If f: R -> R satisfies f(x+y)=f(x)+f(y), for all x, y in R and f(1)=7, then the sum from r=1 to n of f(r) is: A 7n(n+1)/2 B 7n/2 C 7(n+1)/2 D 7n + (n+1)

Let $f: X \rightarrow Y$ be a function. Define a relation $\mathrm{R}$ in $X$ given by $R={(a, b): f(a)=f(b)}$. Examine if $\mathrm{R}$ is an equivalence relation.

• A reflexive

• B symmetric

• C transitive

• D an equivalence relation

A circuit connected to an AC source of EMF $e=e_{0} \sin (100 t)$ with $t$ in seconds gives a phase difference of $\frac{\pi}{4}$ between the EMF $e$ and current $i$. Which of the following circuits will exhibit this:

A. RC circuit with $R=1 k\Omega$ and $C=10 \mu F$
B. RL circuit with $R=1 k\Omega$ and $L=10 mH$
C. RC circuit with $R=1 k\Omega$ and $C=1 \mu F$
D. RL circuit with $R=1 k\Omega$ and $L=1 mH$

If $ e^{|\sin x|} + e^{-|\sin x|} + 4a = 0 $ will have exactly four different solutions in $[0,2\pi]$, then:

• A $ a \in \mathbb{R} $

• B $ a \in \left[-\frac{e}{4}, -\frac{1}{4}\right]$

• C $ a \in \left[\frac{-1-e^{2}}{4e}, \infty\right] $

• D no real value of $ a $ exists

If $f(x)$ is a differentiable function such that $F: \mathbb{R} \rightarrow \mathbb{R}$ and $f\left(\frac{1}{n}\right)=0$ for all $n \geq 1, n \in \mathbb{N}$, then: (A) $f(x)=0$ for all $x \in (0,1)$ (B) $f(0)=0=f'(0)$ (C) $f(0)=0$ but $f(0)$ may or may not be 0 (D) $|f(x)| \leq 1$ for all $x \in (0,1)$

If both the distinct roots of the equation $x^{2}-ax-b=0$ (a, b $\in$ R) are lying in between -2 and 2, then

A $|a|<2-\frac{b}{2}$ B $|a|>2-\frac{b}{2}$ C $|a|<4$ D $|a|>\frac{b}{2}-2$