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1201 |
Determine the values of x for which the following function fails to be continuous or differentiable. 
Justify your answer. a) f(x) is continuous and differentiable b) f(x) is continuous everywhere but not differentiable at
x = 1, 2 c) f(x) is continuous everywhere but not differentiable at x = 2 d) f(x) is neither continuous nor differentiable at x = 1, 2
Determine the values of x for which the following function fails to be continuous or differentiable. Justify your answer. a) f(x) is continuous and differentiable b) f(x) is continuous everywhere but not differentiable at
x = 1, 2 c) f(x) is continuous everywhere but not differentiable at x = 2 d) f(x) is neither continuous nor differentiable at x = 1, 2
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IIT 1997 |
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1202 |
Let   And  where a and b are non-negative real numbers. Determine the composite function gof. If (gof)(x) is continuous for all real x, determine the values of a and b. Is gof differentiable at x = 0? a) a = b = 0 b) a = 0, b = 1 c) a = 1, b = 0 d) a = b = 1
Let And where a and b are non-negative real numbers. Determine the composite function gof. If (gof)(x) is continuous for all real x, determine the values of a and b. Is gof differentiable at x = 0? a) a = b = 0 b) a = 0, b = 1 c) a = 1, b = 0 d) a = b = 1
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IIT 2002 |
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1203 |
Find the equation of the circle touching the line 2x + 3y + 1 = 0 at the point (1, −1) and is orthogonal to the circle which has the line segment having end points (0, −1) and (−2, 3) as diameter.
Find the equation of the circle touching the line 2x + 3y + 1 = 0 at the point (1, −1) and is orthogonal to the circle which has the line segment having end points (0, −1) and (−2, 3) as diameter.
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IIT 2004 |
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1204 |
Show that the value of  wherever defined a) always lies between  and 3 b) never lies between and 3 c) depends on the value of x
Show that the value of wherever defined a) always lies between and 3 b) never lies between and 3 c) depends on the value of x
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IIT 1992 |
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1205 |
Show that f(x) is differentiable at the value of α = 1. Also, a) b2 +c2 = 4 b) 4 b2 = 4 − c2 c) 64 b2 = 4 − c2 d) 64 b2 = 4 + c2
Show that f(x) is differentiable at the value of α = 1. Also, a) b2 +c2 = 4 b) 4 b2 = 4 − c2 c) 64 b2 = 4 − c2 d) 64 b2 = 4 + c2
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IIT 2004 |
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1206 |
The product of r consecutive natural numbers is divisible by r! a) True b) False
The product of r consecutive natural numbers is divisible by r! a) True b) False
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IIT 1985 |
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1207 |
The area bounded by the curve y = f(x), the X–axis and the ordinates x = 1, x = b is (b – 1) sin (3b + 4). Then f(x) is a) (x – 1) cos (3x + b) b) sin (3x + 4) c) sin (3x + 4) + 3 (x – 1) cos (3x + 4) d) none of these
The area bounded by the curve y = f(x), the X–axis and the ordinates x = 1, x = b is (b – 1) sin (3b + 4). Then f(x) is a) (x – 1) cos (3x + b) b) sin (3x + 4) c) sin (3x + 4) + 3 (x – 1) cos (3x + 4) d) none of these
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IIT 2005 |
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1208 |
The sum  where  equals a) i b) i – 1 c) – i d) 0
The sum where equals a) i b) i – 1 c) – i d) 0
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IIT 1998 |
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1209 |
Fill in the blank The value of f (x) =  lies in the interval ……………. a)  b)  c)  d) 
Fill in the blank The value of f (x) = lies in the interval ……………. a) b) c) d)
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IIT 1983 |
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1210 |
Find the area bounded by the curve x2 = 4y and the straight line
x = 4y – 2. a) 3/2 b) 3/4 c) 9/4 d) 9/8
Find the area bounded by the curve x2 = 4y and the straight line
x = 4y – 2. a) 3/2 b) 3/4 c) 9/4 d) 9/8
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IIT 1981 |
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1211 |
If f(x) and g(x) are differentiable functions for 0 ≤ x ≤ 1 such that f(0) = 2, g(0) = 0, f(1) = 6, g(1) = 2 then show that there exists c satisfying 0 < c < 1 and  . a) 0 < c < 1 and b) 0 < c < 1 and  c) 0 < c < 1 and  d) 0 < c < 1 and 
If f(x) and g(x) are differentiable functions for 0 ≤ x ≤ 1 such that f(0) = 2, g(0) = 0, f(1) = 6, g(1) = 2 then show that there exists c satisfying 0 < c < 1 and . a) 0 < c < 1 and b) 0 < c < 1 and c) 0 < c < 1 and d) 0 < c < 1 and
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IIT 1982 |
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1212 |
Let a > 0, b > 0, c > 0 then both the roots of the equation  a) are real and positive b) have negative real parts c) have positive real parts d) none of these
Let a > 0, b > 0, c > 0 then both the roots of the equation a) are real and positive b) have negative real parts c) have positive real parts d) none of these
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IIT 1979 |
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1213 |
If f(x) is a continuous function defined for 1 ≤ x ≤ 3. If f(x) takes rational values for all x and f(2) = 10 then f(1.5) = . . . . a) 2 b) 5 c) 10 d) 20
If f(x) is a continuous function defined for 1 ≤ x ≤ 3. If f(x) takes rational values for all x and f(2) = 10 then f(1.5) = . . . . a) 2 b) 5 c) 10 d) 20
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IIT 1997 |
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1214 |
If x, y, z are real and distinct then  is always a) Non – negative b) Non – positive c) Zero d) None of these
If x, y, z are real and distinct then is always a) Non – negative b) Non – positive c) Zero d) None of these
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IIT 2005 |
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1215 |
Match the following
Let [x] denote the greatest integer less than or equal to x | Column 1 | Column 2 | | i) x|x| | A)continuous in  | | ii)  | B)Differentiable in | | iii) x + [x] | C)Steadily increasing in | | iv) |x – 1| + |x + 1| | D) Not differentiable at least at one point in | a) (i)→ A, B, C, (ii)→ A, D, (iii)→ C, D, (iv)→ A, B b) (i)→ A, (ii)→ A, (iii)→ C, (iv)→ B c) (i)→ B, (ii)→ D, (iii)→ C, (iv)→ A d) (i)→ A, B, (ii)→ A, D, (iii)→ C, D, (iv)→ B
Match the following
Let [x] denote the greatest integer less than or equal to x | Column 1 | Column 2 | | i) x|x| | A)continuous in | | ii) | B)Differentiable in | | iii) x + [x] | C)Steadily increasing in | | iv) |x – 1| + |x + 1| | D) Not differentiable at least at one point in | a) (i)→ A, B, C, (ii)→ A, D, (iii)→ C, D, (iv)→ A, B b) (i)→ A, (ii)→ A, (iii)→ C, (iv)→ B c) (i)→ B, (ii)→ D, (iii)→ C, (iv)→ A d) (i)→ A, B, (ii)→ A, D, (iii)→ C, D, (iv)→ B
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IIT 2007 |
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1216 |
(One or more than one correct answer)
If  are complex numbers such that  and  then the pair of complex numbers  and  satisfy a)  b)  c)  d) None of these
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IIT 1985 |
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1217 |
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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1218 |
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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1219 |
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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1220 |
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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1221 |
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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1222 |
 then tan t =
then tan t =
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IIT 2006 |
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1223 |
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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1224 |
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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1225 |
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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