Exercise 3.7 - Chapter 3 Theory of Equations 12th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

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) $4 \mathrm{i}$
(d) $-4$
Answer:
(d) -4
Hint: $x^3+64=0$
$
\begin{aligned}
& \Rightarrow x^3=-64 \\
& \Rightarrow x^3=(-4)^3 \\
& \Rightarrow x=-4
\end{aligned}
$
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 $\mathrm{h}$ is
(a) $\mathrm{mn}$
(b) $\mathrm{m}+\mathrm{n}$
(c) $\mathrm{m}^{\mathrm{n}}$
(d) $\mathrm{n}^{\mathrm{m}}$
Answer:
(a) $\mathrm{mn}$
Question 3.
A polynomial equation in $\mathrm{x}$ of degree $\mathrm{n}$ always has
(a) $\mathrm{n}$ distinct roots
(b) $\mathrm{n}$ real roots
(c) $\mathrm{n}$ imaginary roots
(d) at most one root.
Answer:
(c) $\mathrm{n}$ imaginary roots (Every real number is also imaginary)

Question 4.
If $\alpha, \beta$ and $\gamma$ are the zeros of $x^3+p x^2+q x+r$, 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:
$
\sum \frac{1}{\alpha}=\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}=\frac{\alpha \beta+\beta \gamma+\gamma \alpha}{\alpha \beta \gamma}=-\frac{q}{r}
$
Question 5.
According to the rational root theorem, which number is not possible rational zero of $4 x^7+2 x^4-$ $10 \mathrm{x}^3-5$ ?
(a) $-1$
(b) $\frac{5}{4}$
(c) $\frac{4}{5}$
(d) 5
Answer:
(c) $\frac{4}{5}$
Hint:
$
\mathrm{a}_{\mathrm{n}}=4 ; \mathrm{a}_0=5
$
Let $\frac{p}{q}$ be the root of $\mathrm{P}(\mathrm{x})$. $\mathrm{P}$ must divide 5, possible values of $\mathrm{P}$ are $\pm 1, \pm 5$
$\mathrm{q}$ must divide 4 , possible values of $\mathrm{q}$ are $\pm 1, \pm 2, \pm 4$
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 $\mathrm{x}^3-\mathrm{kx}^2+9 \mathrm{x}$ has three real zeros if and only if, $\mathrm{k}$ satisfies.
(a) $|\mathrm{k}| \leq 6$
(b) $\mathrm{k}=0$
(c) $|\mathrm{k}|>6$
(d) $|\mathrm{k}| \geq 6$
Answer:
(d) $|\mathrm{k}|>6$

Hint:
$
\mathrm{x}^3-\mathrm{kx}^2+9 \mathrm{x}=0
$
$
\Rightarrow \mathrm{x}\left(\mathrm{x}^2-\mathrm{kx}+9\right)=0
$
$x=0$ is one real root. If the remaining roots to be real if the
$
\begin{aligned}
& \mathrm{b}^2-4 \mathrm{ac} \geq 0 \\
& \Rightarrow \mathrm{k}^2-36 \geq 0 \\
& \Rightarrow \mathrm{k}^2 \geq 36 \\
& \Rightarrow|\mathrm{k}| \geq 6
\end{aligned}
$

Question 7.
The number of real numbers in $[0,2 \pi]$ satisfying $\sin ^4 x-2 \sin ^2 x+1$ is
(a) 2
(b) 4
(c) 1
(d) $\infty$
Answer:
(c) 1
Hint:
$
\begin{aligned}
& \sin ^4 \mathrm{x}-2 \sin ^2 \mathrm{x}+1=0 \\
& \Rightarrow \mathrm{t}^2-2 \mathrm{t}+1=0 \\
& \Rightarrow(\mathrm{t}-1)^2=0 \\
& \Rightarrow \mathrm{t}-1=0 \\
& \Rightarrow \mathrm{t}=1 \\
& \Rightarrow \sin ^2 \mathrm{x}=1 \\
& \Rightarrow \frac{1-\cos 2 x}{2}=1 \\
& \Rightarrow 1-\cos 2 \mathrm{x}=2 \\
& \Rightarrow \cos 2 \mathrm{x}=\cos 0 \\
& \Rightarrow 2 \mathrm{x}=2 \mathrm{n} \pi \\
& \Rightarrow \mathrm{x}=\mathrm{n} \pi \\
& \mathrm{n}=0, \mathrm{x}=0 \\
& \mathrm{n}=1, \mathrm{x}=\pi \\
& \mathrm{n}=2, \mathrm{x}=2 \pi
\end{aligned}
$
Question 8.
If $x^3+12 x^2+10 a x+1999$ definitely has a positive zero, if and only if
(a) $\mathrm{a} \geq 0$
(b) $\mathrm{a}>0$
(c) $a<0$
(d) $\mathrm{a} \leq 0$
Answer:
(c) a $<0$

Hint:
If $a<0$, then $P(x)=x^3+12 x^2+10 a x+1999$ has 2 changes of sign.
$\therefore \mathrm{P}(\mathrm{x})$ has atmost two positive roots. So $\mathrm{a}<0$
Question 9.
The polynomial $\mathrm{x}^3+2 \mathrm{x}+3$ 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:
$\mathrm{P}(\mathrm{x})=\mathrm{x}^3+2 \mathrm{x}+3 ;$ No positive root.
$\mathrm{P}(-\mathrm{x})=-\mathrm{x}^3-2 \mathrm{x}+3 ;$ Only one change in the sign.
$\therefore$ One negative root.
Question 10.
The number of positive zeros of the polynomial $\sum_{j=0}^n{ }^n C_r(-1)^r x^r$
(a) 0
(b) $\mathrm{n}$
(c) $<$ n
(d) $\mathrm{r}$
Answer:
(b) $\mathrm{n}$