A distância entre os pontos $A = (1, 2)$ e $B = (4, 6)$ no plano cartesiano é
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

QUESTION 158
The distance between the points $A(1, 2)$ and $B(4, 6)$ is
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

The parametric equation of line $l$ is $\left\{ \begin{array} { l } x = 1 + 3 t , \\ y = 2 + 4 t \end{array} \right.$ ($t$ is the parameter). The distance from point $(1,0)$ to line $l$ is (A) $\frac { 1 } { 5 }$ (B) $\frac { 2 } { 5 }$ (C) $\frac { 4 } { 5 }$ (D) $\frac { 6 } { 5 }$

The maximum distance from the point $( 0 , - 1 )$ to the line $y = k ( x + 1 )$ is
A. 1
B. $\sqrt { 2 }$
C. $\sqrt { 3 }$
D. 2

Let $P_0 = (0,0)$, $P_1 = (0,4)$, $P_2 = (4,0)$, $P_3 = (-4,-4)$, $P_4 = (2,4)$, $P_5 = (4,6)$ (or similar points). Find the region of all points closer to $P_0$ than to any of $P_1, P_2, P_3, P_4, P_5$, and compute its perimeter.

Let $K$ be the set of all points $(x , y)$ such that $| x | + | y | \leq 1$. Given a point $A$ in the plane, let $F _ { A }$ be the point in $K$ which is closest to $A$. Then the points $A$ for which $F _ { A } = ( 1,0 )$ are
(A) all points $A = ( x , y )$ with $x \geq 1$.
(B) all points $A = ( x , y )$ with $x \geq y + 1$ and $x \geq 1 - y$.
(C) all points $A = ( x , y )$ with $x \geq 1$ and $y = 0$.
(D) all points $A = ( x , y )$ with $x \geq 0$ and $y = 0$.

41. The equation of a plane passing through the line of intersection of the planes $x + 2 y + 3 z = 2$ and $x - y + z = 3$ and at a distance $\frac { 2 } { \sqrt { 3 } }$ from the point $( 3,1 , - 1 )$ is
(A) $5 x - 11 y + z = 17$
(B) $\sqrt { 2 } x + y = 3 \sqrt { 2 } - 1$
(C) $x + y + z = \sqrt { 3 }$
(D) $x - \sqrt { 2 } y = 1 - \sqrt { 2 }$

ANSWER : A

1. Let $P Q R$ be a triangle of area $\triangle$ with $a = 2 , b = \frac { 7 } { 2 }$ and $c = \frac { 5 } { 2 }$, where $a , b$ and $c$ are the lengths of the sides of the triangle opposite to the angles at $P , Q$ and $R$ respectively. Then $\frac { 2 \sin P - \sin 2 P } { 2 \sin P + \sin 2 P }$ equals
   (A) $\frac { 3 } { 4 \Delta }$
   (B) $\frac { 45 } { 4 \Delta }$
   (C) $\left( \frac { 3 } { 4 \Delta } \right) ^ { 2 }$
   (D) $\left( \frac { 45 } { 4 \Delta } \right) ^ { 2 }$

ANSWER : C

1. If $\vec { a }$ and $\vec { b }$ are vectors such that $| \vec { a } + \vec { b } | = \sqrt { 29 }$ and $\vec { a } \times ( 2 \hat { i } + 3 \hat { j } + 4 \hat { k } ) = ( 2 \hat { i } + 3 \hat { j } + 4 \hat { k } ) \times \vec { b }$, then a possible value of $( \vec { a } + \vec { b } ) \cdot ( - 7 \hat { i } + 2 \hat { j } + 3 \hat { k } )$ is
   (A) 0
   (B) 3
   (C) 4
   (D) 8
2. If $P$ is a $3 \times 3$ matrix such that $P ^ { T } = 2 P + I$, where $P ^ { T }$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $X = \left[ \begin{array} { c } x \\ y \\ z \end{array} \right] \neq \left[ \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right]$ such that
   (A) $P X = \left[ \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right]$
   (B) $P X = X$
   (C) $P X = 2 X$
   (D) $P X = - X$

ANSWER : D

1. Let $\alpha$ (a) and $\beta$ (a) be the roots of the equation $( \sqrt [ 3 ] { 1 + a } - 1 ) x ^ { 2 } + ( \sqrt { 1 + a } - 1 ) x + ( \sqrt [ 6 ] { 1 + a } - 1 ) = 0$ where $a > - 1$. Then $\lim _ { a \rightarrow 0 ^ { + } } \alpha ( a )$ and $\lim _ { a \rightarrow 0 ^ { + } } \beta ( a )$ are
   (A) $- \frac { 5 } { 2 }$ and 1
   (B) $- \frac { 1 } { 2 }$ and - 1
   (C) $- \frac { 7 } { 2 }$ and 2
   (D) $- \frac { 9 } { 2 }$ and 3

ANSWER : B

1. Four fair dice $D _ { 1 } , D _ { 2 } , D _ { 3 }$ and $D _ { 4 }$, each having six faces numbered $1,2,3,4,5$ and 6 , are rolled simultaneously. The probability that $D _ { 4 }$ shows a number appearing on one of $D _ { 1 } , D _ { 2 }$ and $D _ { 3 }$ is
   (A) $\frac { 91 } { 216 }$
   (B) $\frac { 108 } { 216 }$
   (C) $\frac { 125 } { 216 }$
   (D) $\frac { 127 } { 216 }$

ANSWER : A

1. The value of the integral $\int _ { - \pi / 2 } ^ { \pi / 2 } \left( x ^ { 2 } + \ln \frac { \pi + x } { \pi - x } \right) \cos x \mathrm {~d} x$ is
   (A) 0
   (B) $\frac { \pi ^ { 2 } } { 2 } - 4$
   (C) $\frac { \pi ^ { 2 } } { 2 } + 4$
   (D) $\frac { \pi ^ { 2 } } { 2 }$

ANSWER : B

1. Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be in harmonic progression with $a _ { 1 } = 5$ and $a _ { 20 } = 25$. The least positive integer $n$ for which $a _ { n } < 0$ is
   (A) 22
   (B) 23
   (C) 24
   (D) 25

SECTION II : Paragraph Type

This section contains $\mathbf { 6 }$ multiple choice questions relating to three paragraphs with two questions on each paragraph. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Paragraph for Questions 49 and 50

Let $a _ { n }$ denote the number of all $n$-digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0 . Let $b _ { n } =$ the number of such $n$-digit integers ending with digit 1 and $c _ { n } =$ the number of such $n$-digit integers ending with digit 0 .

For $a > b > c > 0$, the distance between $( 1,1 )$ and the point of intersection of the lines $a x + b y + c = 0$ and $b x + a y + c = 0$ is less than $2 \sqrt { 2 }$. Then
(A) $a + b - c > 0$
(B) $a - b + c < 0$
(C) $a - b + c > 0$
(D) $a + b - c < 0$

If the point $(1, a)$ lies between the straight lines $x + y = 1$ and $2(x+y) = 3$ then $a$ lies in interval
(1) $\left(\frac{3}{2}, \infty\right)$
(2) $\left(1, \frac{3}{2}\right)$
(3) $(-\infty, 0)$
(4) $\left(0, \frac{1}{2}\right)$

If a line $L$ is perpendicular to the line $5 x - y = 1$, and the area of the triangle formed by the line $L$ and the coordinate axes is 5 sq units, then the distance of the line $L$ from the line $x + 5 y = 0$ is
(1) $\frac { 7 } { \sqrt { 13 } }$ units
(2) $\frac { 7 } { \sqrt { 5 } }$ units
(3) $\frac { 5 } { \sqrt { 13 } }$ units
(4) $\frac { 5 } { \sqrt { 7 } }$ units

A point on the straight line, $3x + 5y = 15$ which is equidistant from the coordinate axes will lie only in:
(1) $1^{\text{st}}$ and $2^{\text{nd}}$ quadrants
(2) $1^{\text{st}}$, $2^{\text{nd}}$ and $4^{\text{th}}$ quadrants
(3) $1^{\text{st}}$ quadrant
(4) $4^{\text{th}}$ quadrant

If the two lines $x + ( a - 1 ) y = 1$ and $2 x + a ^ { 2 } y = 1 , ( a \in R - \{ 0,1 \} )$ are perpendicular, then the distance of their point of intersection from the origin is
(1) $\frac { 2 } { \sqrt { 5 } }$
(2) $\frac { \sqrt { 2 } } { 5 }$
(3) $\frac { 2 } { 5 }$
(4) $\sqrt { \frac { 2 } { 5 } }$

If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines, $x \operatorname { cosec } \alpha - y \sec \alpha = k \cot 2 \alpha$ and $x \sin \alpha + y \cos \alpha = k \sin 2 \alpha$ respectively, then $k ^ { 2 }$ is equal to $:$
(1) $2 p ^ { 2 } + q ^ { 2 }$
(2) $p ^ { 2 } + 2 q ^ { 2 }$
(3) $4 q ^ { 2 } + p ^ { 2 }$
(4) $4 p ^ { 2 } + q ^ { 2 }$

Let the point $P(\alpha, \beta)$ be at a unit distance from each of the two lines $L_1: 3x - 4y + 12 = 0$, and $L_2: 8x + 6y + 11 = 0$. If $P$ lies below $L_1$ and above $L_2$, then $100(\alpha + \beta)$ is equal to
(1) $-14$
(2) 42
(3) $-22$
(4) 14

The distance of the point $( 6 , - 2 \sqrt { 2 } )$ from the common tangent $y = m x + c , m > 0$, of the curves $x = 2 y ^ { 2 }$ and $x = 1 + y ^ { 2 }$ is
(1) $\frac { 1 } { 3 }$
(2) 5
(3) $\frac { 14 } { 3 }$
(4) $5 \sqrt { 3 }$

Let $A$ be the point of intersection of the lines $3 x + 2 y = 14,5 x - y = 6$ and $B$ be the point of intersection of the lines $4 x + 3 y = 8,6 x + y = 5$. The distance of the point $P ( 5 , - 2 )$ from the line $A B$ is
(1) $\frac { 13 } { 2 }$
(2) 8
(3) $\frac { 5 } { 2 }$
(4) 6

The distance of the point $( 2,3 )$ from the line $2 x - 3 y + 28 = 0$, measured parallel to the line $\sqrt { 3 } x - y + 1 = 0$, is equal to
(1) $4 \sqrt { 2 }$
(2) $6 \sqrt { 3 }$
(3) $3 + 4 \sqrt { 2 }$
(4) $4 + 6 \sqrt { 3 }$

Let $A(a, b)$, $B(3, 4)$ and $(-6, -8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2a+3, 7b+5)$ from the line $2x + 3y - 4 = 0$ measured parallel to the line $x - 2y - 1 = 0$ is
(1) $\dfrac{15\sqrt{5}}{7}$
(2) $\dfrac{17\sqrt{5}}{6}$
(3) $\dfrac{17\sqrt{5}}{7}$
(4) $\dfrac{\sqrt{5}}{17}$

Let $\mathrm { A } , \mathrm { B } , \mathrm { C }$ be three points in $xy$-plane, whose position vector are given by $\sqrt { 3 } \hat { i } + \hat { j } , \hat { i } + \sqrt { 3 } \hat { j }$ and $\mathrm { a } \hat { i } + ( 1 - \mathrm { a } ) \hat { j }$ respectively with respect to the origin O. If the distance of the point C from the line bisecting the angle between the vectors $\overrightarrow { \mathrm { OA } }$ and $\overrightarrow { \mathrm { OB } }$ is $\frac { 9 } { \sqrt { 2 } }$, then the sum of all the possible values of $a$ is :
(1) 2
(2) $9/2$
(3) 1
(4) 0

On a coordinate plane, there are two points $A ( - 3,4 ) , B ( 3,2 )$ and a line $L$. Points $A$ and $B$ are on opposite sides of line $L$, and $\vec { n } = ( 4 , - 3 )$ is a normal vector to line $L$. The distance from point $A$ to line $L$ is 5 times the distance from point $B$ to line $L$. Based on the above, answer the following questions.
(1) Find the dot product of vector $\overrightarrow { A B }$ and vector $\vec { n }$. (4 points)
(2) Find the equation of line $L$. (4 points)
(3) Point $P$ is on line $L$ and $\overline { P A } = \overline { P B }$. Find the coordinates of point $P$. (4 points)

There is a shooting game with the launcher placed at the origin of a coordinate plane and three circular target disks with radius 1, centered at $(2,2)$, $(4,6)$, and $(8,1)$ respectively. A player selects a positive number $a$ and presses a button. The launcher then fires a laser beam in the direction of point $(1, a)$ (forming a ray). Assume the laser beam can penetrate through the target after hitting it and continue in the original direction (grazing the edge of the disk also counts as a hit). Select the correct options.
(1) The laser beam lies on a line passing through the origin with slope $a$
(2) If $a = \frac{3}{2}$, the laser beam will hit the disk centered at $(4,6)$
(3) The player can hit all three disks with just one laser beam
(4) The player needs to fire at least three laser beams to hit all three disks
(5) If the player fires one laser beam and hits the disk centered at $(8,1)$, then $a \leq \frac{16}{63}$

In the figure below, three points indicating the positions of apple, pear, and walnut trees in a garden located between a main street and a side street that intersect perpendicularly with each other and have straight edges are shown.
Of the trees in this garden, the one closest to the main street is the apple tree, and the one farthest is the pear tree.
Accordingly, which of the following is the correct ordering from the tree closest to the side street to the one farthest?
A) Pear - Walnut - Apple
B) Pear - Apple - Walnut
C) Walnut - Pear - Apple
D) Apple - Pear - Walnut
E) Apple - Walnut - Pear