Solved papers for JEE Main & Advanced JEE Main Paper (Held On 11 April 2014)

done JEE Main Paper (Held On 11 April 2014) Total Questions - 30

• question_answer1) Let f be an odd function defined on the set of real numbers such that for \[x\ge 0,\]\[f(x)=3sinx+4cosx.\] Then f(x) at \[x=-\frac{11\pi }{6}\]is equal to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[\frac{3}{2}+2\sqrt{3}\]                              
  done clear
  B)
  \[-\frac{3}{2}+2\sqrt{3}\]
  done clear
  C)
  \[\frac{3}{2}-2\sqrt{3}\]                               
  done clear
  D)
  \[-\frac{3}{2}-2\sqrt{3}\]
  done clear
  View Answer play_arrow

• question_answer2) If \[{{z}_{1}},{{z}_{2}}\]and \[{{z}_{3}},{{z}_{4}}\]are 2 pairs of complex conjugate numbers, then \[\arg \left( \frac{{{z}_{1}}}{{{z}_{4}}} \right)+\arg \left( \frac{{{z}_{2}}}{{{z}_{3}}} \right)\]equals:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  0                                             
  done clear
  B)
  \[\frac{\pi }{2}\]
  done clear
  C)
  \[\frac{3\pi }{2}\]                                            
  done clear
  D)
  \[\pi \]
  done clear
  View Answer play_arrow

• question_answer3) If \[\alpha \] and \[\beta \] are roots of the equation, \[{{x}^{2}}-4\sqrt{2}kx+2{{e}^{4\ln k}}-1=0\] for some k, and \[{{\alpha }^{2}}+{{\beta }^{2}}=66,\]then \[{{\alpha }^{3}}+{{\beta }^{2}}\]is equal to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[248\sqrt{2}\]                 
  done clear
  B)
  \[280\sqrt{2}\]
  done clear
  C)
  \[-32\sqrt{2}\]                  
  done clear
  D)
  \[-280\sqrt{2}\]
  done clear
  View Answer play_arrow

• question_answer4) Let A be a \[3\times 3\]matrix such that\[A\left[ \begin{matrix}    1 & 2 & 3  \\    0 & 2 & 3  \\    0 & 1 & 1  \\ \end{matrix} \right]=\left[ \begin{matrix}    0 & 0 & 1  \\    1 & 0 & 0  \\    0 & 1 & 0  \\ \end{matrix} \right]\]Then \[{{A}^{-1}}\]is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[\left[ \begin{matrix}    3 & 1 & 2  \\    3 & 0 & 2  \\    1 & 0 & 1  \\ \end{matrix} \right]\]   
  done clear
  B)
  \[\left[ \begin{matrix}    3 & 2 & 1  \\    3 & 2 & 0  \\    1 & 1 & 0  \\ \end{matrix} \right]\]
  done clear
  C)
  \[\left[ \begin{matrix}    0 & 1 & 3  \\    0 & 2 & 3  \\    1 & 1 & 0  \\ \end{matrix} \right]\]                   
  done clear
  D)
  \[\left[ \begin{matrix}    1 & 2 & 3  \\    0 & 1 & 1  \\    0 & 2 & 3  \\ \end{matrix} \right]\]
  done clear
  View Answer play_arrow

• question_answer5) Let for i = 1, 2, 3, pi(x) be a polynomial of degree 2 in \[x,p{{'}_{i}}(x)\]and \[p'{{'}_{i}}(x)\]be the first and second order derivatives of \[{{p}_{i}}(x)\] respectively. Let,\[A(x)=\left[ \begin{matrix}    {{p}_{1}}(x) & {{p}_{1}}'(x) & {{p}_{1}}''(x)  \\    {{p}_{2}}(x) & {{p}_{2}}'(x) & {{p}_{2}}''(x)  \\    {{p}_{3}}(x) & {{p}_{3}}'(x) & {{p}_{3}}''(x)  \\ \end{matrix} \right]\]and \[B(x)={{[A(x)]}^{T}}A(x).\]Then determinant of B(x):   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  is a polynomial of degree 6 in x.
  done clear
  B)
  is a polynomial of degree 3 in x.
  done clear
  C)
  is a polynomial of degree 2 in x.
  done clear
  D)
  does not depend on x.
  done clear
  View Answer play_arrow

• question_answer6) An eight digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  72 (7!)                  
  done clear
  B)
  18 (7!)
  done clear
  C)
  40 (7!)                  
  done clear
  D)
  36 (7!)
  done clear
  View Answer play_arrow

• question_answer7) The coefficient of \[{{x}^{50}}\]in the binomial expansion of\[{{(1+x)}^{1000}}+x{{(1+x)}^{999}}+{{x}^{2}}\]\[{{(1+x)}^{998}}+....+{{x}^{1000}}\] is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[\frac{\left( 1000 \right)!}{\left( 50 \right)!\left( 950 \right)!}\]                
  done clear
  B)
  \[\frac{\left( 1000 \right)!}{\left( 49 \right)!\left( 951 \right)!}\]
  done clear
  C)
  \[\frac{\left( 1001 \right)!}{\left( 51 \right)!\left( 950 \right)!}\]                
  done clear
  D)
  \[\frac{\left( 1001 \right)!}{\left( 50 \right)!\left( 951 \right)!}\]
  done clear
  View Answer play_arrow

• question_answer8) In a geometric progression, if the ratio of the sum of first 5 terms to the sum of their reciprocals is 49, and the sum of the first and the third term is 35. Then the first term of this geometric progression is:    
  A)
  7                                             
  done clear
  B)
  21
  done clear
  C)
  28                                          
  done clear
  D)
  42
  done clear
  View Answer play_arrow

• question_answer9) The sum of the first 20 terms common between the series 3 + + 11 + 15 + ......... and 1 + 6 + 11 + 16 + ......, is   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  4000                      
  done clear
  B)
  4020
  done clear
  C)
  4200                      
  done clear
  D)
  4220
  done clear
  View Answer play_arrow

• question_answer10) If\[\underset{x\to 2}{\mathop{\lim }}\,\frac{\tan \left( x-2 \right)\left\{ {{x}^{2}}+\left( k-2 \right)x-2k \right\}}{{{x}^{2}}-4x+4}=5,\]then k is equal to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  0                                             
  done clear
  B)
  1
  done clear
  C)
  2                                             
  done clear
  D)
  3
  done clear
  View Answer play_arrow

• question_answer11) Let f(x) = x|x|, g(x) = sin x and h(x) = (gof) (x). Then
  A)
  h(x) is not differentiable at x = 0.
  done clear
  B)
  h(x) is differentiable at x = 0, but h¢(x) is not continuous at x = 0
  done clear
  C)
  h?(x) is continuous at x = 0 but it is not differentiable at x = 0
  done clear
  D)
  h?(x) is differentiable at x = 0
  done clear
  View Answer play_arrow

• question_answer12) For the curve \[y=3\sin \theta \cos \theta ,x={{e}^{\theta }}\sin \theta ,\]\[0\le \theta \le \pi ,\]the tangent is parallel to x-axis when \[\theta \]is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[\frac{3\pi }{4}\]                                            
  done clear
  B)
  \[\frac{\pi }{2}\]
  done clear
  C)
  \[\frac{\pi }{4}\]              
  done clear
  D)
  \[\frac{\pi }{6}\]
  done clear
  View Answer play_arrow

• question_answer13) Two ships A and B are sailing straight away from a fixed point O along routes such that \[\angle AOB\] is always \[120{}^\circ \]. At a certain instance, OA = 8 km, OB = 6 km and the ship A is sailing at the rate of 20 km/hr while the ship B sailing at the rate of 30 km/hr. Then the distance between A and B is changing at the rate (in km/ hr):   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[\frac{260}{\sqrt{37}}\]              
  done clear
  B)
  \[\frac{260}{37}\]
  done clear
  C)
  \[\frac{80}{\sqrt{37}}\]                 
  done clear
  D)
  \[\frac{80}{37}\]
  done clear
  View Answer play_arrow

• question_answer14) The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius \[=\sqrt{3}\] is:
  A)
  \[\frac{4}{3}\sqrt{3}\pi \]                            
  done clear
  B)
  \[\frac{8}{3}\sqrt{3}\pi \]
  done clear
  C)
  \[4\pi \]               
  done clear
  D)
  \[2\pi \]
  done clear
  View Answer play_arrow

• question_answer15) The integral\[\int_{{}}^{{}}{x{{\cos }^{-1}}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)dx(x>0)\]is equal to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[-x+1(1+{{x}^{2}})ta{{n}^{-1}}x+c\]
  done clear
  B)
  \[x-(1+{{x}^{2}})co{{t}^{-1}}x+c\]
  done clear
  C)
  \[-x+(1+{{x}^{2}})co{{t}^{-1}}x+c\]
  done clear
  D)
  \[x-(1+{{x}^{2}})ta{{n}^{-1}}x+c\]
  done clear
  View Answer play_arrow

• question_answer16) If for \[n\ge 1,{{P}_{n}}=\int\limits_{1}^{e}{{{\left( \log x \right)}^{n}}dx,}\]then\[{{P}_{10}}-90{{P}_{8}}\] then is equal to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  -9                                           
  done clear
  B)
  10e
  done clear
  C)
  -9 e                       
  done clear
  D)
  10
  done clear
  View Answer play_arrow

• question_answer17) If the general solution of the differential equation\[y'=\frac{y}{x}+\Phi \left( \frac{x}{y} \right),\]for some function \[\Phi ,\] is given by \[\ln |cx|=x,\]where c is an arbitrary constant, then \[\Phi (2)\]is equal to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  4                                             
  done clear
  B)
  \[\frac{1}{4}\]
  done clear
  C)
  -4                                           
  done clear
  D)
  \[-\frac{1}{4}\]
  done clear
  View Answer play_arrow

• question_answer18) A stair-case of length l rests against a vertical wall and a floor of a room. Let P be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio 1 : 2. If the stair-case begins to slide on the floor, then the locus of P is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  an ellipse of eccentricity\[\frac{1}{2}\]
  done clear
  B)
  an ellipse of eccentricity\[\frac{\sqrt{3}}{2}\]
  done clear
  C)
  a circle of radius\[\frac{1}{2}\]
  done clear
  D)
   a circle of radius\[\frac{\sqrt{3}}{2}l\]
  done clear
  View Answer play_arrow

• question_answer19) The base of an equilateral triangle is along the line given by 3x + 4y = 9. If a vertex of the triangle is (1, 2), then the length of a side of the triangle is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[\frac{2\sqrt{3}}{15}\]                                 
  done clear
  B)
  \[\frac{4\sqrt{3}}{15}\]
  done clear
  C)
  \[\frac{4\sqrt{3}}{5}\]                                   
  done clear
  D)
  \[\frac{2\sqrt{3}}{5}\]
  done clear
  View Answer play_arrow

• question_answer20) The set of all real values of \[\lambda \]for which exactly two common tangents can be drawn to the circles \[{{x}^{2}}+{{y}^{2}}4x4y+6=0\]and \[{{x}^{2}}+{{y}^{2}}10x10y+\lambda =0\]is the interval:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  (12, 32)                                
  done clear
  B)
  (18, 42)
  done clear
  C)
   (12, 24)                               
  done clear
  D)
   (18, 48)
  done clear
  View Answer play_arrow

• question_answer21) Let \[{{L}_{1}}\]be the length of the common chord of the curves \[{{x}^{2}}+{{y}^{2}}=9\] and\[{{y}^{2}}=8x,\] and \[{{L}_{2}}\] be the length of the latus rectum of \[{{y}^{2}}=8x,\] then:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[{{L}_{1}}>{{L}_{2}}\]                   
  done clear
  B)
  \[{{L}_{1}}={{L}_{2}}\]
  done clear
  C)
  \[{{L}_{1}}<{{L}_{2}}\]   
  done clear
  D)
  \[\frac{{{L}_{1}}}{{{L}_{2}}}=\sqrt{2}\]
  done clear
  View Answer play_arrow

• question_answer22) Let \[P(3sec\theta ,2tan\theta )\]and \[Q(3sec\phi ,2tan\phi )\] where \[\theta +\phi =\frac{\pi }{2},\]be two distinct points on the hyperbola \[\frac{{{x}^{2}}}{9}-\frac{{{y}^{2}}}{4}=1.\]Then the ordinate of the point of intersection of the normals at P and Q is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[\frac{11}{3}\]                                
  done clear
  B)
  \[-\frac{11}{3}\]
  done clear
  C)
  \[\frac{13}{2}\]                                
  done clear
  D)
  \[-\frac{13}{2}\]
  done clear
  View Answer play_arrow

• question_answer23) Let A (2, 3, 5), B (? 1, 3, 2) and \[C(\lambda ,5,\mu )\]be the vertices of a DABC. If the median through A is equally inclined to the coordinate axes, then:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[5\lambda -8\mu =0\]                 
  done clear
  B)
  \[8\lambda -5\mu =0\]
  done clear
  C)
  \[10\lambda -7\mu =0\]              
  done clear
  D)
  \[7\lambda -10\mu =0\]
  done clear
  View Answer play_arrow

• question_answer24) The plane containing the line \[\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}\]and parallel to the line\[\frac{x}{1}=\frac{y}{1}=\frac{z}{4}\]passes through the point:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[(1, - 2, 5)\]                           
  done clear
  B)
  \[(1, 0, 5)\]
  done clear
  C)
  \[(0, 3, -5)\]
  done clear
  D)
  \[(-1, -3, 0)\]
  done clear
  View Answer play_arrow

• question_answer25) If \[\overset{\to }{\mathop{{{\left| c \right|}^{2}}}}\,=60\]and \[\overset{\to }{\mathop{c}}\,\times \left( \hat{i}+2\hat{j}+5\hat{k} \right)=\overset{\to }{\mathop{0}}\,,\]then a value of\[\overset{\to }{\mathop{c}}\,.\left( -7\hat{i}+2\hat{j}+3\hat{k} \right)\]is: [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[4\sqrt{2}\]                                      
  done clear
  B)
  12
  done clear
  C)
  24                                          
  done clear
  D)
  122
  done clear
  View Answer play_arrow

• question_answer26) A set S contains 7 elements. A non-empty subset A of S and an element x of S are chosen at random. Then the probability that \[x\in A\]is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[\frac{1}{2}\]                                   
  done clear
  B)
  \[\frac{64}{127}\]
  done clear
  C)
  \[\frac{63}{128}\]                                            
  done clear
  D)
  \[\frac{31}{128}\]
  done clear
  View Answer play_arrow

• question_answer27) If X has a binomial distribution, B(n, p) with parameters n and p such that P(X = 2) = P (X = 3), then E(X), the mean of variable X, is   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  2 ? p                     
  done clear
  B)
  3 ? p
  done clear
  C)
  \[\frac{p}{2}\]                                  
  done clear
  D)
  \[\frac{p}{3}\]
  done clear
  View Answer play_arrow

• question_answer28) If \[2\cos \theta +\sin \theta =1\left( \theta \ne \frac{\pi }{2} \right),\]then \[7\cos \theta +6\sin \theta \]is equal to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[\frac{1}{2}\]                                   
  done clear
  B)
  2
  done clear
  C)
  \[\frac{11}{2}\]                                
  done clear
  D)
  \[\frac{46}{5}\]
  done clear
  View Answer play_arrow

• question_answer29) The angle of elevation of the top of a vertical tower from a point P on the horizontal ground was observed to be \[\alpha .\] After moving a distance 2 metres from P towards the foot of the tower, the angle of elevation changes to \[\beta .\] Then the height (in metres) of the tower is:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  \[\frac{2\sin \alpha \sin \beta }{\sin \left( \beta -\alpha  \right)}\]           
  done clear
  B)
  \[\frac{\sin \alpha \sin \beta }{\cos \left( \beta -\alpha  \right)}\]
  done clear
  C)
  \[\frac{2\sin \left( \beta -\alpha  \right)}{\sin \alpha \sin \beta }\]                           
  done clear
  D)
  \[\frac{\cos \left( \beta -\alpha  \right)}{\sin \alpha \sin \beta }\]
  done clear
  View Answer play_arrow

• question_answer30) The proposition \[\tilde{\ }\left( p\vee \tilde{\ }q \right)\vee \tilde{\ }\left( p\vee q \right)\]is logically equivalent to:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )
  A)
  p                                            
  done clear
  B)
  q
  done clear
  C)
  \[\tilde{\ }p\]    
  done clear
  D)
  \[\tilde{\ }q\]
  done clear
  View Answer play_arrow