124. Suppose $A$ and $B$ are the extreme points of $f(x) = 2x^3 - 3x^2 - 12x + 1$. How many points on the curve $f$ have a tangent line parallel to line $AB$?
(1) zero (2) $1$ (3) $2$ (4) $3$
%% Page 6 121-A Mathematics Page 5

120- At the intersection points of the curves $f(x) = \sin x + \dfrac{1}{2}\cos x$ and $g(x) = \dfrac{3}{2}\sin x$ on the interval $[0, \pi]$, a tangent line to the curve $f(x)$ is drawn. This tangent line intersects the $x$-axis at which interval?
(1) $\dfrac{\pi}{4} - 1$ (2) $\dfrac{\pi}{4} - 2$ (3) $\dfrac{\pi}{4} + \dfrac{1}{\lambda}$ (4) $\dfrac{\pi}{4} + \dfrac{3}{\lambda}$
%% Page 6 121-- Function $f$ is differentiable and periodic with period 5. If $f'(-1)=\dfrac{3}{2}$ and $g(x)=f(x+1)+f(3x+10)$, then $g'(-2)$ is which of the following?
(1) $3$ (2) $\dfrac{7}{2}$ (3) $6$ (4) $\dfrac{13}{2}$

18 -- Line $d$ is tangent to the parabola $y = x^2 + 1$, cuts the $x$-axis at two points, and the tangent lines drawn at those two points are perpendicular to each other. What are the coordinates of the $x$-intercept of line $d$?
(1) $1/25$ (2) $3/25$ (3) $\circ/75$ (4) $2/75$

If $x^{2/3} + y^{2/3} = a^{1/3}$, find the equation of the tangent to the curve at a point, and show that the length of the tangent intercepted between the axes is constant.

3) Considering the similarity of the right triangles ACL and ALM in Figure 4, and recalling the geometric meaning of the derivative, verify that the value of the ordinate $d$ of the centre of the wheel remains constant during motion. Therefore, it seems to the cyclist that they are moving on a flat surface.
[Figure]
Figure 4
\footnotetext{${ } ^ { 1 }$ In general, the length of the arc of a curve with equation $y = \varphi ( x )$ between the abscissae $x _ { 1 }$ and $x _ { 2 }$ is given by $\int _ { x _ { 1 } } ^ { x _ { 2 } } \sqrt { 1 + \left( \varphi ^ { \prime } ( x ) \right) ^ { 2 } } d x$. }

Ministry of Education, University and Research

The graph of the function:
$$f ( x ) = \frac { 2 } { \sqrt { 3 } } - \frac { e ^ { x } + e ^ { - x } } { 2 } , \quad \text { for } x \in \left[ - \frac { \ln ( 3 ) } { 2 } ; \frac { \ln ( 3 ) } { 2 } \right]$$
if replicated several times, can also represent the profile of a platform suitable for being traversed by a bicycle with very particular wheels, having the shape of a regular polygon.

Determine the equation of the tangent line to the curve with equation $y = \sqrt{25 - x^{2}}$ at its point with abscissa 3, using two different methods.

4. Given a function $g$, differentiable on $\mathbb { R }$ and such that $g \left( \frac { \pi } { 4 } \right) = g ^ { \prime } \left( \frac { \pi } { 4 } \right) = 2$, determine the equation of the normal line to the curve $y = g ( x ) \sin ^ { 2 } x$ at its point with abscissa $\frac { \pi } { 4 }$.

25. If the vectors $\vec { a } , \vec { b }$ and $\vec { c }$ form the sides $B C , C A$ and $A B$ respectively of $a$ triangle ABC , then:
(A) $\vec { a } , \vec { b } + \vec { b } , \vec { c } + \vec { c } , \vec { a } = 0$
(B) $\vec { a } \times \vec { b } = \vec { b } \times \vec { c } = \vec { c } \times \vec { a }$
(C) $\vec { a } , \vec { b } = \vec { b } , \vec { c } = \vec { c } , \vec { a }$
(D) $\vec { a } \times \vec { b } + \vec { b } \times \vec { c } + \vec { c } \times \vec { a } = 0$

26. If the normal to the curve $y = f ( x )$ at the point ( 3,4 ) makes an angle $3 p / 4$ with the positive $x$-axis then $\mathrm { f } ^ { \prime } ( 3 ) =$
(A) - 1
(B) $- 3 / 4$
(C) $4 / 3$
(D) 1

24. The point (s) on the curve $y ^ { 3 } + 3 x ^ { 2 } = 12 y$ where the tangent is vertical, is (are)
(A) $\quad ( \pm 4 / \sqrt { } 3 , - 2 )$
(B) $\quad ( \pm \sqrt { } 11 / 3 , - 0 )$
(C) $( 0,0 )$
(D) $\quad ( \pm 4 / \sqrt { } 3,2 )$

25. The equation of the common tangents to the curves $y ^ { 2 } = 8 x$ and $x y = - 1$ is
(A) $\quad 3 y = 9 x + 2$
(B) $\quad y = 2 x + 1$
(C) $\quad 2 y = x + 8$
(D) $\quad y = x + 2$

26. If $y = f ( x )$ and $y \cos x + x \cos y = \Pi$, then the value of $f ^ { \prime } ( 0 )$ is :
(a) $\sqcap$
(b) $- \Pi$
(c) 0
(d) $2 \Pi$

The tangent to the curve $y = e^x$ drawn at the point $(c, e^c)$ intersects the line joining the points $(c-1, e^{c-1})$ and $(c+1, e^{c+1})$
(A) on the left of $x = c$
(B) on the right of $x = c$
(C) at no point
(D) at all points

The tangent $PT$ and the normal $PN$ to the parabola $y^{2}=4ax$ at a point $P$ on it meet its axis at points $T$ and $N$, respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose
(A) vertex is $\left(\frac{2a}{3},0\right)$
(B) directrix is $x=0$
(C) latus rectum is $\frac{2a}{3}$
(D) focus is $(a,0)$

If $$f ( x ) = \begin{cases} - x - \frac { \pi } { 2 } , & x \leq - \frac { \pi } { 2 } \\ - \cos x , & - \frac { \pi } { 2 } < x \leq 0 \\ x - 1 , & 0 < x \leq 1 \\ \ln x , & x > 1 , \end{cases}$$ then
(A) $f ( x )$ is continuous at $x = -\frac{\pi}{2}$
(B) $f ( x )$ is not differentiable at $x = 0$
(C) $f ( x )$ is differentiable at $x = 1$
(D) $f ( x )$ is differentiable at $x = -\frac{3}{2}$

The slope of the tangent to the curve $\left(y - x^5\right)^2 = x\left(1 + x^2\right)^2$ at the point $(1, 3)$ is

The equation of the normal to the parabola, $x ^ { 2 } = 8 y$ at $x = 4$ is
(1) $x + 2 y = 0$
(2) $x + y = 2$
(3) $x - 2 y = 0$
(4) $x + y = 6$

The intercepts on the $x$-axis made by tangents to the curve, $y = \int_0^x |t|\, dt, x \in R$, which are parallel to the line $y = 2x$, are equal to
(1) $\pm 3$
(2) $\pm 4$
(3) $\pm 1$
(4) $\pm 2$

If the function $g(x) = \begin{cases} k\sqrt{x+1}, & 0 \leq x \leq 3 \\ mx + 2, & 3 < x \leq 5 \end{cases}$ is differentiable, then the value of $k + m$ is:
(1) $2$
(2) $\frac{16}{5}$
(3) $\frac{10}{3}$
(4) $4$

Let C be a curve given by $y ( x ) = 1 + \sqrt { 4 x - 3 } , x > \frac { 3 } { 4 }$. If $P$ is a point on C , such that the tangent at $P$ has slope $\frac { 2 } { 3 }$, then a point through which the normal at $P$ passes, is :
(1) $( 1,7 )$
(2) $( 3 , - 4 )$
(3) $( 4 , - 3 )$
(4) $( 2,3 )$

Consider $f(x) = \tan^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right)$, $x \in \left(0, \frac{\pi}{2}\right)$. A normal to $y = f(x)$ at $x = \frac{\pi}{6}$ also passes through the point:
(1) $(0, 0)$
(2) $\left(0, \frac{2\pi}{3}\right)$
(3) $\left(\frac{\pi}{6}, 0\right)$
(4) $\left(\frac{\pi}{4}, 0\right)$

Consider $f(x) = \tan^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right)$, $x \in \left(0, \frac{\pi}{2}\right)$. A normal to $y = f(x)$ at $x = \frac{\pi}{6}$ also passes through the point: (1) $(0, 0)$ (2) $\left(0, \frac{2\pi}{3}\right)$ (3) $\left(\frac{\pi}{6}, 0\right)$ (4) $\left(\frac{\pi}{4}, 0\right)$

The normal to the curve $y ( x - 2 ) ( x - 3 ) = x + 6$ at the point where the curve intersects the $y$-axis passes through the point:
(1) $\left( \frac { 1 } { 2 } , - \frac { 1 } { 3 } \right)$
(2) $\left( \frac { 1 } { 2 } , \frac { 1 } { 3 } \right)$
(3) $\left( - \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$
(4) $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$

The eccentricity of an ellipse whose centre is at the origin is $\frac { 1 } { 2 }$. If one of its directrices is $x = - 4$, then the equation of the normal to it at $\left( 1 , \frac { 3 } { 2 } \right)$ is:
(1) $4 x - 2 y = 1$
(2) $4 x + 2 y = 7$
(3) $x + 2 y = 4$
(4) $2 y - x = 2$

The normal to the curve $y(x - 2)(x - 3) = x + 6$ at the point where the curve intersects the $y$-axis passes through the point:
(1) $\left(-\dfrac{1}{2}, -\dfrac{1}{2}\right)$
(2) $\left(\dfrac{1}{2}, \dfrac{1}{2}\right)$
(3) $\left(\dfrac{1}{2}, -\dfrac{1}{3}\right)$
(4) $\left(\dfrac{1}{2}, \dfrac{1}{3}\right)$