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19 maths questions

Q1 Combinations & Selection Basic Combination Computation View
Let $$\begin{gathered} S _ { 1 } = \{ ( i , j , k ) : i , j , k \in \{ 1,2 , \ldots , 10 \} \} , \\ S _ { 2 } = \{ ( i , j ) : 1 \leq i < j + 2 \leq 10 , i , j \in \{ 1,2 , \ldots , 10 \} \} , \\ S _ { 3 } = \{ ( i , j , k , l ) : 1 \leq i < j < k < l , \quad i , j , k , l \in \{ 1,2 , \ldots , 10 \} \} \end{gathered}$$ and $$S _ { 4 } = \{ ( i , j , k , l ) : i , j , k \text { and } l \text { are distinct elements in } \{ 1,2 , \ldots , 10 \} \} .$$ If the total number of elements in the set $S _ { r }$ is $n _ { r } , r = 1,2,3,4$, then which of the following statements is (are) TRUE ?
(A) $n _ { 1 } = 1000$
(B) $n _ { 2 } = 44$
(C) $n _ { 3 } = 220$
(D) $\frac { n _ { 4 } } { 12 } = 420$
Q2 Sine and Cosine Rules Prove an inequality or ordering relationship in a triangle View
Consider a triangle $P Q R$ having sides of lengths $p , q$ and $r$ opposite to the angles $P , Q$ and $R$, respectively. Then which of the following statements is (are) TRUE ?
(A) $\cos P \geq 1 - \frac { p ^ { 2 } } { 2 q r }$
(B) $\cos R \geq \left( \frac { q - r } { p + q } \right) \cos P + \left( \frac { p - r } { p + q } \right) \cos Q$
(C) $\frac { q + r } { p } < 2 \frac { \sqrt { \sin Q \sin R } } { \sin P }$
(D) If $p < q$ and $p < r$, then $\cos Q > \frac { p } { r }$ and $\cos R > \frac { p } { q }$
Q3 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Let $f : \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right] \rightarrow \mathbb { R }$ be a continuous function such that $$f ( 0 ) = 1 \text { and } \int _ { 0 } ^ { \frac { \pi } { 3 } } f ( t ) d t = 0$$ Then which of the following statements is (are) TRUE ?
(A) The equation $f ( x ) - 3 \cos 3 x = 0$ has at least one solution in $\left( 0 , \frac { \pi } { 3 } \right)$
(B) The equation $f ( x ) - 3 \sin 3 x = - \frac { 6 } { \pi }$ has at least one solution in $\left( 0 , \frac { \pi } { 3 } \right)$
(C) $\lim _ { x \rightarrow 0 } \frac { x \int _ { 0 } ^ { x } f ( t ) d t } { 1 - e ^ { x ^ { 2 } } } = - 1$
(D) $\lim _ { x \rightarrow 0 } \frac { \sin x \int _ { 0 } ^ { x } f ( t ) d t } { x ^ { 2 } } = - 1$
Q4 First order differential equations (integrating factor) View
For any real numbers $\alpha$ and $\beta$, let $y _ { \alpha , \beta } ( x ) , x \in \mathbb { R }$, be the solution of the differential equation $$\frac { d y } { d x } + \alpha y = x e ^ { \beta x } , \quad y ( 1 ) = 1$$ Let $S = \left\{ y _ { \alpha , \beta } ( x ) : \alpha , \beta \in \mathbb { R } \right\}$. Then which of the following functions belong(s) to the set $S$ ?
(A) $f ( x ) = \frac { x ^ { 2 } } { 2 } e ^ { - x } + \left( e - \frac { 1 } { 2 } \right) e ^ { - x }$
(B) $f ( x ) = - \frac { x ^ { 2 } } { 2 } e ^ { - x } + \left( e + \frac { 1 } { 2 } \right) e ^ { - x }$
(C) $f ( x ) = \frac { e ^ { x } } { 2 } \left( x - \frac { 1 } { 2 } \right) + \left( e - \frac { e ^ { 2 } } { 4 } \right) e ^ { - x }$
(D) $f ( x ) = \frac { e ^ { x } } { 2 } \left( \frac { 1 } { 2 } - x \right) + \left( e + \frac { e ^ { 2 } } { 4 } \right) e ^ { - x }$
Q5 Vectors Introduction & 2D True/False or Multiple-Statement Verification View
Let $O$ be the origin and $\overrightarrow { O A } = 2 \hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + \hat { \mathrm { k } } , \quad \overrightarrow { O B } = \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + 2 \hat { \mathrm { k } }$ and $\overrightarrow { O C } = \frac { 1 } { 2 } ( \overrightarrow { O B } - \lambda \overrightarrow { O A } )$ for some $\lambda > 0$. If $| \overrightarrow { O B } \times \overrightarrow { O C } | = \frac { 9 } { 2 }$, then which of the following statements is (are) TRUE ?
(A) Projection of $\overrightarrow { O C }$ on $\overrightarrow { O A }$ is $- \frac { 3 } { 2 }$
(B) Area of the triangle $O A B$ is $\frac { 9 } { 2 }$
(C) Area of the triangle $A B C$ is $\frac { 9 } { 2 }$
(D) The acute angle between the diagonals of the parallelogram with adjacent sides $\overrightarrow { O A }$ and $\overrightarrow { O C }$ is $\frac { \pi } { 3 }$
Q6 Circles Tangent Lines and Tangent Lengths View
Let $E$ denote the parabola $y ^ { 2 } = 8 x$. Let $P = ( - 2,4 )$, and let $Q$ and $Q ^ { \prime }$ be two distinct points on $E$ such that the lines $P Q$ and $P Q ^ { \prime }$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the following statements is (are) TRUE ?
(A) The triangle $P F Q$ is a right-angled triangle
(B) The triangle $Q P Q ^ { \prime }$ is a right-angled triangle
(C) The distance between $P$ and $F$ is $5 \sqrt { 2 }$
(D) $F$ lies on the line joining $Q$ and $Q ^ { \prime }$
Q7 Circles Optimization on a Circle View
Consider the region $R = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x \geq 0 \right.$ and $\left. y ^ { 2 } \leq 4 - x \right\}$. Let $\mathcal { F }$ be the family of all circles that are contained in $R$ and have centers on the $x$-axis. Let $C$ be the circle that has largest radius among the circles in $\mathcal { F }$. Let $( \alpha , \beta )$ be a point where the circle $C$ meets the curve $y ^ { 2 } = 4 - x$. The radius of the circle $C$ is $\_\_\_\_$.
Q8 Circles Optimization on a Circle View
Consider the region $R = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x \geq 0 \right.$ and $\left. y ^ { 2 } \leq 4 - x \right\}$. Let $\mathcal { F }$ be the family of all circles that are contained in $R$ and have centers on the $x$-axis. Let $C$ be the circle that has largest radius among the circles in $\mathcal { F }$. Let $( \alpha , \beta )$ be a point where the circle $C$ meets the curve $y ^ { 2 } = 4 - x$. The value of $\alpha$ is $\_\_\_\_$.
Q9 Stationary points and optimisation Find critical points and classify extrema of a given function View
Let $f _ { 1 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ and $f _ { 2 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ be defined by $$f _ { 1 } ( x ) = \int _ { 0 } ^ { x } \prod _ { j = 1 } ^ { 21 } ( t - j ) ^ { j } d t , \quad x > 0$$ and $$f _ { 2 } ( x ) = 98 ( x - 1 ) ^ { 50 } - 600 ( x - 1 ) ^ { 49 } + 2450 , \quad x > 0$$ where, for any positive integer $n$ and real numbers $a _ { 1 } , a _ { 2 } , \ldots , a _ { n } , \prod _ { i = 1 } ^ { n } a _ { i }$ denotes the product of $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$. Let $m _ { i }$ and $n _ { i }$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f _ { i } , i = 1,2$, in the interval $( 0 , \infty )$. The value of $2 m _ { 1 } + 3 n _ { 1 } + m _ { 1 } n _ { 1 }$ is $\_\_\_\_$.
Q10 Stationary points and optimisation Find critical points and classify extrema of a given function View
Let $f _ { 1 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ and $f _ { 2 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ be defined by $$f _ { 1 } ( x ) = \int _ { 0 } ^ { x } \prod _ { j = 1 } ^ { 21 } ( t - j ) ^ { j } d t , \quad x > 0$$ and $$f _ { 2 } ( x ) = 98 ( x - 1 ) ^ { 50 } - 600 ( x - 1 ) ^ { 49 } + 2450 , \quad x > 0$$ where, for any positive integer $n$ and real numbers $a _ { 1 } , a _ { 2 } , \ldots , a _ { n } , \prod _ { i = 1 } ^ { n } a _ { i }$ denotes the product of $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$. Let $m _ { i }$ and $n _ { i }$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f _ { i } , i = 1,2$, in the interval $( 0 , \infty )$. The value of $6 m _ { 2 } + 4 n _ { 2 } + 8 m _ { 2 } n _ { 2 }$ is $\_\_\_\_$.
Q11 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Let $g _ { i } : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R } , i = 1,2$, and $f : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R }$ be functions such that $$g _ { 1 } ( x ) = 1 , g _ { 2 } ( x ) = | 4 x - \pi | \text { and } f ( x ) = \sin ^ { 2 } x , \text { for all } x \in \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right]$$ Define $$S _ { i } = \int _ { \frac { \pi } { 8 } } ^ { \frac { 3 \pi } { 8 } } f ( x ) \cdot g _ { i } ( x ) d x , \quad i = 1,2$$ The value of $\frac { 16 S _ { 1 } } { \pi }$ is $\_\_\_\_$.
Q12 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Let $g _ { i } : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R } , i = 1,2$, and $f : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R }$ be functions such that $$g _ { 1 } ( x ) = 1 , g _ { 2 } ( x ) = | 4 x - \pi | \text { and } f ( x ) = \sin ^ { 2 } x , \text { for all } x \in \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right]$$ Define $$S _ { i } = \int _ { \frac { \pi } { 8 } } ^ { \frac { 3 \pi } { 8 } } f ( x ) \cdot g _ { i } ( x ) d x , \quad i = 1,2$$ The value of $\frac { 48 S _ { 2 } } { \pi ^ { 2 } }$ is $\_\_\_\_$.
Q13 Circles Infinite Series or Sequences Involving Circles View
Let $$M = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x ^ { 2 } + y ^ { 2 } \leq r ^ { 2 } \right\} ,$$ where $r > 0$. Consider the geometric progression $a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } , n = 1,2,3 , \ldots$. Let $S _ { 0 } = 0$ and, for $n \geq 1$, let $S _ { n }$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$, let $C _ { n }$ denote the circle with center ( $S _ { n - 1 } , 0$ ) and radius $a _ { n }$, and $D _ { n }$ denote the circle with center ( $S _ { n - 1 } , S _ { n - 1 }$ ) and radius $a _ { n }$. Consider $M$ with $r = \frac { 1025 } { 513 }$. Let $k$ be the number of all those circles $C _ { n }$ that are inside $M$. Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then
(A) $k + 2 l = 22$
(B) $2 k + l = 26$
(C) $2 k + 3 l = 34$
(D) $3 k + 2 l = 40$
Q14 Circles Infinite Series or Sequences Involving Circles View
Let $$M = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x ^ { 2 } + y ^ { 2 } \leq r ^ { 2 } \right\} ,$$ where $r > 0$. Consider the geometric progression $a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } , n = 1,2,3 , \ldots$. Let $S _ { 0 } = 0$ and, for $n \geq 1$, let $S _ { n }$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$, let $C _ { n }$ denote the circle with center ( $S _ { n - 1 } , 0$ ) and radius $a _ { n }$, and $D _ { n }$ denote the circle with center ( $S _ { n - 1 } , S _ { n - 1 }$ ) and radius $a _ { n }$. Consider $M$ with $r = \frac { \left( 2 ^ { 199 } - 1 \right) \sqrt { 2 } } { 2 ^ { 198 } }$. The number of all those circles $D _ { n }$ that are inside $M$ is
(A) 198
(B) 199
(C) 200
(D) 201
Q15 Stationary points and optimisation Prove an inequality using calculus-based optimisation View
Let $\psi _ { 1 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad \psi _ { 2 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad f : [ 0 , \infty ) \rightarrow \mathbb { R }$ and $g : [ 0 , \infty ) \rightarrow \mathbb { R }$ be functions such that $f ( 0 ) = g ( 0 ) = 0$, $$\begin{gathered} \psi _ { 1 } ( x ) = e ^ { - x } + x , \quad x \geq 0 \\ \psi _ { 2 } ( x ) = x ^ { 2 } - 2 x - 2 e ^ { - x } + 2 , \quad x \geq 0 \\ f ( x ) = \int _ { - x } ^ { x } \left( | t | - t ^ { 2 } \right) e ^ { - t ^ { 2 } } d t , \quad x > 0 \end{gathered}$$ and $$g ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \sqrt { t } e ^ { - t } d t , \quad x > 0$$ Which of the following statements is TRUE ?
(A) $f ( \sqrt { \ln 3 } ) + g ( \sqrt { \ln 3 } ) = \frac { 1 } { 3 }$
(B) For every $x > 1$, there exists an $\alpha \in ( 1 , x )$ such that $\psi _ { 1 } ( x ) = 1 + \alpha x$
(C) For every $x > 0$, there exists a $\beta \in ( 0 , x )$ such that $\psi _ { 2 } ( x ) = 2 x \left( \psi _ { 1 } ( \beta ) - 1 \right)$
(D) $f$ is an increasing function on the interval $\left[ 0 , \frac { 3 } { 2 } \right]$
Q16 Stationary points and optimisation Prove an inequality using calculus-based optimisation View
Let $\psi _ { 1 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad \psi _ { 2 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad f : [ 0 , \infty ) \rightarrow \mathbb { R }$ and $g : [ 0 , \infty ) \rightarrow \mathbb { R }$ be functions such that $f ( 0 ) = g ( 0 ) = 0$, $$\begin{gathered} \psi _ { 1 } ( x ) = e ^ { - x } + x , \quad x \geq 0 \\ \psi _ { 2 } ( x ) = x ^ { 2 } - 2 x - 2 e ^ { - x } + 2 , \quad x \geq 0 \\ f ( x ) = \int _ { - x } ^ { x } \left( | t | - t ^ { 2 } \right) e ^ { - t ^ { 2 } } d t , \quad x > 0 \end{gathered}$$ and $$g ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \sqrt { t } e ^ { - t } d t , \quad x > 0$$ Which of the following statements is TRUE ?
(A) $\psi _ { 1 } ( x ) \leq 1$, for all $x > 0$
(B) $\psi _ { 2 } ( x ) \leq 0$, for all $x > 0$
(C) $f ( x ) \geq 1 - e ^ { - x ^ { 2 } } - \frac { 2 } { 3 } x ^ { 3 } + \frac { 2 } { 5 } x ^ { 5 }$, for all $x \in \left( 0 , \frac { 1 } { 2 } \right)$
(D) $g ( x ) \leq \frac { 2 } { 3 } x ^ { 3 } - \frac { 2 } { 5 } x ^ { 5 } + \frac { 1 } { 7 } x ^ { 7 }$, for all $x \in \left( 0 , \frac { 1 } { 2 } \right)$
Q17 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
A number is chosen at random from the set $\{ 1,2,3 , \ldots , 2000 \}$. Let $p$ be the probability that the chosen number is a multiple of 3 or a multiple of 7 . Then the value of $500 p$ is $\_\_\_\_$.
Q18 Circles Chord Length and Chord Properties View
Let $E$ be the ellipse $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$. For any three distinct points $P , Q$ and $Q ^ { \prime }$ on $E$, let $M ( P , Q )$ be the mid-point of the line segment joining $P$ and $Q$, and $M \left( P , Q ^ { \prime } \right)$ be the mid-point of the line segment joining $P$ and $Q ^ { \prime }$. Then the maximum possible value of the distance between $M ( P , Q )$ and $M \left( P , Q ^ { \prime } \right)$, as $P , Q$ and $Q ^ { \prime }$ vary on $E$, is $\_\_\_\_$.
Q19 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
For any real number $x$, let $[ x ]$ denote the largest integer less than or equal to $x$. If $$I = \int _ { 0 } ^ { 10 } \left[ \sqrt { \frac { 10 x } { x + 1 } } \right] d x$$ then the value of $9 I$ is $\_\_\_\_$.