April 7, 1997 Math 262 Exam 2 Name(10) 1. The vectors (1, 2, 1), (3, 4, 5), (2, 0,k) are linearly dependent ifA. k =1B. k =6C. k 6=6D. k =0E. k 6=0(10) 2. If T : P3−→ P3is a linear transformation such that T (x2− 1) = x2+x − 3,T(3x)=6xand T (2x +1)=4x +4,thenT (x2)isA. x2B. x2+ x − 2C. x2+ x − 1D. x2+ xE. x2+ x +11(10) 3. Use Cramer’s Rule to solve the system below for the unknown functions u1(x)andu2(x).u1sin x + u2cos x =0u1cos x − u2sin x = ex(10) 4. What is the correct form of ypto use when finding a particular solution to the equationy00+ y = x cos x using the method of undetermined coefficients?Do not compute the coefficients. Just write down the FORM of the particularsolution. (For example, if the right hand side were x2, the correct form of ypwouldbe Ax2+ Bx + C.)(20) 5. LetA =11−201 a24−3a) for what value(s) of a is det A 6=0.b) Find all a such that the equation Ax = 0 has a nontrivial solution.2(20) 6. Find the general solution y(x) to the differential equationy00+3y0+2y =10sinx.3(20) 7. Let T : R4−→ R3be defined by Tx = Ax whereA =11−1 −301 1−422−2 −6.Find a basis for ker(T ). What is the dimension of ker(T
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