\( \displaystyle \frac{\cos(2a)}{1\cdot 3}+\frac{\cos(4a)}{3\cdot 5}+\frac{\cos(6a)}{5\cdot 7}+\cdots \cdots  \)

Let \( R(x) \) be remainder when

\( P(x)=x^{73}-2x^6+3x^3 \) is divided by

\( x^4(x^2-1) \).Then the number of real

roots of the equation \( R''(x)=0 \), is

If  \( a,b,c>0 \) and \(  a+b+c=1 \). Then maximum value of \( (1-a)(2-b)(3-c) \)

The expression \[ \displaystyle \sum^{2017}_{n=0}\bigg[\sin(2^{n+1}x)\prod^{n}_{k=0}\cos^2(2^kx)\bigg] \]

The number of positive integers \[ x,y,z \] satisfying \[ x^{y^{z}}\cdot y^{z^{x}}\cdot z^{x^{y}}=5xyz \]

Finding all natural number ordered pair \[ (x,y) \] in \[ \displaystyle \binom{x}{y}=2023 \]

Evaluation of \[ \sum^{n-1}_{k=1}\sin\bigg(\frac{k\pi}{n}\bigg) \] and using that result to find

\[ \displaystyle \int^{\frac{\pi}{2}}_0\ln(\sin(x))dx \]

If \[ \displaystyle a_{k}=\bigg(\frac{k!}{1\cdot 3\cdot 5\cdots (2k+1)}\bigg)^2 \].

And  \[ m=\lim_{n\rightarrow \infty}\bigg(a_1+a_2+a_3+\cdots a_n\bigg) \]. Then

Options :

(a) : \[ \displaystyle m>\frac{4}{27} \]

(b) : \[ \displaystyle m < \frac{4}{27} \]

(c) : DNE (does not exists)

(d) : None

If  \( \displaystyle p,q,r \) are roots of \( x^3+3x^2-1=0 \).

Then \( p^2q+q^2r+r^2p= \)

Evaluation of \( \displaystyle \int^\infty_0\frac{x+1}{x+2}\cdot \frac{x+3}{x+4}\cdot \frac{x+5}{x+6}\cdots\cdots dx \)