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## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 3 Theory of Equations Ex 3.7

Choose the correct or the most suitable answer from the given four alternatives:

Question 1.

A zero of x^{3} + 64 is _______

(a) 0

(b) 4

(c) 4i

(d) -4

Answer:

(d) -4

Hint: x^{3} + 64 = 0

⇒ x^{3} = -64

⇒ x^{3} = (-4)^{3}

⇒ x = -4

Question 2.

If f and g are polynomials of degrees m and n respectively, and if h(x) = (f 0 g) (x), then the degree of h is ______

(a) mn

(b) m + n

(c) m^{n}

(d) n^{m}

Answer:

(a) mn

Question 3.

A polynomial equation in x of degree n always has _______

(a) n distinct roots

(b) n real roots

(c) n imaginary roots

(d) at most one root.

Answer:

(c) n imaginary roots (Every real number is also imaginary)

Question 4.

If α, β and γ are the zeros of x^{3} + px^{2} + qx + r, then \(\sum \frac{1}{\alpha}\) is ______

(a) \(-\frac{q}{r}\)

(b) \(\frac{q}{p}\)

(c) \(\frac{q}{r}\)

(d) \(-\frac{q}{p}\)

Answer:

(a) \(-\frac{q}{r}\)

Hint:

Question 5.

According to the rational root theorem, which number is not possible rational zero of 4x^{7} + 2x^{4} – 10x^{3} – 5?

(a) -1

(b) \(\frac{5}{4}\)

(c) \(\frac{4}{5}\)

(d) 5

Answer:

(c) \(\frac{4}{5}\)

Hint:

a_{n} = 4; a_{0} = 5

Let \(\frac{p}{q}\) be the root of P (x). P must divide 5, possible values of P are ±1, ±5

q must divide 4, possible values of q are ±1, ±2, ±4

Possible roots are \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{4}\)

Question 6.

The polynomial x^{3} – kx^{2} + 9x has three real zeros if and only if, k satisfies.

(a) |k| ≤ 6

(b) k = 0

(c) |k| > 6

(d) |k| ≥ 6

Answer:

(d) |k| ≥ 6

Hint:

x^{3} – kx^{2} + 9x = 0

⇒ x (x^{2} – kx + 9) = 0

x = 0 is one real root. If the remaining roots to be real if the

b^{2} – 4ac ≥ 0

⇒ k^{2} – 36 ≥ 0

⇒ k^{2} ≥ 36

⇒ |k| ≥ 6

Question 7.

The number of real numbers in [0, 2π] satisfying sin^{4} x – 2sin^{2} x + 1 is ______

(a) 2

(b) 4

(c) 1

(d) ∞

Answer:

(c) 1

Hint:

sin^{4} x – 2sin^{2} x + 1 = 0

⇒ t^{2} – 2t + 1 = 0

⇒ (t – 1)^{2} = 0

⇒ t – 1 = 0

⇒ t = 1

⇒ sin^{2} x = 1

⇒ \(\frac{1-\cos 2 x}{2}=1\)

⇒ 1 – cos 2x = 2

⇒ cos 2x = cos 0

⇒ 2x = 2nπ

⇒ x = nπ

n = 0, x = 0

n = 1, x = π

n = 2, x = 2π

Question 8.

If x^{3} + 12x^{2} + 10ax + 1999 definitely has a positive zero, if and only if _______

(a) a ≥ 0

(b) a > 0

(c) a < 0

(d) a ≤ 0

Answer:

(c) a < 0

Hint:

If a < 0, then P(x) = x^{3} + 12x^{2} + 10ax + 1999 has 2 changes of sign.

∴ P (x) has atmost two positive roots. So a < 0

Question 9.

The polynomial x^{3} + 2x + 3 has _______

(a) one negative and two imaginary zeros

(b) one positive and two imaginary zeros

(c) three real zeros

(d) no zeros

Answer:

(a) one negative and two imaginary zeros

Hint:

P(x) = x^{3} + 2x + 3; No positive root.

P(-x) = -x^{3} – 2x + 3; Only one change in the sign.

∴ One negative root.

Question 10.

The number of positive zeros of the polynomial is ______

(a) 0

(b) n

(c) <n

(d) r

Answer:

(b) n