776 |
Let and where O, A and B are non-collinear points. Let p denote the area of the quadrilateral OABC and let q denote the area of the quadrilateral with OA and OC as adjacent sides. If p = kq then k = . . . . .
Let and where O, A and B are non-collinear points. Let p denote the area of the quadrilateral OABC and let q denote the area of the quadrilateral with OA and OC as adjacent sides. If p = kq then k = . . . . .
|
IIT 1997 |
|
777 |
If M is a 3 x 3 matrix where det (M) = 1 and MMT = I, then prove that det (M – I) = 0.
If M is a 3 x 3 matrix where det (M) = 1 and MMT = I, then prove that det (M – I) = 0.
|
IIT 2004 |
|
778 |
Let A = If U1, U2, U3 are column matrices satisfying AU1 = , AU2 = and AU3 = and U is a 3 x 3 matrix whose columns are U1, U2, U3 then the value of [ 3 2 0 ] U is a) b) c) d)
Let A = If U1, U2, U3 are column matrices satisfying AU1 = , AU2 = and AU3 = and U is a 3 x 3 matrix whose columns are U1, U2, U3 then the value of [ 3 2 0 ] U is a) b) c) d)
|
IIT 2006 |
|
779 |
Let a, b, c, d be real numbers in geometric progression. If u, v, w satisfy the system of equations Then show that the roots of the equation and are reciprocal of each other.
|
IIT 1999 |
|
780 |
The interior angles of a polygon are in Arithmetic Progression. The smallest angle is 120° and the common difference is 5. Find the number of sides of the polygon.
The interior angles of a polygon are in Arithmetic Progression. The smallest angle is 120° and the common difference is 5. Find the number of sides of the polygon.
|
IIT 1980 |
|
781 |
Let a1, a2, … an be positive real numbers in Geometric Progression. For each n let An, Gn, Hn be respectively the arithmetic mean, geometric mean and harmonic mean of a1, a2, . . . ., an. Find the expressions for the Geometric mean of G1, G2, . . . .Gn in terms of A1, A2, . . . .,An; H1, H2, . . . .Hn
Let a1, a2, … an be positive real numbers in Geometric Progression. For each n let An, Gn, Hn be respectively the arithmetic mean, geometric mean and harmonic mean of a1, a2, . . . ., an. Find the expressions for the Geometric mean of G1, G2, . . . .Gn in terms of A1, A2, . . . .,An; H1, H2, . . . .Hn
|
IIT 2001 |
|
782 |
If total number of runs scored in n matches is where n > 1 and the runs scored in the kth match are given by k.2n + 1 – k where 1 ≤ k ≤ n. Find n.
If total number of runs scored in n matches is where n > 1 and the runs scored in the kth match are given by k.2n + 1 – k where 1 ≤ k ≤ n. Find n.
|
IIT 2005 |
|
783 |
In a triangle ABC if cotA, cotB, cotC are in Arithmetic Progression then a, b, c are in . . . . . Progression.
In a triangle ABC if cotA, cotB, cotC are in Arithmetic Progression then a, b, c are in . . . . . Progression.
|
IIT 1985 |
|
784 |
For any odd integer n ≥ 1, n3 – (n – 1)3 + . . . + (−)n – 1 13 = . . .
For any odd integer n ≥ 1, n3 – (n – 1)3 + . . . + (−)n – 1 13 = . . .
|
IIT 1996 |
|
785 |
The edges of a parallelepiped are of unit length and are parallel to non-coplanar unit vectors such that . Then the volume of the parallelepiped is a) b) c) d)
The edges of a parallelepiped are of unit length and are parallel to non-coplanar unit vectors such that . Then the volume of the parallelepiped is a) b) c) d)
|
IIT 2008 |
|
786 |
Consider three planes P1 : x – y + z = 1 P2 : x + y – z = −1 P3 : x – 3y + 3z = 2 Let L1, L2, L3 be lines of intersection of planes P2 and P3, P3 and P1, and P1 and P2 respectively. Statement 1 – At least two of the lines L1, L2, L3 are non parallel Statement 2 – The three planes do not have a common point. a) Statement 1 is true. Statement 2 is true. Statement 2 is a correct explanation of statement 1. b) Statement 1 is true. Statement 2 is true. Statement 2 is not a correct explanation of statement 1. c) Statement 1 is true. Statement 2 is false. d) Statement 1 is false. Statement 2 is true.
Consider three planes P1 : x – y + z = 1 P2 : x + y – z = −1 P3 : x – 3y + 3z = 2 Let L1, L2, L3 be lines of intersection of planes P2 and P3, P3 and P1, and P1 and P2 respectively. Statement 1 – At least two of the lines L1, L2, L3 are non parallel Statement 2 – The three planes do not have a common point. a) Statement 1 is true. Statement 2 is true. Statement 2 is a correct explanation of statement 1. b) Statement 1 is true. Statement 2 is true. Statement 2 is not a correct explanation of statement 1. c) Statement 1 is true. Statement 2 is false. d) Statement 1 is false. Statement 2 is true.
|
IIT 2008 |
|
787 |
The solution of primitive equation is . If and then is a) b) c) d)
|
IIT 2005 |
|
788 |
If then prove that
If then prove that
|
IIT 1983 |
|
789 |
Solution of the differential equation is
Solution of the differential equation is
|
IIT 2006 |
|
790 |
The parameter on which the value of the determinant Δ = does not depend upon is a) a b) p c) d d) x
The parameter on which the value of the determinant Δ = does not depend upon is a) a b) p c) d d) x
|
IIT 1997 |
|
791 |
For what value of m does the system of equations 3x + my = m, 2x − 5y = 20 have a solution satisfying the condition x > 0, y > 0. a) m (−∞, ∞) b) m (−∞, −15) ∪ (30, ∞) c) d)
For what value of m does the system of equations 3x + my = m, 2x − 5y = 20 have a solution satisfying the condition x > 0, y > 0. a) m (−∞, ∞) b) m (−∞, −15) ∪ (30, ∞) c) d)
|
IIT 1979 |
|
792 |
If α is a repeated root of a quadratic equation f(x) = 0 and A(x), B(x), C(x) be polynomials of degree 3, 4, 5 respectively, Then show that is divisible by f(x) where prime denotes the derivatives.
If α is a repeated root of a quadratic equation f(x) = 0 and A(x), B(x), C(x) be polynomials of degree 3, 4, 5 respectively, Then show that is divisible by f(x) where prime denotes the derivatives.
|
IIT 1984 |
|
793 |
If , for every real number x, then the minimum value of f a) does not exist because f is unbounded b) is not attained even though f is bounded c) is equal to 1 d) is equal to –1
If , for every real number x, then the minimum value of f a) does not exist because f is unbounded b) is not attained even though f is bounded c) is equal to 1 d) is equal to –1
|
IIT 1998 |
|
794 |
The function is defined by then is a) b) c) d) None of these
The function is defined by then is a) b) c) d) None of these
|
IIT 1999 |
|
795 |
Suppose for x ≥ . If g(x) is the function whose graph is the reflection of f(x) with respect to the line y = x then g(x) equals a) b) c) d)
Suppose for x ≥ . If g(x) is the function whose graph is the reflection of f(x) with respect to the line y = x then g(x) equals a) b) c) d)
|
IIT 2002 |
|
796 |
Domain of definition of the function for real values of x is a) b) c) d)
Domain of definition of the function for real values of x is a) b) c) d)
|
IIT 2003 |
|
797 |
Let f be a one–one function with domain {x, y, z} and range {1, 2, 3}. It is given that exactly one of the following statements is true and remaining statements are false f (1) = 1, f (y) ≠ 1, f (z) ≠ 2. Determine
Let f be a one–one function with domain {x, y, z} and range {1, 2, 3}. It is given that exactly one of the following statements is true and remaining statements are false f (1) = 1, f (y) ≠ 1, f (z) ≠ 2. Determine
|
IIT 1982 |
|
798 |
Let {x} and [x] denote the fractional and integral part of a real number x respectively. Solve 4{x} = x + [x]
Let {x} and [x] denote the fractional and integral part of a real number x respectively. Solve 4{x} = x + [x]
|
IIT 1994 |
|
799 |
a) True b) False
a) True b) False
|
IIT 1978 |
|
800 |
Find the range of values of t for which a) (−, −) b) ( , ) c) (− , − ) U ( , ) d) (−, )
|
IIT 2005 |
|