12th Class Mathematics Sample Paper

done Maths Olympiad Model Paper-1 Total Questions - 50   

• question_answer1)
  The determinate \[\left| \begin{matrix}    xp+y & x & y  \\    yp+z & y & z  \\    0 & xp+y & yp+z  \\ \end{matrix} \right|=0,\] if:
  A)
   x, y, z are in A.P.
  done clear
  B)
   x, y, z are in G..P.
  done clear
  C)
   x, y, z are in H.P.
  done clear
  D)
   xy, yz, zx are in A.P.
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer2)
  A student appears for test I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in test I, II and III are p, q and \[\frac{1}{2}\]respectively. If the probability that the student is successful is,\[\frac{1}{2}\]then:
  A)
   \[p=q=1\]                   
  done clear
  B)
   \[p=q=1\]
  done clear
  C)
   \[p=1,\,\,q=0\] 
  done clear
  D)
   \[p=1,\,\,q=\frac{1}{2}\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer3)
  lf \[y={{x}^{n}}\,\,\log x+x\,\,{{(\log \,\,x)}^{n}},\] then\[\frac{dy}{dx}=\]
  A)
   \[{{x}^{n-1}}\,\,(1+n\,\,\log \,\,x)+{{(\log \,\,x)}^{n-1}}[n+\log \,\,x]\]
  done clear
  B)
   \[{{x}^{n-2}}\,\,(1+n\,\,\log \,\,x)+{{(\log \,\,x)}^{n-1}}[n+\log \,\,x]\]
  done clear
  C)
   \[{{x}^{n-1}}\,\,(1+n\,\,\log \,\,x)+{{(\log \,\,x)}^{n-1}}[n-\log \,\,x]\]
  done clear
  D)
   \[{{x}^{n-2}}(1-n\,\,\log \,\,x)+{{(\log \,\,x)}^{n-1}}[n-\log \,\,x]\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer4)
  lf \[P=\left[ \begin{matrix}    \sqrt{3/2} & 1/2  \\    -1/2 & \sqrt{3/2}  \\ \end{matrix} \right],\] \[A=\left[ \begin{matrix}    1 & 1  \\    0 & 1  \\ \end{matrix} \right]\] and \[Q=PA{{P}^{T}},\] \[\,X={{P}^{T}}{{Q}^{2005}}P,\] then find the value of X.
  A)
   \[\left[ \begin{matrix}    1 & 2005  \\    0 & 1  \\ \end{matrix} \right]\]      
  done clear
  B)
   \[\left[ \begin{matrix}    1 & 2005  \\    2005 & 1  \\ \end{matrix} \right]\]
  done clear
  C)
   \[\left[ \begin{matrix}    1 & 0  \\    2005 & 1  \\ \end{matrix} \right]\]                  
  done clear
  D)
   \[\left[ \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right]\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer5)
  If \[f(x)=\cos (\text{In}\,x),\] then \[f(x).f(y)-\frac{1}{2}\left[ f\left( \frac{x}{y} \right)+f(xy) \right]\] has the value:
  A)
   \[-1\]                           
  done clear
  B)
   \[\frac{1}{2}\]
  done clear
  C)
   \[-2\]                           
  done clear
  D)
   0
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer6)
  For a positive integer n, let \[{{f}_{n}}\,\,(\theta )=\left( \tan \,\,\frac{\theta }{2} \right)\,\,(1+\sec \,\,\theta )\,\,(1+\sec 2\,\,\theta )\,\,(1+\sec 4\,\,\theta )\]\[.....(1+\sec {{2}^{n}}\,\,\theta )\]
  A)
   \[{{f}_{2}}\,\,\left( \frac{\pi }{16} \right)=1\]                 
  done clear
  B)
   \[{{f}_{3}}\,\,\left( \frac{\pi }{32} \right)=1\]
  done clear
  C)
   \[{{f}_{4}}\left( \frac{\pi }{64} \right)=1\]                     
  done clear
  D)
   \[{{f}_{5}}\left( \frac{\pi }{128} \right)=1\]
  done clear
  E)
   All of these
  done clear
  View Answer play_arrow

• question_answer7)
  Let [x] denote the greatest integer less than or equal to x. If \[f(x)=[x\sin \pi x],\] then f (x) is:
  A)
   continuous at x = 0
  done clear
  B)
   continuous in (-1, 0)
  done clear
  C)
   differentiable in (-1, 1)
  done clear
  D)
   All of these
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer8)
  The value of the integral \[\int{\frac{{{\cos }^{3}}x+{{\cos }^{5}}x}{{{\sin }^{2}}x+{{\sin }^{4}}x}}dx\] is:
  A)
   \[\sin x-6{{\tan }^{-1}}(\sin x)+c\]
  done clear
  B)
   \[\sin x-2{{(\sin x)}^{-1}}+c\]
  done clear
  C)
   \[\sin x-2\,\,{{(\sin \,\,x)}^{-1}}-6{{\tan }^{-1}}\,\,(\sin \,\,x)+c\]
  done clear
  D)
   \[\sin x-2\,\,{{(\sin \,\,x)}^{-1}}+5{{\tan }^{-1}}\,\,(\sin \,\,x)+c\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer9)
  Let \[g\,\,(x)=\int_{0}^{x}{f\,\,(t)\,\,\,dt,}\] where f is such that \[\frac{1}{2}\,\,\underline{<}\,\,f\,\,(t)\,\,\underline{<}\,\,1\] for \[t\in [0,1]\] and \[0\,\,\underline{<}\,\,f\,\,(t)\,\,\frac{1}{2}\] for\[t\in [1,2]\]. Then g(2) satisfies the inequality:
  A)
   \[-\frac{3}{2}\,\,\underline{<}\,\,g\,\,(2)<\frac{1}{2}\]    
  done clear
  B)
   \[0\underline{<}\,\,g\,\,(2)<2\]
  done clear
  C)
   \[\frac{3}{2}\,\,\underline{<}\,\,g\,\,(2)<\frac{5}{2}\]     
  done clear
  D)
   \[2\underline{<}\,\,g(2)<4\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer10)
  The area bounded by the curves \[y={{(x-1)}^{2}},\]\[y={{(x+1)}^{2}}\] and \[y=\frac{1}{4}\]is:
  A)
   \[\frac{1}{3}\] sq units
  done clear
  B)
   \[\frac{2}{3}\] sq units
  done clear
  C)
   \[\frac{1}{4}\] sq units              
  done clear
  D)
   \[\frac{1}{5}\] sq units
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer11)
  The order of the differential equation whose general solution is given by \[y=({{c}_{1}}+{{c}_{2}})\,\,\cos \,\,(x+{{c}_{3}})\,\,{{c}_{4}}{{e}^{x+c5}}\] where \[{{c}_{1}},\]\[{{c}_{2}},\]\[{{c}_{3}},\]\[{{c}_{4}},\]\[{{c}_{5}},\] are arbitrary constants, is:
  A)
   5                                
  done clear
  B)
   4
  done clear
  C)
   3        
  done clear
  D)
   2
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer12)
  Let f (x) be differentiable on the interval \[(0,\infty )\] such that \[f(1)=1,\] and \[\underset{t\to x}{\mathop{\lim }}\,\frac{{{t}^{2}}f\,\,(x)-{{x}^{2}}f\,\,(t)}{t-x}=1\] for each x > 0. Then f (x) is:
  A)
   \[\frac{1}{3x}+\frac{2{{x}^{2}}}{3}\]
  done clear
  B)
   \[-\frac{1}{3x}+\frac{4{{x}^{2}}}{3}\]
  done clear
  C)
   \[-\frac{1}{x}+\frac{2}{{{x}^{2}}}\] 
  done clear
  D)
   \[\frac{1}{x}\] 
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer13)
  If P (1, 2), Q (4, 6), R (5, 7) and S (a, b) are the vertices of a parallelogram PQRS, then:
  A)
   a = 2, b= 4                 
  done clear
  B)
   a = 3, b = 4
  done clear
  C)
   a = 2, b= 3     
  done clear
  D)
   a = 3, b = 5
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer14)
  If \[{{\sin }^{-1}}\left( x-\frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{4}-..... \right)+{{\cos }^{-1}}\left( {{x}^{2}}-\frac{{{x}^{4}}}{2}+\frac{{{x}^{6}}}{4}-..... \right)=\frac{\pi }{2},\] for \[0<|x|<\sqrt{2},\] then x equals:
  A)
   \[\frac{1}{2}\]                         
  done clear
  B)
   1
  done clear
  C)
   \[\frac{-1}{2}\]                        
  done clear
  D)
   \[-1\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer15)
  A man from the top of a 100 metres high tower sees a car moving towards the tower at an angle of depression of\[30{}^\circ \]. After some time, the angle of depression becomes\[60{}^\circ \]. The distance (in metres) travelled by the car during this time is:
  A)
   \[100\sqrt{3}\]              
  done clear
  B)
   \[\frac{200\sqrt{3}}{3}\]
  done clear
  C)
   \[\frac{100\sqrt{3}}{3}\]                       
  done clear
  D)
   200
  done clear
  E)
   None of these    
  done clear
  View Answer play_arrow

• question_answer16)
                      Let \[\overrightarrow{a}={{a}_{1}}\hat{i}+{{a}_{2}}\hat{j}+{{a}_{3}}\hat{k},\] \[\overrightarrow{b}={{b}_{1}}\hat{i}+{{b}_{2}}\hat{j}+{{b}_{3}}\hat{k}\] and \[\overrightarrow{c}={{c}_{1}}\hat{i}+{{c}_{2}}\hat{j}+{{c}_{3}}\hat{k}\] be three non-zero vectors such that \[\overrightarrow{c}\]is unit vector perpendicular to both the vectors \[\overrightarrow{a}\]and \[\overrightarrow{b}\]. If the angle between \[\overrightarrow{a}\]and \[\overrightarrow{b}\]is \[\frac{\pi }{6},\] then \[{{\left| \begin{matrix}    {{a}_{1}} & {{a}_{2}} & {{a}_{3}}  \\    {{b}_{1}} & {{b}_{2}} & {{b}_{3}}  \\    {{c}_{1}} & {{c}_{2}} & {{c}_{3}}  \\ \end{matrix} \right|}^{2}}\]is equal to :
  A)
   0                                
  done clear
  B)
   1
  done clear
  C)
   \[\frac{1}{4}\,\,(a_{1}^{2}+a_{2}^{2}+a_{3}^{3})\,\,(b_{1}^{2}+b_{2}^{2}+b_{3}^{2})\]
  done clear
  D)
   \[\frac{3}{4}\,\,(a_{1}^{2}+a_{2}^{2}+a_{3}^{3})\,\,(b_{1}^{2}+b_{2}^{2}+b_{3}^{2})\,\,(c_{1}^{2}+c_{2}^{2}+c_{3}^{2})\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer17)
  The number of distinct real roots of \[\left| \begin{matrix}    \sin x & \cos x & \cos x  \\    \cos x & \sin x & \cos x  \\    \cos x & \cos x & \sin x  \\ \end{matrix} \right|=0\] in the interval \[-\frac{\pi }{4}\underline{<}\,\,x\,\,\underline{<}\frac{\pi }{4}\] is :
  A)
   0                                
  done clear
  B)
   2
  done clear
  C)
   1
  done clear
  D)
   3
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer18)
  Let a, b, c be non-zero real numbers such that \[\int_{0}^{1}{(1+{{\cos }^{8}}x)\,\,(a{{x}^{2}}+bx+c)\,\,dx}=\]\[\int_{0}^{2}{(1+co{{s}^{8}}\,\,x)\,\,(a{{x}^{2}}+bx+c)\,\,dx}\] Then, the quadratic equation \[a{{x}^{2}}+bx+c=0\]has:
  A)
   no root in (0, 2)
  done clear
  B)
   at least one root in (1, 2)
  done clear
  C)
   a double root in (0, 2)
  done clear
  D)
   two imaginary roots
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer19)
  The triangle formed by the tangent to the curve \[f(x)={{x}^{2}}+bx-b\] at the point (1, 1) and the coordinate axes, lies in the first quadrant. If its area is 2, then the value of b is:
  A)
   \[-1\]                           
  done clear
  B)
   3
  done clear
  C)
   \[-3\]   
  done clear
  D)
   1
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer20)
  Let \[f(x)=\left| \begin{matrix}    {{x}^{3}} & \sin x & \cos x  \\    6 & -1 & 0  \\    p & {{p}^{2}} & {{p}^{3}}  \\ \end{matrix} \right|,\] where p is constant. Then \[\frac{{{d}^{3}}}{d{{x}^{3}}}\,\,f\,\,(x)\] at \[x=0\]is:
  A)
   \[p\]                            
  done clear
  B)
   \[p+{{p}^{2}}\]
  done clear
  C)
   \[p+{{p}^{3}}\]                    
  done clear
  D)
   independent of p
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer21)
  There exists a function f (x) satisfying \[f\,\,(0)=1,\]\[f'(0)=-1,\]\[f(x)>0\] for all x and
  A)
   \[f''(x)<0\]for all x
  done clear
  B)
   \[-1<f''(x)<0\]for all x
  done clear
  C)
   \[-2\,\underline{<}\,\,f''(x)\,\,\underline{<}-1\]for all x
  done clear
  D)
   \[f''(x)<-2\] for all x
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer22)
  The function \[f\,\,(x)=\frac{\text{In}\,\,(1+ax)-\text{In}\,\,(1-bx)}{x}\] is not defined at \[x=0\]. The value which should be assigned to f at \[x=0\] so that it is continuous at \[x=0\], is:
  A)
   \[a-b\]              
  done clear
  B)
   \[a+b\]
  done clear
  C)
   In \[a+\text{ln }b\]                   
  done clear
  D)
   Both (b) and (c)
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer23)
  Let \[g(x)=1+x-[x]\] and \[f(x)=\left\{ \begin{matrix}    -1 & x<0  \\    0 & x=0,  \\    1 & x>0  \\ \end{matrix} \right.\]  then for all x,  f[g(x)] is equal to:
  A)
   x                                
  done clear
  B)
   1
  done clear
  C)
   f(x)                             
  done clear
  D)
   g(x)
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer24)
  If \[A=\left[ \begin{matrix}    1 & 0 & 0  \\    0 & 1 & 1  \\    0 & -2 & 4  \\ \end{matrix} \right],\] \[6{{A}^{-1}}={{A}^{2}}+cA+dl,\] then \[(c,d)\]is:
  A)
   (-6, 11) 
  done clear
  B)
   (-11, 6)
  done clear
  C)
   (11, 6)                         
  done clear
  D)
   (6, 11)
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer25)
  The equation  \[{{x}^{\frac{3}{4}{{({{\log }_{2}}x)}^{2}}+{{\log }_{2}}x-\frac{5}{4}}}=\sqrt{2}\]has:
  A)
   at least one real solution
  done clear
  B)
   exactly three real solutions
  done clear
  C)
   exactly one irrational solution
  done clear
  D)
   All of these
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer26)
  If the system of equations \[x-ky-z=0,\]\[kx-y-z=0,\]\[x+y-z=0\] has a non-zero solution, then possible values of k are:
  A)
   \[-1,\text{ }2\]              
  done clear
  B)
   1, 2
  done clear
  C)
   0, 1                             
  done clear
  D)
   \[-1,\text{ }1\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer27)
  For non-zero vectors \[\overrightarrow{a},\] \[\overrightarrow{b},\] \[\overrightarrow{c},\]\[|(\overrightarrow{a}\times \overrightarrow{b}).\overrightarrow{c}|=\overrightarrow{a}||\overrightarrow{b}||\overrightarrow{c}|\] holds,  if and only if:
  A)
   \[\overrightarrow{a}.\overrightarrow{b}=0,\]\[\overrightarrow{b}.\overrightarrow{c}=0\]      
  done clear
  B)
   \[\overrightarrow{b}.\overrightarrow{c}=0,\]\[\overrightarrow{c}.\overrightarrow{a}=0\]
  done clear
  C)
   \[\overrightarrow{c}.\overrightarrow{a}=0,\]\[\overrightarrow{a}.\overrightarrow{b}=0\]      
  done clear
  D)
   \[\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b.}\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{a}=0\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer28)
  The point of intersection of the lines \[\overrightarrow{r}=7\hat{i}+10\hat{j}+13\hat{k}+s\,\,(2\hat{i}+3\hat{j}+4\hat{k})\] and \[\overrightarrow{r}=3\hat{i}+5\hat{j}+7\hat{k}+t\,\,(\hat{i}+2\hat{j}+3\hat{k})\] is
  A)
   \[\hat{i}+\hat{j}-\hat{k}\]                    
  done clear
  B)
   \[2\hat{i}-\hat{j}+4\hat{k}\]
  done clear
  C)
   \[\hat{i}-\hat{j}+\hat{k}\]                    
  done clear
  D)
   \[\hat{i}+\hat{j}+\hat{k}\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer29)
  A tetrahedron has vertices at 0 (0, 0, 0), A (1, 2, 1), B (2, 1, 3) and C\[\left( -1,\text{ }1,\text{ }2 \right)\] . Then the angle between the faces OAB and ABC will be
  A)
   \[{{\cos }^{-1}}\,\,\left( \frac{19}{35} \right)\]   
  done clear
  B)
   \[{{\cos }^{-1}}\,\,\left( \frac{17}{31} \right)\]
  done clear
  C)
   \[30{}^\circ \]                         
  done clear
  D)
   \[90{}^\circ \]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer30)
  The equation of a curve is y = f (x). The tangents at (1, f (1)), (2, f (2)) and (3, f (3)) make angles \[\frac{\pi }{6},\]\[\frac{\pi }{3}\] and \[\frac{\pi }{4}\]respectively with the positive direction of the x-axis. Then the value of \[\int_{2}^{3}{f'\,\,(x)f''\,\,(x)\,\,dx+\int_{1}^{3}{f''\,\,(x)\,\,dx}}\] is equal to
  A)
   \[-\frac{1}{\sqrt{3}}\]                           
  done clear
  B)
   \[\frac{1}{\sqrt{3}}\]
  done clear
  C)
   1                                
  done clear
  D)
   \[\frac{2}{\sqrt{3}}\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer31)

  Directions: Study the information below and answer questions based on it. Radhe, Alam, Prem, Minakshi, Prakash and Kanika are six members of a family. Each one is engaged in a different profession i.e. Doctor, Lawyer, Teacher, Engineer, Nurse and a Banker.

  1. Each one of them remains at home on a different day of the week from Monday to Saturday.

  2. The lawyer in the family stays at home on Thursday.

  3. Prem stays at home on Tuesday.

  4. Radhe, a doctor, does not remain at home either on Wednesday or Saturday.

  5. Minakshi is neither a lawyer nor the teacher, but she remains at home on Friday.

  6. Alam is the engineer.

  7. Prakash is the bank manager..

  Who is the nurse in the family?
  A)
   Minakshi
  done clear
  B)
   Radhe
  done clear
  C)
   Kanika             
  done clear
  D)
   Prakash
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer32)

  Directions: Study the information below and answer questions based on it. Radhe, Alam, Prem, Minakshi, Prakash and Kanika are six members of a family. Each one is engaged in a different profession i.e. Doctor, Lawyer, Teacher, Engineer, Nurse and a Banker.

  1. Each one of them remains at home on a different day of the week from Monday to Saturday.

  2. The lawyer in the family stays at home on Thursday.

  3. Prem stays at home on Tuesday.

  4. Radhe, a doctor, does not remain at home either on Wednesday or Saturday.

  5. Minakshi is neither a lawyer nor the teacher, but she remains at home on Friday.

  6. Alam is the engineer.

  7. Prakash is the bank manager.

  Who among them stays at home on the following day on which Prem stays at home?
  A)
   Radhe             
  done clear
  B)
   Minakshi
  done clear
  C)
   Alam or Prakash           
  done clear
  D)
   Kanika
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer33)

  Directions: Study the information below and answer questions based on it. Radhe, Alam, Prem, Minakshi, Prakash and Kanika are six members of a family. Each one is engaged in a different profession i.e. Doctor, Lawyer, Teacher, Engineer, Nurse and a Banker.

  1. Each one of them remains at home on a different day of the week from Monday to Saturday.

  2. The lawyer in the family stays at home on Thursday.

  3. Prem stays at home on Tuesday.

  4. Radhe, a doctor, does not remain at home either on Wednesday or Saturday.

  5. Minakshi is neither a lawyer nor the teacher, but she remains at home on Friday.

  6. Alam is the engineer.

  7. Prakash is the bank manager.

  Which of the following combinations is correct?
  A)
   Prem-Teacher   
  done clear
  B)
   Minakshi-Nurse
  done clear
  C)
   Kanika-Thursday           
  done clear
  D)
   All of these
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer34)

  Directions: Study the information below and answer questions based on it.

  (i) There are three houses on each side of the road.

  (ii) These six houses are labeled as P, Q, R, S, T and U.

  (iii) The houses are of different colours, namely. Red, Blue, Green, Orange, Yellow and White.

  (iv) The houses are of different heights.

  (v) T, the tallest house, is exactly opposite to the Red coloured house.

  (vi) The shortest house is exactly opposite to the Green coloured house.

  (vii) U, the Orange coloured house, is located between P and S.

  (viii) R, the Yellow coloured house, is exactly opposite to P.

  (ix) Q, the Green coloured house, is exactly opposite to U.

  (x) P, the White coloured house, is taller than R, but shorter than S and Q.

  What is the colour of the house diagonally opposite to the Yellow coloured house?
  A)
   White              
  done clear
  B)
   Blue
  done clear
  C)
   Green              
  done clear
  D)
   Red
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer35)

  Directions: Study the information below and answer questions based on it.

  (i) There are three houses on each side of the road.

  (ii) These six houses are labeled as P, Q, R, S, T and U.

  (iii) The houses are of different colours, namely. Red, Blue, Green, Orange, Yellow and White.

  (iv) The houses are of different heights.

  (v) T, the tallest house, is exactly opposite to the Red coloured house.

  (vi) The shortest house is exactly opposite to the Green coloured house.

  (vii) U, the Orange coloured house, is located between P and S.

  (viii) R, the Yellow coloured house, is exactly opposite to P.

  (ix) Q, the Green coloured house, is exactly opposite to U.

  (x) P, the White coloured house, is taller than R, but shorter than S and Q.

  Which is the second tallest house?
  A)
   P                                
  done clear
  B)
   S
  done clear
  C)
   Q                                
  done clear
  D)
   R
  done clear
  E)
   Cannot determine
  done clear
  View Answer play_arrow

• question_answer36)

  Directions: Study the information below and answer questions based on it.

  (i) There are three houses on each side of the road.

  (ii) These six houses are labeled as P, Q, R, S, T and U.

  (iii) The houses are of different colours, namely. Red, Blue, Green, Orange, Yellow and White.

  (iv) The houses are of different heights.

  (v) T, the tallest house, is exactly opposite to the Red coloured house.

  (vi) The shortest house is exactly opposite to the Green coloured house.

  (vii) U, the Orange coloured house, is located between P and S.

  (viii) R, the Yellow coloured house, is exactly opposite to P.

  (ix) Q, the Green coloured house, is exactly opposite to U.

  (x) P, the White coloured house, is taller than R, but shorter than S and Q.

  What is the colour of the tallest house?
  A)
   Red                             
  done clear
  B)
   Blue
  done clear
  C)
   Green                          
  done clear
  D)
   Yellow
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer37)

  Direction: In each of the questions below three or four statements followed by three conclusions numbered I II and III are given. You have to take the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.

  Statements:

  All trees are books.

  Some books are tables.

  All tables are pencils.

  Conclusions:

  I. Some Pencil are table.

  II. Some books are trees.

  III. Some tables are trees.

  A)
   Only I and II follow
  done clear
  B)
   Only II and III follow
  done clear
  C)
   Only III follow
  done clear
  D)
   All follow
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer38)

  Direction: In each of the questions below three or four statements followed by three conclusions numbered I II and III are given. You have to take the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.

  Statements:

  All chairs are table.

  All tables are telephones.

  All telephones are cellphones.

  No cellphone is computer.

  Conclusions:

  I. All cellphones are tables.

  II. Some chairs are computers.

  III. No computer is chairs.

  A)
   All follow
  done clear
  B)
   Only I and II follow
  done clear
  C)
   Either II or III follow
  done clear
  D)
   Only III follow
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer39)
  Choose the correct mathematical statement from the following.
  A)
   \[\frac{13}{147}\div \frac{104}{35}=\frac{13}{21}+\frac{43}{42}-\frac{17}{23}\times \frac{69}{136}\]
  done clear
  B)
   \[\frac{13}{147}+\frac{104}{35}\div \frac{13}{21}=\frac{43}{42}-\frac{17}{23}\times \frac{69}{136}\]
  done clear
  C)
   \[\frac{13}{147}\div \frac{104}{35}+\frac{13}{21}=\frac{43}{42}-\frac{17}{23}\times \frac{69}{136}\]
  done clear
  D)
   \[\frac{13}{147}\times \frac{104}{35}+\frac{13}{21}=\frac{43}{42}-\frac{17}{23}\div \frac{69}{136}\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer40)

  Five people A, B, C, D and E are seated around a round table. Every chair is spaced equidistant from adjacent chairs.

  (i) C is seated next to A.

  (ii) A is seated two seats from D.

  (iii) B is not seated next to A.

  On the basis of above information, which of the following must be true?

  1. D is seated next to B.

  2. E is seated next to A.

  3. D and C are separated by two seats.

  Select the correct answer using the code given below:
  A)
   1 only             
  done clear
  B)
   1 and 2 only
  done clear
  C)
   3 only
  done clear
  D)
   Neither 1 nor 2 nor 3
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer41)
  In calculating the mean and variance of 10 readings, a student wrongly used the figure 52 for the correct figure of 25. He obtained the mean and variance as 45.0 and 16.0 respectively. Determine the correct mean and variance.
  A)
   42.3, 43.81     
  done clear
  B)
   43.81, 42.3
  done clear
  C)
   33.81, 32.3     
  done clear
  D)
   38.3, 43.81
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer42)
  The position vectors of the vertices A, B and C of a tetrahedron ABCD are \[\hat{i}+\hat{j}+\hat{k},\] \[\hat{i}\]and 3\[\hat{i}\], respectively. The altitude from vertex D to the opposite face ABC meets the median line through A of the triangle ABC at a point E. If the length of the side AD is 4 and the volume of the tetrahedron is \[\frac{2\sqrt{2}}{3},\] find the position vector of the point E for all its possible positions.
  A)
   \[\left( -1,\text{ }2,\text{ }3 \right)\text{ }or\text{ }\left( 3,-2,-1 \right)\]
  done clear
  B)
   \[\left( -1,-3,\text{ }3 \right)\text{ }or\text{ }\left( 3,\text{ }3,\text{ }1 \right)\]
  done clear
  C)
   \[\left( -1,\text{ }3,\text{ }3 \right)\text{ }or\text{ }\left( 3,\,-1,\,-1 \right)\]
  done clear
  D)
   \[\left( 3,\text{ }3,-1 \right)\text{ }or\text{ }\left( -1,\text{ }3,-3 \right)\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer43)
  If \[f(x-y)=f(x).g(y)-f(y).g(x)\] and \[g(x-y)=g(x).g(y)-f(x)\] for all \[x,\]\[y\in R.\] If right hand derivative at x = 0 exists for f(x). Find derivative of g(x) at x = 0.
  A)
   4                                
  done clear
  B)
   3
  done clear
  C)
   1                                
  done clear
  D)
   0
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer44)
  Find the values of a and b so that the function \[f(x)=\left\{ \begin{matrix}    x+a\sqrt{2}\sin x, & 0\underline{<}\,\,x<\pi /4  \\    2x\,\,\cot x+b, & \pi /4\,\,\underline{<}x\,\,\underline{<}\,\,\pi /2  \\    a\,\,\cos 2x-b\sin x, & \pi /2<\,\,x\,\,\underline{<}\,\pi   \\ \end{matrix} \right.\] is continuous for \[0\underline{<}\,\,x<\pi \text{.}\]
  A)
   \[\frac{\pi }{6},\]\[-\frac{\pi }{6}\]                     
  done clear
  B)
   \[\frac{\pi }{6},\,\,\,-\frac{\pi }{12}\]
  done clear
  C)
   \[\frac{\pi }{4},\]\[-\frac{\pi }{12}\]                   
  done clear
  D)
   \[\frac{\pi }{4},\]\[-\frac{\pi }{6}\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer45)
  There is circular plot of land of diameter 24 metres. At the centre of the plot, a pit in the form of a frustum of a cone is dug up. The depth of the pit is 3 meters. The diameters of the top and the bottom of the pit are 4 metres and 2 metres respectively. The earth dug out from the pit is spread uniformly over the remaining area of the plot. Find the height to which the level of the plot rises.
  A)
   8 cm                           
  done clear
  B)
   7cm
  done clear
  C)
   6cm                            
  done clear
  D)
   5cm
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer46)
  A curve ?C? passes through (2, 0) and the slope at (x, y) is \[\frac{{{(x+1)}^{2}}+(y-3)}{x+1}\]. Find the area bounded by curve and x-axis in fourth quadrant.
  A)
   \[\frac{4}{3}sq\,\,unit\]
  done clear
  B)
   \[\frac{8}{3}sq\,\,unit\]
  done clear
  C)
   \[\frac{3}{5}sq\,\,unit\]             
  done clear
  D)
   \[\frac{3}{4}sq\,\,unit\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer47)
  Let f(x) be a continuous function given by\[f(x)=\left\{ \begin{matrix}    2x, & |x|\underline{<}\,\,1  \\    {{x}^{2}}+ax+b, & |x|\,\,>1  \\ \end{matrix} \right\}\]. Find the area of the region in the third quadrant bounded by the curves \[x=-2{{y}^{2}}\] and \[y=f\left( x \right)\]lying on the left of the line \[8x+1=0.\]
  A)
   \[\frac{257}{192}sq\,\,unit\]      
  done clear
  B)
   \[\frac{389}{175}sq\,\,unit\]
  done clear
  C)
   \[\frac{382}{125}sq\,\,unit\]                  
  done clear
  D)
   \[\frac{157}{192}sq\,\,unit\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer48)
  A person goes to office either by car, scooter, bus or train, the probability of which being \[\frac{1}{7},\]\[\,\frac{3}{7},\] \[\frac{2}{7}\]and \[\frac{1}{7}\] respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is \[\frac{2}{9},\] \[\frac{1}{9},\] \[\frac{4}{9}\] and \[\frac{1}{9}\] respectively. Given that he reached office in time, then what is the probability that he travelled by a car.
  A)
   \[\frac{9}{7}\]                         
  done clear
  B)
   \[\frac{2}{7}\]
  done clear
  C)
   \[\frac{3}{7}\]                         
  done clear
  D)
   \[\frac{5}{7}\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer49)
  If \[{{x}_{1}},\] \[{{x}_{2}},\] \[{{x}_{3}},\] \[{{x}_{4}}\] are roots of the equation \[{{x}^{4}}-{{x}^{3}}\sin 2\beta +{{x}^{2}}\cos 2\beta -x\,\,\cos \beta -\sin \beta =0,\] then \[{{\tan }^{-1}}{{x}_{1}}+{{\tan }^{-1}}{{x}_{2}}+{{\tan }^{-1}}{{x}_{3}}+{{\tan }^{-1}}{{x}_{4}}=\]
  A)
  \[\beta \]                                   
  done clear
  B)
   \[\frac{\pi }{0}-\beta \]
  done clear
  C)
   \[\pi -\beta \]                 
  done clear
  D)
   \[-\beta \]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow

• question_answer50)
  If \[P(A\cap B\cap \overline{C)}=\frac{1}{3}\] and\[P(\overline{A}\cap B\cap \overline{C})=\frac{1}{3},\] then \[P(B\cap C)\]is:
  A)
   \[\frac{1}{12}\]
  done clear
  B)
   \[\frac{1}{6}\]
  done clear
  C)
   \[\frac{1}{15}\]                        
  done clear
  D)
   \[\frac{1}{9}\]
  done clear
  E)
   None of these
  done clear
  View Answer play_arrow