1. Problem 17: Find $k$ if $k+7$, $2k-2$, and $2k+6$ are three consecutive terms of an AP. Step 1: In an arithmetic progression (AP), the difference between consecutive terms is constant. Step 2: Set the common difference equal: $$ (2k - 2) - (k + 7) = (2k + 6) - (2k - 2) $$ Step 3: Simplify each side: $$ (2k - 2 - k - 7) = (2k + 6 - 2k + 2) $$ $$ (k - 9) = 8 $$ Step 4: Solve for $k$: $$ k - 9 = 8 \\ k = 17 $$ 2. Problem 18: Given zeros $\\alpha$ and $\beta$ of polynomial $3x^2 + 6x + k$ satisfy: $$ \alpha + \beta + \alpha\beta = - \frac{2}{3} $$ Step 1: From the quadratic, sum of zeros $\alpha + \beta = -\frac{b}{a} = -\frac{6}{3} = -2$. Step 2: Product of zeros $\alpha\beta = \frac{c}{a} = \frac{k}{3}$. Step 3: Substitute into given relation: $$ (-2) + \frac{k}{3} = - \frac{2}{3} $$ Step 4: Solve for $k$: $$ \frac{k}{3} = - \frac{2}{3} + 2 = - \frac{2}{3} + \frac{6}{3} = \frac{4}{3} \\ k = 4 $$ 3. Problem 19: Assertion (A): Common difference $d$ of AP with $a_n = 4n - 70$ is 4. Reason (R): $d = a_n - a_{n-1}$. Step 1: Find $d$ from $a_n$: $$ d = a_n - a_{n-1} = (4n - 70) - [4(n-1) - 70] = 4n - 70 - 4n +4 + 70 = 4 $$ Step 2: Both assertion and reason are true and reason correctly explains assertion. 4. Problem 20: Assertion (A): $$ \frac{2 \tan 30^\circ}{1 + \tan^2 30^\circ} = \sin 60^\circ $$ Reason (R): $\tan 30^\circ = \frac{\sqrt{3}}{3}$. Step 1: Calculate LHS: $$ \tan 30^\circ = \frac{\sqrt{3}}{3}, \tan^2 30^\circ = \frac{1}{3} $$ $$ \text{LHS} = \frac{2 (\frac{\sqrt{3}}{3})}{1 + \frac{1}{3}} = \frac{2 \frac{\sqrt{3}}{3}}{\frac{4}{3}} = \frac{2 \frac{\sqrt{3}}{3}}{\frac{4}{3}} = \frac{2 \sqrt{3}}{3} \cdot \frac{3}{4} = \frac{2 \sqrt{3}}{4} = \frac{\sqrt{3}}{2} $$ Step 2: RHS is $\sin 60^\circ = \frac{\sqrt{3}}{2}$. Step 3: Hence, both assertion and reason are true; reason is the correct explanation of assertion. Final Answers: 17. $k=17$ (option b) 18. $k=4$ (option c) 19. (a) Both assertion and reason true and reason correct explanation. 20. (a) Both assertion and reason true and reason correct explanation.