Mathematics Study Material

Mathematics

Question 1:
Solve the inequality
2x2 + 3x > 2
2x2 + 3x – 2 > 0
(x + 2) (2x – 1) > 0
or, x + 2 > 0 or, 2x – 1 > 0
x > -2 or, x > ½
so, -2 < x < ½
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Question 3
Solve the quadratic equation
(a) Given,
f(x) = ½ x2 – 4x + 6
(i) Completing square -
f(x) = 0
½ x2 – 4x + 6 = 0
x2 – 8x + 12
x2 – 8x + (4)2 + 12 – (4)2
(x – 4)2 + 12 – 16
(x – 4)2 – 4
or, ½ (x – 4)2 – 2
(ii) To draw the graph, above obtained equation can be represented as –
y = ½ (x – 4)2 – 2
(as compared with parabolic equation y = ax2)
here, a > 0 therefore, parabola of this equation will open up in upward direction.
(iii) Interval form of given equation is
(x – 4)2 – 4 ≥ 0
(x – 4)2 ≥ 4
0 ≤ x ≤ 4
(b) g = ½ x2 – 4x + 6
(i) Given equation, ½x2 – 4x + 6
can be written in interval form as 0 ≤ x ≤ 4
therefore,
0 1 2 3 4 5 6
(ii) Image set of g can be identified in following way –
x2 – 8x + 12
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