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Practice Questions

1.

 Which of the following expression is a polynomial? A.X2 + 2/X + 3 B.X2 - 2X + 3 C.1/X2 - 2X + 3 D.X2 - 2 √X - 3

Clearly, (X2 - 2X + 3) is a polynomial since in options (a), (c) and (d), X does not have integral powers.

2.

 If f(X) = X4 - 2X3 + 2X2 - X - 1, Find f(-1/2). A.9/16 B.5/16 C.3/16 D.21/16

f(-1/2) = (-1/2)4 - 2(-1/2)3 + 2(-1/2)2 - (-1/2) - 1
= 1/16 + 1/4 + 1/2 + 1/2 - 1 = 5/16

3.

 Which of the following is a root of the polynomial f(X) = X3 - 2X2 - X + 2 A.X = - 2 B.X = 1 C.X = 3 D.X = -3

From the given options
f(1) = (1)3 - 2 (1)2 - 1 + 2
= 1 - 2 - 1 + 2
= 0
Hence, X = 1 is a root of the polynomial f(X).

4.

 If X = 2 is a root of the polynomial f(X) = 5X - 3X2 + KX - 10, find the value of K. A.- 9 B.3 C.9 D.- 3

If X = 2 is a root of the polynomial f(X), then f(2) = 0.
now, f(2) = 5X (2)2 - 3X (2)2 + KX2 - 10 = 0
40 - 12 + 2K - 10 = 0
18 + 2K = 0
K = - 9

5.

 If a/b = c/d = e/f = 3, then [(2a2 + 3c2 + 4e2)/(2b2 + 3d2 + 4f2)] = ? A.2 B.3 C.4 D.9

a/b = c/d = e/f = 3
a = 3b, c = 3d, and e = 3f
(2a2 + 3c2 + 4e2)/(2b2 + 3d2 + 4f2)
= (2 x 9b2 + 3 x 9d2 + 4 x 9f2)/(2b2 + 3d2 + 4f2)
= 9(2b2 + 3d2 + 4f2)/(2b2 + 3d2 + 4f2)
= 9

6.

 X + 1/X = 1, Find X6 + X3 + 1 = ? A.0 B.1 C.2 D.3

X + 1/X = 1
X2 + 1 = X
X2 - X + 1 = 0
= X6 + X3 + 1
= X3(X3 + 1) + 1
= X3(X + 1)(X2 - X + 1) + 1
= 0 + 1
= 1

7.

 X + 1/X = 5, then value of (X4 + 1/X2)/(X2 - 3X + 1) = ? A.14 B.26 C.55 D.74

Given that, (X4 + 1/X2)/(X2 - 3X + 1)
= (X4 + 1/X2)/(X2 + 1 - 3X)
= [X4 + 1/X2)]/[X(X + 1/X) - 3X]
= (X4 + 1/X2)/(5X - 3X)
= (X4 + 1/X2)/2X
= 1/2 [ X4/X + 1/X2.X]
= 1/2 [ X3 + 1/X3]
= 1/2 [ (X + 1/X)3 - 3X.1/X (X + 1/X)]
= 1/2 [(5)3 - (3 x 5)]
= 1/2 [ 125 - 15 ]
= 1/2 [ 110 ] = 55

8.

 If X = 2 + √3, Y = 2 - √3, then value of [(X2 + Y2)/(X3 + Y3)] is A.17/13 B.5/16 C.1/13 D.7/26

[(X2 + Y2)/(X3 + Y3)] = [(X + Y)2 - 2XY/(X + Y)3 - 3XY(X + Y)]
X + Y = 2 + √3 + 2 - √3 = 4
X x Y = (2 + √3) x (2 - √3) = 1
= (4)2 -2/(4)3 - 3 x 1(4)
= [16 - 2/64 - 12]
= 14/52
= 7/26