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Let the vector space V=R^3 and W={(a,b,c);a+b+c=0} i.e. W consists of those vectors each with the property that the sum of its components is zero. Is W a subspace of V
x+(λ-3)y=0
(λ-3)x+y=0
Have a non-trivial solution
B1 and B2 are two types of boats which are to be used to ferry 800 troops and 90 tons of equipment across a lake. Each B1 can carry 200 men and 15 tons of equipment while each B2 can carry 100 men and 15 tons of equipment. If each B1 costs Rs. 90 to operate and each B2 costs Rs. 44 to operate, find the number of each boat that should be used if the cost is to be minimum.
Give a geometric description of a single linear equation in three variables. Then give a geometric description of the solution set of a system of 3 linear equations in 3 variables if the system
(a) is inconsistent.
(b) is consistent and has no free variables.
(c) is consistent and has exactly one free variable.
(d) is consistent and has two free variables
Find the standard matrix A for the linearly transformation T :R^2_R^2
5)Solve the following linear programming problem using two phase method.
Minimize z = -3x1 + x2 - 2x3
Subject to
x1 +3x2 +x3 ≤5
2x1 –x2 +x3 ≥2
4x1 + 3x2 - 2x3 = 5
x1, x2, x3 ≥ 0
a. Find the orthogonal and normal canonical forms of 2y^2-2yz+2zx-2xy.
b. The operation,* defined by a*b= sin(ab), is a binary operation on N
True or false with full explanation
terms of a parameter θ, are (a cos θ, b sin θ). Obtain the equations of the tangents at θ = θ1
and θ = θ1 + π/2.
Find the coordinates of the points of intersection of the two tangents, and deduce the
Cartesian equation of its locus.
(b) Find the equations of the tangents to the hyperbola x^2 − 9y^2 = 9 that are drawn from the
point (3, 2) and the points of contact. Find the area of the triangle that these tangents
form with their chord of contact
Suppose that A, B, C are 3×3 matrices with det (A) = 2, det (B) = 3 and det (C) = 5. Compute the following determinants:
(a) det (AB)
(b) det (3AB-2C2)
(c) det (A2CTB-1)
The sum of three numbers is 20. If we multiply the first number by 2 and add the second
number and subtract the third number, then we get 23. If we multiply the first number by 3
and add second and third number to it, then we get 46. Let x be the first number, y be the
second number and z
be the third number.
(a) Obtain a system of linear equations to represent the given information.
(b) Write down the system in (a) as a matrix equation.
(c) Use inverse matrix to solve for x , y and z .
#340153 on Dec 2023