# Quiz at GanitCharcha

Welcome to GanitCharcha's Quiz page. Our quizzes are not made to test one or to help one test how much Mathematics on a given topic he/she knows, rather it is purposefully designed to help people feel motivated to learn. Our quizzes will help to paint the construction of one's own mathematical understanding while instilling love for the subject. Under the broad name of Quiz, one can also find different types of Math problems to challenge his/her math mind.

## Quiz : Higher Secondary Level Quiz January 2015

1
Let $N$ be the number such that it is comprised of $81$ ones and $2$ zeros between each pair of ones. That is $N = 1001001001001\ldots1001$. The number $N$ is divisible by

2
Let $N = 80pq2pq$ be a $7$ digit number. If $N$ is exactly divisible by $90$ then the sum of the digits in N is equal to

3
If $2(y-a)$ is the Harmonic Mean (H.M) between $y - x$, $y - z$, then $x - a$, $y - a$ and $z - a$ are in

4
The sum of three distinct positive integers such that the sum of their reciprocals is an integer is

5
The sum of $$\frac{1}{2\sqrt{1} + 1\sqrt{2}} + \frac{1}{3\sqrt{2} + 2\sqrt{3}} + \ldots + \frac{1}{100\sqrt{99} + 99\sqrt{100}}$$ is

6
Find the remainder when $2^{75}$ is divided by $37$.

7
Given $f: N -> R$ such that $f(1) = 1$ and $$f(1) + 2f(2) + 3f(3) + \ldots + nf(n) = n(n+1)f(n),$$ then $f(2000)$ is

8
The coefficients of $x^{99}$ in $(x-1)(x-2)(x-3)\ldots(x-100)$ is

9
If $a^{x} = bc$, $b^{y} = ac$ and $c^{z} = ab$, then $$\frac{1}{1+x} + \frac{1}{1+y} + \frac{1}{1+z}$$ equals to

10
The value of $a$ for which the roots of the equation $2x^{2} - (a^{2} + 8a - 1)x + (a^{2} - 4a) = 0$ are of opposite signs is
(a) 1,    (b) -4,    (c) $\frac{\sqrt{65}}{2}$ and (d) None of these

11
If $f(x) = \sqrt{2}$ if $x$ is rational and $f(x) = 1$ if $x$ is irrtional, then $\phi(x) = [f(x)]$
(a) is discontinuous for all $x$
(b) is continuous for all $x$
(c) is differentiable for all $x$
(d) is a periodic function.

12
The values of $p$ for which $$f(x) = (\frac{\sqrt{p+4}}{1-p} -1)x^{5} - 3x + log 5$$ decreases for all real values of $x$ are
(a) $(-\infty, \infty)$, (b)$[-4, -1] \cup (1, \infty)$
(c) $[-3, 0) \cup (2, \infty)$ and (d)$(-4, \infty)$.

13
If $$(x + y)^{m+n} = x^{m}.y^{n}$$, then

(a) $\frac{dy}{dx}$ is independent of $m$ but dependent of $n$
(b) $\frac{dy}{dx} = \frac{m}{n}$
(c) $\frac{dy}{dx}$ is independent of $n$ but dependent of $m$
(d) $\frac{dy}{dx}$ does not dependent on $m$ or $n$.

$$\lim_{x \to 2} \frac{f(4) - f(x^{2}}{x - 2}$$ is (given $$f(x)$$ is differentiable and $f'(4) = 5$) equal to