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Q37 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

Let $P = \left[ \begin{array} { c c c } 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{array} \right]$ and $I$ be the identity matrix of order 3. If $Q = \left[ q _ { i j } \right]$ is a matrix such that $P ^ { 50 } - Q = I$, then $\frac { q _ { 31 } + q _ { 32 } } { q _ { 21 } }$ equals
(A) 52
(B) 103
(C) 201
(D) 205

Q38 Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View

Let $b _ { i } > 1$ for $i = 1,2 , \ldots , 101$. Suppose $\log _ { e } b _ { 1 } , \log _ { e } b _ { 2 } , \ldots , \log _ { e } b _ { 101 }$ are in Arithmetic Progression (A.P.) with the common difference $\log _ { e } 2$. Suppose $a _ { 1 } , a _ { 2 } , \ldots , a _ { 101 }$ are in A.P. such that $a _ { 1 } = b _ { 1 }$ and $a _ { 51 } = b _ { 51 }$. If $t = b _ { 1 } + b _ { 2 } + \cdots + b _ { 51 }$ and $s = a _ { 1 } + a _ { 2 } + \cdots + a _ { 51 }$, then
(A) $s > t$ and $a _ { 101 } > b _ { 101 }$
(B) $s > t$ and $a _ { 101 } < b _ { 101 }$
(C) $s < t$ and $a _ { 101 } > b _ { 101 }$
(D) $s < t$ and $a _ { 101 } < b _ { 101 }$

Q39 Addition & Double Angle Formulae Telescoping Sum of Trigonometric Terms View

The value of $\sum _ { k = 1 } ^ { 13 } \frac { 1 } { \sin \left( \frac { \pi } { 4 } + \frac { ( k - 1 ) \pi } { 6 } \right) \sin \left( \frac { \pi } { 4 } + \frac { k \pi } { 6 } \right) }$ is equal to
(A) $3 - \sqrt { 3 }$
(B) $2 ( 3 - \sqrt { 3 } )$
(C) $2 ( \sqrt { 3 } - 1 )$
(D) $2 ( 2 + \sqrt { 3 } )$

Q40 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

The value of $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { x ^ { 2 } \cos x } { 1 + e ^ { x } } d x$ is equal to
(A) $\frac { \pi ^ { 2 } } { 4 } - 2$
(B) $\frac { \pi ^ { 2 } } { 4 } + 2$
(C) $\pi ^ { 2 } - e ^ { \frac { \pi } { 2 } }$
(D) $\pi ^ { 2 } + e ^ { \frac { \pi } { 2 } }$

Q41 Areas by integration View

Area of the region $\left\{ ( x , y ) \in \mathbb { R } ^ { 2 } : y \geq \sqrt { | x + 3 | } , 5 y \leq x + 9 \leq 15 \right\}$ is equal to
(A) $\frac { 1 } { 6 }$
(B) $\frac { 4 } { 3 }$
(C) $\frac { 3 } { 2 }$
(D) $\frac { 5 } { 3 }$

Q42 Vectors 3D & Lines Normal Vector and Plane Equation View

Let $P$ be the image of the point $( 3,1,7 )$ with respect to the plane $x - y + z = 3$. Then the equation of the plane passing through $P$ and containing the straight line $\frac { x } { 1 } = \frac { y } { 2 } = \frac { z } { 1 }$ is
(A) $x + y - 3 z = 0$
(B) $3 x + z = 0$
(C) $x - 4 y + 7 z = 0$
(D) $2 x - y = 0$

Q43 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View

Let $f ( x ) = \lim _ { n \rightarrow \infty } \left( \frac { n ^ { n } ( x + n ) \left( x + \frac { n } { 2 } \right) \cdots \left( x + \frac { n } { n } \right) } { n ! \left( x ^ { 2 } + n ^ { 2 } \right) \left( x ^ { 2 } + \frac { n ^ { 2 } } { 4 } \right) \cdots \left( x ^ { 2 } + \frac { n ^ { 2 } } { n ^ { 2 } } \right) } \right) ^ { \frac { x } { n } }$, for all $x > 0$. Then
(A) $f \left( \frac { 1 } { 2 } \right) \geq f ( 1 )$
(B) $f \left( \frac { 1 } { 3 } \right) \leq f \left( \frac { 2 } { 3 } \right)$
(C) $f ^ { \prime } ( 2 ) \leq 0$
(D) $\frac { f ^ { \prime } ( 3 ) } { f ( 3 ) } \geq \frac { f ^ { \prime } ( 2 ) } { f ( 2 ) }$

Q44 Tangents, normals and gradients Find tangent line equation at a given point View

Let $a , b \in \mathbb { R }$ and $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by $f ( x ) = a \cos \left( \left| x ^ { 3 } - x \right| \right) + b | x | \sin \left( \left| x ^ { 3 } + x \right| \right)$. Then $f$ is
(A) differentiable at $x = 0$ if $a = 0$ and $b = 1$
(B) differentiable at $x = 1$ if $a = 1$ and $b = 0$
(C) NOT differentiable at $x = 0$ if $a = 1$ and $b = 0$
(D) NOT differentiable at $x = 1$ if $a = 1$ and $b = 1$

Q45 Stationary points and optimisation Find critical points and classify extrema of a given function View

Let $f : \mathbb { R } \rightarrow ( 0 , \infty )$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be twice differentiable functions such that $f ^ { \prime \prime }$ and $g ^ { \prime \prime }$ are continuous functions on $\mathbb { R }$. Suppose $f ^ { \prime } ( 2 ) = g ( 2 ) = 0 , \quad f ^ { \prime \prime } ( 2 ) \neq 0$ and $g ^ { \prime } ( 2 ) \neq 0$. If $\lim _ { x \rightarrow 2 } \frac { f ( x ) g ( x ) } { f ^ { \prime } ( x ) g ^ { \prime } ( x ) } = 1$, then
(A) $f$ has a local minimum at $x = 2$
(B) $f$ has a local maximum at $x = 2$
(C) $f ^ { \prime \prime } ( 2 ) > f ( 2 )$
(D) $f ( x ) - f ^ { \prime \prime } ( x ) = 0$ for at least one $x \in \mathbb { R }$

Q46 Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View

Let $f : \left[ - \frac { 1 } { 2 } , 2 \right] \rightarrow \mathbb { R }$ and $g : \left[ - \frac { 1 } { 2 } , 2 \right] \rightarrow \mathbb { R }$ be functions defined by $f ( x ) = \left[ x ^ { 2 } - 3 \right]$ and $g ( x ) = | x | f ( x ) + | 4 x - 7 | f ( x )$, where $[ y ]$ denotes the greatest integer less than or equal to $y$ for $y \in \mathbb { R }$. Then
(A) $f$ is discontinuous exactly at three points in $\left[ - \frac { 1 } { 2 } , 2 \right]$
(B) $f$ is discontinuous exactly at four points in $\left[ - \frac { 1 } { 2 } , 2 \right]$
(C) $g$ is NOT differentiable exactly at four points in $\left( - \frac { 1 } { 2 } , 2 \right)$
(D) $g$ is NOT differentiable exactly at five points in $\left( - \frac { 1 } { 2 } , 2 \right)$

Q47 Complex Numbers Argand & Loci Circle Equation and Properties via Complex Number Manipulation View

Let $a , b \in \mathbb { R }$ and $a ^ { 2 } + b ^ { 2 } \neq 0$. Suppose $S = \left\{ z \in \mathbb { C } : z = \frac { 1 } { a + i b t } , t \in \mathbb { R } , t \neq 0 \right\}$, where $i = \sqrt { - 1 }$.
If $z = x + i y$ and $z \in S$, then $( x , y )$ lies on
(A) the circle with radius $\frac { 1 } { 2 a }$ and centre $\left( \frac { 1 } { 2 a } , 0 \right)$ for $a > 0 , b \neq 0$
(B) the circle with radius $- \frac { 1 } { 2 a }$ and centre $\left( - \frac { 1 } { 2 a } , 0 \right)$ for $a < 0 , b \neq 0$
(C) the $x$-axis for $a \neq 0 , b = 0$
(D) the $y$-axis for $a = 0 , b \neq 0$

Q48 Circles Optimization on a Circle View

Let $P$ be the point on the parabola $y ^ { 2 } = 4 x$ which is at the shortest distance from the center $S$ of the circle $x ^ { 2 } + y ^ { 2 } - 4 x - 16 y + 64 = 0$. Let $Q$ be the point on the circle dividing the line segment $S P$ internally. Then
(A) $S P = 2 \sqrt { 5 }$
(B) $S Q : Q P = ( \sqrt { 5 } + 1 ) : 2$
(C) the $x$-intercept of the normal to the parabola at $P$ is 6
(D) the slope of the tangent to the circle at $Q$ is $\frac { 1 } { 2 }$

Q49 Simultaneous equations View

Let $a , \lambda , \mu \in \mathbb { R }$. Consider the system of linear equations
$$\begin{aligned} & a x + 2 y = \lambda \\ & 3 x - 2 y = \mu \end{aligned}$$
Which of the following statement(s) is(are) correct?
(A) If $a = - 3$, then the system has infinitely many solutions for all values of $\lambda$ and $\mu$
(B) If $a \neq - 3$, then the system has a unique solution for all values of $\lambda$ and $\mu$
(C) If $\lambda + \mu = 0$, then the system has infinitely many solutions for $a = - 3$
(D) If $\lambda + \mu \neq 0$, then the system has no solution for $a = - 3$

Q50 Vectors 3D & Lines Vector Algebra and Triple Product Computation View

Let $\hat { u } = u _ { 1 } \hat { i } + u _ { 2 } \hat { j } + u _ { 3 } \hat { k }$ be a unit vector in $\mathbb { R } ^ { 3 }$ and $\hat { w } = \frac { 1 } { \sqrt { 6 } } ( \hat { i } + \hat { j } + 2 \hat { k } )$. Given that there exists a vector $\vec { v }$ in $\mathbb { R } ^ { 3 }$ such that $| \hat { u } \times \vec { v } | = 1$ and $\hat { w } \cdot ( \hat { u } \times \vec { v } ) = 1$. Which of the following statement(s) is(are) correct?
(A) There is exactly one choice for such $\vec { v }$
(B) There are infinitely many choices for such $\vec { v }$
(C) If $\hat { u }$ lies in the $x y$-plane then $\left| u _ { 1 } \right| = \left| u _ { 2 } \right|$
(D) If $\hat { u }$ lies in the $x z$-plane then $2 \left| u _ { 1 } \right| = \left| u _ { 3 } \right|$

Q51 Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View

Football teams $T _ { 1 }$ and $T _ { 2 }$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T _ { 1 }$ winning, drawing and losing a game against $T _ { 2 }$ are $\frac { 1 } { 2 } , \frac { 1 } { 6 }$ and $\frac { 1 } { 3 }$, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T _ { 1 }$ and $T _ { 2 }$, respectively, after two games.
$P ( X > Y )$ is
(A) $\frac { 1 } { 4 }$
(B) $\frac { 5 } { 12 }$
(C) $\frac { 1 } { 2 }$
(D) $\frac { 7 } { 12 }$

Q52 Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View

Football teams $T _ { 1 }$ and $T _ { 2 }$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T _ { 1 }$ winning, drawing and losing a game against $T _ { 2 }$ are $\frac { 1 } { 2 } , \frac { 1 } { 6 }$ and $\frac { 1 } { 3 }$, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T _ { 1 }$ and $T _ { 2 }$, respectively, after two games.
$P ( X = Y )$ is
(A) $\frac { 11 } { 36 }$
(B) $\frac { 1 } { 3 }$
(C) $\frac { 13 } { 36 }$
(D) $\frac { 1 } { 2 }$

Q53 Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View

Let $F _ { 1 } \left( x _ { 1 } , 0 \right)$ and $F _ { 2 } \left( x _ { 2 } , 0 \right)$, for $x _ { 1 } < 0$ and $x _ { 2 } > 0$, be the foci of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 8 } = 1$. Suppose a parabola having vertex at the origin and focus at $F _ { 2 }$ intersects the ellipse at point $M$ in the first quadrant and at point $N$ in the fourth quadrant.
The orthocentre of the triangle $F _ { 1 } M N$ is
(A) $\left( - \frac { 9 } { 10 } , 0 \right)$
(B) $\left( \frac { 2 } { 3 } , 0 \right)$
(C) $\left( \frac { 9 } { 10 } , 0 \right)$
(D) $\left( \frac { 2 } { 3 } , \sqrt { 6 } \right)$

Q54 Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View

Let $F _ { 1 } \left( x _ { 1 } , 0 \right)$ and $F _ { 2 } \left( x _ { 2 } , 0 \right)$, for $x _ { 1 } < 0$ and $x _ { 2 } > 0$, be the foci of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 8 } = 1$. Suppose a parabola having vertex at the origin and focus at $F _ { 2 }$ intersects the ellipse at point $M$ in the first quadrant and at point $N$ in the fourth quadrant.
If the tangents to the ellipse at $M$ and $N$ meet at $R$ and the normal to the parabola at $M$ meets the $x$-axis at $Q$, then the ratio of area of the triangle $M Q R$ to area of the quadrilateral $M F _ { 1 } N F _ { 2 }$ is
(A) $3 : 4$
(B) $4 : 5$
(C) $5 : 8$
(D) $2 : 3$

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