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Q1 Circles Intersection of Circles or Circle with Conic View

Consider the two curves $$\begin{aligned} & C _ { 1 } : y ^ { 2 } = 4 x \\ & C _ { 2 } : x ^ { 2 } + y ^ { 2 } - 6 x + 1 = 0 \end{aligned}$$ Then,
(A) $C _ { 1 }$ and $C _ { 2 }$ touch each other only at one point
(B) $C _ { 1 }$ and $C _ { 2 }$ touch each other exactly at two points
(C) $C _ { 1 }$ and $C _ { 2 }$ intersect (but do not touch) at exactly two points
(D) $C _ { 1 }$ and $C _ { 2 }$ neither intersect nor touch each other

Q2 Reciprocal Trig & Identities View

If $0 < x < 1$, then $$\sqrt { 1 + x ^ { 2 } } \left[ \left\{ x \cos \left( \cot ^ { - 1 } x \right) + \sin \left( \cot ^ { - 1 } x \right) \right\} ^ { 2 } - 1 \right] ^ { \frac { 1 } { 2 } } =$$ (A) $\frac { x } { \sqrt { 1 + x ^ { 2 } } }$
(B) $x$
(C) $x \sqrt { 1 + x ^ { 2 } }$
(D) $\sqrt { 1 + x ^ { 2 } }$

Q3 Vector Product and Surfaces View

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $\hat { a } , \hat { b } , \hat { c }$ such that $$\hat { a } \cdot \hat { b } = \hat { b } \cdot \hat { c } = \hat { c } \cdot \hat { a } = \frac { 1 } { 2 }$$ Then, the volume of the parallelopiped is
(A) $\frac { 1 } { \sqrt { 2 } }$
(B) $\frac { 1 } { 2 \sqrt { 2 } }$
(C) $\frac { \sqrt { 3 } } { 2 }$
(D) $\frac { 1 } { \sqrt { 3 } }$

Q4 Circles Circle Identification and Classification View

Let $a$ and $b$ be non-zero real numbers. Then, the equation $$\left( a x ^ { 2 } + b y ^ { 2 } + c \right) \left( x ^ { 2 } - 5 x y + 6 y ^ { 2 } \right) = 0$$ represents
(A) four straight lines, when $c = 0$ and $a , b$ are of the same sign
(B) two straight lines and a circle, when $a = b$, and $c$ is of sign opposite to that of $a$
(C) two straight lines and a hyperbola, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$
(D) a circle and an ellipse, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$

Q5 Differentiating Transcendental Functions Determine parameters from function or curve conditions View

Let $g ( x ) = \frac { ( x - 1 ) ^ { n } } { \log \cos ^ { m } ( x - 1 ) } ; 0 < x < 2 , m$ and $n$ are integers, $m \neq 0 , n > 0$, and let $p$ be the left hand derivative of $| x - 1 |$ at $x = 1$.
If $\lim _ { x \rightarrow 1 + } g ( x ) = p$, then
(A) $n = 1 , m = 1$
(B) $n = 1 , m = - 1$
(C) $n = 2 , m = 2$
(D) $n > 2 , m = n$

Q6 Stationary points and optimisation Composite or piecewise function extremum analysis View

The total number of local maxima and local minima of the function $$f ( x ) = \begin{cases} ( 2 + x ) ^ { 3 } , & - 3 < x \leq - 1 \\ x ^ { 2 / 3 } , & - 1 < x < 2 \end{cases}$$ is
(A) 0
(B) 1
(C) 2
(D) 3

Q7 Circles Chord Length and Chord Properties View

A straight line through the vertex $P$ of a triangle $P Q R$ intersects the side $Q R$ at the point $S$ and the circumcircle of the triangle $P Q R$ at the point $T$. If $S$ is not the centre of the circumcircle, then
(A) $\frac { 1 } { P S } + \frac { 1 } { S T } < \frac { 2 } { \sqrt { Q S \times S R } }$
(B) $\frac { 1 } { P S } + \frac { 1 } { S T } > \frac { 2 } { \sqrt { Q S \times S R } }$
(C) $\frac { 1 } { P S } + \frac { 1 } { S T } < \frac { 4 } { Q R }$
(D) $\frac { 1 } { P S } + \frac { 1 } { S T } > \frac { 4 } { Q R }$

Q8 Conic sections Confocal or Related Conic Construction View

Let $P \left( x _ { 1 } , y _ { 1 } \right)$ and $Q \left( x _ { 2 } , y _ { 2 } \right) , y _ { 1 } < 0 , y _ { 2 } < 0$, be the end points of the latus rectum of the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$. The equations of parabolas with latus rectum $P Q$ are
(A) $x ^ { 2 } + 2 \sqrt { 3 } \quad y = 3 + \sqrt { 3 }$
(B) $x ^ { 2 } - 2 \sqrt { 3 } \quad y = 3 + \sqrt { 3 }$
(C) $x ^ { 2 } + 2 \sqrt { 3 } \quad y = 3 - \sqrt { 3 }$
(D) $x ^ { 2 } - 2 \sqrt { 3 } \quad y = 3 - \sqrt { 3 }$

Q9 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Let $$S _ { n } = \sum _ { k = 1 } ^ { n } \frac { n } { n ^ { 2 } + k n + k ^ { 2 } } \quad \text { and } \quad T _ { n } = \sum _ { k = 0 } ^ { n - 1 } \frac { n } { n ^ { 2 } + k n + k ^ { 2 } } ,$$ for $n = 1,2,3 , \cdots$. Then,
(A) $\quad S _ { n } < \frac { \pi } { 3 \sqrt { 3 } }$
(B) $\quad S _ { n } > \frac { \pi } { 3 \sqrt { 3 } }$
(C) $T _ { n } < \frac { \pi } { 3 \sqrt { 3 } }$
(D) $T _ { n } > \frac { \pi } { 3 \sqrt { 3 } }$

Q10 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

Let $f ( x )$ be a non-constant twice differentiable function defined on $( - \infty , \infty )$ such that $f ( x ) = f ( 1 - x )$ and $f ^ { \prime } \left( \frac { 1 } { 4 } \right) = 0$. Then,
(A) $f ^ { \prime \prime } ( x )$ vanishes at least twice on $[ 0,1 ]$
(B) $f ^ { \prime } \left( \frac { 1 } { 2 } \right) = 0$
(C) $\quad \int _ { - 1 / 2 } ^ { 1 / 2 } f \left( x + \frac { 1 } { 2 } \right) \sin x d x = 0$
(D) $\int _ { 0 } ^ { 1 / 2 } f ( t ) e ^ { \sin \pi t } d t = \int _ { 1 / 2 } ^ { 1 } f ( 1 - t ) e ^ { \sin \pi t } d t$

Q11 Product & Quotient Rules View

Let $f$ and $g$ be real valued functions defined on interval $( - 1,1 )$ such that $g ^ { \prime \prime } ( x )$ is continuous, $g ( 0 ) \neq 0 , g ^ { \prime } ( 0 ) = 0 , g ^ { \prime \prime } ( 0 ) \neq 0$, and $f ( x ) = g ( x ) \sin x$.
STATEMENT-1 : $\lim _ { x \rightarrow 0 } [ g ( x ) \cot x - g ( 0 ) \operatorname { cosec } x ] = f ^ { \prime \prime } ( 0 )$. and STATEMENT-2 : $\quad f ^ { \prime } ( 0 ) = g ( 0 )$.
(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

Q12 Vectors: Lines & Planes Coplanarity and Relative Position of Planes View

Consider three planes $$\begin{aligned} & P _ { 1 } : x - y + z = 1 \\ & P _ { 2 } : x + y - z = - 1 \\ & P _ { 3 } : x - 3 y + 3 z = 2 . \end{aligned}$$ Let $L _ { 1 } , L _ { 2 } , L _ { 3 }$ be the lines of intersection of the planes $P _ { 2 }$ and $P _ { 3 } , P _ { 3 }$ and $P _ { 1 }$, and $P _ { 1 }$ and $P _ { 2 }$, respectively.
STATEMENT-1: At least two of the lines $L _ { 1 } , L _ { 2 }$ and $L _ { 3 }$ are non-parallel. and STATEMENT-2 : The three planes do not have a common point.
(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

Q13 3x3 Matrices Linear System Existence and Uniqueness via Determinant View

Consider the system of equations $$\begin{aligned} & x - 2 y + 3 z = - 1 \\ & - x + y - 2 z = k \\ & x - 3 y + 4 z = 1 . \end{aligned}$$ STATEMENT-1 : The system of equations has no solution for $k \neq 3$. and STATEMENT-2 : The determinant $\left| \begin{array} { c c c } 1 & 3 & - 1 \\ - 1 & - 2 & k \\ 1 & 4 & 1 \end{array} \right| \neq 0$, for $k \neq 3$.
(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

Q14 Probability Definitions Verifying Statements About Probability Properties View

Consider the system of equations $a x + b y = 0 , c x + d y = 0$, where $a , b , c , d \in \{ 0,1 \}$. STATEMENT-1 : The probability that the system of equations has a unique solution is $\frac { 3 } { 8 }$. and STATEMENT-2 : The probability that the system of equations has a solution is 1.
(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

Q15 Circles Circle Equation Derivation View

A circle $C$ of radius 1 is inscribed in an equilateral triangle $P Q R$. The points of contact of $C$ with the sides $P Q , Q R , R P$ are $D , E , F$, respectively. The line $P Q$ is given by the equation $\sqrt { 3 } x + y - 6 = 0$ and the point $D$ is $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.
The equation of circle $C$ is
(A) $\quad ( x - 2 \sqrt { 3 } ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$
(B) $( x - 2 \sqrt { 3 } ) ^ { 2 } + \left( y + \frac { 1 } { 2 } \right) ^ { 2 } = 1$
(C) $\quad ( x - \sqrt { 3 } ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 1$
(D) $( x - \sqrt { 3 } ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$

Q16 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View

A circle $C$ of radius 1 is inscribed in an equilateral triangle $P Q R$. The points of contact of $C$ with the sides $P Q , Q R , R P$ are $D , E , F$, respectively. The line $P Q$ is given by the equation $\sqrt { 3 } x + y - 6 = 0$ and the point $D$ is $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.
Points $E$ and $F$ are given by
(A) $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right) , ( \sqrt { 3 } , 0 )$
(B) $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right) , ( \sqrt { 3 } , 0 )$
(C) $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right) , \left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right)$
(D) $\left( \frac { 3 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right) , \left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right)$

Q17 Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View

A circle $C$ of radius 1 is inscribed in an equilateral triangle $P Q R$. The points of contact of $C$ with the sides $P Q , Q R , R P$ are $D , E , F$, respectively. The line $P Q$ is given by the equation $\sqrt { 3 } x + y - 6 = 0$ and the point $D$ is $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.
Equations of the sides $Q R , R P$ are
(A) $y = \frac { 2 } { \sqrt { 3 } } x + 1 , y = - \frac { 2 } { \sqrt { 3 } } x - 1$
(B) $y = \frac { 1 } { \sqrt { 3 } } x , y = 0$
(C) $y = \frac { \sqrt { 3 } } { 2 } x + 1 , y = - \frac { \sqrt { 3 } } { 2 } x - 1$
(D) $y = \sqrt { 3 } x , y = 0$

Q18 Implicit equations and differentiation Second derivative via implicit differentiation View

Consider the functions defined implicitly by the equation $y ^ { 3 } - 3 y + x = 0$ on various intervals in the real line. If $x \in ( - \infty , - 2 ) \cup ( 2 , \infty )$, the equation implicitly defines a unique real valued differentiable function $y = f ( x )$. If $x \in ( - 2,2 )$, the equation implicitly defines a unique real valued differentiable function $y = g ( x )$ satisfying $g ( 0 ) = 0$.
If $f ( - 10 \sqrt { 2 } ) = 2 \sqrt { 2 }$, then $f ^ { \prime \prime } ( - 10 \sqrt { 2 } ) =$
(A) $\frac { 4 \sqrt { 2 } } { 7 ^ { 3 } 3 ^ { 2 } }$
(B) $- \frac { 4 \sqrt { 2 } } { 7 ^ { 3 } 3 ^ { 2 } }$
(C) $\frac { 4 \sqrt { 2 } } { 7 ^ { 3 } 3 }$
(D) $- \frac { 4 \sqrt { 2 } } { 7 ^ { 3 } 3 }$

Q19 Areas Between Curves Select Correct Integral Expression View

Consider the functions defined implicitly by the equation $y ^ { 3 } - 3 y + x = 0$ on various intervals in the real line. If $x \in ( - \infty , - 2 ) \cup ( 2 , \infty )$, the equation implicitly defines a unique real valued differentiable function $y = f ( x )$. If $x \in ( - 2,2 )$, the equation implicitly defines a unique real valued differentiable function $y = g ( x )$ satisfying $g ( 0 ) = 0$.
The area of the region bounded by the curve $y = f ( x )$, the $x$-axis, and the lines $x = a$ and $x = b$, where $- \infty < a < b < - 2$, is
(A) $\int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x + b f ( b ) - a f ( a )$
(B) $\quad - \int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x + b f ( b ) - a f ( a )$
(C) $\int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x - b f ( b ) + a f ( a )$
(D) $- \int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x - b f ( b ) + a f ( a )$

Q20 Indefinite & Definite Integrals Recovering Function Values from Derivative Information View

Consider the functions defined implicitly by the equation $y ^ { 3 } - 3 y + x = 0$ on various intervals in the real line. If $x \in ( - \infty , - 2 ) \cup ( 2 , \infty )$, the equation implicitly defines a unique real valued differentiable function $y = f ( x )$. If $x \in ( - 2,2 )$, the equation implicitly defines a unique real valued differentiable function $y = g ( x )$ satisfying $g ( 0 ) = 0$.
$\int _ { - 1 } ^ { 1 } g ^ { \prime } ( x ) d x =$
(A) $2 g ( - 1 )$
(B) 0
(C) $- 2 g ( 1 )$
(D) $2 g ( 1 )$

Q21 Complex Numbers Argand & Loci Intersection of Loci and Simultaneous Geometric Conditions View

Let $A , B , C$ be three sets of complex numbers as defined below $$\begin{aligned} & A = \{ z : \operatorname { Im } z \geq 1 \} \\ & B = \{ z : | z - 2 - i | = 3 \} \\ & C = \{ z : \operatorname { Re } ( ( 1 - i ) z ) = \sqrt { 2 } \} \end{aligned}$$ The number of elements in the set $A \cap B \cap C$ is
(A) 0
(B) 1
(C) 2
(D) $\infty$

Q22 Complex Numbers Argand & Loci Distance and Region Optimization on Loci View

Let $A , B , C$ be three sets of complex numbers as defined below $$\begin{aligned} & A = \{ z : \operatorname { Im } z \geq 1 \} \\ & B = \{ z : | z - 2 - i | = 3 \} \\ & C = \{ z : \operatorname { Re } ( ( 1 - i ) z ) = \sqrt { 2 } \} \end{aligned}$$ Let $z$ be any point in $A \cap B \cap C$. Then, $| z + 1 - i | ^ { 2 } + | z - 5 - i | ^ { 2 }$ lies between
(A) 25 and 29
(B) 30 and 34
(C) 35 and 39
(D) 40 and 44

Q23 Complex Numbers Argand & Loci Distance and Region Optimization on Loci View

Let $A , B , C$ be three sets of complex numbers as defined below $$\begin{aligned} & A = \{ z : \operatorname { Im } z \geq 1 \} \\ & B = \{ z : | z - 2 - i | = 3 \} \\ & C = \{ z : \operatorname { Re } ( ( 1 - i ) z ) = \sqrt { 2 } \} \end{aligned}$$ Let $z$ be any point in $A \cap B \cap C$ and let $w$ be any point satisfying $| w - 2 - i | < 3$. Then, $| z | - | w | + 3$ lies between
(A) -6 and 3
(B) - 3 and 6
(C) - 6 and 6
(D) - 3 and 9

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