976 |
A rectangle with sides (2m – 1) and (2n – 1) is divided into squares of unit length by drawing parallel lines. Then the number of rectangles possible with odd side lengths is a) mn (m + 1)(n + 1) b) c) d)
A rectangle with sides (2m – 1) and (2n – 1) is divided into squares of unit length by drawing parallel lines. Then the number of rectangles possible with odd side lengths is a) mn (m + 1)(n + 1) b) c) d)
|
IIT 2005 |
|
977 |
Find the values of a and b, so that the functions Is continuous for 0 ≤ x ≤ π a) b) c) d)
Find the values of a and b, so that the functions Is continuous for 0 ≤ x ≤ π a) b) c) d)
|
IIT 1989 |
|
978 |
Let α ε ℝ, then a function f : ℝ → ℝ is differentiable at α if and only if there is a function g : ℝ → ℝ which is continuous at α and satisfies f(x) – f(α) = g(x) (x – α) for all x ε ℝ. a) True b) False
Let α ε ℝ, then a function f : ℝ → ℝ is differentiable at α if and only if there is a function g : ℝ → ℝ which is continuous at α and satisfies f(x) – f(α) = g(x) (x – α) for all x ε ℝ. a) True b) False
|
IIT 2001 |
|
979 |
If two functions f and g satisfy the given conditions x, y ε ℝ, f(x – y) = f(x)g(y) – f(y)g(x) and g(x – y) = g(x) . g(y) + f(x) . f(y). If the RHD at x = 0 exists for f(x) then find the derivative of g(x) at x = 0.
If two functions f and g satisfy the given conditions x, y ε ℝ, f(x – y) = f(x)g(y) – f(y)g(x) and g(x – y) = g(x) . g(y) + f(x) . f(y). If the RHD at x = 0 exists for f(x) then find the derivative of g(x) at x = 0.
|
IIT 2005 |
|
980 |
Let be a real valued function. The set of points where f(x) is not differentiable are a) {0} b) {1} c) {0, 1} d) {∅}
Let be a real valued function. The set of points where f(x) is not differentiable are a) {0} b) {1} c) {0, 1} d) {∅}
|
IIT 1981 |
|
981 |
Let f(x) = [x] where [.] denotes the greatest integer function. Then the domain of f is . . . ., points of discontinuity of f are . . . . a) ∀ x ε I b) ∀ x ε I − {0} c) ∀ x ε I – {0, 1} d) ∀ x ε I – {0, 1, 2}
Let f(x) = [x] where [.] denotes the greatest integer function. Then the domain of f is . . . ., points of discontinuity of f are . . . . a) ∀ x ε I b) ∀ x ε I − {0} c) ∀ x ε I – {0, 1} d) ∀ x ε I – {0, 1, 2}
|
IIT 1996 |
|
982 |
PQ and PR are two infinite rays, QAR is an arc. Points lying in the shaded region excluding the boundary satisfies a) |z + 1| > 2; |arg(z + 1)| < b) |z + 1| < 2; |arg(z + 1)| < c) d)
PQ and PR are two infinite rays, QAR is an arc. Points lying in the shaded region excluding the boundary satisfies a) |z + 1| > 2; |arg(z + 1)| < b) |z + 1| < 2; |arg(z + 1)| < c) d)
|
IIT 2005 |
|
983 |
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. The positive value of k for which has only one root is a) b) 1 c) e d) ln2
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. The positive value of k for which has only one root is a) b) 1 c) e d) ln2
|
IIT 2007 |
|
984 |
Let the complex numbers are vertices of an equilateral triangle. If be the circumcentre of the triangle, then prove that
|
IIT 1981 |
|
985 |
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 .
|
IIT 1997 |
|
986 |
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
|
IIT 1983 |
|
987 |
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 .
|
IIT 2004 |
|
988 |
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
|
IIT 2005 |
|
989 |
The area enclosed within the curve |x| + |y| = 1 is . . . a) 1 b) c) d) 2
The area enclosed within the curve |x| + |y| = 1 is . . . a) 1 b) c) d) 2
|
IIT 1981 |
|
990 |
Let g(x) be a function of x defined on (−1, 1). If the area of the equilateral triangle with two of its vertices as (0, 0) and [x, g(x)] is , then the function g(x) is a) b) c) d) None of the above
Let g(x) be a function of x defined on (−1, 1). If the area of the equilateral triangle with two of its vertices as (0, 0) and [x, g(x)] is , then the function g(x) is a) b) c) d) None of the above
|
IIT 1989 |
|
991 |
Show that the integral of is
Show that the integral of is
|
IIT 1979 |
|
992 |
= a) True b) False
= a) True b) False
|
IIT 1986 |
|
993 |
a) – 1 b) 2 c) 1 + e−1 d) None of these
a) – 1 b) 2 c) 1 + e−1 d) None of these
|
IIT 1981 |
|
994 |
Let f(x) be differentiable on the interval (0, ∞) such that f (1) = 1 and for each x > 0. Then f(x) is a) b) c) d)
Let f(x) be differentiable on the interval (0, ∞) such that f (1) = 1 and for each x > 0. Then f(x) is a) b) c) d)
|
IIT 2007 |
|
995 |
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
|
IIT 2007 |
|
996 |
Find all possible values of b > 0, so that the area of the bounded region enclosed between the parabolas and is maximum. a) b = 1 b) b ≥ 1 c) b ≤ 1 d) 0 < b < 1
Find all possible values of b > 0, so that the area of the bounded region enclosed between the parabolas and is maximum. a) b = 1 b) b ≥ 1 c) b ≤ 1 d) 0 < b < 1
|
IIT 1997 |
|
997 |
Let C1 and C2 be the graph of the function y = x2 and y = 2x respectively. Let C3 be the graph of the function y = f (x), 0 ≤ x ≤ 1, f (0) = 0. Consider a point P on C1. Let the lines through P, parallel to the axes meet C2 and C3 at Q and R respectively (see figure). If for every position of P (on C1) the area of the shaded regions OPQ and OPR are equal, determine the function f(x). a) x2 – 1 b) x3 – 1 c) x3 – x2 d) 1 + x2 + x3
Let C1 and C2 be the graph of the function y = x2 and y = 2x respectively. Let C3 be the graph of the function y = f (x), 0 ≤ x ≤ 1, f (0) = 0. Consider a point P on C1. Let the lines through P, parallel to the axes meet C2 and C3 at Q and R respectively (see figure). If for every position of P (on C1) the area of the shaded regions OPQ and OPR are equal, determine the function f(x). a) x2 – 1 b) x3 – 1 c) x3 – x2 d) 1 + x2 + x3
|
IIT 1998 |
|
998 |
|
IIT 1978 |
|
999 |
Let P be a point on the ellipse . Let the line parallel to Y–axis passing through P meets the circle at the point Q such that P and Q are on the same side of the X–axis. For two positive real numbers r and s find the locus of the point R on PQ such that PˆR : RˆQ = r : s and P varies over the ellipse.
Let P be a point on the ellipse . Let the line parallel to Y–axis passing through P meets the circle at the point Q such that P and Q are on the same side of the X–axis. For two positive real numbers r and s find the locus of the point R on PQ such that PˆR : RˆQ = r : s and P varies over the ellipse.
|
IIT 2001 |
|
1000 |
If (1 + x)n = C0 + C1x + C2x2 + . . . + Cnxn, then show that the sum of the products of the Cj’s is taken two at a time represented by is equal to
If (1 + x)n = C0 + C1x + C2x2 + . . . + Cnxn, then show that the sum of the products of the Cj’s is taken two at a time represented by is equal to
|
IIT 1983 |
|