Sequences & Series1 passages

Sequences & Series — JEE Maths Comprehension Passages & Solutions

1 reading-comprehension sets on Sequences & Series with full step-by-step solutions, including past-year (PYQ) passages. Free to practice.

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Let {tn}n∈N\{t_n\}_{n \in \mathbb{N}} be an arithmetic progression with non-zero common difference dd, and let Sn=∑i=1ntiS_n = \displaystyle\sum_{i=1}^n t_i denote the sum of its first nn terms. The partial sums satisfy the strict inequalities S6>S7>S5S_6 > S_7 > S_5, and the terms satisfy the identity 2(t6+t7)=d2(t_6 + t_7) = d. Let kk be the uniquely determined positive integer satisfying SkSk+1<0S_k S_{k+1} < 0. Consider a real-valued function f:R→Rf:\mathbb{R} \to \mathbb{R} defined as: f(x)={(sin⁡ ⁣(kx22)+cos⁡(ax)ebx2) ⁣1x2,x≠0L,x=0f(x) = \begin{cases} \left(\dfrac{\sin\!\left(\dfrac{kx^2}{2}\right)+\cos(ax)}{e^{bx^2}}\right)^{\!\frac{1}{x^2}}, & x \neq 0 \\[8pt] L, & x = 0 \end{cases} where a∈(0,∞)a \in (0,\infty) and b∈Rb \in \mathbb{R}. The function f(x)f(x) is continuous at x=0x = 0. Furthermore, let g:[1,e]→Rg:[1,e] \to \mathbb{R} be a twice-differentiable function satisfying g(1)=0g(1) = 0, g(e)=L2g(\sqrt{e}) = \dfrac{L}{2}, and g(e)=Lg(e) = L.
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