Mathematics
Answer
Factorizing,
⇒ x2 + 3x + 2
⇒ x2 + 2x + x + 2
⇒ x(x + 2) + 1(x + 2)
⇒ (x + 1)(x + 2)
Given,
x2 + 3x + 2 is a factor of 3x3 + x2 - ax - b.
∴ (x + 1) and (x + 2) are factors of 3x3 + x2 - ax - b.
Therefore, on substituting x = -1, in 3x3 + x2 - ax - b, remainder will be equal to 0.
⇒ 3(-1)3 + (-1)2 - a(-1) - b = 0
⇒ 3(-1) + 1 + a - b = 0
⇒ -3 + 1 + a - b = 0
⇒ -2 + a - b = 0
⇒ a = b + 2 ………….(1)
Also, on substituting x = -2, in 3x3 + x2 - ax - b, remainder will be equal to 0.
⇒ 3(-2)3 + (-2)2 - a(-2) - b = 0
⇒ 3(-8) + 4 + 2a - b = 0
⇒ -24 + 4 + 2a - b = 0
⇒ -20 + 2a - b = 0
⇒ 2a - b = 20
Substituting value of a from equation (1) in above equation, we get :
⇒ 2(b + 2) - b = 20
⇒ 2b + 4 - b = 20
⇒ b + 4 = 20
⇒ b = 16.
⇒ a = b + 2 = 18.
Hence, a = 18 and b = 16.
Related Questions
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(i) Plot the points A(1, 1), B(5, 3) and C(2, 7).
(ii) Construct the locus of points equidistant from A and B.
(iii) Construct the locus of points equidistant from AB and AC.
(iv) Construct the point P such that PA = PB and P is equidistant from AB and AC.
Mohit draws a card from a well-shuffled deck of 52 cards. Find the probability of getting :
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Use graph paper for this question. Take 2 cm = 1 unit along both the axes.
(i) Plot the points A(2, 3), B(2, 2) and C(3, 2).
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(iv) Join A, B, C, C'', B'', A'', A''', B''', C''', C', B', A' and A to make a closed figure.