JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(\quad f:(-\infty, \infty)-\{0\} \rightarrow R\) be a differentiable function such that \(f^{\prime}(1)=\lim _{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right)\). Then \(\lim _{a \rightarrow \infty} \frac{a(a+1)}{2} \tan ^{-1}\left(\frac{1}{a}\right)+a^2-2 \log _e a\) is equal to
- A \(\frac{3}{2}+\frac{\pi}{4}\)
- B \(\frac{3}{8}+\frac{\pi}{4}\)
- C \(\frac{5}{2}+\frac{\pi}{8}\)
- D \(\frac{3}{4}+\frac{\pi}{8}\)
Answer & Solution
Correct Answer
(C) \(\frac{5}{2}+\frac{\pi}{8}\)
Step-by-step Solution
Detailed explanation
\( f:(-\infty, \infty)-\{0\} \rightarrow R \) \( f^{\prime}(1)=\lim _{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right) \) \( \lim _{a \rightarrow \infty} \frac{a(a+1)}{2} \tan ^{-1}\left(\frac{1}{a}\right)+a^2-2 \ln (a) \)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The number of distinct solutions of the equation \(\log _{\frac{1}{2}}|\sin x|=2-\log _{\frac{1}{2}}|\cos x|\) in the interval \([0,2 \pi],\) isJEE Mains 2020 Hard
- Let \( f(x)=[x]^{2}-[x+3]-3, x\in\mathbb{R} \) where \( [\bullet]\) is the greatest integer function. ThenJEE Mains 2026 Easy
- For \(0 \le x \le \frac{\pi }{2}\), the value of \(\int\limits_0^{{{\sin }^2}\,x} {{{\sin }^{ - 1}}\,\left( {\sqrt t } \right)} dt + \int\limits_0^{{{\cos }^2}\,x} {{{\cos }^{ - 1}}\,\left( {\sqrt t } \right)}\, dt\) equalsJEE Mains 2013 Hard
- Let \(f(x)\) and \(g(x)\) be two functions satisfying \(f\left(x^{2}\right)\) \(+g(4-x)=4 x^{3}\) and \(g(4-x)+g(x)=0\), then the value of \(\int_{-4}^{4} f(x)^{2} d x\) isJEE Mains 2021 Hard
- If the sum of the squares of the reciprocals of the roots \(\alpha\) and \(\beta\) of the equation \(3 x^{2}+\lambda x-1=0\) is 15 , then \(6\left(\alpha^{3}+\beta^{3}\right)^{2}\) is equal toJEE Mains 2022 Hard
- If the equation of a plane \(P ,\) passing through the intesection of the planes, \(x+4 y-z+7=0\) and \(3 x+y+5 z=8\) is \(ax +b y+6 z=15\) for some \(a, b \in R,\) then the distance of the point \((3,2,-1)\) from the plane \(P\) isJEE Mains 2020 Medium
More PYQs from JEE Mains
- Let \(\alpha, \beta\) be the roots of the equation \(x^2-a x-b=0\) with \(\operatorname{Im}(\alpha) \lt \operatorname{Im}(\beta)\). Let \(P_n=\alpha^n-\beta^n\). If \(\mathrm{P}_3=-5 \sqrt{7} i, \mathrm{P}_4=-3 \sqrt{7} i, \mathrm{P}_5=11 \sqrt{7} i\) and \(\mathrm{P}_6=45 \sqrt{7} i\), then \(\left|\alpha^4+\beta^4\right|\) is equal to __________.JEE Mains 2025 Medium
- Let \(\mathrm{E}\) be an ellipse whose axes are parallel to the co-ordinates axes, having its center at \((3,-4)\), one focus at \((4,-4)\) and one vertex at \((5,-4) .\) If \(m x-y=4, m\,>\,0\) is a tangent to the ellipse \(\mathrm{E}\), then the value of \(5 \mathrm{~m}^{2}\) is equal to \(.....\)JEE Mains 2021 Hard
- The values of \('a'\) for which one root of the equation \(x^2 - (a +1)\,x + a^2 + a - 8 = 0\) exceeds \(2\) and the other is lesser than \(2\), are given byJEE Mains 2013 Hard
- A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines
\(\mathrm{L}_1: 2 \mathrm{x}+\mathrm{y}+6=0\) and \(\mathrm{L}_2: 4 \mathrm{x}+2 \mathrm{y}-\mathrm{p}=0, \mathrm{p} \gt 0\), at the points \(A\) and \(B\), respectively. If \(A B=\frac{9}{\sqrt{2}}\) and the foot of the perpendicular from the point A on the line \(L_2\) is \(M\), then \(\frac{A M}{B M}\) is equal toJEE Mains 2025 Easy - Let \(A\) be the area bounded by the curve \(y=x|x-3|\), the \(x\)-axis and the ordinates \(x=-1\) and \(x=2\). Then \(12\,A\) is equal to \(...........\).JEE Mains 2023 Hard
- Let \(\vec{a}=6 \hat{i}+9 \hat{j}+12 \hat{k}, \vec{b}=\alpha \hat{i}+11 \hat{j}-2 \hat{k}\) and \(\vec{c}\) be vectors such that \(\vec{a} \times \vec{c}=\vec{a} \times \vec{b}\). If \(\vec{a} \cdot \vec{c}=-12\), \(\vec{c} .(\hat{i}-2 \hat{j}+\hat{k})=5\), then \(\vec{c} \cdot(\hat{i}+\hat{j}+\hat{k})\) is equal to \(.............\).JEE Mains 2023 Hard