3x + my = m, 2x – 5y = 20 have solutions satisfying x > 0, y > 0? a) m ε ( b) m ε ( c) m ε ( ∪ ( d) m ε (
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IIT 1980 |
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1125 |
Given 
and f(x) is a quadratic polynomial. V is a point of maximum of f(x) and ‘A’ is the point where f(x) cuts the X–axis. ‘B’ is a point such that AB subtends a right angle at V. Find the area between chord AB and f(x).
a) 125 b) 125/2 c) 125/3 d) 125/6
Given
and f(x) is a quadratic polynomial. V is a point of maximum of f(x) and ‘A’ is the point where f(x) cuts the X–axis. ‘B’ is a point such that AB subtends a right angle at V. Find the area between chord AB and f(x). a) 125 b) 125/2 c) 125/3 d) 125/6
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IIT 2005 |
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