Maths

Suppose y = y(x) be the solution curve to the differential equation $\frac{dy}{dx}-y=2-e^{-x}$ such that $\underset{x\to \infty }{lim}y\left(x\right)$ is finite. If a and b are respectively the x and y intercepts of the tangent to the curve at x = 0, then the value of a 4b is equal to…………

The statement $\left(\sim \left(p\iff \sim q\right)\right)\wedge qis:$

The plane passing through the line L : $\mathcal{l}x-y+3\left(1-\mathcal{l}\right)z=1,x+2y-z=2$ and perpendicular to the plane 3x + 2y + z = 6 is 3x 8y + 7z = 4. If is the acute angle between the line L and the y-axis, then 415 cos2 is equal to……….

 $tan\left(2ta{n}^{-1}\frac{1}{5}+se{c}^{-1}\frac{\sqrt[]{5}}{2}+2ta{n}^{-1}\frac{1}{8}\right)$ is equal to:

The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of these observations 12 and 10 were wrongly recorded. If $\sigma$ is the standard deviation of the data after omitting the two wrong observations from the data, then $38{\sigma }^2$ is equal to………

Let S = $\left\{\theta \in \left[0,2\pi \right]:8^{2si{n}^2\theta }+8^{2co{s}^2\theta }=16\right\}.$ Then $n\left(S\right)+{\displaystyle \sum _{\theta \in S}^{}\left(sec\left(\frac{\pi }{4}+2\theta \right)cosec\left(\frac{\pi }{4}+2\theta \right)\right)}$ is equal to:

If the sum of solutions of system of equations 2sin²q - cos2q = 0 and 2 cos²q + 3sinq = 0 in the interval [0, 2p] is kp, then k is equal to…….

  ${\displaystyle \sum _{k=1}^{10}\frac{k}{k^4+k^2+1}=\frac{m}{n};}$ where m and n are co-prime, then m + n is equal to…….

Let S = $\left\{\theta \in \left[0,2\pi \right]:8^{2si{n}^2\theta }+8^{2co{s}^2\theta }=16\right\}.$ Then $n\left(S\right)+{\displaystyle \sum _{\theta \in S}^{}\left(sec\left(\frac{\pi }{4}+2\theta \right)cosec\left(\frac{\pi }{4}+2\theta \right)\right)}$ is equal to:

Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits 1,2,3,4,5 and 6 without repetition of digits. Then the total number of such numbers is………