Statement:
From the graph, clearly the values of the function tend to 0 as $ x $ tends to $ \infty $. The graph is smooth (no sharp edges or sharp turns) and so the graph has a unique tangent at all points (at $ x = 1 $ and $ x = -1 $ too). The function values are decreasing (slope of the curve is negative) in $[-0.5, 0]$ (Option 7).
Solution:
Based on the graph description:
Option 2: Values tend to 0 as $ x \to \infty $.
Option 5: Graph is smooth and has a unique tangent at all points.
Option 7: Function is decreasing in $[-0.5, 0]$ (slope is negative).
Question 2: Convergence of Sequences
Statement:
Statement 1: If ${a_n}$ converges to $a$ and ${b_n}$ converges to $b$, then ${a_n b_n}$ converges to $ab$.
Statement 2: If $a_n = (-1)^n$, $b_n = (-2)^n$, then ${a_n}$ and ${b_n}$ do not converge. But ${a_n b_n^3} = {(-1)^n (-2)^{3n}} = {8^n}$, which diverges. (Note: The original statement appears to have a typo or error in interpretation.)
Statement 3: $\lim a_n = \lim C = C$ where $C$ is a constant.
Solution:
Statement 1: True.
Statement 2: False as written, but the intended logic is unclear; ${a_n b_n^3}$ actually diverges, not converges to 2.
Statement 3: True.
Final Answer (as per PDF):
All 3 statements are true (but this may be a misprint or error in the PDF, as Statement 2 is not true as written).
Question 3: Subsequences and Limits
Statement:
Note that $\left{\frac{2n+1}{(2n+1)!}\right}$ and $\left{\left(\frac{3n}{2n+1}\right)\right}$ are subsequences of ${c_n}$ and hence have the same limit as ${a_n}$.
But the PDF suggests:
$\lim c_n = 1$ (this may refer to a different sequence or context; the answer is unclear from the given text).
Question 4: Subsequence Identification
Statement:
$a_n = \frac{1}{n}$ is a subsequence of ${ \ldots }$. Only one sequence is a subsequence of ${ \ldots }$. $c_n = 0$ cannot be a subsequence of ${ \ldots }$.
Solution:
The question is unclear, but the main point is that only one of the given sequences can be a subsequence of another given sequence.
But the PDF says the answer is 18, which does not match.
Question 7: Unique Tangent Condition
Statement:
If there is no sharp turn at $a$, then the graph has a unique tangent at $a$. Curves 1, 3, and 4 have sharp turns at 0 (Curve 4 has a jump at 0). Hence these three curves do not have a unique tangent at 0.
Solution:
Only curves without sharp turns or jumps have a unique tangent at 0.
Question 8: Cost Calculation
Statement:
$P_1(2.6) = (100 \times 2) + 200 = 400$
$P_2(2.6) = (100 \times 3) + 200 = 500$
$P_3(2.6) = (100 \times 2.6) + 200 = 460$
The lowest cost is Rs 400 by availing scheme A.
Solution:
Lowest cost is Rs 400 (Scheme A).
Question 9: Cost Schemes and Tangents
Statement:
Scheme A, B, C costs are shown. Clearly, tangents of $p_1(t)$ and $p_2(t)$ do not exist at $t = 1, 2, 3, \ldots$ (jumps). Tangent exists for all values of $t$ for $p_3(t)$.
Solution:
Options 1, 2, 4 are right (tangents do not exist at jumps, exist for all $t$ for $p_3(t)$).
Question 10: Piecewise Function
Statement:
Figure 2: $p_3(0) = 200$, $p_3(0.5) = 300$, etc. (No question is explicitly asked here.)
Solution:
No explicit question or solution is provided for this.
Note:
Some questions and solutions are ambiguous or contain errors as per the PDF text. The above is a faithful extraction based on the provided content1.
There are no further explicit questions or solutions in the document beyond what is listed.
