[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-398808-105":59,"doc-detail-398808-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","acet-december-2023-mathematics-question-paper","ACET December 2023 - Mathematics - Question Paper","","ACET December 2023 Mathematics paper for the Institute of Actuaries of India. It includes multiple-choice questions covering core topics such as equation solving, function properties and continuity, sets and set identities, inverse trigonometric evaluation, series summations, conjugates and complex arithmetic, calculus with integrals and areas under curves, linear algebra vectors and angles, numerical methods with Newton-Raphson, interpolation, and forward interpolation from tabulated values. 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The solution(s) of the equation |2􀝔 + 23|= |􀝔 + 11| is (are)  \nA. −14 .  \nB. − 343 and 4.  \nC. −12 and − 343.  \nD. 6 and − 343.  \n1 mark  \n2. If 􀝂(􀝔) = 4 (􀝔 − 3) and 􀝃 (􀝔) = 􀝔 2 − 7􀝔 + 12, then the value of 􀝔 for which the ratio 􀯚􀯙((􀯫􀯫))  \nis undefined is  \nA. 1.  \nB. 2.  \nC. 3.  \nD. 4.  \n1 mark  \n3. For any two sets 􀜣 and 􀜤 if 􀜣 ∩ 􀜤 = 􀜣 ∪ 􀜤 , then  \nA. 􀜣 = 􀜤 .  \nB. 􀜣 = Ω (the universal set) .  \nC. 􀜣 = ∅ (the null set) .  \nD. 􀜣 = 􀜤 􀯖 .  \n1 mark  \n4. If 􀝂(􀝔) = 2􀝔2 + 5􀝔 − 24 and 􀝃 (􀝔) = 3􀝔 + 2 (􀝔 real), then the roots of 􀝂 (􀝔) +  \n􀝃 (􀝔) are  \nA. real and equal.  \nB. real and distinct.  \nC. complex conjugate pair of non-real roots.  \n[D. one](D. one) is real and the other is non-real.  \n1 mark  \n5. The value of (√3 + 1)4 − (√3 − 1)4 is  \nA. 32√3 .  \nB. 16√3 .  \nC. 8√3 .  \nD. 96.  \n1 mark  \n6. The value of cos −1 45 + tan−1 17 is (assuming all the angles to be in the first quadrant)􀰗  \nA. ~~ ~~  \n2 .  \n􀰗  \nB. ~~ ~~  \n3 .  \n􀰗  \nC. ~~ ~~  \n4.  \nD. 􀰗 6 .  \n1 mark  \n7. The value of ∑􀯡􀯝= 1 (3􀝆 − 1)2 is  \nA. 􀯡2 [6􀝊2 − 3􀝊 + 1] .  \nB. 􀯡2 [6􀝊2 + 3􀝊 + 1] .  \nC. 􀯡2 [6􀝊2 − 3􀝊 − 1] .  \nD. 􀯡2 [6􀝊2 + 3􀝊 − 1] .  \n2 marks  \n8. The conjugate of ((13􀯜􀯜))((22􀯜􀯜)) is  \nA. ~~ ~~63169 − 􀝅 16169.  \nB.  63  + 􀝅  16   \n169 169.  \nC. −  63  − 􀝅  16   \n169 169.  \nD. −  63  + 􀝅  16   \n169 169.  \n2 marks  \n9. Let  \n􀝂 (􀝔) = {7􀝉􀝔18 + 􀝊 iiifff􀝔3􀝔3􀝔1, \u003C2, 12,  \n􀝉, 􀝊 real. The function 􀝂 (􀝔) is continuous everywhere if (􀝉, 􀝊) is  \nA. (~~ ~~1110 , 103) .  \nB. (~~ ~~119 , 103) .  \nC. (~~ ~~119 , 13) .  \nD. (~~ ~~119 , − 103) .  \n1 mark  \n10. If 􀝔 3 + 􀝕 3 = cos 􀝔 + 􀝕 , then 􀯗􀯗􀯬􀯫 is equal to  \nA.  3􀯫2 −sin􀯫  .  \n1−3􀯬2  \n3􀯫2 −sin 􀯫  \nB. ~~ ~~  \n1+3􀯬2 .  \nC.  3􀯫2 +sin􀯫 1+3􀯬2 .  \nD.  3􀯫2 +sin􀯫 1−3􀯬2 .  \n2 marks  \n11. Let 􀝂 (􀝔) = −2􀝔3 − 9􀝔2 − 12􀝔 + 1 be a real function. The interval in which 􀝂 (􀝔) is  \nincreasing in 􀝔 is  \nA. (−∞, −2) .  \nB. (−1, ∞) .  \nC. (−2, −1) .  \nD. (−2, 0) .  \n3 marks  \n12. The value of ∫5 5scino􀯫s~~ ~~􀯫 􀝀􀝔 is  \nA. 0.  \nB. 5.  \nC. log 􀯘 2.  \nD. 􀟨 .  \n1 mark  \n13. The area bounded by the curve 􀝕 = 􀯫22~~ ~~ , the lines 􀝕 = 0, 􀝔 = 1 and 􀝔 = 3 is  \nA. 13.  \nB. 4 13.  \nC. 18 2  \n3.  \nD. 4.  \n1 mark  \n14. The value of the integral ∫ 􀝔 log 3􀝔 􀝀􀝔 is  \nA. 􀯫22 log 3􀝔 − 􀯫22 + 􀜥 .  \nB. 􀯫22 log 3􀝔 + 􀯫42 + 􀜥 .  \nC. 􀝔 2 log 3􀝔 − 􀯫42 + 􀜥 .  \nD. 􀯫22 log 3􀝔 − 􀯫42 + 􀜥 .  \n2 marks  \n15. If  \n􀜯 = [ −sci1ons􀟠􀟠 c1􀟠 sin11 􀟠 ], then |􀜯| is equal to  \nA. 1.  \nB. −2 .  \nC. −1 .  \nD. 0.  \n1 mark  \n16. If  \n[􀝖 4 􀝔  􀝕 125] = [  􀜽121 1􀜾􀝀], then 􀝔 + 􀝕 + 􀝖 is equal to  \nA. −16 .  \nB. 0.  \nC. 16.  \nD. 24.  \n1 mark  \n17. The approximate positive root of 􀝔 2 − 2 = 0, obtained after three iterations by Newton  \nRaphson method starting from the initial value 􀝔0 = 1, is  \nA. 1.41426.  \nB. 1.92326.  \nC. 1.81526.  \nD. 1.41667.  \n2 marks  \n18. The following table gives values of 􀝂(􀝔) for some value of 􀝔 .  \n\n| 􀝔 | 50 | 60 | 70 | 80 |\n| --- | --- | --- | --- | --- |\n| 􀝂 (􀝔) | 205 | 225 | 248 | 274 |\n\nThe value of 􀝂 (54) by the Newton’s forward interpolation formula is  \nA. 202.23.  \nB. 227.78.  \nC. 207.12.  \nD. 212.64.  \n3 marks  \n19. Let ⃑􀜣 = ⃑􀝔⃑⃑⃑ + 􀝕⃑ + 􀝖⃑ , ⃑⃑􀜤 = − ⃑􀝔⃑⃑⃑ + 􀝕⃑ − 􀝖⃑ and ⃑􀜥 = 􀝕⃑ − 􀝖⃑ . The area of the triangle ∆􀜣􀜤􀜥 (in  \nsq.units) is  \nA. 3.  \nB. 2.  \nC. 1.  \nD. 0.  \n2 marks  \n20. Let ⃑􀜣 = 2⃑􀝔⃑⃑⃑ + 4􀝕⃑ + 􀝖⃑ , ⃑⃑􀜤 = 3 ⃑􀝔⃑⃑⃑ − 􀝕⃑ + 2􀝖⃑ . The angle between ⃑􀜣 and ⃑⃑􀜤 is  \nA. cos −1 73.  \nB. cos −1 76.  \nC. cos −1  4   \n√6 .  \nD. cos −1  4   \n7√5 .  \n1 mark  \nStatistics  \n21. The mean of 20 observations is 25. On checking it was found that two observations are wrongly recorded as 15 and 17, while the correct observations were 25 and 27. Then the correct mean is  \nA. 30.  \nB. 27.  \nC. 26.  \nD. 25.5.  \n1 mark  \n22. The weighted arithmetic mean of 12, 16 and 20 with weights 􀝓1, 􀝓2 and 􀝓3 , respectively, is known to be 18. If the weights are changed to 3􀝓1, 3􀝓2 and 3􀝓3 , then the weighted arithmetic mean will be  \nA. 6.  \nB. 18.  \nC. 24.  \nD. 54.  \n1 mark  \n23. The standard deviation of 20 observations is 2.1. ","cbCaiul3QPHPEC82","https://ap.wps.com/l/cbCaiul3QPHPEC82","pdf",463119,"English","# Mathematics\n## Questions 1-20\n# Statistics\n## Questions 21-30","[{\"question\":\"What subjects are covered in this ACET December 2023 paper?\",\"answer\":\"It covers Mathematics and Statistics, including equations, functions and continuity, inverse trigonometry, series, complex conjugates, calculus integrals and areas, vectors and angles, numerical methods like Newton-Raphson, interpolation, and core statistics and probability topics.\"},{\"question\":\"How does the paper assess calculus and geometry?\",\"answer\":\"It includes questions requiring evaluation of integrals, finding areas bounded by curves and lines, and solving integrals with logarithmic expressions, alongside vector geometry for triangle area and the angle between vectors.\"},{\"question\":\"What probability skills are tested?\",\"answer\":\"It tests probability concepts such as independence and “exactly one succeeds,” binomial-style reasoning, draws from a bag with replacement, and hypergeometric-style selection without replacement for defective items.\"}]","ACET December 2023 - Mathematics - Question Paper | PDF",1790489088]