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Additional problems address projection length using direction cosines, angle between diagonals in a cube, probabilities based on last-digit constraints, tournament outcomes, and identifying conics through discriminant analysis of a generalized quadratic equation.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/aptitude-and-practice-questions-mathematics-ap-eamcet/451389/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/aptitude-and-practice-questions-mathematics-ap-eamcet/451389.png","ImageObject",300,407,{"name":42,"@type":43},"Blitz","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-04","2026-09-30",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",6,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"How is the intersection of the two sets A and B determined?","Question",{"text":63,"@type":64},"Substitute the given circle conditions into the other relation, rewrite the equation in normalized form, and solve for common points. The solution states that A ∩ B consists of four points.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What method checks whether the parameter system of linear equations has a unique solution or is inconsistent?",{"text":68,"@type":64},"Compute the determinant D of the coefficient matrix and analyze when D is non-zero or zero. The solution concludes inconsistency when |a| equals √3.",{"name":70,"@type":61,"acceptedAnswer":71},"How is the minimum of |z| + |z − 1| for a complex number found?",{"text":72,"@type":64},"Use the triangle inequality bounds and the fact that |z| = |−z|, then compare |z| + |1 − z| with |z + (1 − z)| = |1|. The minimum value is 1.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},451389,1791317231,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,104,109,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":93,"slug":94},70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":101,"show_sort_weight":102,"slug":103},"Technology",50,"technology",{"id":105,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},7,"Healthcare",40,"healthcare",{"id":110,"doc_module":4,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},8,"Research & Report",30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":96,"language":139,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":12,"update_tm":143,"read_time":144,"preview_page_count":4},24464137899374,"https://us-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","1. If A = [(x, y) : x2 + y2 = 25] and B = [(x, y) : x2 + 9y2 = 144], then A ∩ B contains   points.  \nSolution:  \nA = Set of all values (x, y) : x2 + y2 = 25 = 52 B = [x2 / 144] + [y2 / 16] = 1  \ni.e. , [x2 / ( 12)2] + [y2 / (4)2] = 1.  \nA ∩ B consists of four points.  \n2. In a college of 300 students, every student reads 5 newspapers, and every newspaper  \nis read by 60 students. The number of newspapers is   Solution:  \nLet the number of newspapers be x.  \nIf every student reads one newspaper, the number of students would be x (60) = 60x Since every student reads 5 newspapers, the number of students = [x * 60] / [5] = 300 x = 25  \nConsider the system of equations x + y + z = 1 , 2x + 3y + 2z = 1 , 2x + 3y + (a2 – 1)z = a + 1 then  \n(a) System has a unique solution for |a | = √3  \n(b) System is inconsistence for |a | = √3  \n(c) System is inconsistence for a = 4  \n(d) System is inconsistence for a = 3 Answer: (b)  \nSolution:  \nGiven a system of linear equations: x + y + z = 1   ( 1)  \n2x + 3y + 2z = 1   (2)  \n2x + 3y + (a2 – 1)z = a + 1   (3) D= | 11123223a2−1|  \n= 1[3(a2 – 1)–(3)(2)] – 1[2(a2 – 1)–(2)(2)] + 1[2(3)– 2(3)]= 3a2 – 3 – 6 – 2a2 + 2 + 4 + 0  \n= a2 – 3 Consider D ≠ 0 So, a2 – 3 ≠ 0 ⇒ a2 ≠ 3  \n⇒ |a | ≠ ±√3  \nThat means the given linear equation system is inconsistent for |a | = ±√3 .  \nSo option (b) is correct.  \n3. If the system of linear equations are 2x + 2ay + az = 0  \n2x + 3by + bz = 0 and  \n2x + 4cy + cz = 0 ,  \nwhere a, b, c ∈ R are non-zero and distinct ; has a non-zero solution, then (a) a + b + c = 0  \n(b) 1/a, 1/b, 1/c are in A. P.  \n(c) a, b, c are in A. P.  \n(d) a, b, c are in G. P.  \nAnswer: (b)  \nSolution:  \nGiven a system of linear equations 2x + 2ay + az = 0  \n2x + 3by + bz = 0 and  \n2x + 4cy + cz = 0 ,  \nAs the given system of equations has a non-zero solution, we have D = 0.  \n⇒ (3b − 2a)(c − a) − (4c − 2a)(b − a) = 0  \n⇒ 3bc − 2ac − 3ab + 2a2 − [4bc − 4ac − 2ab + 2a2] = 0 ⇒ −bc + 2ac − ab = 0  \n⇒ ab + bc = 2ac ⇒ 1/c + 1/a = 2/b  \nWhich shows that 1/a, 1/b, and 1/c are in A. P.  \nIf ( 1 + i) ( 1 + 2i) ( 1 + 3i)   ( 1 + ni) = a + ib, then what is 2 * 5 * 10   ( 1 + n2) is equal to? Solution:  \nWe have ( 1 + i) ( 1 + 2i) ( 1 + 3i)   ( 1 + ni) = a + ib   (i)  \n( 1 − i) ( 1 − 2i) ( 1 − 3i)   ( 1 − ni) = a − ib   (ii) Multiplying (i) and (ii) ,  \nwe get 2 * 5 * 10   ( 1 + n2) = a2 + b2  \n4. If z is a complex number, then the minimum value of |z | + |z − 1 | is    \nSolution:  \nFirst, note that |−z |= |z | and |z1 + z2| ≤ |z1| + |z2| Now |z | + |z − 1 | = |z | + | 1 − z | ≥ |z + ( 1 − z) |= | 1 |  \n= 1  \nHence, the minimum value of |z | + |z − 1 | is 1.  \nA letter lock consists of three rings, each marked with ten different letters. In how many ways is it possible to make an unsuccessful attempt to open the lock?  \nSolution:  \nTwo rings may have the same letter at a time, but the same ring cannot have two letters at a time. Therefore, we must proceed ring-wise. Each of the three rings can have any one of the 10 different letters in 10 ways.  \nTherefore, the total number of attempts = 10 × 10 × 10 = 1000. But out of these 1000 attempts, only one attempt is successful.  \nTherefore, the required number of unsuccessful attempts = 1000 – 1 = 999.  \n| 5. The total number of positive integral solutions for x, y, z such that x* y * z = 24 , is\u003Cbr>  .\u003Cbr>Solution:\u003Cbr>We have, x* y * z = 24\u003Cbr>x* y * z = 23 × 31\u003Cbr>The number of ways of distributing ‘n’ identical balls into ‘r’ different boxes is (n + r − 1)C(r − 1) Here we have to group 4 numbers into three groups\u003Cbr>Number of integral positive solutions = (3 + 3 − 1)C(3 − 1) × ( 1+ 3 − 1)C(3 − 1)= 5C2 × 3C2\u003Cbr>= 30 |\n| --- |\n| 6. The projection of any line on coordinate axes be, respectively 3 , 4 , 5 then its length is   .\u003Cbr>Solution:\u003Cbr>Let the line segment be AB, then as given AB cos α = 3 , AB cos β = 4 , AB cos γ = 5 ⇒ AB2 (cos2α + cos2β + cos2γ) = 32 + 42 + 52\u003Cbr>AB = √[9 + 16 + 25] = 5√2 ,\u003Cbr>where α , β and γ are the angles made by the line with the axes.","cbCaijrQ9Dclny09","https://ap.wps.com/l/cbCaijrQ9Dclny09","pdf",182592,"English","# Practice Questions\n## Algebra and Linear Systems\n## Complex Numbers\n## Geometry and Conics\n## Probability","[{\"question\":\"How is the intersection of the two sets A and B determined?\",\"answer\":\"Substitute the given circle conditions into the other relation, rewrite the equation in normalized form, and solve for common points. The solution states that A ∩ B consists of four points.\"},{\"question\":\"What method checks whether the parameter system of linear equations has a unique solution or is inconsistent?\",\"answer\":\"Compute the determinant D of the coefficient matrix and analyze when D is non-zero or zero. The solution concludes inconsistency when |a| equals √3.\"},{\"question\":\"How is the minimum of |z| + |z − 1| for a complex number found?\",\"answer\":\"Use the triangle inequality bounds and the fact that |z| = |−z|, then compare |z| + |1 − z| with |z + (1 − z)| = |1|. The minimum value is 1.\"}]","Aptitude and Practice Questions - Mathematics (AP EAMCET) | PDF",1790735017,13]