NSEJS-2018 (Maths Section)

Problem 1 of 20

Let $AB$ be a diameter of a circle $C_1$ of radius $30\ cm$ and with center $O$. Two circles $C_2$ and $C_3$ of radii $15\ cm$

and $10\ cm$ touch $C_1$ internally at $A$ and $B$ respectively. $A$ fourth circle $C_4$ touches $C_1$, $C_2$ and $C_3$. What is

the largest possible radius of $C_4$

Problem 2 of 20

A $5 \times 5 \times 5$ cube is built using unit cubes. How many different cuboids (that differ in at least one unit cube)

can be formed using the same number of unit cubes ?

Problem 3 of 20

What is the largest value of the positive integer $k$ such that $k$ divides $n^2 (n^2 – 1) (n^2 – n – 2)$ for every natural number $n$?

Problem 4 of 20

A person kept rolling a regular (six faced) die until one of the numbers appeared third time on the top. This happened in $12\xasuper{th}$ throw and the sum of all the numbers in $12$ throws was $46$. Which number appeared least number of times?

(Select all possible values)

Problem 5 of 20

In a square $ABCD$, a point $P$ is inside the square such that $ABP$ is an equilateral triangle. The segment $AP$

cuts the diagonal $BD$ in $E$. Suppose $AE = 2$. The area of $ABCD$ is:

Problem 6 of 20

Let $n$ be a positive integer not divisible by $6$. Suppose $n$ has $6$ positive divisors. The number of positive

divisors of $9n$ is:

Problem 7 of 20

The value of

$\dfrac{\sqrt{a + x} - \sqrt{a - x}}{\sqrt{a + x} + \sqrt{a - x}}$

when $x = \dfrac{2ab}{b^2 + 1}$

Problem 8 of 20

Two regular polygons of different number of sides are taken. In one of them, its sides are coloured red and diagonals are coloured green; in the other, sides are coloured green and diagonals are coloured red. Suppose there are $103$ red lines and $80$ green lines. The total number of sides the two polygons together have is:

Problem 9 of 20

A box contains some red and some yellow balls. If one red ball is removed, one seventh of the remaining

balls would be red ; if one yellow ball is removed, one-sixth of the remaining balls would be red. If $n$

denotes the total number of balls in the box, then the sum of the digits of $n$ is:

Problem 10 of 20

Let $ABCD$ be a rectangle. Let $X$ and $Y$ be points respectively on $AB$ and $CD$ such that

$AX : XB = 1 : 2 = CY : YD$. Join $AY$ and $CX$; let $BY$ intersect $CX$ in $K$; let $DX$ intersect $AY$ in $L$.

If $\dfrac{m}{n}$ denotes the ratio of the area $XKYL$ to that of $ABCD$, then $m + n$ equals:

Problem 11 of 20

Let $ABC$ be an equilateral triangle. The bisector of $\angle BAC$ meets the circumcircle of $ABC$ in $D$. Suppose

$DB + DC = 4$. The diameter of the circumcircle of $ABC$ is:

Problem 12 of 20

Let $T_k$ denote the $k\xasuper{th}$ term of an arithmetic progression. Suppose there are positive integers $m \ne n$ such

that $T_m = \dfrac{1}{n}$

and $T_n = \dfrac{1}{m}$

Then $T_{mn}$ equals:

Problem 13 of 20

In a triangle $ABC$, let $AD$ be the median from $A$; Let $E$ be a point on $AD$ such that $AE : ED = 1 : 2$; and let

$BE$ extended meets $AC$ in $F$. The ratio of $\dfrac{AF}{FC}$ is:

Problem 14 of 20

If $sin \theta$ and $cos \theta$ are roots of the equation $px^2 + qx +r = 0$, then:

Problem 15 of 20

For a regular $k$-sided polygon, let $\alpha (k)$ denotes its interior angle. Suppose $n \gt 4$ is such that $\alpha (n – 2)$,

$\alpha (n)$, $\alpha (n + 3)$ forms an arithmetic progression. The sum of digits of $n$ is

Problem 16 of 20

The sum of $5$ numbers in geometric progression is $24$. The sum of their reciprocals is $6$. The product of the

terms of the geometric progression is:

Problem 17 of 20

Digits $a$ and $b$ are such that the product $4a1 \times 25b$ is divisible by $36$ (in base $10$). The number of ordered

pairs $(a, b)$ is:

Problem 18 of 20

The integer closest to $\sqrt{111...1– 222...2}$ , where there are $2018$ ones and $1009$ twos, is:

Problem 19 of 20

In a triangle $ABC$, a point $D$ on $AB$ is such that $AD : AB = 1 : 4$ and $DE$ is parallel to $BC$ with $E$ on $AC$. Let $M$ and $N$ be the mid points of $DE$ and $BC$ respectively. What is the ratio of the area of the quadrilateral $BNMD$ to that of triangle $ABC$?

Problem 20 of 20

The number of distinct integers in the collection

$\left\lfloor\dfrac{10^2}{1} \right\rfloor$, $\left\lfloor\dfrac{10^2}{2} \right\rfloor$, $\left\lfloor\dfrac{10^2}{3} \right\rfloor$, $...$, $\left\lfloor\dfrac{10^2}{20} \right\rfloor$

where $\left\lfloor x \right\rfloor$ denotes the largest integer not exceeding $x$, is: