Solved papers for JEE Main & Advanced AIEEE Solved Paper-2012

done AIEEE Solved Paper-2012 Total Questions - 30

• question_answer1) The equation \[{{e}^{\sin x}}-{{e}^{-\sin x}}-4=0\] has:   AIEEE  Solved  Paper-2012
  A)
  infinite number of real roots
  done clear
  B)
  no real roots
  done clear
  C)
  exactly one real root
  done clear
  D)
  exactly four real roots
  done clear
  View Answer play_arrow

• question_answer2) Let \[\hat{a}\] and \[\hat{b}\] be two unit vectors. If the vectors \[\vec{c}=\hat{a}+2\hat{b}\] and \[\vec{d}=5\hat{a}-4\hat{b}\] are perpendicular to each other, then the angle between \[\hat{a}\] and \[\hat{b}\] is:   AIEEE  Solved  Paper-2012
  A)
  \[\frac{\pi }{6}\]                                  
  done clear
  B)
  \[\frac{\pi }{2}\]
  done clear
  C)
  \[\frac{\pi }{3}\]                                  
  done clear
  D)
  \[\frac{\pi }{4}\]
  done clear
  View Answer play_arrow

• question_answer3) A spherical balloon is filled with \[4500\pi \] cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of \[72\pi \] cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is:   AIEEE  Solved  Paper-2012
  A)
  \[\frac{9}{7}\]                                      
  done clear
  B)
  \[\frac{7}{9}\]
  done clear
  C)
  \[\frac{2}{9}\]                                      
  done clear
  D)
  \[\frac{9}{2}\]
  done clear
  View Answer play_arrow

• question_answer4) Statement-1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + .... + (361 + 380 + 400) is 8000. Statement-2: \[\sum\limits_{k=1}^{n}{({{k}^{3}}-{{(k-1)}^{3}}={{n}^{3}}}\], for any natural number n.   AIEEE  Solved  Paper-2012
  A)
  Statement-1 is false, Statement-2 is true.
  done clear
  B)
  Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.
  done clear
  C)
  Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1.
  done clear
  D)
  Statement-1 is true, statement-2 is false.
  done clear
  View Answer play_arrow

• question_answer5) The negation of the statement ?If I become a teacher, then I will open a school? is:   AIEEE  Solved  Paper-2012
  A)
  I will become a teacher and I will not open a school.
  done clear
  B)
  Either I will not become a teacher or I will not open a school.
  done clear
  C)
  Neither I will become a teacher nor I will open a school
  done clear
  D)
  I will not become a teacher or I will open a school.
  done clear
  View Answer play_arrow

• question_answer6) If the integral \[\int{\frac{5\tan x}{\tan x-2}dx=x+a\,\ell n\left| \sin x-2\cos x \right|+k}\], then a is equal to:   AIEEE  Solved  Paper-2012
  A)
               \[-1\]                                       
  done clear
  B)
               \[-2\] 
  done clear
  C)
               1                                                
  done clear
  D)
               2
  done clear
  View Answer play_arrow

• question_answer7) Statement-1: An equation of a common tangent to the parabola \[{{y}^{2}}=16\sqrt{3}x\] and the ellipse \[2{{x}^{2}}+{{y}^{2}}=4\] is \[y=2x+2\sqrt{3}\]. Statement-2: If the line \[mx+\frac{4\sqrt{3}}{m},(m\ne 0)\] is a common tangent to the parabola \[{{y}^{2}}=16\sqrt{3}x\] and the ellipse \[2{{x}^{2}}+{{y}^{2}}=4\], then m satisfies\[{{m}^{4}}+2{{m}^{2}}=24\].   AIEEE  Solved  Paper-2012
  A)
  Statement-1 is false, Statement-2 is true.
  done clear
  B)
  Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.
  done clear
  C)
  Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1.
  done clear
  D)
               Statement-1 is true, statement-2 is false.
  done clear
  View Answer play_arrow

• question_answer8) Let \[A=\left( \begin{matrix}   1 & 0 & 0  \\   2 & 1 & 0  \\    3 & 2 & 1  \\ \end{matrix} \right)\]. If \[{{\mu }_{1}}\] and \[{{\mu }_{2}}\] are column matrices such that \[A{{u}_{1}}\left( \begin{align}   & 1 \\  & 0 \\  & 0 \\ \end{align} \right)\] and \[A{{u}_{2}}\left( \begin{align}   & 0 \\  & 1 \\  & 0 \\ \end{align} \right)\], then \[{{u}_{1}}+{{u}_{2}}\] is equal to:   AIEEE  Solved  Paper-2012
  A)
  \[\left( \begin{align}   & -1 \\  & 1 \\  & 0 \\ \end{align} \right)\]                                          
  done clear
  B)
  \[\left( \begin{align}   & -1 \\  & 1 \\  & -1 \\ \end{align} \right)\]
  done clear
  C)
  \[\left( \begin{align}   & -1 \\  & -1 \\  & 0 \\ \end{align} \right)\]                                          
  done clear
  D)
  \[\left( \begin{align}   & 1 \\  & -1 \\  & -1 \\ \end{align} \right)\]
  done clear
  View Answer play_arrow

• question_answer9) If n is a positive integer, then \[{{\left( \sqrt{3}+1 \right)}^{2n}}-{{\left( \sqrt{3}-1 \right)}^{2n}}\] is:   AIEEE  Solved  Paper-2012
  A)
  an irrational number
  done clear
  B)
  an odd positive integer
  done clear
  C)
  an even positive integer
  done clear
  D)
  a rational number other than positive integers
  done clear
  View Answer play_arrow

• question_answer10) If 100 times the 100th term of an AP with non zero common  difference equals the 50 times its 50th term, then the 150th term of this AP is :   AIEEE  Solved  Paper-2012
  A)
  -150                                          
  done clear
  B)
  150 times its 50th term
  done clear
  C)
   150                                          
  done clear
  D)
  zero
  done clear
  View Answer play_arrow

• question_answer11) In a \[\Delta PQR\], if \[3\sin P+4\cos Q=6\] and \[4\sin Q+3\cos P=1\], then the angle R is equal to:   AIEEE  Solved  Paper-2012
  A)
  \[\frac{5\pi }{6}\]                               
  done clear
  B)
  \[\frac{\pi }{6}\]
  done clear
  C)
  \[\frac{\pi }{4}\]                                  
  done clear
  D)
  \[\frac{3\pi }{4}\]
  done clear
  View Answer play_arrow

• question_answer12) A equation of a plane parallel to the plane \[x-2y+2z-5=0\]and at a unit distance from the origin is:     AIEEE  Solved  Paper-2012
  A)
  \[x-2y+2z-3=0\]  
  done clear
  B)
  \[x-2y+2z+1=0\] 
  done clear
  C)
  \[x-2y+2z-1=0\]  
  done clear
  D)
  \[x-2y+2z+5=0\]
  done clear
  View Answer play_arrow

• question_answer13) If the line \[2x+y=k\] passes through the point which divides the line segment joining the points (1, 1) and (2, 4) in the ratio 3 : 2, then k equals :   AIEEE  Solved  Paper-2012
  A)
  \[\frac{29}{5}\]                                    
  done clear
  B)
  5
  done clear
  C)
  6                                                
  done clear
  D)
  \[\frac{11}{5}\]
  done clear
  View Answer play_arrow

• question_answer14) Let \[{{x}_{1}},{{x}_{2}},\,....\,,{{x}_{n}}\] be n observations, and let \[\overline{x}\] be their arithmetic mean and \[{{\sigma }^{2}}\] be the variance Statement-1: Variance of \[2{{x}_{1}},2{{x}_{2}},\,......,\,2{{x}_{n}}\] is \[4{{\sigma }^{2}}\]. Statement-2: Arithmetic mean \[2{{x}_{1}},2{{x}_{2}},\,......,\,2{{x}_{n}}\] is \[4\overline{x}\].   AIEEE  Solved  Paper-2012
  A)
  Statement-1 is false, Statement-2 is true.
  done clear
  B)
  Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for  Statement-1.
  done clear
  C)
  Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1.
  done clear
  D)
  Statement-1 is true, statement-2 is false.
  done clear
  View Answer play_arrow

• question_answer15) The population p(t) at time t of a certain mouse species satisfies the differential equation\[\frac{dp(t)}{dt}=0.5\,p(t)-450\]If \[p(0)=850\], then the time at which the population becomes zero is :   AIEEE  Solved  Paper-2012
  A)
  \[2\,\ell n18\]                                      
  done clear
  B)
  \[\ell n\,9\]
  done clear
  C)
  \[\frac{1}{2}\ell n\,18\]                    
  done clear
  D)
  \[\ell n\,18\]
  done clear
  View Answer play_arrow

• question_answer16) Let a, \[b\in R\] be such that the function f given by \[f(x)=\ell n\left| x \right|+b{{x}^{2}}+ax,\,x\ne 0\] has extreme values at \[x=-1\] and \[x=2\]. Statement-1: f has local maximum at \[x=-1\] and at \[x=2\]. Statement-2: \[a=\frac{1}{2}\] and \[b=\frac{-1}{4}\].   AIEEE  Solved  Paper-2012
  A)
  Statement-1 is false, Statement-2 is true.
  done clear
  B)
  Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.
  done clear
  C)
  Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1.
  done clear
  D)
  Statement-1 is true, statement-2 is false.
  done clear
  View Answer play_arrow

• question_answer17) The area bounded between the parabolas \[{{x}^{2}}=\frac{y}{4}\] and \[{{x}^{2}}=9y\] and the straight line \[y=2\]is:   AIEEE  Solved  Paper-2012
  A)
  \[20\sqrt{2}\]                       
  done clear
  B)
  \[\frac{10\sqrt{2}}{3}\]
  done clear
  C)
  \[\frac{20\sqrt{2}}{3}\]                    
  done clear
  D)
  \[10\sqrt{2}\]
  done clear
  View Answer play_arrow

• question_answer18) Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls is:   AIEEE  Solved  Paper-2012
  A)
  880                                           
  done clear
  B)
  629 
  done clear
  C)
  630                                           
  done clear
  D)
  879
  done clear
  View Answer play_arrow

• question_answer19) If f : \[R\to R\] is a function defined by \[f(x)=[x]\cos \left( \frac{2x-1)}{2} \right)\pi \], where \[[x]\] denotes the greatest integer function, then f is:   AIEEE  Solved  Paper-2012
  A)
  continuous for every real \[x\].
  done clear
  B)
  discontinuous only at \[x=0\].
  done clear
  C)
  discontinuous only at non-zero integral values of \[x\].
  done clear
  D)
               continuous only at \[x=0\].
  done clear
  View Answer play_arrow

• question_answer20) If the line \[\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}\] and \[\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}\] intersect, then k is equal to :   AIEEE  Solved  Paper-2012
  A)
  \[-1\]                                       
  done clear
  B)
  \[\frac{2}{9}\]
  done clear
  C)
  \[\frac{9}{2}\]                                      
  done clear
  D)
  0
  done clear
  View Answer play_arrow

• question_answer21) Three numbers are chosen at random without replacement from {1, 2, 3, ..., 8}. The probability that their minimum is 3, given that their maximum is 6, is :   AIEEE  Solved  Paper-2012
  A)
  \[\frac{3}{8}\]                                      
  done clear
  B)
  \[\frac{1}{5}\]
  done clear
  C)
  \[\frac{1}{4}\]                                      
  done clear
  D)
  \[\frac{2}{5}\]
  done clear
  View Answer play_arrow

• question_answer22) If \[z\ne 1\] and \[\frac{{{z}^{2}}}{z-1}\] is real, then the point represented by the complex number z lies :   AIEEE  Solved  Paper-2012
  A)
  either on the real axis or on a circle passing through the origin.
  done clear
  B)
  on a circle with centre at the origin.
  done clear
  C)
  either on the real axis or on a circle not passing through the origin.
  done clear
  D)
  on the imaginary axis.
  done clear
  View Answer play_arrow

• question_answer23) Let P and Q be \[3\times 3\] matrices \[P\ne Q\]. If \[{{P}^{3}}={{Q}^{3}}\] and \[{{P}^{2}}={{Q}^{2}}\], then determinant of \[({{P}^{2}}={{Q}^{2}})\] is equal to :   AIEEE  Solved  Paper-2012
  A)
  \[-2\]                                       
  done clear
  B)
  1
  done clear
  C)
  0                                                
  done clear
  D)
  \[-1\]
  done clear
  View Answer play_arrow

• question_answer24) If \[g(x)=\int\limits_{0}^{x}{\cos 4t\,\,dt}\], then \[g(x+\pi )\] equals   AIEEE  Solved  Paper-2012
  A)
  \[\frac{g(x)}{g(\pi )}\]                                      
  done clear
  B)
  \[g(x)+g(\pi )\]
  done clear
  C)
  \[g(x)-g(\pi )\]        
  done clear
  D)
  \[g(x).\,g(\pi )\]
  done clear
  View Answer play_arrow

• question_answer25) The length of the diameter of the circle which touches the x-axis at the point (1, 0) and passes through the point (2, 3) is :   AIEEE  Solved  Paper-2012
  A)
  \[\frac{10}{3}\]                                    
  done clear
  B)
  \[\frac{3}{5}\]
  done clear
  C)
  \[\frac{6}{5}\]                                      
  done clear
  D)
  \[\frac{5}{3}\]
  done clear
  View Answer play_arrow

• question_answer26) Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can formed such that \[Y\subseteq X,Z\subseteq X\] and \[Y\cap Z\] is empty, is :   AIEEE  Solved  Paper-2012
  A)
  \[{{5}^{2}}\]                                          
  done clear
  B)
  \[{{3}^{5}}\]
  done clear
  C)
  \[{{2}^{5}}\]                                          
  done clear
  D)
  \[{{5}^{3}}\]
  done clear
  View Answer play_arrow

• question_answer27) An ellipse is drawn by taking a diameter of the circle \[{{(x-1)}^{2}}+{{y}^{2}}=1\] as its semi-minor axis and a diameter of the circle \[{{x}^{2}}+{{(y-2)}^{2}}=4\] is semi-major axis. If the centre of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is :   AIEEE  Solved  Paper-2012
  A)
  \[4{{x}^{2}}+{{y}^{2}}=4\]                              
  done clear
  B)
  \[{{x}^{2}}+4{{y}^{2}}=8\]
  done clear
  C)
  \[4{{x}^{2}}+{{y}^{2}}=8\]                              
  done clear
  D)
  \[{{x}^{2}}+4{{y}^{2}}=16\]
  done clear
  View Answer play_arrow

• question_answer28) Consider the function, \[f(x)=\left| x-2 \right|+\left| x-5 \right|,x\in R\]. Statement-1: \[f'(4)=0\] Statement-2: \[f\] is continuous in [2, 5], differentiable in (2, 5) and \[f(2)=f(5)\].   AIEEE  Solved  Paper-2012
  A)
  Statement-1 is false, Statement-2 is true.
  done clear
  B)
  Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.
  done clear
  C)
  Statement-1 is true, statement-2 is true; statement-2 is not a corect explanation for Statement-1.
  done clear
  D)
  Statement-1 is true, statement-2 is false.
  done clear
  View Answer play_arrow

• question_answer29) A line is drawn through the point (1, 2) to meet the coordinate axes at P and Q such that it forms a triangle OPQ, where O is the origin. if the area of the triangle OPQ is least, then the slope of the line PQ is :   AIEEE  Solved  Paper-2012
  A)
  \[-\frac{1}{4}\]                                    
  done clear
  B)
  \[-4\]
  done clear
  C)
  \[-2\]                                       
  done clear
  D)
  \[-\frac{1}{2}\]
  done clear
  View Answer play_arrow

• question_answer30) Let ABCD be a parallelogram such that \[\overrightarrow{AB}=\vec{q},\,\,\overrightarrow{AD}=\vec{p}\] and \[\angle BAD\] be an acute angle. If \[\vec{r}\] is the vector that coincides with the altitude directed from the vertex B to the side AD, then \[\vec{r}\] is given by :   AIEEE  Solved  Paper-2012
  A)
  \[\vec{r}=3\vec{q}-\frac{3(\vec{p}.\,\vec{q})}{(\vec{p}.\,\vec{p})}\vec{p}\]          
  done clear
  B)
  \[\vec{r}=-\vec{q}+\left( \frac{\vec{p}.\,\vec{q}}{\vec{p}.\,\vec{p}} \right)\vec{p}\]
  done clear
  C)
  \[\vec{r}=\vec{q}-\left( \frac{\vec{p}.\,\vec{q}}{\vec{p}.\,\vec{p}} \right)\vec{p}\]           
  done clear
  D)
  \[\vec{r}=-3\vec{q}+\frac{3(\vec{p}.\,\vec{q})}{(\vec{p}.\,\vec{p})}\vec{p}\]
  done clear
  View Answer play_arrow