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2012 paper2

2 maths questions

Q41 Straight Lines & Coordinate Geometry Point-to-Line Distance Computation View
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
    \setcounter{enumi}{41}
  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
    \setcounter{enumi}{42}
  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
    \setcounter{enumi}{44}
  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
    \setcounter{enumi}{45}
  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
    \setcounter{enumi}{46}
  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
    \setcounter{enumi}{47}
  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 .
Q49 Sequences and series, recurrence and convergence Direct term computation from recurrence View
49. The value of $b _ { 6 }$ is
(A) 7
(B) 8
(C) 9
(D) 11
ANSWER : B
    \setcounter{enumi}{49}
  1. Which of the following is correct?
    (A) $a _ { 17 } = a _ { 16 } + a _ { 15 }$
    (B) $c _ { 17 } \neq c _ { 16 } + c _ { 15 }$
    (C) $b _ { 17 } \neq b _ { 16 } + c _ { 16 }$
    (D) $a _ { 17 } = c _ { 17 } + b _ { 16 }$

ANSWER:A
Paragraph for Questions 51 and 52
Let $f ( x ) = ( 1 - x ) ^ { 2 } \sin ^ { 2 } x + x ^ { 2 }$ for all $x \in \mathbb { R }$, and let $g ( x ) = \int _ { 1 } ^ { x } \left( \frac { 2 ( t - 1 ) } { t + 1 } - \ln t \right) f ( t ) d t$ for all $x \in ( 1 , \infty )$. 51. Which of the following is true?
(A) $g$ is increasing on $( 1 , \infty )$
(B) $g$ is decreasing on $( 1 , \infty )$
(C) $g$ is increasing on $( 1,2 )$ and decreasing on $( 2 , \infty )$
(D) $g$ is decreasing on $( 1,2 )$ and increasing on $( 2 , \infty )$
ANSWER : B
    \setcounter{enumi}{51}
  1. Consider the statements : $\mathbf { P }$ : There exists some $x \in \mathbb { R }$ such that $f ( x ) + 2 x = 2 \left( 1 + x ^ { 2 } \right)$ Q: There exists some $x \in \mathbb { R }$ such that $2 f ( x ) + 1 = 2 x ( 1 + x )$ Then
    (A) both $\mathbf { P }$ and $\mathbf { Q }$ are true
    (B) $\mathbf { P }$ is true and $\mathbf { Q }$ is false
    (C) $\mathbf { P }$ is false and $\mathbf { Q }$ is true
    (D) both $\mathbf { P }$ and $\mathbf { Q }$ are false

ANSWER : C
MATHEMATICS
Paragraph for Questions 53 and 54
A tangent $P T$ is drawn to the circle $x ^ { 2 } + y ^ { 2 } = 4$ at the point $P ( \sqrt { 3 } , 1 )$. A straight line $L$, perpendicular to $P T$ is a tangent to the circle $( x - 3 ) ^ { 2 } + y ^ { 2 } = 1$. 53. A possible equation of $L$ is
(A) $x - \sqrt { 3 } y = 1$
(B) $x + \sqrt { 3 } y = 1$
(C) $x - \sqrt { 3 } y = - 1$
(D) $x + \sqrt { 3 } y = 5$
ANSWER : A
    \setcounter{enumi}{53}
  1. A common tangent of the two circles is
    (A) $x = 4$
    (B) $y = 2$
    (C) $x + \sqrt { 3 } y = 4$
    (D) $x + 2 \sqrt { 2 } y = 6$

ANSWER : D
SECTION III : Multiple Correct Answer(s) Type
This section contains 6 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct. 55. For every integer $n$, let $a _ { n }$ and $b _ { n }$ be real numbers. Let function $f : \mathbb { R } \rightarrow \mathbb { R }$ be given by $f ( x ) = \left\{ \begin{array} { l l } a _ { n } + \sin \pi x , & \text { for } x \in [ 2 n , 2 n + 1 ] \\ b _ { n } + \cos \pi x , & \text { for } x \in ( 2 n - 1,2 n ) \end{array} \right.$, for all integers $n$. If $f$ is continuous, then which of the following hold(s) for all $n$ ?
(A) $a _ { n - 1 } - b _ { n - 1 } = 0$
(B) $a _ { n } - b _ { n } = 1$
(C) $a _ { n } - b _ { n + 1 } = 1$
(D) $a _ { n - 1 } - b _ { n } = - 1$
ANSWER : BD
    \setcounter{enumi}{55}
  1. If $f ( x ) = \int _ { 0 } ^ { x } e ^ { t ^ { 2 } } ( t - 2 ) ( t - 3 ) d t$ for all $x \in ( 0 , \infty )$, then
    (A) $f$ has a local maximum at $x = 2$
    (B) $f$ is decreasing on $( 2,3 )$
    (C) there exists some $c \in ( 0 , \infty )$ such that $f ^ { \prime \prime } ( c ) = 0$
    (D) $f$ has a local minimum at $x = 3$

ANSWER : ABCD 57. If the straight lines $\frac { x - 1 } { 2 } = \frac { y + 1 } { k } = \frac { z } { 2 }$ and $\frac { x + 1 } { 5 } = \frac { y + 1 } { 2 } = \frac { z } { k }$ are coplanar, then the plane(s) containing these two lines is(are)
(A) $y + 2 z = - 1$
(B) $y + z = - 1$
(C) $y - z = - 1$
(D) $y - 2 z = - 1$
ANSWER : BC
    \setcounter{enumi}{57}
  1. Let $X$ and $Y$ be two events such that $P ( X \mid Y ) = \frac { 1 } { 2 } , P ( Y \mid X ) = \frac { 1 } { 3 }$ and $P ( X \cap Y ) = \frac { 1 } { 6 }$. Which of the following is (are) correct?
    (A) $P ( X \cup Y ) = \frac { 2 } { 3 }$
    (B) $X$ and $Y$ are independent
    (C) $X$ and $Y$ are not independent
    (D) $P \left( X ^ { \mathrm { c } } \cap Y \right) = \frac { 1 } { 3 }$

ANSWER : AB
MATHEMATICS
    \setcounter{enumi}{58}
  1. If the adjoint of a $3 \times 3$ matrix $P$ is $\left[ \begin{array} { l l l } 1 & 4 & 4 \\ 2 & 1 & 7 \\ 1 & 1 & 3 \end{array} \right]$, then the possible value(s) of the determinant of $P$ is (are)
    (A) - 2
    (B) - 1
    (C) 1
    (D) 2

ANSWER : AD
    \setcounter{enumi}{59}
  1. Let $f : ( - 1,1 ) \rightarrow \mathbb { R }$ be such that $f ( \cos 4 \theta ) = \frac { 2 } { 2 - \sec ^ { 2 } \theta }$ for $\theta \in \left( 0 , \frac { \pi } { 4 } \right) \cup \left( \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$. Then the value(s) of $f \left( \frac { 1 } { 3 } \right)$ is (are)
    (A) $1 - \sqrt { \frac { 3 } { 2 } }$
    (B) $1 + \sqrt { \frac { 3 } { 2 } }$
    (C) $1 - \sqrt { \frac { 2 } { 3 } }$
    (D) $1 + \sqrt { \frac { 2 } { 3 } }$

Zero Marks to all