Limits Past Paper PDF OCR July 2022

# Limits ## Available * **Time:** 9:00 AM, 15th July 2022 (Friday) * **Duration:** 75 min * **Questions:** 30 * **Marks:** 100 ## Important Instructions Please read the following test related instructions: * This is a chapter test of Limits. * There are 30 questions in this test which contains b...

# Limits ## Available * **Time:** 9:00 AM, 15th July 2022 (Friday) * **Duration:** 75 min * **Questions:** 30 * **Marks:** 100 ## Important Instructions Please read the following test related instructions: * This is a chapter test of Limits. * There are 30 questions in this test which contains both multiple choice questions and numerical questions. * The duration of the paper is 75 minutes. ## Sections ### Mathematics Single Correct * 4 will be awarded for correct attempt. * -1 will be awarded for incorrect attempt. * 0 will be awarded for no response. ### Mathematics Numerical * The maximum number of questions that can be attempted in this section is 5 questions. * 4 will be awarded for correct attempt. * -1 will be awarded for incorrect attempt. * 0 will be awarded for no response. # LIMITS ## Time Left: 01:14:58 ## Question Paper ## Instructions ### Mathematics Single Correct | Question No. | Section Details | Marking Scheme | |---|---|---| | 1 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 2 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 3 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 4 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 5 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 6 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 7 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 8 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 9 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 10 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 11 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 12 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 13 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 14 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 15 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 16 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 17 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 18 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 19 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | | 20 | Mathematics Single Correct (Maximum Marks: 80) Only One Option Correct Type | > | ## Question No. 1 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * $lim_{x \to 0}(\frac{x}{tan(1/2x)})$ is equal to: * 0 * 1/2 * 1 * ∞ ## Question No. 2 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * $lim_{x \to 0}(\frac{x tan(2x) - 2x tan(x)}{(1 - cos (2x))^2})$ is equal to: * 2 * - 2 * 1/2 * - 1/2 ## Question No. 3 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * $lim_{x \to \infty}(\sqrt{x + \sqrt {x + \sqrt{x}}} - \sqrt{x})$ is given by: * 0 * 1/2 * log(2) * none of these ## Question No. 4 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * $lim_{x \to 0}[(3x + \frac{ 1}{x})^2 - (2x - \frac{1}{x})^2]$ is equal to: * 5 * 2 * 0 * 10 ## Question No. 5 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * If $lim_{x \to \infty} f(x)$ exists and is finite and $lim_{x \to \infty}(f(x) + \frac{3f(x) - 1}{(f(x))^2})$ = 3, then the value of $lim_{x \to \infty}(f(x)$ is: * 1 * - 1 * 2 * none of these ## Question No. 6 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * $lim_{x \to \infty}(\frac{x^3 - x}{\sqrt{1 + x^6}})$ is equal to: * - 1 * 1 * 0 * 3 ## Question No. 7 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * $lim_{x \to 0}{[log_e(1 + x) + \frac{x - 1}{x}]}$ is equal to: * ∞ * 1/2 * - 1/2 * none of these ## Question No. 8 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * $lim_{x \to 0} (\frac{sin(cos^2(x))}{x^2})$ equals: * π * π/2 * 1 ## Question No. 9 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * $lim_{x \to 2}(\frac{(cos(a)) + (sin(a)) - 1}{x - 2})$ is equal to: * ∞ * (cos(a))^2 + (sin(a))^2 * 0 * none of these ## Question No. 10 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * The value of $lim_{x \to 0} (\frac{1}{x} sin^{-1}(\frac{2x}{1 + x^2}))$ is: * 2 * ∞ * does not exist * none of these ## Question No. 11 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * $lim_{x \to 0}(\frac{\sqrt{1 - cos(2x)}}{x})$ is equal to: * - 1 * 1 * 0 * none of these ## Question No. 12 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * $lim_{x \to ∞} (\frac{x^2 +5x + 3}{x^2 + x + 3})^x$ is: * e^4 * e^2 * e^3 * e ## Question No. 13 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * $lim_{n \to ∞}[\frac{1}{1 - n^2} + \frac{2}{1 - n^2} + ... + \frac{n}{1 - n^2}]$ is equal to: * 0 * - 1/2 * 1/2 * 1 ## Question No. 14 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * $lim_{n \to ∞} (3^n + 5^n + 7^n)^{1/n} $ is equal to: * e^3 * e^5 * 5 * 7 ## Question No. 15 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * If $f(x) = (\frac{x}{x + 3})^{3x}$, then: * $lim_{x \to ∞} f(x) = e^-3$ * $lim_{x \to ∞} f(x) = e^-9$ * $lim_{x \to 2} f(x) = \frac{64}{625}$ * none of these ## Question No. 16 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * $lim_{x \to 0}(\frac{e^x + e^-x - 2}{x^2})$ is equal to: * 1 * - 1 * 1/2 * - 1/2 ## Question No. 17 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * $lim_{x \to ∞}(\frac{x^n}{e^x})$ = 0 is true, only when: * no value of n * n is any whole number * n = 0 only * n = 2 only ## Question No. 18 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * If $lim_{x \to a} (\frac{f(x)}{x})$ = l1 and $lim_{x \to a} (\frac{f(λx)}{x})$ = l2 then λ is equal to: * 1 * l1/l2 * l2/l1 * none of these ## Question No. 19 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * The value of $lim_{x \to 0}([\frac{11x}{sin(x)}] + [\frac{21sin(x)}{x}])$, where [x] is the greatest integer less than or equal to x is: * 32 * 31 * 11 * 21 ## Question No. 20 This section contains 20 questions. 20 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. * $lim_{x \to a^-}[(\frac{|x|^3}{x} - \frac{|x|^3}{a})^3]$ (a > 0), where [x] denotes the greatest integer less than or equal to x is: * a^2 - 3 * a^2 - 1 * a^2 * none of these ## Mathematics Numerical ### Section Details * Mathematics Numerical (Maximum Marks: 20) * Numerical Type This section contains 10 questions. 1 are multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 9 are numerical questions. The answer has to be filled into the input box provided below. ## Question No. 1 If f(x) is a polynomial satisfying f(x)f($\frac{1}{x}$) = f(x) + f($\frac{1}{x}$) and f(2) > 1, then $lim_{x \to 1}f(x)$ is: ## Question No. 2 The value of $lim_{x \to 1^-}(\frac{[sin^{-1}(x)] - 1}{x})$ is (where [.] denotes greatest integer function) ## Question No. 3 $lim_{n \to ∞} ( \frac{2}{3} + \frac{2}{3} + \frac{2}{3} + ....+ \frac{2}{3})^n$ is equal to ## Question No. 4 If $lim_{x \to ∞} (\sqrt{ax^2 - x + 1} - ax - b) = 0$ then for k ≥ 2, $lim_{n \to ∞} sec^2(k!πb)$ is equal to: ## Question No. 5 Let f : R → R be a positive increasing function with $lim_{x \to ∞}(\frac{f(3x)}{f(x)})$ = 1. Then $lim_{x \to ∞} (\frac{f(2x)}{f(x)})$ ## Question No. 6 Find all possible value of a, if $lim_{x \to a} (\frac{x^9 - a^9 }{x - 9}) = lim_{x \to 3}(6 + x)$ ## Question No. 7 The value of $lim_{x \to 0} (cos(x))^{cot(x^2)}$ is $\frac{π}{\sqrt{e}}$ then n = ## Question No. 8 If n > m are positive integers, then $lim_{α \to 0}(\frac{sin(α^n)}{(sin(α))^m})$ = ## Question No. 9 The least positive integer n for which the $lim_{x \to 0}(\frac{(cosx - 1)(cosx - e^x)}{x^n})$ is a finite non-zero number is ## Question No. 10 Let f : R → [0, ∞) be such that $lim_{x \to 5} f(x)$ exists and $lim_{x \to 5} \frac{(f(x))^2 -9}{\sqrt{x - 5}}$ = 0 Then $lim_{x \to 5} f(x)$ equals