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Can the magnitude of the resultant of the two vectors be greater than the sum of magnitude of individual vector?

Three of the best-selling albums of 2013 were Justin Timberlake's The 20120 Experience, Eminem's The Marshall Mathers LP2, and Imagine Dragons' Night Visions. Eminem's album sold 0.33 million more copies than Imagine Dragons'. The number of Justin Timberlake albums sold was 0.37 million less than twice the number of Imagine Dragons' sales. The three recording artists sold a total of 5.56 million albums. How many albums did each artist sell?

a) Check whether the following system of equations has a solution.

x+y+3z+w = 5

−x+y+z−5w = 7

x+2y+5z−w = 5 (6)

b) Let T : P2 → P1 be defined by

T(a+bx+cx2

) = b+c+ (a−c)x.

Check that T is a linear transformation. Find the matrix of the transformation with

respect to the ordered bases B1 = {x

2

, x

2 +x, x

2 +x+1} and B2 = {1, x}. Find the

kernel of T.

x+y+3z+w = 5

−x+y+z−5w = 7

x+2y+5z−w = 5 (6)

b) Let T : P2 → P1 be defined by

T(a+bx+cx2

) = b+c+ (a−c)x.

Check that T is a linear transformation. Find the matrix of the transformation with

respect to the ordered bases B1 = {x

2

, x

2 +x, x

2 +x+1} and B2 = {1, x}. Find the

kernel of T.

The admission fee at an amusement park is $ 1.50 for children and $ 4.00 for adults. On a certain day, 299 people entered the park, and the admission fees collected totaled $936 . How many children and how many adults were admitted?

) Let V = R

2

. Define addition + on V by (x1, y1) + (x2, y2) = (x1 +x2, y1 +y2) and

scalar multiplication · by r·(a,b) = (ra,0). Check whether V satisfies all the

conditions for it to be a vector space over R with respect to these operations.

2

. Define addition + on V by (x1, y1) + (x2, y2) = (x1 +x2, y1 +y2) and

scalar multiplication · by r·(a,b) = (ra,0). Check whether V satisfies all the

conditions for it to be a vector space over R with respect to these operations.

For any four vectors a^ vector, b ^ vector, C ^ vector and d ^ vector determine : (a^ vector × b^ vector)(c ^ vector × d ^ vector) + (b^ vector × c^ vector) (a^vector × d^vector) + (c^vector × a^ vector) ( b^vector × d^ vector).

Obtain a unit vector perpendicular to the plane of the vectors A^vector = 3i^cap - 4j^cap +2k^cap and B ^vector = i^cap + j ^ cap +3 k^ cap.

Check whether the following system of equations has a solution

x + y + 3z + w = 5

-x + y + z - 5w = 7

x + 2y + 5z - w = 5

x + y + 3z + w = 5

-x + y + z - 5w = 7

x + 2y + 5z - w = 5

Find the radius and the center of the circular section of the sphere |r| = 17 cut off by

the plane r . (i + 2j + 2k) = 24.

the plane r . (i + 2j + 2k) = 24.

Find the equation of the line passing through the point A(1, 0, -1) and parallel to the

line joining B(1, 2, 3) and C( -1, 2, 0). Is it perpendicular to the line

(1 + 3α , 0 ,1 - 2α ).

line joining B(1, 2, 3) and C( -1, 2, 0). Is it perpendicular to the line

(1 + 3α , 0 ,1 - 2α ).