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\(\rm d\left(x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!} \ldots\right) \)/dx किसका समतुल्य है?

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  1. sin(x)
  2. ex
  3. cos(x)
  4. tan(x)

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Option 3 : cos(x)
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More Analysis Questions

Q1.What is the cardinality of the set of real solutions of ex + x = 1?
Q2.मान लीजिए u = u(x, t) निम्न प्रारंभिक मान समस्या का हल है \(\rm \left\{\begin{matrix}u_t+2024u_x=0,&x ∈ R, t>0\\\ u(x, 0)=u_0(x), &x ∈ R\end{matrix}\right.\) जहाँ u0 : ℝ → ℝ एक स्वेच्छ C1 फलन है। निम्नलिखित कथनों पर विचार करें: S1 : यदि At = {x ∈ ℝ : u(x, t) < 1} और |At| प्रत्येक t ≥ 0 के लिए A का लेबेग माप दर्शाता है, तो |At| = |A0|, ∀t > 0 S2 : यदि u0 लेबेग समाकलनीय है, तो प्रत्येक t > 0 के लिए, फलन x → u(x, t) लेबेग समाकलनीय है।
Q3.Let (an)n≥1 be a bounded sequence in ℝ. Which of the following statements is FALSE? 
Q4.Consider the set A = {x ∈ ℚ : 0 < (√2 - 1)x < √2 + 1} as a subset of ℝ. Which of the following statements is true?
Q5.For each n ≥ 1 define fn : ℝ → ℝ by \(\rm f_n(x)=\frac{x^2}{√{x^2+\frac{1}{n}}}, \) x ∈ ℝ where √ denotes the non-negative square root. Wherever \(\rm \lim_{n \rightarrow \infty}f_n(x)\) exists, denote it by f(x). Which of the following statements is true? 
Q6.Let \(\rm S =\left\{x \in R:x >1 \ and \ \frac{1-x^4}{1-x^3}>22\right\}\) Which of the following is true about S? 
Q7.Let S be a dense subset of R and f : ℝ → ℝ a given function. Define g : S → ℝ by g(x) = f(x). Which of the following statements is necessarily true? 
Q8.Let C be the collection of all sets S such that the power set of S is countably infinite. Which of the following statements is true? 
Q9.Let 𝑐00 = {(𝑥1, 𝑥2, 𝑥3, … ) ∶ 𝑥𝑖 ∈ ℝ, 𝑖 ∈ ℕ, 𝑥𝑖 ≠ 0 only for finitely many indices 𝑖}. For (𝑥1, 𝑥2, 𝑥3, … ) ∈ 𝑐00, let ‖(𝑥1, 𝑥2, 𝑥3, … )‖∞ = sup{|𝑥𝑖 | ∶ 𝑖 ∈ ℕ}. Define 𝐹, 𝐺 ∶ (𝑐00, ‖⋅‖∞) → (𝑐00, ‖⋅‖∞) by 𝐹((𝑥1, 𝑥2, … , 𝑥𝑛, … )) = \(\rm \left((1+x)_{x_1}, (2+\frac{1}{2})x_2,..., (n+\frac{1}{n})x_n,...\right)\) 𝐺((𝑥1, 𝑥2, … , 𝑥𝑛, … )) = \(\rm \left(\frac{x_1}{1+1},\frac{x_2}{2+\frac{1}{2}},..., \frac{x_n}{n+\frac{1}{n}},..\right)\) for all (𝑥1, 𝑥2, … , 𝑥𝑛, … ) ∈ 𝑐00. Then
Q10.Let ℓ2 = {(𝑥1, 𝑥2, 𝑥3, … ) ∶ 𝑥𝑛 ∈ ℝ for all 𝑛 ∈ ℕ and \(\rm \Sigma_{n=1}^\infty x_n^2<\infty \}\) For a sequence (𝑥1, 𝑥2, 𝑥3, … ) ∈ ℓ2 , define ‖(𝑥1, 𝑥2, 𝑥3, … )‖2 = \(\rm (\Sigma_{n=1}^\infty x_n^2)^{\frac{1}{2}}\) Let 𝑆 ∶ (ℓ2 , ‖⋅‖2) → (ℓ2 , ‖⋅‖2) and 𝑇 ∶ (ℓ2 , ‖⋅‖2) → (ℓ2 , ‖⋅‖2) be defined by 𝑆(𝑥1, 𝑥2, 𝑥3, … ) = (𝑦1, 𝑦2, 𝑦3, … ), where 𝑦𝑛 = \(\rm \left\{\begin{matrix}0,&n=1\\\ x_{n-1},& n\ge2\end{matrix}\right.\) 𝑇(𝑥1, 𝑥2, 𝑥3, … ) = (𝑦1, 𝑦2, 𝑦3, … ), where 𝑦𝑛 = \(\rm \left\{\begin{matrix}0,&n\ is\ odd\\\ x_{n},& n\ is\ even\end{matrix}\right.\) Then
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More Mathematical Science Questions

Q1.The only function among the following that satisfies Cauchy's Riemann (C-R) equations is : 
Q2.What is the cardinality of the set of real solutions of ex + x = 1?
Q3.Let A be a 10 x 10 real matrix. Assume that the rank of A is 7. Which of the following statements is necessarily true? 
Q4.Let a point P be chosen at random on the line segment AB of length α. Let Z1 and Z2 denote the lengths of line segments AP and BP respectively. Then the value of E(|Z1 - Z2|) Is 
Q5.Consider the contour γ given by \(\rm \gamma (\theta)=\left\{\begin{matrix}e^{2i \theta}&for\ \theta \in [0, \pi /2]\\\ 1+2e^{2i\theta}&for\ \theta \in [\pi/2, 3\pi/2]\\\ e^{2i\theta}&for\ \theta \in [3\pi/2, 2\pi]\end{matrix}\right.\) Then what is the value of \(\rm \int_{\gamma}\frac{dz}{z(z-2)}\)?
Q6.Let X1....X10 be a random sample from a distribution with the probability density function \(\rm f(x|θ)=\left\{\begin{matrix}θ x^{θ-1}, & if \:0<x<1\\\ 0, & otherwise,\end{matrix}\right.\) where θ > 0 Is an unknown parameter. The prior distribution of θ is given by   \(\rm \pi (θ)=\left\{\begin{matrix}θ e^{-θ}, & if\ θ>0\\\ 0, & otherwise,\end{matrix}\right.\) The Bayes estimator of θ under squared error loss is  
Q7.For a quadratic form f(x, y, z) ∈ ℝ[x, y z], we say that (a, b, c) ∈ ℝ3 is a zero of f if f(a, b, c) = 0. Which of the following quadratic forms has at least one zero different from (0, 0, 0)? 
Q8.Let A : ℝm → ℝn be a non-zero linear transformation. Which of the following statements is true? 
Q9.Let X1, X2 be a random sample from a population having probability density function f ∈ { f0, f1} where  \(\rm f_0(x)=\left\{\begin{matrix}\frac{1}{2}& if\ 0\le x\le 2\\\ 0&otherwise\end{matrix}\right.\ and \ \rm f_1(x)=\left\{\begin{matrix}\frac{1}{4}& if\ 0\le x\le 4\\\ 0&otherwise\end{matrix}\right.\) For testing the null hypothesis H0 : f = f0 against the alternate hypothesis H1 : f = f1, the power of a most powerful test of size α = 0.05 is equal to  
Q10.Let V be the real vector space of 2 x 2 matrices with entries in ℝ. Let T : V → V denote the linear transformation defined by T(B) = AB for all B ∈ V, where \(\rm A=\begin{pmatrix}2&0\\\ 0&1\end{pmatrix}\). What is the characteristic polynomial of T? 
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