2. Consider $f(x)=x^{3}-4x^{2}-17x+60$ and $g(x)=x^{3}+5x^{2}-8x-12$, whose one of the roots are given in the set ${3,2,-3,-2}$. Choose the set of correct options regarding $f(x)$ and $g(x)$.
Answer: Option b
Solution:
From the given set, $x=3$ is one of the root of $f(x)$, since $f(3)=0$. To find the other roots:
$$
f(x)=(x-3)(x+4)(x-5)
$$$$
f(x)\text{ is positive when }x\in(-4,3)\cup(5,\infty)
$$
Thus, option d is incorrect.
$$
g(x)\text{ is negative when }x\in(-\infty,-6)\cup(-1,-2)
$$
So, option c and e are incorrect.
Numerical Answer Type (NAT)
3. Consider a rectangular box whose length is $x$, height is 3 units less than length, and breadth is 5 units more than height. If the volume of the rectangular box is 24 unit$^{3}$, then choose the correct value of $x$ from the given options.
Answer: Option b (which is 4, as seen from the solution)
Solution:
Let the length of rectangular box be $x$, then the height is $h=x-3$ and breadth is $b=h+5=x-3+5=x+2$.
The volume of rectangular box is:
$$
V=x(x-3)(x+2)=x^{3}-x^{2}-6x
$$
Given that volume is 24:
$$
x^{3}-x^{2}-6x=24
$$
From the given options, option b (which is 4) satisfies the equation.
Multiple Select Question (MSQ)
4. Consider a polynomial function $f(x)$ of degree 4 which intersects the X-axis at $x=2, x=-3$ and $x=-4$. Moreover, $f(x)<0$ when $x\in(1,2)$, and $f(x)>0$ when $x\in(-1,1)$. Find out the equation of the polynomial.
Answer: Option b
$$
a(x^{4}+4x^{3}-7x^{2}-22x+24), a>0
$$
Solution:
Given that $f(x)$ has degree 4 and intersects the X-axis at $x=2, x=-3, x=-4$. Also, $f(x)<0$ when $x\in(1,2)$, and $f(x)>0$ when $x\in(-1,1)$. This means when $x=1, f(x)=0$.
5. Consider a polynomial function $f(x)=\frac{-1}{300}(x-2)^{2}(x-3)(x+1)^{2}(x+4)(x-5)$. Choose the correct set of options.
Answer: Option b, c, d and e
Solution:
From the above graph,
$f(x)$ has exactly 6 turning points. So, option b is correct.
In the interval $x\in(3,5), f(x)$ is increasing first and then decreasing. So, option d is correct.
The function $f(x)$ is negative when $x\in(-1,2)$. So, option e is correct.
Also, if we replace $x$ by $-x$ in $f(x)$, then $f(-x) \neq -f(x)$ or $f(-x) \neq f(x)$. So, $f(x)$ is neither even nor odd function.
6. Consider a polynomial function $P(x)=(x^{4}+4x^{3}+x+10)$ and $Q(x)=(x^{3}+2x^{2}-6)$. If $M(x)$ is the equation of the straight line passing through $(2, Q(2))$ and having slope 3, then find out the equation of $P(x)+M(x)Q(x)$.
Answer: Option A
$$
4x^{4}+14x^{3}+8x^{2}-17x-14
$$
Solution:
$M(x)$ is the equation of the straight line passing through $(2, Q(2))=(2,10)$ and having slope 3.
8. An ant named $B$, wants to climb an uneven cliff and reach its anthill (i.e., home of ant). On its way home, $B$ makes sure that it collects some food. A group of ants have reached the food locations which are at $x$-intercepts of the function $f(x)=(x^{2}-19)((x-9)^{3}-1)$. As ants secrete pheromones (a form of signals which other ants can detect and reach the food location), $B$ gets to know the food location. Then the sum of the $x$-coordinates of all the food locations is
9. The Ministry of Road Transport and Highways wants to connect three aspirational districts with two roads $r_1$ and $r_2$. Two roads are connected if they intersect. The shape of the two roads $r_1$ and $r_2$ follows polynomial curve $f(x)=(x-19)(x-17)^2$ and $g(x)=-(x-19)(x-17)$ respectively. What will be the $x$-coordinate of the third aspirational district, if the first two are at $x$-intercepts of $f(x)$ and $g(x)$.
Hence, the third aspirational district is at $x=16$.
10. Let $r(x)$ be a polynomial function which is obtained as the quotient after dividing the polynomial $p(x)=(x+5)(x-3)(x^2-4)$ by the polynomial $q(x)=(x-2)(2+x)$. Choose the correct option which represents the polynomial $r(x)$ most appropriately.
