Exams › JEE Advanced › Maths

Let \(a\) be the minimum value of the function \(x^2+2x+3\) for real \(x\), and let \(b=\lim_{\theta\to 0}\frac{1-\cos\theta}{\theta^2}\). What is the value of the summation \(\sum b^n r^{-1}\)?

1. \((2^{n+1}-1)/(3\cdot 2^n)\)
2. \((2^{n+1}+1)/(3\cdot 2^n)\)
3. \((4^{n+1}-1)/(3\cdot 2^n)\)
4. None of these

Correct answer: \((2^{n+1}+1)/(3\cdot 2^n)\)

Solution

The quadratic \(x^2+2x+3=(x+1)^2+2\) has minimum value \(a=2\). Also, \(\lim_{\theta\to 0}\frac{1-\cos\theta}{\theta^2}=\frac12\), so the resulting summation simplifies to the stated expression. The correct option is \((2^{n+1}+1)/(3\cdot 2^n)\).

Related JEE Advanced Maths questions

• If the limit \(\lim_{x\to ?}\big(-x+c\lfloor x-1\rfloor+d\lfloor 1+x\rfloor\big)\) exists, where \(\lfloor\,\cdot\,\rfloor\) denotes the floor function, what are the possible values of \(c\) and \(d\)?
• Evaluate \(f(x)=\lim_{n\to\infty}\frac{x^{2n}-1}{x^{2n}+1}\). Which of the following is true?
• Determine the value of \(L\) if \(L=\lim_{x\to 0}\frac{\sin x+ae^x+be^{-x}+c\ln(1+x)}{x^3}\), given that \(L\) is finite.
• What is the value of the limit $\lim_{\theta\to 0^+}\{AQ\}$?
• Consider the function \(f(x)\) defined by \[ f(x)=\lim_{n\to\infty}\left\{\frac{n(x+n)(x+n/2)\cdots(x+n/n)}{n!\,(x^2+n^2)(x^2+n^2/4)\cdots(x^2+n^2/n^2)}\right\}^{x/n},\quad x>0. \] Which of the following is true?
• For a positive integer $n$, let \[ f(n)=n+\frac{16+5n-3n^2}{4n+3n^2}+\frac{32+n-3n^2}{8n+3n^2}+\frac{48-3n-3n^2}{12n+3n^2}+\cdots+\frac{25n-7n^2}{7n^2}. \] Determine the value of $\lim_{n\to\infty} f(n)$.

⚔️ Practice JEE Advanced Maths free + battle 1v1 →