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For the following questions answer them individually
If $$0 > \alpha, \beta < \frac{\pi}{4}, \cos(\alpha + \beta) = \frac{4}{5}$$ and $$\sin(\alpha - \beta) = \frac{5}{13}$$ then $$\tan 2 \alpha =$$
If $$x = a \cos \theta + b \sin \theta$$ and $$y = a \sin \theta - b \cos \theta$$, then $$x^2 + y^2 =$$
$$\frac{\tan 10^\circ + \tan 50^\circ}{1 - \tan 10^\circ \tan 50^\circ} =$$
A man in a boat rowing away from the cliff 150 meters high, observed that it takes 2 minutes to change the angleofelevation of the top of the cliff from $$60^\circ$$ to $$45^\circ$$, then the speed of the boat in km/hour is
If the polynomial $$-a^2 x^3 - 2ax^2 + b^2 x + 1$$ is divisible by $$x + 1$$, then $$a - 1 =$$
A polynomial in x leaves remainders —1 and 7 when it is divided by x + 1 and x - 3 respectively. If that polynomial is divided by $$x^2 - 2x - 3$$, then the remainder is
If $$73 \times 74 \times 75 \times 76$$ is divided by 14, then the remainder is
The sum of the first 10 terms of the series 1, 3, 5, 7, ......... is
The coefficient of x in the expansion of $$\left(3x^2 - \frac{1}{2x}\right)^5$$ is
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