jee-advanced 2007 Q50

jee-advanced · India · paper2 Vectors Introduction & 2D Vector Properties and Identities (Conceptual)
50. Let $\vec { a } , \vec { b } , \vec { c }$ be unit vectors such that $\vec { a } + \vec { b } + \vec { c } = \overrightarrow { 0 }$. Which one of the following is correct?
(A) $\vec { a } \times \vec { b } = \vec { b } \times \vec { c } = \vec { c } \times \vec { a } = \overrightarrow { 0 }$
(B) $\vec { a } \times \vec { b } = \vec { b } \times \vec { c } = \vec { c } \times \vec { a } \neq \overrightarrow { 0 }$
(C) $\vec { a } \times \vec { b } = \vec { b } \times \vec { c } = \vec { a } \times \vec { c } \neq \overrightarrow { 0 }$
(D) $\vec { a } \times \vec { b } , \vec { b } \times \vec { c } , \vec { c } \times \vec { a }$ are mutually perpendicular
Answer [Figure] [Figure] ◯ ◯
(A)
(B)
(C)
(D) 51. Let $A B C D$ be a quadrilateral with area 18, with side $A B$ parallel to the side $C D$ and $A B = 2 C D$. Let $A D$ be perpendicular to $A B$ and $C D$. If a circle is drawn inside the quadrilateral $A B C D$ touching all the sides, then its radius is
(A) 3
(B) 2
(C) $\frac { 3 } { 2 }$
(D) 1
Answer [Figure] [Figure] [Figure] [Figure]
(A)
(B)
(C)
(D) 52. Let $f ( x ) = \frac { x } { \left( 1 + x ^ { n } \right) ^ { 1 / n } }$ for $n \geq 2$ and $g ( x ) = \underbrace { f \circ f \circ \cdots \circ f ) } _ { f \text { occurs } n \text { times } } ( x )$. Then $\int x ^ { n - 2 } g ( x ) d x$ equals
(A) $\frac { 1 } { n ( n - 1 ) } \left( 1 + n x ^ { n } \right) ^ { 1 - \frac { 1 } { n } } + K$
(B) $\frac { 1 } { n - 1 } \left( 1 + n x ^ { n } \right) ^ { 1 - \frac { 1 } { n } } + K$
(C) $\frac { 1 } { n ( n + 1 ) } \left( 1 + n x ^ { n } \right) ^ { 1 + \frac { 1 } { n } } + K$
(D) $\frac { 1 } { n + 1 } \left( 1 + n x ^ { n } \right) ^ { 1 + \frac { 1 } { n } } + K$ Answer [Figure] ◯ ◯
(A)
(B)
(C)
(D) 53. The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is
(A) 360
(B) 192
(C) 96
(D) 48
Answer [Figure]
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) 54. Consider the planes $3 x - 6 y - 2 z = 15$ and $2 x + y - 2 z = 5$.
STATEMENT-1 : The parametric equations of the line of intersection of the given planes are $x = 3 + 14 t , y = 1 + 2 t , z = 15 t$.
because
STATEMENT-2 : The vector $14 \hat { i } + 2 \hat { j } + 15 \hat { k }$ is parallel to the line of intersection of given planes.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
Answer
[Figure]
(A)
(B)
(C)
(D) 55. STATEMENT-1 : The curve $y = \frac { - x ^ { 2 } } { 2 } + x + 1$ is symmetric with respect to the line $x = 1$. because STATEMENT-2 : A parabola is symmetric about its axis.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True Answer
(A) [Figure] [Figure] [Figure]
(A)
(B)
(C)
(D) 56. Let $f ( x ) = 2 + \cos x$ for all real $x$.
STATEMENT-1 : For each real $t$, there exists a point $c$ in $[ t , t + \pi ]$ such that $f ^ { \prime } ( c ) = 0$. because STATEMENT-2 : $f ( t ) = f ( t + 2 \pi )$ for each real $t$.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
Answer
[Figure]
(A)
[Figure]
(B)
[Figure]
(C)
◯
(D) 57. Lines $L _ { 1 } : y - x = 0$ and $L _ { 2 } : 2 x + y = 0$ intersect the line $L _ { 3 } : y + 2 = 0$ at $P$ and $Q$, respectively. The bisector of the acute angle between $L _ { 1 }$ and $L _ { 2 }$ intersects $L _ { 3 }$ at $R$.
STATEMENT-1 : The ratio $P R : R Q$ equals $2 \sqrt { 2 } : \sqrt { 5 }$.
because
STATEMENT-2 : In any triangle, bisector of an angle divides the triangle into two similar triangles.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True Answer
[Figure]
(A)
[Figure]
(B)
(B)
[Figure]
(D)
  1. Which one of the following statements is correct?
    (A) $G _ { 1 } > G _ { 2 } > G _ { 3 } > \cdots$
    (B) $G _ { 1 } < G _ { 2 } < G _ { 3 } < \cdots$
    (C) $G _ { 1 } = G _ { 2 } = G _ { 3 } = \cdots$
    (D) $G _ { 1 } < G _ { 3 } < G _ { 5 } < \cdots$ and $G _ { 2 } > G _ { 4 } > G _ { 6 } > \cdots$ Answer ◯
    (A) [Figure]
    (B) [Figure]
    (C) ◯
    (D)
  2. Which one of the following statements is correct?
    (A) $A _ { 1 } > A _ { 2 } > A _ { 3 } > \cdots$
    (B) $A _ { 1 } < A _ { 2 } < A _ { 3 } < \cdots$
    (C) $A _ { 1 } > A _ { 3 } > A _ { 5 } > \cdots$ and $A _ { 2 } < A _ { 4 } < A _ { 6 } < \cdots$
    (D) $A _ { 1 } < A _ { 3 } < A _ { 5 } < \cdots$ and $A _ { 2 } > A _ { 4 } > A _ { 6 } > \cdots$

Answer
[Figure] ◯ ◯
(A)
(B)
(C)
(D) 60. Which one of the following statements is correct?
(A) $H _ { 1 } > H _ { 2 } > H _ { 3 } > \cdots$
(B) $H _ { 1 } < H _ { 2 } < H _ { 3 } < \cdots$
(C) $H _ { 1 } > H _ { 3 } > H _ { 5 } > \cdots$ and $H _ { 2 } < H _ { 4 } < H _ { 6 } < \cdots$
(D) $H _ { 1 } < H _ { 3 } < H _ { 5 } < \cdots$ and $H _ { 2 } > H _ { 4 } > H _ { 6 } > \cdots$
M61-63: Paragraph for Question Nos. 61 to 63
If a continuous function $f$ defined on the real line $\mathbf { R }$, assumes positive and negative values in $\mathbf { R }$ then the equation $f ( x ) = 0$ has a root in $\mathbf { R }$. For example, if it is known that a continuous function $f$ on $\mathbf { R }$ is positive at some point and its minimum value is negative then the equation $f ( x ) = 0$ has a root in $\mathbf { R }$. Consider $f ( x ) = k e ^ { x } - x$ for all real $x$ where $k$ is a real constant. Answer ◯
(A) [Figure]
(B) O
(C) [Figure]
(D) 61. The line $y = x$ meets $y = k e ^ { x }$ for $k \leq 0$ at
(A) no point
(B) one point
(C) two points
(D) more than two points
Answer
[Figure]
(A)
[Figure]
(B)
[Figure]
(C)
[Figure]
(D)
  1. The positive value of $k$ for which $k e ^ { x } - x = 0$ has only one root is
    (A) $\frac { 1 } { e }$
    (B) 1
    (C) $e$
    (D) $\log _ { e } 2$

Answer [Figure] [Figure] [Figure]
(A)
(B)
(C)
(D) 63. For $k > 0$, the set of all values of $k$ for which $k e ^ { x } - x = 0$ has two distinct roots is
(A) $\left( 0 , \frac { 1 } { e } \right)$
(B) $\left( \frac { 1 } { e } , 1 \right)$
(C) $\left( \frac { 1 } { e } , \infty \right)$
(D) $( 0,1 )$
Answer ◯ ◯ ◯
(A)
(B)
(C)
(D) 64. Let $f ( x ) = \frac { x ^ { 2 } - 6 x + 5 } { x ^ { 2 } - 5 x + 6 }$.
Match the expressions/statements in Column I with expressions/statements in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.
Column I
(A) If $- 1 < x < 1$, then $f ( x )$ satisfies
(B) If $1 < x < 2$, then $f ( x )$ satisfies
(C) If $3 < x < 5$, then $f ( x )$ satisfies
(D) If $x > 5$, then $f ( x )$ satisfies
Column II
(p) $0 < f ( x ) < 1$
(q) $f ( x ) < 0$
(r) $f ( x ) > 0$
(s) $f ( x ) < 1$
Answer [Figure] 65. Let $( x , y )$ be such that
$$\sin ^ { - 1 } ( a x ) + \cos ^ { - 1 } ( y ) + \cos ^ { - 1 } ( b x y ) = \frac { \pi } { 2 }$$
Match the statements in Column I with statements in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.
Column I
(A) If $a = 1$ and $b = 0$, then $( x , y )$
(B) If $a = 1$ and $b = 1$, then $( x , y )$
(C) If $a = 1$ and $b = 2$, then $( x , y )$
(D) If $a = 2$ and $b = 2$, then $( x , y )$
(p) lies on the circle $x ^ { 2 } + y ^ { 2 } = 1$
(q) lies on $\left( x ^ { 2 } - 1 \right) \left( y ^ { 2 } - 1 \right) = 0$
(r) lies on $y = x$
(s) lies on $\left( 4 x ^ { 2 } - 1 \right) \left( y ^ { 2 } - 1 \right) = 0$
Column II
  1. Match the statements in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

Column I
(A) Two intersecting circles
(B) Two mutually external circles
(C) Two circles, one strictly inside the other
(D) Two branches of a hyperbola
Column II
(p) have a common tangent
(q) have a common normal
(r) do not have a common tangent
(s) do not have a common normal
Answer [Figure]
Let $\alpha , \beta$ be the roots of the equation $\mathrm { x } ^ { 2 } - \mathrm { px } + \mathrm { r } = 0$ and $\frac { \alpha } { 2 } , 2 \beta$ be the roots of the equation $\mathrm { x } ^ { 2 } - \mathrm { qx } + \mathrm { r } = 0$. Then the value of $r$ is 
50. Let $\vec { a } , \vec { b } , \vec { c }$ be unit vectors such that $\vec { a } + \vec { b } + \vec { c } = \overrightarrow { 0 }$. Which one of the following is correct?\\
(A) $\vec { a } \times \vec { b } = \vec { b } \times \vec { c } = \vec { c } \times \vec { a } = \overrightarrow { 0 }$\\
(B) $\vec { a } \times \vec { b } = \vec { b } \times \vec { c } = \vec { c } \times \vec { a } \neq \overrightarrow { 0 }$\\
(C) $\vec { a } \times \vec { b } = \vec { b } \times \vec { c } = \vec { a } \times \vec { c } \neq \overrightarrow { 0 }$\\
(D) $\vec { a } \times \vec { b } , \vec { b } \times \vec { c } , \vec { c } \times \vec { a }$ are mutually perpendicular

Answer\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-21_93_93_1463_335}\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-21_128_122_1439_474}\\
◯\\
◯\\
(A)\\
(B)\\
(C)\\
(D)\\
51. Let $A B C D$ be a quadrilateral with area 18, with side $A B$ parallel to the side $C D$ and $A B = 2 C D$. Let $A D$ be perpendicular to $A B$ and $C D$. If a circle is drawn inside the quadrilateral $A B C D$ touching all the sides, then its radius is\\
(A) 3\\
(B) 2\\
(C) $\frac { 3 } { 2 }$\\
(D) 1

Answer\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-21_102_99_2058_333}\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-21_100_100_2058_489}\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-21_110_104_2058_640}\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-21_93_94_2062_794}\\
(A)\\
(B)\\
(C)\\
(D)\\
52. Let $f ( x ) = \frac { x } { \left( 1 + x ^ { n } \right) ^ { 1 / n } }$ for $n \geq 2$ and $g ( x ) = \underbrace { f \circ f \circ \cdots \circ f ) } _ { f \text { occurs } n \text { times } } ( x )$. Then $\int x ^ { n - 2 } g ( x ) d x$ equals\\
(A) $\frac { 1 } { n ( n - 1 ) } \left( 1 + n x ^ { n } \right) ^ { 1 - \frac { 1 } { n } } + K$\\
(B) $\frac { 1 } { n - 1 } \left( 1 + n x ^ { n } \right) ^ { 1 - \frac { 1 } { n } } + K$\\
(C) $\frac { 1 } { n ( n + 1 ) } \left( 1 + n x ^ { n } \right) ^ { 1 + \frac { 1 } { n } } + K$\\
(D) $\frac { 1 } { n + 1 } \left( 1 + n x ^ { n } \right) ^ { 1 + \frac { 1 } { n } } + K$\\
Answer\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-22_95_104_638_485}\\
◯ ◯\\
(A)\\
(B)\\
(C)\\
(D)\\
53. The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is\\
(A) 360\\
(B) 192\\
(C) 96\\
(D) 48

Answer\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-22_98_93_1177_335}\\
(A)\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-22_93_93_1177_494}\\
(B)\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-22_98_98_1177_644}\\
(C)\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-22_91_91_1179_797}\\
(D)\\
54. Consider the planes $3 x - 6 y - 2 z = 15$ and $2 x + y - 2 z = 5$.

STATEMENT-1 : The parametric equations of the line of intersection of the given planes are $x = 3 + 14 t , y = 1 + 2 t , z = 15 t$.

\section*{because}
STATEMENT-2 : The vector $14 \hat { i } + 2 \hat { j } + 15 \hat { k }$ is parallel to the line of intersection of given planes.\\
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1\\
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1\\
(C) Statement-1 is True, Statement-2 is False\\
(D) Statement-1 is False, Statement-2 is True

\section*{Answer}
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-22_92_91_2328_337}\\
(A)\\
(B)\\
(C)\\
(D)\\
55. STATEMENT-1 : The curve $y = \frac { - x ^ { 2 } } { 2 } + x + 1$ is symmetric with respect to the line $x = 1$.\\
because\\
STATEMENT-2 : A parabola is symmetric about its axis.\\
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1\\
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1\\
(C) Statement-1 is True, Statement-2 is False\\
(D) Statement-1 is False, Statement-2 is True\\
Answer\\
(A)\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-23_91_93_913_494}\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-23_91_98_913_644}\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-23_93_94_913_794}\\
(A)\\
(B)\\
(C)\\
(D)\\
56. Let $f ( x ) = 2 + \cos x$ for all real $x$.

STATEMENT-1 : For each real $t$, there exists a point $c$ in $[ t , t + \pi ]$ such that $f ^ { \prime } ( c ) = 0$.\\
because\\
STATEMENT-2 : $f ( t ) = f ( t + 2 \pi )$ for each real $t$.\\
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1\\
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1\\
(C) Statement-1 is True, Statement-2 is False\\
(D) Statement-1 is False, Statement-2 is True

Answer

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{e79946a0-735d-438d-ac42-5265ab72284d-23_99_97_2086_335}
\captionsetup{labelformat=empty}
\caption{(A)}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{e79946a0-735d-438d-ac42-5265ab72284d-23_102_95_2083_494}
\captionsetup{labelformat=empty}
\caption{(B)}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{e79946a0-735d-438d-ac42-5265ab72284d-23_97_96_2088_646}
\captionsetup{labelformat=empty}
\caption{(C)}
\end{center}
\end{figure}

◯

(D)\\
57. Lines $L _ { 1 } : y - x = 0$ and $L _ { 2 } : 2 x + y = 0$ intersect the line $L _ { 3 } : y + 2 = 0$ at $P$ and $Q$, respectively. The bisector of the acute angle between $L _ { 1 }$ and $L _ { 2 }$ intersects $L _ { 3 }$ at $R$.

STATEMENT-1 : The ratio $P R : R Q$ equals $2 \sqrt { 2 } : \sqrt { 5 }$.

\section*{because}
STATEMENT-2 : In any triangle, bisector of an angle divides the triangle into two similar triangles.\\
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1\\
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1\\
(C) Statement-1 is True, Statement-2 is False\\
(D) Statement-1 is False, Statement-2 is True\\
Answer

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{e79946a0-735d-438d-ac42-5265ab72284d-24_81_82_1211_337}
\captionsetup{labelformat=empty}
\caption{(A)}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{e79946a0-735d-438d-ac42-5265ab72284d-24_81_89_1211_481}
\captionsetup{labelformat=empty}
\caption{(B)\\
(B)}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{e79946a0-735d-438d-ac42-5265ab72284d-24_83_85_1211_756}
\captionsetup{labelformat=empty}
\caption{(D)}
\end{center}
\end{figure}

\begin{enumerate}
  \setcounter{enumi}{57}
  \item Which one of the following statements is correct?\\
(A) $G _ { 1 } > G _ { 2 } > G _ { 3 } > \cdots$\\
(B) $G _ { 1 } < G _ { 2 } < G _ { 3 } < \cdots$\\
(C) $G _ { 1 } = G _ { 2 } = G _ { 3 } = \cdots$\\
(D) $G _ { 1 } < G _ { 3 } < G _ { 5 } < \cdots$ and $G _ { 2 } > G _ { 4 } > G _ { 6 } > \cdots$\\
Answer ◯\\
(A)\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-24_76_83_1774_476}\\
(B)\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-24_76_80_1774_614}\\
(C)\\
◯\\
(D)
  \item Which one of the following statements is correct?\\
(A) $A _ { 1 } > A _ { 2 } > A _ { 3 } > \cdots$\\
(B) $A _ { 1 } < A _ { 2 } < A _ { 3 } < \cdots$\\
(C) $A _ { 1 } > A _ { 3 } > A _ { 5 } > \cdots$ and $A _ { 2 } < A _ { 4 } < A _ { 6 } < \cdots$\\
(D) $A _ { 1 } < A _ { 3 } < A _ { 5 } < \cdots$ and $A _ { 2 } > A _ { 4 } > A _ { 6 } > \cdots$
\end{enumerate}

\section*{Answer}
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-24_85_85_2358_485}\\
◯\\
◯\\
(A)\\
(B)\\
(C)\\
(D)\\
60. Which one of the following statements is correct?\\
(A) $H _ { 1 } > H _ { 2 } > H _ { 3 } > \cdots$\\
(B) $H _ { 1 } < H _ { 2 } < H _ { 3 } < \cdots$\\
(C) $H _ { 1 } > H _ { 3 } > H _ { 5 } > \cdots$ and $H _ { 2 } < H _ { 4 } < H _ { 6 } < \cdots$\\
(D) $H _ { 1 } < H _ { 3 } < H _ { 5 } < \cdots$ and $H _ { 2 } > H _ { 4 } > H _ { 6 } > \cdots$

\section*{M61-63: Paragraph for Question Nos. 61 to 63}
If a continuous function $f$ defined on the real line $\mathbf { R }$, assumes positive and negative values in $\mathbf { R }$ then the equation $f ( x ) = 0$ has a root in $\mathbf { R }$. For example, if it is known that a continuous function $f$ on $\mathbf { R }$ is positive at some point and its minimum value is negative then the equation $f ( x ) = 0$ has a root in $\mathbf { R }$.\\
Consider $f ( x ) = k e ^ { x } - x$ for all real $x$ where $k$ is a real constant.\\
Answer ◯\\
(A)\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-25_93_97_769_494}\\
(B)\\
O\\
(C)\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-25_89_94_773_794}\\
(D)\\
61. The line $y = x$ meets $y = k e ^ { x }$ for $k \leq 0$ at\\
(A) no point\\
(B) one point\\
(C) two points\\
(D) more than two points

Answer

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{e79946a0-735d-438d-ac42-5265ab72284d-25_91_100_1276_328}
\captionsetup{labelformat=empty}
\caption{(A)}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{e79946a0-735d-438d-ac42-5265ab72284d-25_89_95_1276_494}
\captionsetup{labelformat=empty}
\caption{(B)}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{e79946a0-735d-438d-ac42-5265ab72284d-25_91_104_1276_640}
\captionsetup{labelformat=empty}
\caption{(C)}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{e79946a0-735d-438d-ac42-5265ab72284d-25_89_96_1276_794}
\captionsetup{labelformat=empty}
\caption{(D)}
\end{center}
\end{figure}

\begin{enumerate}
  \setcounter{enumi}{61}
  \item The positive value of $k$ for which $k e ^ { x } - x = 0$ has only one root is\\
(A) $\frac { 1 } { e }$\\
(B) 1\\
(C) $e$\\
(D) $\log _ { e } 2$
\end{enumerate}

Answer\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-25_92_93_1778_494}\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-25_89_91_1778_646}\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-25_89_91_1778_797}\\
(A)\\
(B)\\
(C)\\
(D)\\
63. For $k > 0$, the set of all values of $k$ for which $k e ^ { x } - x = 0$ has two distinct roots is\\
(A) $\left( 0 , \frac { 1 } { e } \right)$\\
(B) $\left( \frac { 1 } { e } , 1 \right)$\\
(C) $\left( \frac { 1 } { e } , \infty \right)$\\
(D) $( 0,1 )$

Answer ◯\\
◯\\
◯\\
(A)\\
(B)\\
(C)\\
(D)\\
64. Let $f ( x ) = \frac { x ^ { 2 } - 6 x + 5 } { x ^ { 2 } - 5 x + 6 }$.

Match the expressions/statements in Column I with expressions/statements in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

\section*{Column I}
(A) If $- 1 < x < 1$, then $f ( x )$ satisfies\\
(B) If $1 < x < 2$, then $f ( x )$ satisfies\\
(C) If $3 < x < 5$, then $f ( x )$ satisfies\\
(D) If $x > 5$, then $f ( x )$ satisfies

\section*{Column II}
(p) $0 < f ( x ) < 1$\\
(q) $f ( x ) < 0$\\
(r) $f ( x ) > 0$\\
(s) $f ( x ) < 1$

Answer\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-26_442_564_1170_328}\\
65. Let $( x , y )$ be such that

$$\sin ^ { - 1 } ( a x ) + \cos ^ { - 1 } ( y ) + \cos ^ { - 1 } ( b x y ) = \frac { \pi } { 2 }$$

Match the statements in Column I with statements in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

\section*{Column I}
(A) If $a = 1$ and $b = 0$, then $( x , y )$\\
(B) If $a = 1$ and $b = 1$, then $( x , y )$\\
(C) If $a = 1$ and $b = 2$, then $( x , y )$\\
(D) If $a = 2$ and $b = 2$, then $( x , y )$\\
(p) lies on the circle $x ^ { 2 } + y ^ { 2 } = 1$\\
(q) lies on $\left( x ^ { 2 } - 1 \right) \left( y ^ { 2 } - 1 \right) = 0$\\
(r) lies on $y = x$\\
(s) lies on $\left( 4 x ^ { 2 } - 1 \right) \left( y ^ { 2 } - 1 \right) = 0$

\section*{Column II}
\begin{enumerate}
  \setcounter{enumi}{65}
  \item Match the statements in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.
\end{enumerate}

\section*{Column I}
(A) Two intersecting circles\\
(B) Two mutually external circles\\
(C) Two circles, one strictly inside the other\\
(D) Two branches of a hyperbola

\section*{Column II}
(p) have a common tangent\\
(q) have a common normal\\
(r) do not have a common tangent\\
(s) do not have a common normal

Answer\\
\includegraphics[max width=\textwidth, alt={}, center]{e79946a0-735d-438d-ac42-5265ab72284d-28_439_566_1078_326}
Paper Questions
Q3 Q4 Q5 Q8 Q9 Q10 Q11 Q13 Q14 Q16 Q17 Q18 Q19 Q21 Q22 Q24 Q25 Q26 Q27 Q28 Q29 Q30 Q31 Q33 Q34 Q35 Q36 Q37 Q38 Q39 Q40 Q41 Q42 Q43 Q44 Q45 Q46 Q48 Q49 Q50