NSEJS-2017 (Maths Section)

Problem 1 of 20

How many four digit numbers are there such that when they are divided by $101$, they have $99$ as remainder?

Problem 2 of 20

$1\dfrac{1}{2} + 1\dfrac{1}{6} + 1\dfrac{1}{12} + 1\dfrac{1}{20} + 1\dfrac{1}{30} + ... + 1\dfrac{1}{380} = $________.

Problem 3 of 20

Diagonals of a quadrilateral bisect each other. Therefore, the quadrilateral must be:

Problem 4 of 20

By which smallest number should we divide $198396198$ to get a perfect square?

Problem 5 of 20

If $x^2 + xy + xz = 135$, $y^2 + yz + xy = 351$ and $z^2 + xz + yz = 243$, then $x^2 + y^2 + z^2 = $__________.

Problem 6 of 20

If $p + q + r = 2$, $p^2 + q^2 + r^2 = 30$ and $pqr = 10$, the value of $(1-p)(1-q)(1-r) = $______.

Problem 7 of 20

$\left(x + \dfrac{1}{x}\right) = 5$, then

$\left(x^3 + \dfrac{1}{x^3}\right) - 5\left(x^2 + \dfrac{1}{x^2}\right) + \left(x + \dfrac{1}{x}\right) = $_________.

Problem 8 of 20

If $x = (\sqrt{21} - \sqrt{20})$ and $y = (\sqrt{18} - \sqrt{17})$, then:

Problem 9 of 20

A train is travelling at a speed of $54\ km/hr$. It is not stopping at a certain station. It crosses the person showing green flag in $20$ seconds and crosses the platform in $36$ seconds. What is the length of the train?

Problem 10 of 20

If $(a + b + c + d) = 4$, then

$\dfrac{1}{(1-a)(1-b)(1-c)} + \dfrac{1}{(1-b)(1-c)(1-d)}$ $ + \dfrac{1}{(1-c)(1-d)(1-a)} + \dfrac{1}{(1-d)(1-a)(1-b)} = $_________.

Problem 11 of 20

What will be the remainder if the number $(7)^{2017}$ is divided by $25$?

Problem 12 of 20

What is the radius of the circumcircle of a triangle whose sides are $30\ cm$, $36\ cm$ and $30\ cm$.

Problem 13 of 20

The mean of the following frequency distribution is ____________.

$Class\ Interval$	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$
$Frequency$	$4$	$6$	$8$	$10$	$12$

Problem 14 of 20

If $x^2 – 3x + 2$ is a factor of $x^4 – px^2 + q$, then $p,\ q$ are:

Problem 15 of 20

What is the sum of all odd numbers between $500$ and $600$?

Problem 16 of 20

In $\triangle ABC$, segment $AD$, segment $BE$ and segment $CF$ are altitudes. If $AB \times AC = 172.8\ cm^2$ and $BE × CF = 108.3\ cm^2$ then $AD \times BC =$

Problem 17 of 20

The sum of two numbers is $13$ and the sum of their cubes is $1066$. Find the product of those two numbers.

Problem 18 of 20

If $ABCD$ is a cyclic quadrilateral, $AB = 204$, $BC = 104$, $CD= 195$, $DA = 85$ and $BD = 221$, then $AC =$_________.

Problem 19 of 20

The seventy first $Independence\ day$ was on a $Tuesday$. After how many years $Independence\ day$ will again be on a $Tuesday$?

(Note for international students: India became Independent on $15\xasuper{th}\ August, 1947$)

Problem 20 of 20

If the roots of the equation

$\dfrac{x^2 - bx}{ax - c} = \dfrac{m - 1}{m + 1}$ are equal and of opposite signs; then the value of $m$ is: