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Q1 Complex Numbers Argand & Loci Distance and Region Optimization on Loci View

Let $S$ be the set of all complex numbers $z$ satisfying $| z - 2 + i | \geq \sqrt { 5 }$. If the complex number $z _ { 0 }$ is such that $\frac { 1 } { \left| z _ { 0 } - 1 \right| }$ is the maximum of the set $\left\{ \frac { 1 } { | z - 1 | } : z \in S \right\}$, then the principal argument of $\frac { 4 - z _ { 0 } - \overline { z _ { 0 } } } { z _ { 0 } - \overline { z _ { 0 } } + 2 i }$ is
(A) $- \frac { \pi } { 2 }$
(B) $\frac { \pi } { 4 }$
(C) $\frac { \pi } { 2 }$
(D) $\frac { 3 \pi } { 4 }$

Q2 Matrices Matrix Algebra and Product Properties View

Let $$M = \left[ \begin{array} { c c } \sin ^ { 4 } \theta & - 1 - \sin ^ { 2 } \theta \\ 1 + \cos ^ { 2 } \theta & \cos ^ { 4 } \theta \end{array} \right] = \alpha I + \beta M ^ { - 1 }$$ where $\alpha = \alpha ( \theta )$ and $\beta = \beta ( \theta )$ are real numbers, and $I$ is the $2 \times 2$ identity matrix. If $\alpha ^ { * }$ is the minimum of the set $\{ \alpha ( \theta ) : \theta \in [ 0,2 \pi ) \}$ and $\beta ^ { * }$ is the minimum of the set $\{ \beta ( \theta ) : \theta \in [ 0,2 \pi ) \}$, then the value of $\alpha ^ { * } + \beta ^ { * }$ is
(A) $- \frac { 37 } { 16 }$
(B) $- \frac { 31 } { 16 }$
(C) $- \frac { 29 } { 16 }$
(D) $- \frac { 17 } { 16 }$

Q3 Circles Chord Length and Chord Properties View

A line $y = m x + 1$ intersects the circle $( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25$ at the points $P$ and $Q$. If the midpoint of the line segment $P Q$ has $x$-coordinate $- \frac { 3 } { 5 }$, then which one of the following options is correct?
(A) $\quad - 3 \leq m < - 1$
(B) $2 \leq m < 4$
(C) $4 \leq m < 6$
(D) $6 \leq m < 8$

Q4 Areas by integration View

The area of the region $\left\{ ( x , y ) : x y \leq 8,1 \leq y \leq x ^ { 2 } \right\}$ is
(A) $16 \log _ { e } 2 - \frac { 14 } { 3 }$
(B) $8 \log _ { e } 2 - \frac { 14 } { 3 }$
(C) $16 \log _ { e } 2 - 6$
(D) $8 \log _ { e } 2 - \frac { 7 } { 3 }$

Q5 Sequences and series, recurrence and convergence Multiple-choice on sequence properties View

Let $\alpha$ and $\beta$ be the roots of $x ^ { 2 } - x - 1 = 0$, with $\alpha > \beta$. For all positive integers $n$, define $$\begin{aligned} & a _ { n } = \frac { \alpha ^ { n } - \beta ^ { n } } { \alpha - \beta } , \quad n \geq 1 \\ & b _ { 1 } = 1 \text { and } \quad b _ { n } = a _ { n - 1 } + a _ { n + 1 } , \quad n \geq 2 . \end{aligned}$$ Then which of the following options is/are correct?
(A) $\quad a _ { 1 } + a _ { 2 } + a _ { 3 } + \cdots + a _ { n } = a _ { n + 2 } - 1$ for all $n \geq 1$
(B) $\quad \sum _ { n = 1 } ^ { \infty } \frac { a _ { n } } { 10 ^ { n } } = \frac { 10 } { 89 }$
(C) $b _ { n } = \alpha ^ { n } + \beta ^ { n }$ for all $n \geq 1$
(D) $\quad \sum _ { n = 1 } ^ { \infty } \frac { b _ { n } } { 10 ^ { n } } = \frac { 8 } { 89 }$

Q6 3x3 Matrices Determinant of Parametric or Structured Matrix View

Let $$M = \left[ \begin{array} { l l l } 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{array} \right] \quad \text { and } \quad \operatorname { adj } M = \left[ \begin{array} { r r r } - 1 & 1 & - 1 \\ 8 & - 6 & 2 \\ - 5 & 3 & - 1 \end{array} \right]$$ where $a$ and $b$ are real numbers. Which of the following options is/are correct?
(A) $a + b = 3$
(B) $\quad ( \operatorname { adj } M ) ^ { - 1 } + \operatorname { adj } M ^ { - 1 } = - M$
(C) $\operatorname { det } \left( \operatorname { adj } M ^ { 2 } \right) = 81$
(D) If $M \left[ \begin{array} { l } \alpha \\ \beta \\ \gamma \end{array} \right] = \left[ \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right]$, then $\alpha - \beta + \gamma = 3$

Q7 Conditional Probability Bayes' Theorem with Production/Source Identification View

There are three bags $B _ { 1 } , B _ { 2 }$ and $B _ { 3 }$. The bag $B _ { 1 }$ contains 5 red and 5 green balls, $B _ { 2 }$ contains 3 red and 5 green balls, and $B _ { 3 }$ contains 5 red and 3 green balls. Bags $B _ { 1 } , B _ { 2 }$ and $B _ { 3 }$ have probabilities $\frac { 3 } { 10 } , \frac { 3 } { 10 }$ and $\frac { 4 } { 10 }$ respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?
(A) Probability that the chosen ball is green, given that the selected bag is $B _ { 3 }$, equals $\frac { 3 } { 8 }$
(B) Probability that the chosen ball is green equals $\frac { 39 } { 80 }$
(C) Probability that the selected bag is $B _ { 3 }$, given that the chosen ball is green, equals $\frac { 5 } { 13 }$
(D) Probability that the selected bag is $B _ { 3 }$ and the chosen ball is green equals $\frac { 3 } { 10 }$

Q8 Sine and Cosine Rules Multi-step composite figure problem View

In a non-right-angled triangle $\triangle P Q R$, let $p , q , r$ denote the lengths of the sides opposite to the angles at $P , Q , R$ respectively. The median from $R$ meets the side $P Q$ at $S$, the perpendicular from $P$ meets the side $Q R$ at $E$, and $R S$ and $P E$ intersect at $O$. If $p = \sqrt { 3 } , q = 1$, and the radius of the circumcircle of the $\triangle P Q R$ equals 1, then which of the following options is/are correct?
(A) Length of $R S = \frac { \sqrt { 7 } } { 2 }$
(B) Area of $\triangle S O E = \frac { \sqrt { 3 } } { 12 }$
(C) Length of $O E = \frac { 1 } { 6 }$
(D) Radius of incircle of $\triangle P Q R = \frac { \sqrt { 3 } } { 2 } ( 2 - \sqrt { 3 } )$

Q9 Circles Infinite Series or Sequences Involving Circles View

Define the collections $\left\{ E _ { 1 } , E _ { 2 } , E _ { 3 } , \ldots \right\}$ of ellipses and $\left\{ R _ { 1 } , R _ { 2 } , R _ { 3 } , \ldots \right\}$ of rectangles as follows: $E _ { 1 } : \frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1 ;$ $R _ { 1 }$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _ { 1 }$; $E _ { n }$ : ellipse $\frac { x ^ { 2 } } { a _ { n } ^ { 2 } } + \frac { y ^ { 2 } } { b _ { n } ^ { 2 } } = 1$ of largest area inscribed in $R _ { n - 1 } , n > 1$; $R _ { n }$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _ { n } , n > 1$. Then which of the following options is/are correct?
(A) The eccentricities of $E _ { 18 }$ and $E _ { 19 }$ are NOT equal
(B) $\quad \sum _ { n = 1 } ^ { N } \left( \right.$ area of $\left. R _ { n } \right) < 24$, for each positive integer $N$
(C) The length of latus rectum of $E _ { 9 }$ is $\frac { 1 } { 6 }$
(D) The distance of a focus from the centre in $E _ { 9 }$ is $\frac { \sqrt { 5 } } { 32 }$

Q10 Stationary points and optimisation Composite or piecewise function extremum analysis View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be given by $$f ( x ) = \left\{ \begin{aligned} x ^ { 5 } + 5 x ^ { 4 } + 10 x ^ { 3 } + 10 x ^ { 2 } + 3 x + 1 , & x < 0 \\ x ^ { 2 } - x + 1 , & 0 \leq x < 1 \\ \frac { 2 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 7 x - \frac { 8 } { 3 } , & 1 \leq x < 3 \\ ( x - 2 ) \log _ { e } ( x - 2 ) - x + \frac { 10 } { 3 } , & x \geq 3 \end{aligned} \right.$$ Then which of the following options is/are correct?
(A) $f$ is increasing on $( - \infty , 0 )$
(B) $f ^ { \prime }$ has a local maximum at $x = 1$
(C) $f$ is onto
(D) $f ^ { \prime }$ is NOT differentiable at $x = 1$

Q11 Differential equations Finding a DE from a Limit or Implicit Condition View

Let $\Gamma$ denote a curve $y = y ( x )$ which is in the first quadrant and let the point $( 1,0 )$ lie on it. Let the tangent to $\Gamma$ at a point $P$ intersect the $y$-axis at $Y _ { P }$. If $P Y _ { P }$ has length 1 for each point $P$ on $\Gamma$, then which of the following options is/are correct?
(A) $y = \log _ { e } \left( \frac { 1 + \sqrt { 1 - x ^ { 2 } } } { x } \right) - \sqrt { 1 - x ^ { 2 } }$
(B) $\quad x y ^ { \prime } + \sqrt { 1 - x ^ { 2 } } = 0$
(C) $y = - \log _ { e } \left( \frac { 1 + \sqrt { 1 - x ^ { 2 } } } { x } \right) + \sqrt { 1 - x ^ { 2 } }$
(D) $x y ^ { \prime } - \sqrt { 1 - x ^ { 2 } } = 0$

Q12 Vectors 3D & Lines Parametric Representation of a Line View

Let $L _ { 1 }$ and $L _ { 2 }$ denote the lines $$\begin{aligned} & \vec { r } = \hat { i } + \lambda ( - \hat { i } + 2 \hat { j } + 2 \hat { k } ) , \lambda \in \mathbb { R } \text { and } \\ & \vec { r } = \mu ( 2 \hat { i } - \hat { j } + 2 \hat { k } ) , \mu \in \mathbb { R } \end{aligned}$$ respectively. If $L _ { 3 }$ is a line which is perpendicular to both $L _ { 1 }$ and $L _ { 2 }$ and cuts both of them, then which of the following options describe(s) $L _ { 3 }$?
(A) $\vec { r } = \frac { 2 } { 9 } ( 4 \hat { i } + \hat { j } + \hat { k } ) + t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$
(B) $\vec { r } = \frac { 2 } { 9 } ( 2 \hat { i } - \hat { j } + 2 \hat { k } ) + t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$
(C) $\vec { r } = \frac { 1 } { 3 } ( 2 \hat { i } + \hat { k } ) + t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$
(D) $\vec { r } = t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$

Q13 Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions View

Let $\omega \neq 1$ be a cube root of unity. Then the minimum of the set $$\left\{ \left| a + b \omega + c \omega ^ { 2 } \right| ^ { 2 } : a , b , c \text { distinct non-zero integers } \right\}$$ equals $\_\_\_\_$

Q14 Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View

Let $A P ( a ; d )$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d > 0$. If $$A P ( 1 ; 3 ) \cap A P ( 2 ; 5 ) \cap A P ( 3 ; 7 ) = A P ( a ; d )$$ then $a + d$ equals $\_\_\_\_$

Q15 Modelling and Hypothesis Testing View

Let $S$ be the sample space of all $3 \times 3$ matrices with entries from the set $\{ 0,1 \}$. Let the events $E _ { 1 }$ and $E _ { 2 }$ be given by $$\begin{aligned} & E _ { 1 } = \{ A \in S : \operatorname { det } A = 0 \} \text { and } \\ & E _ { 2 } = \{ A \in S : \text { sum of entries of } A \text { is } 7 \} . \end{aligned}$$ If a matrix is chosen at random from $S$, then the conditional probability $P \left( E _ { 1 } \mid E _ { 2 } \right)$ equals $\_\_\_\_$

Q16 Circles Tangent Lines and Tangent Lengths View

Let the point $B$ be the reflection of the point $A ( 2,3 )$ with respect to the line $8 x - 6 y - 23 = 0$. Let $\Gamma _ { A }$ and $\Gamma _ { B }$ be circles of radii 2 and 1 with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma _ { A }$ and $\Gamma _ { B }$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$, then the length of the line segment $A C$ is $\_\_\_\_$

Q17 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

If $$I = \frac { 2 } { \pi } \int _ { - \pi / 4 } ^ { \pi / 4 } \frac { d x } { \left( 1 + e ^ { \sin x } \right) ( 2 - \cos 2 x ) }$$ then $27 I ^ { 2 }$ equals

Q18 Vectors 3D & Lines Line-Plane Intersection View

Three lines are given by $$\begin{aligned} & \vec { r } = \lambda \hat { i } , \lambda \in \mathbb { R } \\ & \vec { r } = \mu ( \hat { i } + \hat { j } ) , \quad \mu \in \mathbb { R } \text { and } \\ & \vec { r } = v ( \hat { i } + \hat { j } + \hat { k } ) , \quad v \in \mathbb { R } \end{aligned}$$ Let the lines cut the plane $x + y + z = 1$ at the points $A , B$ and $C$ respectively. If the area of the triangle $A B C$ is $\triangle$ then the value of $( 6 \Delta ) ^ { 2 }$ equals

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