Quantitative Techniques Practice Paper on Mensuration for CLAT 2022

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Quantitative Techniques Practice Paper on MensurationQuestion 1: A hollow metal pipe with the length 21cm and inner radius 5cm and outer radius 6.5cm. Answer the following questions based on given information only.The Lateral surface area of the pipe is approximately1500cm2​​  (b) 1518cm2  (c) 1650cm2  (d) 1620cm2The volume of the metal of the pipe is1138.5cm3 (b)1100cm3  (c)​​ 1239cm3  (d) 1139.5cm3If the pipe wasn’t hollow and the metal was inside then what is the volume of the cylinder?​​ Consider the outer radius for the whole cylinder.2788.5cm3 (b)2888.5cm3  (c) 2898.5cm3  (d) 2988cm3What would be the volume of a cone with the same height and inner radius as the radius of the cone?540cm3 (b) 560cm3  (c) 550 cm3  (d) 530cm3If the hollow pipe is melted and remoulded in a sphere, then what is the radius of the new sphere?12cm​​   (b) 4.9cm  (c) 5.5cm  (d) 6.47cmQuestion 2: A chocolate box is 50cm long, 30cm wide and 15cm high. If 1 cubic cm of a cardboard weights 0.5gms then answer the questions based on given data.2.1 How much chocolate that box can hold in terms of volume not weight(assume​​ 1cm thickness in calculation)? ​​ ​​ ​​ ​​ ​​ ​​​​ (a) 225m3  (b) 18816cm3  (c) 22.5m3  (d)22.5cm32.2 If the thickness of the box is 1cm then, What is the total weight of the box? ​​ ​​ ​​ ​​ ​​ ​​​​ (a) 1300 gms (b) 1270gms  (c) 1.84kg  (d) 11.7kgs2.3 What is the outer surface area of the box if it is open from the top(exclude thickness)?​​  ​​ ​​ ​​ ​​ ​​ ​​​​ (a) 0.39m2  (b) 390cm2  (c) 39m2  (d) 3900m2​​ 2.4 Suppose the box is filled with chocolate in cube shape with side 2cm. How many choclates the box will hold? ​​ ​​ ​​ ​​ ​​ ​​​​ (a) 2500  (b) 2540  (c) 2400  (d) 23522.5 If the length of the box is reduced by 10%, then what is the total surface area of the box? ​​ ​​ ​​ ​​ ​​​​ (a) 6938cm2 (b) 6900cm2  (c) 6950cm2  (d) 6900cm2 Answer and Solutions:Ans 1.1 (b) 1625cm2Solution: Inner Radius= r = 5cm  Outer Radius = R = 6.5cm ​​ ​​ ​​​​ Lateral surface area of Pipe = External Surface area + Internal surface area ​​​​     ​​ = ​​ 2(pi) * ( R+r) *h    ​​ = 2*22*(6.5+5) 21/7 = 1518cm2Ans 1.2 (a) 1138.5cm2Solution:​​ Volume of hollow​​ cylinder = (pi) (R2​​ - r2) * h = 22*(6.52​​ -52) * 21/7​​ ​​ = 1138.5cm2Ans 1.3 (a) 2788.5cm2Solution:​​ Volume of the cylinder if not hollow = (pi) R2h       ​​ ​​ ​​​​ = 22*6.5*6.5*21/7 = 2788.5cm2Ans 1.4 (c) 550cm2Solution:​​ Volume of a cone with the same height and inner r = 1/3 * (pi) r2h         ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ = 1/3 * 22/7 *5*5*21         ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ = 550cm2Ans 1.5 (d) 6.47cm2Solution:​​ When one solid is moulded into another one, the volume remains the same. Hence,  Volume of sphere = Volume of a hollow cylinder  4/3*(pi)*radius3​​ = (pi) (R2​​ – r2) h  radius3 = (6.52-52)*21*3/4  radius = 6.47cmAns 2.1 (b) 20160cm3Solution:​​ As the thickness is 1 cm we will deduce it with actual length and breadth​​ Volume of inner box = (50-1)*(30-1)*(15-1) = 48*28*14= 18816 cm3Ans 2.2 (c) 1.17kgsSolution:​​ Volume of whole Cuboid = l*b*h = 50*30*15 = 22500cm3​​ Volume of material = Volumeof whole box – Volume of inner box   ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ = 22500 –18816 cm3​​ ​​ =​​ 3684 cm31cubic cm = 0.5gm, Hence Weight = 3684*0.5 gms = 1840 gms = 1.840 kgsAns 2.3 (a) 0.39m2Solution:​​ As the upper surface of the box is open hence we only have 5 surfaces whose area we need to calculate.​​ Surface Area of outer box = base area(l*b) + side wall(2*b*h) + side wall(2*l*b)    ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ = 50*30 + 2*15*30 + 2*50*15    ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ = 3900 cm2   ​​  ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ = 0.39m2 ​​ [ since 100cm = 1m]​​ Ans 2.4 (d) 2352Solution:​​ The number of chocolates in a box can be equal to total volume of box by volume of one chocolate as the volume as to be equated.N*volume of chocolate = Volume of the box​​ N = 18816/(23) = 2352Ans 2.5 (a) 6938cm2Solution:​​ If the length is reduced by 10% then new length = 50*0.9 = 45cmTotal surface Area = Outer area+ Inner surface area + Area of top surface(thickness)= [(45*30)+2(45*15)+2(30*15)] + [(43*28)+2(43*14)+2(28*14)] +[2*1*45+2*1*28]= [1350+1350+900] + [1204+1204*784] + 146= 3600+3192+146 = 6938 cm2(function( $ ) { //footnotes $('.defaultNote').children(":first").css('cursor', 'pointer'); $('.defaultNote').children(":first").click(function(){ var gotoID = '#' + $(this).parent().attr('id'); $('html, body').animate( { scrollTop: $('a[href="' + gotoID + '"]').offset().top - 60, }, 250, 'linear' ) }); }(jQuery)); var numFormat = ",."; (function( $ ) { "use strict"; //do some cleaning on load var sortCells = $('td[data-sorting], th[data-sorting]'); sortCells.each(function(){ var first = $(this).text().trim().charAt(0); if (first == '@'){ var oldHTML = $(this).html(); var newHTML = oldHTML.replace('@', ''); $(this).html(newHTML); } }); function sortingTable(node, pos, ordBy){ var rows = node.children('tbody').children('tr').get(); rows.sort(function(a, b) { var X = $(a).children('td').eq(pos).text().toUpperCase(); var Y = $(b).children('td').eq(pos).text().toUpperCase(); if (numFormat != '.,'){ //we are dealing with a different locale and we need to do some //transformation in numbers X = convert2number(X); Y = convert2number(Y); } else { X = X.replace(/,/g, '').replace(/ /g, ''); Y = Y.replace(/,/g, '').replace(/ /g, ''); } var currencies = ['$', '¥', '£', '€']; var arrayLength = currencies.length; var XX = X; var YY = Y; for (var i = 0; i < arrayLength; i++) { XX = XX.replace(currencies[i], ''); YY = YY.replace(currencies[i], ''); } if (!isNaN(XX) && !isNaN(YY)){ var floatXX = parseFloat(XX); var floatYY = parseFloat(YY); return -(floatYY-floatXX) * ordBy; } else if (!isNaN(XX) && isNaN(YY)){ return -1 * ordBy; } else if (isNaN(XX) && !isNaN(YY)){ return ordBy; } else { if(X < Y) { return -1 * ordBy; } if(X > Y) { return 1 * ordBy; } return 0; } }); $.each(rows, function(index, row) { node.children('tbody').append(row); }); } function removeTableCellDecoration() { $("td[data-sorting], th[data-sorting]").removeClass('sortDecorator'); } //unbind the click event just in case we are loading more than one document $("td[data-sorting], th[data-sorting]").off('click'); $("td[data-sorting], th[data-sorting]").click(function(){ removeTableCellDecoration(); $(this).addClass('sortDecorator'); //get desired ordering var ordering = $(this).attr('data-sortOrdering'); if (ordering == 'ASC'){ var ordBy = -1; $(this).attr('data-sortOrdering', 'DESC'); } else { var ordBy = 1; $(this).attr('data-sortOrdering', 'ASC'); } var cellPosition = $(this).prevAll().length; var table = $(this).parents('table:first'); sortingTable(table, cellPosition, ordBy); }); function convert2number(str) { str = str.trim(); if (str.match(/^\d+/)) { //we have to take special care when the thousand separators are '.' var result_0 = str.replace(/\./g, '_@_'); var result_1 = result_0.replace(window.numFormat[0], '.'); var result = result_1.replace(/_@_/g, ''); result .replace(/ /g, ''); return result; } else { return str; } } }(jQuery));