July 2015

PIEAS Aggregate Calculator

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PIEAS Aggregate Calculator

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NUST Aggregate Calculator

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NUST Aggregate Calculator

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COMSATS Aggregate Calculator

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COMSATS Aggregate Calculator

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GIKI Aggregate Calculator

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GIKI Aggregate Calculator

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Variation Of Gravitational Acceleration 'g'

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<b><span style="font-family: Times, Times New Roman, serif; font-size: large;">At poles or longitude:</span></b></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">                                          It is ‘constant’ and ‘maximum’ i.e (9.8 m/s.s)<br /><br /><b>At equator or latitude:</b></span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">                                          It ‘decreases’ with increasing distance from center of Earth.<br /><br /><b>At some altitude (height):</b>    </span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">                                               It decreases exponentially , and is inversely proportional to the square of height H above the surface of earth. The formula for g at the surface of earth is given by:</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">                                                            <span style="color: #990000;"> g = GM/R<sup style="margin: 0px; padding: 0px;">2</sup></span></span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">Here ‘g’ is the acceleration due to gravity, ‘G’ is the gravitational constant, ‘M’ is the mass of Earth and ‘R’ is the mean radius of Earth.</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">Let ‘g<sub style="margin: 0px; padding: 0px;">o</sub>’ be the value of acceleration due to gravity at radius ‘R’ from center of Earth. ‘g<sub style="margin: 0px; padding: 0px;">H</sub>’ be the value of acceleration due to gravity at a height ‘H’ from Earth’s surface.</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">So,</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">                                                          <span style="color: #cc0000;">  g<sub style="margin: 0px; padding: 0px;">o</sub>= GM/R<sup style="margin: 0px; padding: 0px;">2</sup> </span></span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">And</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">                                                       <span style="color: #cc0000;"> g<sub style="margin: 0px; padding: 0px;">H </sub>= GM / [(R+H)] <sup style="margin: 0px; padding: 0px;">2</sup></span> </span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">Dividing the equations we get :</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">                                                      <span style="color: #cc0000;">  g<sub style="margin: 0px; padding: 0px;">H </sub>= g<sub style="margin: 0px; padding: 0px;">o</sub> (R / (R+H))<sup style="margin: 0px; padding: 0px;">2</sup></span></span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;"><b><span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>In Depth:</b><br />                        It decreases linearly, and is inversely proportional to the depth in Earth. Let ‘g<sub style="margin: 0px; padding: 0px;">o</sub>' be the value of gravitational acceleration on Earth’s surface from center of Earth and the radius be ‘R’ and 'g<sub style="margin: 0px; padding: 0px;">x</sub>' be the value of gravitational acceleration at a depth of ‘(R – x)’ , So, a formula can be derived which  gives</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">                                                        <span style="color: #cc0000;"> g<sub style="margin: 0px; padding: 0px;">x</sub> = g<sub style="margin: 0px; padding: 0px;">o</sub> (1- x/R  )         </span> </span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">When x = R i.e. center of Earth g<sub style="margin: 0px; padding: 0px;">x</sub> = 0. So at center of Earth acceleration due to gravity is zero.</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">Its value at sun is <b><span style="color: #38761d;">274 m/s<sup style="margin: 0px; padding: 0px;">2</sup></span></b> . At moon it is <b><span style="color: #38761d;">1.6 m/s<sup style="margin: 0px; padding: 0px;">2</sup></span></b></span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;"><b><span style="margin: 0px; padding: 0px;">Some important concepts:-</span></b></span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;"><span style="color: purple;"><span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Just imagine the Earth starts rotating very fast. What will happen to the person standing on it?</span> Obviously, he will fly off tangentially to the surface of the Earth. It means the gravitational force on that person has decreased. In this case the value of g will be ‘<span style="color: #38761d;">MINIMUM</span> ‘.</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">The variation of gravitational acceleration with rotation of earth is given by </span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">                                                       <span style="color: #cc0000;"> g<sup style="margin: 0px; padding: 0px;">' </sup>= g - R<sub style="margin: 0px; padding: 0px;">e</sub>ω^2 cos^2θ</span></span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">When ‘ω’ increases g’ becomes more and more less.</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;"><span style="color: purple;"><span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Now what happens if the Earth stops suddenly? </span><br />The value of g will become maximum or infinity in that case.</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">In this case ‘ω’ becomes zero so does the factor <span style="color: #38761d;">R<sub style="margin: 0px; padding: 0px;">e</sub>ω^2cos^2θ</span> so g<sup style="margin: 0px; padding: 0px;">' </sup>= g, which is maximum value.</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;"><span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>As we know gravitational force of attraction is the force that attracts a body towards the center of Earth. Now if a body is placed at the center of the Earth what will be the gravitational force of attraction in this case. It will be zero as the body is already placed at the center of the Earth. (already discussed in variation of ‘g’ with depth.)</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;"><span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Now, if a body starts moving from equators to poles value of g will increase continuously increases till it reaches its maximum value at poles. And if a body starts moving from poles to equators, value of g will decrease till it reaches its minimum value at equators. </span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">This can also be explained on the basis of this formula</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">                                                     <span style="color: #cc0000;">    g<sup style="margin: 0px; padding: 0px;">' </sup>= g - R<sub style="margin: 0px; padding: 0px;">e</sub>ω^2 cos^2θ</span></span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">==> At poles the value of θ is 90 which make the factor <span style="color: #38761d;">R<sub style="margin: 0px; padding: 0px;">e</sub>ω^2cos^2θ </span>equal to zero so gravitation acceleration is maximum i.e g’ = g.</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">==> At equator the value of θ is 0 which gives the factor <span style="color: #38761d;">R<sub style="margin: 0px; padding: 0px;">e</sub>ω^2cos^2θ</span> its maximum value so g<sup style="margin: 0px; padding: 0px;">' </sup>< g.</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">==> Now, as we move from equator to poles the value of θ increases and cosine value gets lowers making g' greater and vice verse.</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">This effect can also be simply under by Newton’s law of gravitation which gives</span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;">                                                               g= GM/R^2 </span></div>
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<span style="font-family: Times, Times New Roman, serif; font-size: large;"><img src="http://www.open.edu/openlearnworks/pluginfile.php/66047/mod_page/content/1/S103_3_003i.jpg" /></span></div>
<span style="font-family: Times, Times New Roman, serif; font-size: large;">At poles the radius is smaller so gravitational acceleration is large and at equator radius is large so value of acceleration due to gravity gets smaller.</span></div>
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Variation Of Gravitational Acceleration 'g'

At poles or longitude: It is ‘constant’ and ‘maximum’ i.e (9.8 m/s.s)At equator or latitude: It ‘decreases’ with in…

FAST Past Paper (2012)

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<b><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">2) Find equation of tangent to</span><span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;"> <span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">y^2= 4ax<o:p></o:p></span></span></b></div>
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<span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;"><b><br /></b></span></div>
<div class="MsoNormal" style="background-color: white; font-family: Arial, San-Serif; line-height: 29px; margin: 0px; padding: 0px;">
<span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;"><b>3) A Solenoid bent into a circle is called?</b></span><span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;"><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a) Solenoid<br />b) Toroid<br />c) both an and b<br />d) none of the above</span></span></div>
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<b><span style="line-height: 15.3333320617676px; margin: 0px; padding: 0px;"><span style="font-family: Arial, sans-serif; margin: 0px; padding: 0px;"><br />4) Unit vector is use to determine</span></span><span class="apple-converted-space" style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;"> </span></b></div>
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<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a) direction<br />b) magnitude<br />c) resultant<br />d) none of the above<br /><b><br />5) From which of the following dc cannot flow continuously?</b></span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a) Inductor</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">b) Capacitor</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">c) Resistor</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">d) All</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">  <b><br />6) tan75=?</b></span><b><br /></b></span><b><span style="font-family: Arial, sans-serif; margin: 0px; padding: 0px;"><br /></span><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">7) Which of them is odd: </span></b><span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;"><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a) circle</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">b) pentagon</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">c) triangle</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">d) rectangle</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">e) square</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;"><b><br /><br />8) the value of angular momentum is for;</b></span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a. Spinning bodies motion</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">b. Orbital motion<span class="apple-converted-space" style="margin: 0px; padding: 0px;"> </span></span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">c. All types of bodies</span></span><b><span style="font-family: Arial, sans-serif; margin: 0px; padding: 0px;"><br /></span><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">9) in a triangle if b=5 c=10 and angle alpha=60 degree</span></b><span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;"><b>, <span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">where angle a has opposite side a angle b has opposite side b and angle c has opposite side c</span>, <span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">find side a</span></b><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a.10</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">b.5 sq. root 3</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">c. sq.root125</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;"><br /><b><br />10) if two forces of 10 N each applied simultaneously on a same point with 60 degree and 120 degree respectively then their resultant will be?</b><br /><br /><b>11) if a tap fills a tank in 8 Hours and another tap takes 16 Hours to empty the same tank...If both taps are used how long will it take to fill the tank? </b><br /><b><br />12) in order to increase resistance resistors are connected in: </b></span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a. Parallel</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">b. Series<br /><b><br />13) for what value of x tangent to the curve of this equation is parallel to y axis y=5 sq. root x +11?<span class="apple-converted-space" style="margin: 0px; padding: 0px;"> </span></b></span><b><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;"><br />14) If an object moves away from surface of earth g:.</span></b><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a. Remains same</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">b. Varies inversely with square of distance</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">c. Varies directly with square of distance</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">d. None<br /><b><br />15) which of the following z not included in SAARC</b></span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a) Bangladesh</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">b) Iran</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">c) India</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">d) silence  </span><br /><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;"><b>16) she is vain___ her beauty</b></span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a. To</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">b. Of</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">c. In</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">d. By</span></span><b><span style="font-family: Arial, sans-serif; margin: 0px; padding: 0px;"><br /></span><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">17) frequency of radio waves?</span></b><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">a. Very high</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">b. High</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">c. Low</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">d. None<br /><b><br />18) resistance of a good fuse would be</b></span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">a. 1000 ohm</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">b. Minimum</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">c. Maximum</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">d. None<br />  </span><b><br /><span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">19) terminal potential difference will be equal to emf when</span></b><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">a. Circuit is close</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">b. Circuit open</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">c. Both a and b</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">d. R is zero<br /><br />  ..</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;"><b>20) If sin (alpha -beta) = sq. root 3 by 2 and alpha = 15 then beta =?</b></span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">a) 75</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">b) 45</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">c) -75</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">d) -45<br /><br /><b>21) Source of electrical energy</b></span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">a. Resistor</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">b. Battery</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">c. Capacitor</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">d. N0ne<br /><br />  </span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;"><b>22) which one is odd</b></span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">a. Speak</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">b. Record</span><br />
<span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;">c. State</span><br />
<span style="margin: 0px; padding: 0px;"><span style="font-family: Arial, sans-serif; margin: 0px; padding: 0px;"><span style="line-height: 11.5px; margin: 0px; padding: 0px;">d. Report</span></span><br /><br /><span style="font-family: Arial, sans-serif; margin: 0px; padding: 0px;"><b><span style="line-height: 11.5px; margin: 0px; padding: 0px;">23) </span><span style="line-height: 14px; margin: 0px; padding: 0px;">Synonym</span><span style="line-height: 11.5px; margin: 0px; padding: 0px;"> of revolving?</span></b></span></span><span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;"><br /><b>24) f(x)=tanx in interval (pie/2,pie)</b><br />a. increases<br />b. Decrease<br />c. Increases logarithmically<br />d. Neither increase nor decrease</span><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;"><br />  </span><span style="font-family: Arial, sans-serif; line-height: 15.3333320617676px; margin: 0px; padding: 0px;"><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;"><b>25) if 52% students failed in math and 42% in eng and 17% in both. What would be the % of students passed in both?  </b></span><br /><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;"><b>26) People disliked him because he is addicted ___ bad habits</b></span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a. To</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">b. By</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">c. For</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">d. From</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;"><b>27) she entrusted all her baby duties___ me</b></span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a. To</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">b.on</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">c. Upon</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">d. Of<br /><br />  </span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;"><b>28) in the graph of f(x), the point of maxima is</b></span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a. always higher point</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">b. Always lower point</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">c. May be high or lower</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">d. Neither lower nor high<br /><br />  </span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;"><b>29) Bulb glows by</b></span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a. A.c</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">b. D.c</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">c. Both</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">d. N0ne<br /><br />  </span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;"><b>30) Potential energy difference between A and B is</b></span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a. Ua-Ub=Wab</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">b. Ub-Ua=Wab</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">c. Ub-Ua=-Wab<br /><br />  </span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;"><b>31) Ohms law is obeyed by devices</b></span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">a. Linear</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">b. N0n linear</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">c. Exponential</span><br /><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin: 0px; padding: 0px;">d. Logarithmic</span></span></div>
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1) The height of close satellite to earth isa. 400kmb.300kmc. 250kmd. 500km 2) Find equation of tangent to y^2= 4ax 3) A Solenoid bent into a circle is called?a) Solenoidb) Toroidc) both an and bd) n…

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IST Aggregate Calculator

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Intelligence Questions From NET-1 2014

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<span style="font-family: Times, Times New Roman, serif; font-size: large;"><span style="color: #38761d;">Assalam O Alaikum !</span><br />                                  There are some intelligence questions given below which were in NET-1 (NUST Entrance Test) held in December,2013. Well, these questions are not much difficult so try to solve them. Meanwhile you can discuss the answers below in comments. This practice will be helpful for you I.A.</span></div>
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<span style="color: #660000; font-size: large;"><b>Intelligence :</b></span></div>
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<span style="font-size: large;"><br /></span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><b>1)</b> Which of these rest is different from others?</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">a)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Physics</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">b)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Chemistry</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">c)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Geography</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">d)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Zoology</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px;">
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><br /></span></div>
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><div style="margin: 0px; padding: 0px; text-align: justify;">
<b>2)</b> Choose the odd one out</div>
</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">a)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Chocolate</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">b)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Bread</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">c)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Cake</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">d)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Biscuits</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px;">
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><br /></span></div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><b>3)</b><span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Pick the odd one out</span></div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">a)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>cot</span></div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">b)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>sheet</span></div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">c)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>blanket</span></div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">d)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>pillow</span></div>
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<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px;">
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><br /></span></div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><b>4)</b><span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Pick the odd one out</span></div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">a)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Study</span></div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">b)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Knowledge</span></div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">c)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Write</span></div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">d)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Learn</span></div>
</div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px;">
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><br /></span></div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><b>5)</b><span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Pick the odd one</span></div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">a)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>green </span></div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">b)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>violet</span></div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">c)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>white</span></div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">d)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>yellow</span></div>
</div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px;">
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><br /></span></div>
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><div style="margin: 0px; padding: 0px; text-align: justify;">
<b>6)</b> Pick the odd one out</div>
</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">a)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>SAARC</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">b)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>WHO</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">c)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>IMF</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">d)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>UNICEF</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px;">
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><br /></span></div>
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><div style="margin: 0px; padding: 0px; text-align: justify;">
<b>7)</b> Pick the odd one out</div>
</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">a)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Calendar</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">b)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Month</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">c)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Day </span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">d)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>Time</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><br /></span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><b>8)</b> <span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>32 31 30 </span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">     33 33 29</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">     34 35 ?</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">a)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>27</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">b)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>30</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">c)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>31</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">d)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>40</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px;">
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><br /></span></div>
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><div style="margin: 0px; padding: 0px; text-align: justify;">
<b>9)</b> Complete the series:</div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
3760, 1880, 360, 180, 45,? </div>
</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">a)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>13 </span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">b)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>15 </span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">c)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>11.5</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">d)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>22.5</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px;">
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><br /></span></div>
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><div style="margin: 0px; padding: 0px; text-align: justify;">
<b>10)</b> What will be the next term in the sequence? </div>
<div style="margin: 0px; padding: 0px; text-align: justify;">
25/50, 50/100, 150/300, 600/1200, ?</div>
</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">a)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>3000/6000</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">b)<span class="Apple-tab-span" style="margin: 0px; padding: 0px; white-space: pre;"> </span>1200/2400</span></div>
<div style="background-color: white; font-family: Arial, San-Serif; font-size: 16px; line-height: 29px; margin: 0px; padding: 0px;">
<div style="margin: 0px; padding: 0px; text-align: justify;">
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">c)</span><span class="Apple-tab-span" style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px; white-space: pre;"> </span><span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;">750/1500</span></div>
<span style="font-family: Arial, Helvetica, sans-serif; margin: 0px; padding: 0px;"><div style="margin: 0px; padding: 0px; text-align: justify;">
d)    none of these</div>
</span></div>
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Intelligence Questions From NET-1 2014

Assalam O Alaikum ! There are some intelligence questions given below which were in NET-1 (NUST Entrance Test) held in December,2013. Well, these questions are not mu…

How To Solve Code Word MCQs

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<span style="font-family: Times, Times New Roman, serif; font-size: large;">Especially In NUST Entry Test, there are always 3 or 4 code word MCQs. When I appeared in NET-2, I wasted too much time to solve these MCQs because I didn't know how to tackle this type of question. After some period, I'd found a trick to solve code word question so I am going to share that trick with you. I hope, you will like this.</span></div>
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How To Solve Code Word MCQs

Especially In NUST Entry Test, there are always 3 or 4 code word MCQs. When I appeared in NET-2, I wasted too much time to solve these MCQs because I didn't know how to tackle this type of question. …

Shortcut Method To Find Derivative of Implicit Functions

<div dir="ltr" style="text-align: left;" trbidi="on"><div dir="ltr" style="text-align: left;" trbidi="on"><div class="separator" style="clear: both; text-align: center;"></div><div style="text-align: left;"><span style="color: purple; font-family: "times" , "times new roman" , serif; font-size: large;"><b>Assalam O Alaikum !</b></span></div><div style="text-align: left;"><span style="font-family: "times" , "times new roman" , serif; font-size: large;">                                 Hi Guys, How are you ? I hope you are enjoying a good health. I am back with a new lesson for you. Well, I know it is quite easy to find the derivative of Implicit functions but it takes too much time. I am gonna tell you a shortcut method to find derivative of implicit functions which will surely save your time. </span></div><br />
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Shortcut Method To Find Derivative of Implicit Functions

Assalam O Alaikum ! Hi Guys, How are you ? I hope you are enjoying a good health. I am back with a new lesson for you. Well, I know it is quite easy to find the derivat…

How To Find The Rank of Matrix

<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #990000;"><b>The Rank of a Matrix :</b></span></span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;">You can think of an <i style="font-weight: normal;">r</i> x <i style="font-weight: normal;">c</i> matrix as a set of <i style="font-weight: normal;">r</i> row vectors, each having <i style="font-weight: normal;">c</i> elements; or you can think of it as a set of <i style="font-weight: normal;">c</i> column vectors, each having <i style="font-weight: normal;">r</i> elements.</span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><br />
<b>The rank of a matrix is defined as :</b><br />
(a) the maximum number of linearly independent <i style="font-weight: normal;">column</i> vectors in the matrix<br />
or<br />
(b) the maximum number of linearly independent <i style="font-weight: normal;">row</i> vectors in the matrix.<br />
Both definitions are equivalent.</span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><br />
For an <i style="font-weight: normal;">r</i> x <i style="font-weight: normal;">c</i> matrix,</span></div><ul style="text-align: left;"><li><span style="font-family: Arial, Helvetica, sans-serif;">If <i>r</i> is less than <i>c</i>, then the maximum rank of the matrix is <i>r</i>.</span></li>
<li><span style="font-family: Arial, Helvetica, sans-serif;">If <i>r</i> is greater than <i>c</i>, then the maximum rank of the matrix is <i>c</i>.</span></li>
</ul><span style="font-family: Arial, Helvetica, sans-serif;">The rank of a matrix would be zero only if the matrix had no elements. If a matrix had even one element, its minimum rank would be one.<br />
<span style="color: #990000;"><b><br />
</b></span></span><br />
<div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #990000;"><b>How to Find Matrix Rank :</b></span></span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #990000;"><b><br />
</b></span>In this section, we describe a method for finding the rank of any matrix. This method assumes familiarity with echelon matrices and echelon transformations.<br />
The maximum number of linearly independent vectors in a matrix is equal to the number of non-zero rows in its row echelon matrix. Therefore, to find the rank of a matrix, we simply transform the matrix to its row echelon form and count the number of non-zero rows.<br />
Consider matrix <strong>A</strong> and its row echelon matrix, <b>A<sub>ref</sub>. </b></span><table style="border-collapse: collapse; border-spacing: 0px; color: black; font-weight: normal; line-height: 20px; margin-left: auto; margin-right: auto; margin-top: 20px; padding: 0px;"><tbody>
<tr><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdrightbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">7</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">8</span></td></tr>
</tbody></table></td><td style="margin: 0px; padding: 0px;"><span style="font-family: Arial, Helvetica, sans-serif;">   ⇒   </span></td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdrightbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td></tr>
</tbody></table></td></tr>
<tr><td style="margin: 0px; padding: 0px; text-align: center; vertical-align: bottom;"><strong><span style="font-family: Arial, Helvetica, sans-serif;">A</span></strong></td><td style="margin: 0px; padding: 0px;"></td><td style="margin: 0px; padding: 0px; text-align: center; vertical-align: bottom;"><strong><span style="font-family: Arial, Helvetica, sans-serif;">A<sub>ref</sub></span></strong></td></tr>
</tbody></table><span style="font-family: Arial, Helvetica, sans-serif; font-weight: normal;">Because the row echelon form <strong>A<sub>ref</sub></strong> has two non-zero rows, we know that matrix <strong>A</strong> has two independent row vectors; and we know that the rank of matrix <strong>A</strong> is 2.<br />
You can verify that this is correct. Row 1 and Row 2 of matrix <strong>A</strong> are linearly independent. However, Row 3 is a linear combination of Rows 1 and 2. Specifically, Row 3 = 3*( Row 1 ) + 2*( Row 2). Therefore, matrix <strong>A</strong> has only two independent row vectors.</span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><b><br />
</b><span style="color: #990000;"><b>Full Rank Matrices :</b></span></span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #990000;"><b><br />
</b></span>When all of the vectors in a matrix are linearly independent, the matrix is said to be <strong style="font-weight: normal;">full rank</strong>. Consider the matrices <strong style="font-weight: normal;">A</strong> and <strong style="font-weight: normal;">B</strong> below.</span><table style="border-collapse: collapse; border-spacing: 0px; color: black; font-weight: normal; line-height: 20px; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td style="margin: 0px; padding: 0px 0px 10px;"><span style="font-family: Arial, Helvetica, sans-serif;"><strong>A = </strong> </span></td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; margin-bottom: 10px; margin-left: 10px; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="2" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">3</span></td><td class="tdrightbracket" rowspan="2" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">4</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">6</span></td></tr>
</tbody></table></td><td style="margin: 0px; padding: 0px; width: 60px;"><span style="font-family: Arial, Helvetica, sans-serif;"> </span></td><td style="margin: 0px; padding: 0px 0px 10px;"><span style="font-family: Arial, Helvetica, sans-serif;"><strong>B</strong> =  </span></td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; margin-bottom: 10px; margin-left: 10px; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdrightbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">3</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td></tr>
</tbody></table></td></tr>
</tbody></table><span style="font-family: Arial, Helvetica, sans-serif; font-weight: normal;">Notice that row 2 of matrix <strong>A</strong> is a scalar multiple of row 1; that is, row 2 is equal to twice row 1. Therefore, rows 1 and 2 are linearly dependent. Matrix <strong>A</strong> has only one linearly independent row, so its rank is 1. Hence, matrix <strong>A</strong> is not full rank.<br />
Now, look at matrix <strong>B</strong>. All of its rows are linearly independent, so the rank of matrix <strong>B</strong> is 3. Matrix <strong>B</strong> is full rank.</span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><br />
<span style="color: #38761d;"><b>Test Your Understanding:</b></span></span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #38761d; font-weight: normal;"><b><br />
</b></span><span style="color: red; font-weight: normal;"><b>Problem 1 :</b></span>Consider the matrix <strong style="font-weight: normal;">X</strong>, shown below.</span><table style="border-collapse: collapse; border-spacing: 0px; color: black; font-weight: normal; line-height: 20px; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td style="margin: 0px; padding: 0px;"><span style="font-family: Arial, Helvetica, sans-serif;"><strong>X</strong> =    </span></td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="2" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">4</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">4</span></td><td class="tdrightbracket" rowspan="2" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">3</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">4</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">8</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td></tr>
</tbody></table></td></tr>
</tbody></table><span style="font-family: Arial, Helvetica, sans-serif; font-weight: normal;">What is its rank?<br />
(A) 0<br />
(B) 1<br />
(C) 2<br />
(D) 3<br />
(E) 4<br />
<strong><span style="color: red;"><br />
</span></strong></span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif; font-weight: normal;"><strong><span style="color: red;">Solution :</span></strong>The correct answer is (C). Since the matrix has more than zero elements, its rank must be greater than zero. And since it has fewer rows than columns, its maximum rank is equal to the maximum number of linearly independent rows. And because neither row is linearly dependent on the other row, the matrix has 2 linearly independent rows; so its rank is 2.<strong><br />
</strong><strong><span style="color: red;"><br />
</span></strong></span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif; font-weight: normal;"><strong><span style="color: red;">Problem 2</span></strong>Consider the matrix <strong>Y</strong>, shown below.</span><table style="border-collapse: collapse; border-spacing: 0px; color: black; font-weight: normal; line-height: 20px; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td style="margin: 0px; padding: 0px;"><span style="font-family: Arial, Helvetica, sans-serif;"><strong>Y</strong> =    </span></td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="4" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">3</span></td><td class="tdrightbracket" rowspan="4" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">3</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">5</span></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">3</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">4</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">7</span></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">4</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">5</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">9</span></td></tr>
</tbody></table></td></tr>
</tbody></table><span style="font-family: Arial, Helvetica, sans-serif; font-weight: normal;">What is its rank?<br />
(A) 0<br />
(B) 1<br />
(C) 2<br />
(D) 3<br />
(E) 4<br />
<strong><span style="color: red;"><br />
</span></strong></span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif; font-weight: normal;"><strong><span style="color: red;">Solution</span></strong>The correct answer is (C). Since the matrix has more than zero elements, its rank must be greater than zero. And since it has fewer columns than rows, its maximum rank is equal to the maximum number of linearly independent columns.<br />
Columns 1 and 2 are independent, because neither can be derived as a scalar multiple of the other. However, column 3 is linearly dependent on columns 1 and 2, because column 3 is equal to column 1 plus column 2. That leaves the matrix with a maximum of two linearly independent columns; that is, column 1 and column 2. So the matrix rank is 2.</span></div><div style="font-weight: normal; text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><br />
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How To Find The Rank of Matrix

The Rank of a Matrix :You can think of an r x c matrix as a set of r row vectors, each having c elements; or you can think of it as a set of c column vectors, each having r elements. The rank of a mat…

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