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Q41 Complex Numbers Arithmetic Trigonometric/Polar Form and De Moivre's Theorem View

For any integer $k$, let $\alpha _ { k } = \cos \left( \frac { k \pi } { 7 } \right) + i \sin \left( \frac { k \pi } { 7 } \right)$, where $i = \sqrt { - 1 }$. The value of the expression $\frac { \sum _ { k = 1 } ^ { 12 } \left| \alpha _ { k + 1 } - \alpha _ { k } \right| } { \sum _ { k = 1 } ^ { 3 } \left| \alpha _ { 4 k - 1 } - \alpha _ { 4 k - 2 } \right| }$ is

Q42 Arithmetic Sequences and Series Find Common Difference from Given Conditions View

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6:11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

Q43 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View

The coefficient of $x ^ { 9 }$ in the expansion of $( 1 + x ) \left( 1 + x ^ { 2 } \right) \left( 1 + x ^ { 3 } \right) \ldots \left( 1 + x ^ { 100 } \right)$ is

Q44 Circles Tangent Lines and Tangent Lengths View

Suppose that the foci of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 5 } = 1$ are $\left( f _ { 1 } , 0 \right)$ and $\left( f _ { 2 } , 0 \right)$ where $f _ { 1 } > 0$ and $f _ { 2 } < 0$. Let $P _ { 1 }$ and $P _ { 2 }$ be two parabolas with a common vertex at ( 0,0 ) and with foci at ( $f _ { 1 } , 0$ ) and ( $2 f _ { 2 } , 0$ ), respectively. Let $T _ { 1 }$ be a tangent to $P _ { 1 }$ which passes through ( $2 f _ { 2 } , 0$ ) and $T _ { 2 }$ be a tangent to $P _ { 2 }$ which passes through $\left( f _ { 1 } , 0 \right)$. If $m _ { 1 }$ is the slope of $T _ { 1 }$ and $m _ { 2 }$ is the slope of $T _ { 2 }$, then the value of $\left( \frac { 1 } { m _ { 1 } ^ { 2 } } + m _ { 2 } ^ { 2 } \right)$ is

Q45 Chain Rule Limit Evaluation Involving Composition or Substitution View

Let $m$ and $n$ be two positive integers greater than 1 . If
$$\lim _ { \alpha \rightarrow 0 } \left( \frac { e ^ { \cos \left( \alpha ^ { n } \right) } - e } { \alpha ^ { m } } \right) = - \left( \frac { e } { 2 } \right)$$
then the value of $\frac { m } { n }$ is

Q46 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View

If
$$\alpha = \int _ { 0 } ^ { 1 } \left( e ^ { 9 x + 3 \tan ^ { - 1 } x } \right) \left( \frac { 12 + 9 x ^ { 2 } } { 1 + x ^ { 2 } } \right) d x$$
where $\tan ^ { - 1 } x$ takes only principal values, then the value of $\left( \log _ { e } | 1 + \alpha | - \frac { 3 \pi } { 4 } \right)$ is

Q47 Indefinite & Definite Integrals Finding a Function from an Integral Equation View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous odd function, which vanishes exactly at one point and $f ( 1 ) = \frac { 1 } { 2 }$. Suppose that $F ( x ) = \int _ { - 1 } ^ { x } f ( t ) d t$ for all $x \in [ - 1,2 ]$ and $G ( x ) = \int _ { - 1 } ^ { x } t | f ( f ( t ) ) | d t$ for all $x \in [ - 1,2 ]$. If $\lim _ { x \rightarrow 1 } \frac { F ( x ) } { G ( x ) } = \frac { 1 } { 14 }$, then the value of $f \left( \frac { 1 } { 2 } \right)$ is

Q48 Vectors Introduction & 2D Expressing a Vector as a Linear Combination View

Suppose that $\vec { p } , \vec { q }$ and $\vec { r }$ are three non-coplanar vectors in $\mathbb { R } ^ { 3 }$. Let the components of a vector $\vec { s }$ along $\vec { p } , \vec { q }$ and $\vec { r }$ be 4,3 and 5 , respectively. If the components of this vector $\vec { s }$ along $( - \vec { p } + \vec { q } + \vec { r } ) , ( \vec { p } - \vec { q } + \vec { r } )$ and $( - \vec { p } - \vec { q } + \vec { r } )$ are $x , y$ and $z$, respectively, then the value of $2 x + y + z$ is

Q49 Discriminant and conditions for roots Parameter range for specific root conditions (location/count) View

Let $S$ be the set of all non-zero real numbers $\alpha$ such that the quadratic equation $\alpha x ^ { 2 } - x + \alpha = 0$ has two distinct real roots $x _ { 1 }$ and $x _ { 2 }$ satisfying the inequality $\left| x _ { 1 } - x _ { 2 } \right| < 1$. Which of the following intervals is(are) a subset(s) of $S$ ?
(A) $\left( - \frac { 1 } { 2 } , - \frac { 1 } { \sqrt { 5 } } \right)$
(B) $\left( - \frac { 1 } { \sqrt { 5 } } , 0 \right)$
(C) $\left( 0 , \frac { 1 } { \sqrt { 5 } } \right)$
(D) $\left( \frac { 1 } { \sqrt { 5 } } , \frac { 1 } { 2 } \right)$

Q50 Trig Graphs & Exact Values View

If $\alpha = 3 \sin ^ { - 1 } \left( \frac { 6 } { 11 } \right)$ and $\beta = 3 \cos ^ { - 1 } \left( \frac { 4 } { 9 } \right)$, where the inverse trigonometric functions take only the principal values, then the correct option(s) is(are)
(A) $\quad \cos \beta > 0$
(B) $\quad \sin \beta < 0$
(C) $\quad \cos ( \alpha + \beta ) > 0$
(D) $\quad \cos \alpha < 0$

Q51 Circles Tangent Lines and Tangent Lengths View

Let $E _ { 1 }$ and $E _ { 2 }$ be two ellipses whose centers are at the origin. The major axes of $E _ { 1 }$ and $E _ { 2 }$ lie along the $x$-axis and the $y$-axis, respectively. Let $S$ be the circle $x ^ { 2 } + ( y - 1 ) ^ { 2 } = 2$. The straight line $x + y = 3$ touches the curves $S , E _ { 1 }$ and $E _ { 2 }$ at $P , Q$ and $R$, respectively. Suppose that $P Q = P R = \frac { 2 \sqrt { 2 } } { 3 }$. If $e _ { 1 }$ and $e _ { 2 }$ are the eccentricities of $E _ { 1 }$ and $E _ { 2 }$, respectively, then the correct expression(s) is(are)
(A) $e _ { 1 } ^ { 2 } + e _ { 2 } ^ { 2 } = \frac { 43 } { 40 }$
(B) $\quad e _ { 1 } e _ { 2 } = \frac { \sqrt { 7 } } { 2 \sqrt { 10 } }$
(C) $\left| e _ { 1 } ^ { 2 } - e _ { 2 } ^ { 2 } \right| = \frac { 5 } { 8 }$
(D) $e _ { 1 } e _ { 2 } = \frac { \sqrt { 3 } } { 4 }$

Q52 Stationary points and optimisation Geometric or applied optimisation problem View

Consider the hyperbola $H : x ^ { 2 } - y ^ { 2 } = 1$ and a circle $S$ with center $N \left( x _ { 2 } , 0 \right)$. Suppose that $H$ and $S$ touch each other at a point $P \left( x _ { 1 } , y _ { 1 } \right)$ with $x _ { 1 } > 1$ and $y _ { 1 } > 0$. The common tangent to $H$ and $S$ at $P$ intersects the $x$-axis at point $M$. If ( $l , m$ ) is the centroid of the triangle $\triangle P M N$, then the correct expression(s) is(are)
(A) $\frac { d l } { d x _ { 1 } } = 1 - \frac { 1 } { 3 x _ { 1 } ^ { 2 } }$ for $x _ { 1 } > 1$
(B) $\frac { d m } { d x _ { 1 } } = \frac { x _ { 1 } } { 3 \left( \sqrt { x _ { 1 } ^ { 2 } - 1 } \right) }$ for $x _ { 1 } > 1$
(C) $\frac { d l } { d x _ { 1 } } = 1 + \frac { 1 } { 3 x _ { 1 } ^ { 2 } }$ for $x _ { 1 } > 1$
(D) $\frac { d m } { d y _ { 1 } } = \frac { 1 } { 3 }$ for $y _ { 1 } > 0$

Q53 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

The option(s) with the values of $a$ and $L$ that satisfy the following equation is(are)
$$\frac { \int _ { 0 } ^ { 4 \pi } e ^ { t } \left( \sin ^ { 6 } a t + \cos ^ { 4 } a t \right) d t } { \int _ { 0 } ^ { \pi } e ^ { t } \left( \sin ^ { 6 } a t + \cos ^ { 4 } a t \right) d t } = L ?$$
(A) $\quad a = 2 , L = \frac { e ^ { 4 \pi } - 1 } { e ^ { \pi } - 1 }$
(B) $\quad a = 2 , L = \frac { e ^ { 4 \pi } + 1 } { e ^ { \pi } + 1 }$
(C) $\quad a = 4 , L = \frac { e ^ { 4 \pi } - 1 } { e ^ { \pi } - 1 }$
(D) $\quad a = 4 , L = \frac { e ^ { 4 \pi } + 1 } { e ^ { \pi } + 1 }$

Q54 Stationary points and optimisation Existence or properties of extrema via abstract/theoretical argument View

Let $f , g : [ - 1,2 ] \rightarrow \mathbb { R }$ be continuous functions which are twice differentiable on the interval $( - 1,2 )$. Let the values of $f$ and $g$ at the points $- 1,0$ and 2 be as given in the following table:

	$x = - 1$	$x = 0$	$x = 2$
$f ( x )$	3	6	0
$g ( x )$	0	1	- 1

In each of the intervals $( - 1,0 )$ and $( 0,2 )$ the function $( f - 3 g ) ^ { \prime \prime }$ never vanishes. Then the correct statement(s) is(are)
(A) $\quad f ^ { \prime } ( x ) - 3 g ^ { \prime } ( x ) = 0$ has exactly three solutions in $( - 1,0 ) \cup ( 0,2 )$
(B) $f ^ { \prime } ( x ) - 3 g ^ { \prime } ( x ) = 0$ has exactly one solution in $( - 1,0 )$
(C) $f ^ { \prime } ( x ) - 3 g ^ { \prime } ( x ) = 0$ has exactly one solution in $( 0,2 )$
(D) $f ^ { \prime } ( x ) - 3 g ^ { \prime } ( x ) = 0$ has exactly two solutions in ( $- 1,0$ ) and exactly two solutions in ( 0,2 )

Q55 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

Let $f ( x ) = 7 \tan ^ { 8 } x + 7 \tan ^ { 6 } x - 3 \tan ^ { 4 } x - 3 \tan ^ { 2 } x$ for all $x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$. Then the correct expression(s) is(are)
(A) $\quad \int _ { 0 } ^ { \pi / 4 } x f ( x ) d x = \frac { 1 } { 12 }$
(B) $\quad \int _ { 0 } ^ { \pi / 4 } f ( x ) d x = 0$
(C) $\int _ { 0 } ^ { \pi / 4 } x f ( x ) d x = \frac { 1 } { 6 }$
(D) $\int _ { 0 } ^ { \pi / 4 } f ( x ) d x = 1$

Q56 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

Let $f ^ { \prime } ( x ) = \frac { 192 x ^ { 3 } } { 2 + \sin ^ { 4 } \pi x }$ for all $x \in \mathbb { R }$ with $f \left( \frac { 1 } { 2 } \right) = 0$. If $m \leq \int _ { 1 / 2 } ^ { 1 } f ( x ) d x \leq M$, then the possible values of $m$ and $M$ are
(A) $m = 13 , M = 24$
(B) $\quad m = \frac { 1 } { 4 } , M = \frac { 1 } { 2 }$
(C) $m = - 11 , M = 0$
(D) $m = 1 , M = 12$

Q57 Probability Definitions Conditional Probability and Bayes' Theorem View

One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box II is $\frac { 1 } { 3 }$, then the correct option(s) with the possible values of $n _ { 1 } , n _ { 2 } , n _ { 3 }$ and $n _ { 4 }$ is(are)
(A) $n _ { 1 } = 3 , n _ { 2 } = 3 , n _ { 3 } = 5 , n _ { 4 } = 15$
(B) $n _ { 1 } = 3 , n _ { 2 } = 6 , n _ { 3 } = 10 , n _ { 4 } = 50$
(C) $n _ { 1 } = 8 , n _ { 2 } = 6 , n _ { 3 } = 5 , n _ { 4 } = 20$
(D) $n _ { 1 } = 6 , n _ { 2 } = 12 , n _ { 3 } = 5 , n _ { 4 } = 20$

Q58 Probability Definitions Conditional Probability and Bayes' Theorem View

A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is $\frac { 1 } { 3 }$, then the correct option(s) with the possible values of $n _ { 1 }$ and $n _ { 2 }$ is(are)
(A) $\quad n _ { 1 } = 4$ and $n _ { 2 } = 6$
(B) $\quad n _ { 1 } = 2$ and $n _ { 2 } = 3$
(C) $n _ { 1 } = 10$ and $n _ { 2 } = 20$
(D) $n _ { 1 } = 3$ and $n _ { 2 } = 6$

Q59 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View

Let $F : \mathbb { R } \rightarrow \mathbb { R }$ be a thrice differentiable function. Suppose that $F ( 1 ) = 0 , F ( 3 ) = - 4$ and $F ^ { \prime } ( x ) < 0$ for all $x \in ( 1 / 2,3 )$. Let $f ( x ) = x F ( x )$ for all $x \in \mathbb { R }$. The correct statement(s) is(are)
(A) $f ^ { \prime } ( 1 ) < 0$
(B) $f ( 2 ) < 0$
(C) $f ^ { \prime } ( x ) \neq 0$ for any $x \in ( 1,3 )$
(D) $f ^ { \prime } ( x ) = 0$ for some $x \in ( 1,3 )$

Q60 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

If $\int _ { 1 } ^ { 3 } x ^ { 2 } F ^ { \prime \prime } ( x ) d x = - 12$ and $\int _ { 1 } ^ { 3 } x ^ { 3 } F ^ { \prime \prime } ( x ) d x = 40$, then the correct expression(s) is(are)
(A) $9 f ^ { \prime } ( 3 ) + f ^ { \prime } ( 1 ) - 32 = 0$
(B) $\int _ { 1 } ^ { 3 } f ( x ) d x = 12$
(C) $9 f ^ { \prime } ( 3 ) - f ^ { \prime } ( 1 ) + 32 = 0$
(D) $\int _ { 1 } ^ { 3 } f ( x ) d x = - 12$

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