Solved question paper for Math Mar-2018 (PSEB 12th)

Question paper 1

1. 1. (i) If is a binary operation such that a * b = a² +b² then 3 * 5 is  
       (a) 34           (b) 9             (c) 8              (d) 25 

   Answer:

2. (ii) If cos^-1  x = y  then 
    
   (a) −𝜋/2 <= y <=  𝜋/2               (b)  -π <=y <= π     

   (c) 0 <= y <= 𝜋/2                      (d)  0 <= y <= π 

   Answer:

3. (iii) If A is a matrix of order 3x3 and |A| =10 then |adj• A। is 
    
        (a) 0           (b) 10              (c) 100           d) 1000

   Answer:

4. (iv) If y = sin (sin^-1 x + cos^-1 x),  x € [-1, 1] then  dy/dx  is 
        

          (a) 𝜋/2           (b) −𝜋/2           (c) 0          d) 1

   Answer:

6. (v) If 
   f(x) = {  \( {{{sinx\over x}\over k-1} , {x! = 0 \over x=0}}\)  } ,𝑥 = 0 is continuous at x=0 then 

   
         (a) 2        (b) 0            (c) -1              (d) 1 

   Answer:

7. (vi) \(\int e^x ( log x {1\over x}) \) dx is equal to 

   (a) e^x + c     (b) e^x logx + c    (c) e^x/x + c     (d) log x + c

   Answer:

8. (vii) Integrating factor of differential equation dy/dx + 𝑦 = 3 is                                               
    
   (a) x       (b) e      (c) e^x     (d) logx 

   Answer:

9. (vii)  This inequality   |a . b|   <= |a| |b| 5 is called  
    
         (a) Cauchy-Schwartz inequality         (b) Triangle inequality  
    
         (c) Rolle's Theorem                               (d) Lagrange's Mean Value theorem 

   Answer:

11. (ix) Distance between plane 3x +4y-20 = 0 and point (0, 0,-7) is 
     
           (a) 4 units         (b) 3 units       (c) 2 units         (d) 1 unit 

    Answer:

12. (x) If P(E) denotes probability of occurrence of event E then 
     
         (a) P (E)  € [-1, 1]         (b) P (E) € (1, 2)          (c) P (E) € (0, 1)          (d) P (E) €  [0, 1] 

    Answer:

13. 2. If matrix A = [aij]_{3 x 2} , and aij = (3i-2j)² or matrix A find them 

    Answer:

14. 3. Check whether Lagrange's mean value theorem is applicable on f(x) = sin x + cos x Interval [0, 𝜋/2 ]

    Answer:

16. 4. Evaluate  \(\int\limits_0^{x/2} sin^3X / sin^3X + cos^3X { sin^3x \over sin^3x + cos^3x }dx\)   

    Answer:

17. 5. Evaluate \(\int {7dx \over x(x^7-1)}\)

    Answer:

18. 6. Find particular solution of differential equation  \({dy \over dx} {1+y^2 \over 1+x^2}\)  given that  x=0 or y= 1 

    Answer:

    \({dy \over dx} {1+y^2 \over 1+x^2}\)  given that y(0) = 1 

    \({dy \over 1+y^2} = {dx \over 1+ x^2}\)

    Integrating both side, we get :

    \(\int{dy \over 1+y^2} =\int {dx \over 1+ x^2} + c\)

    \(tan^{-1}y = tan^{-1}x + c\)

    \(tan^{-1}(1) = tan^{-1}(0) + c\)

    \(tan^{-1}tan ({\pi \over 4}) = tan^{-1}tan(0) + c\)

    \(c = {\pi \over 4}\)

    \(tan^{-1}y = tan^{-1}x + {\pi \over 4}\)

19. 7. Form differential equation representing the family of lines making equal intercepts on the co-ordinate axes. 

    Answer:

21. 8. Find the angle between the plane 2x+3 y-5z= 10 and the line passing from the points (2, 3,-1)       2 Or (1, 2, 1)

    Answer:

22. 9. If P (A) = 7/13, P (B) = 9/13 and P (AUB) = 12/13 then find P(A|B) 

    Answer:

    P (A) = 7/13, P (B) = 9/13 

    P (AUB) = 12/13  P(A|B) = ? 

    P(A|B) = \(P(A \bigcap B) \over P(B)\)

    \(P (A \bigcup B) = P(A) + P(B) - P (A \bigcap B)\)

    \({12 \over 13 } = {7 \over 13 } + {9 \over 13} - P(A \bigcap B)\)

    \({12 \over 13 } = {16 \over 13} - P(A \bigcap B)\)

    \(P(A \bigcap B) = {16 -12 \over 13} = {4 \over 3}\)

    \(P(A | B) = {{4 \over 13} \over {9 \over 13}}\)

    \(P(A | B) = {4 \over 9}\)

23. 10  Prove that function f : R --> R,  f(x) =  \(3−2𝑥 \over 7\)   in one-one and onto. Also find  f^-1

    Answer:

     f(x) =  \(3−2𝑥 \over 7\)    

    7 : R --> R         x € R

    7(x) is 1- 7:

    For x₁ x₂ € R

    7(x₁) = f(x₂)

     \(3−2𝑥_1 \over 7\)    =   \(3−2𝑥_2 \over 7\)    

    2x₁   =  2x₂  

    x₁    =    x₂

    .·. f   is 1 - 1

    f  is  onto :  Let K  €  R

    f(x) = K

     \(3−2𝑥 \over 7\)   = K     =>   3 - 2x = 7K

    x =  \(3−7K \over 2 \)    =>  R

    .·. 7(x)  is onto 

    Now,  7^-1

    For   x € R,    Let f^-1 (x) = K

    => 7(K) = x

    =>  \(3-2K \over 7 \)  = x

    3 - 2K = 7x

    -2K = 7x - 3

    K = \(3 - 7x \over 2 \)

    For x € R , 7^-1 (x)   =    \(3 - 7x \over 2 \)

     

24. 11. Prove that : sin^-1 ( \(5\over13\) ) +cos^-1 (\(4\over 5\))  =  \(1\over 2\)  sin^-1 \(3696 \over 4225\)

    Answer:

26. 12.  Express  \( \begin{bmatrix} 2 & 5 & -1 \\ 3 & 1 & 5 \\ 7 & 6 & 9 \end{bmatrix}\)  as sum of symmetric and skew-symmetric matrices. 

    Answer:

    A = \( \begin{bmatrix} 2 & 5 & -1 \\ 3 & 1 & 5 \\ 7 & 6 & 9 \end{bmatrix}\)

    \(A^{t} = \) \( \begin{bmatrix} 2 & 3 & -7 \\ 5 & 1 & 6 \\ -1 & 5 & 9 \end{bmatrix}\)

    For Symmetric Matrix : \(A + A^T \over 2\)

    \(A + A^T \over 2\) = \(1 \over 2\) =  \( \begin{bmatrix} 2 & 5 & -1 \\ 3 & 1 & 5 \\ 7 & 6 & 9 \end{bmatrix}\)  +    \( \begin{bmatrix} 2 & 3 & -7 \\ 5 & 1 & 6 \\ -1 & 5 & 9 \end{bmatrix}\)

    \(A + A^T \over 2\) = \(1 \over 2\) =   \( \begin{bmatrix} 4 & 8 & 6 \\ 8 & 2 & 11 \\ 6 & 11 & 18 \end{bmatrix}\)

    For Skew-symmetric Matrix = \(A + A^T \over 2\)

    \(A + A^T \over 2\) = \(1 \over 2\) =  \( \begin{bmatrix} 2 & 5 & -1 \\ 3 & 1 & 5 \\ 7 & 6 & 9 \end{bmatrix}\)  +    \( \begin{bmatrix} 2 & 3 & -7 \\ 5 & 1 & 6 \\ -1 & 5 & 9 \end{bmatrix}\)

    \(A + A^T \over 2\) = \(1 \over 2\) =   \( \begin{bmatrix} 0 & 2 & -8 \\ -2 & 0 & -1 \\ 8 & 1 & 0 \end{bmatrix}\)

27.   Or 

    If x,y,z are different and  \( \begin{vmatrix} x & x^2 & 1 + x^3 \\ y & y^2 & 1 + y^3 \\ z & z^2 & 1 + z^3 \end{vmatrix}\)  =   0   then prove that  xyz =-1  
     

    Answer:

    \( \begin{vmatrix} x & x^2 & 1 + x^3 \\ y & y^2 & 1 + y^3 \\ z & z^2 & 1 + z^3 \end{vmatrix}\)  = 0

    \( \begin{vmatrix} x & x^2 & 1 \\ y & y^2 & 1 \\ z & z^2 & 1 \end{vmatrix}\)    +    \( \begin{vmatrix} x & x^2 & x^3 \\ y & y^2 & y^3 \\ z & z^2 & z^3 \end{vmatrix}\)  = 0

    => A₁  + A₂ = 0

    A₂  =  \( \begin{vmatrix} x & x^2 & x^3 \\ y & y^2 & y^3 \\ z & z^2 & z^3 \end{vmatrix}\)

    A₂   =  xyz \( \begin{vmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \end{vmatrix}\)

    A₂   =  xyz \( \begin{vmatrix} x & x^2 & 1 \\ y & y^2 & 1 \\ z & z^2 & 1 \end{vmatrix}\)

    A₂   = xyz A₁   

    A₁   + A₂   =  0

    A₁  + xyz A₁  =  0

    A₁ ( 1 + xyz ) = 0

    A₁  = xyz \( \begin{vmatrix} x & x^2 & 1 \\ y & y^2 & 1 \\ z & z^2 & 1 \end{vmatrix}\)

    R₂ +  R₂ -  R1   &  R3 + R3 - R1

    A₁  = xyz \( \begin{vmatrix} x & x^2 & 1 \\ y-x & y^2-x^2 & 0 \\ z-x & z^2-x^2 & 0 \end{vmatrix}\)

     =  (y-x) (z-x) =   \( \begin{vmatrix} x & x^2 & 1 \\ 1 & y^2-x^2 & 0 \\ 1 & z^2-x^2 & 0 \end{vmatrix}\)

     =  (y-x) (z-x) (3 + x - y - x)

    A₁  = (y - x) (z - x) (z - y)

     =  (y - x) (z - x)(z - y) (1 + xyz) = 0 

    =>  1 + x y z = 0

    xyz = -1

28. 13. If y = (x)^tanx + (tanx)^x then find 𝑑𝑦/𝑑𝑥

     

    Answer:

    y = (x)^tanx + (tanx)^x

    Let u = x^tanx ,  v =  tanx^x

    Now x = u

    log u  =  tanx.logx

    \({1 \over u} . {du \over dx} = {tan x \over x }+ logx.sec^2 x\)

    \({du \over dx} = u [ {tanx\over x} + sec^2x . logx]\)

    \({du \over dx} = x^{tanx} [{tanx \over x} + sec^2 x .logx]\)

    \( v = (tanx)^x\)

    \(logv = x log (tanx)\)

    \({1 \over v} .{dv \over dx} = x.{1 \over tan}.sec^2x + 1 .log(tanx)\)

    \({dv \over dx} = (tanx)^x [x .{cosx \over sin x}.{1 \over cos ^2x} + log(tanx)] \)

     \({dv \over dx } = (tanx)^x [{1 \over sinx.cosx} + log(tanx)]\)

    \({dy \over dx} = {du \over dx} + {dv \over dx}\)

    \({dy\over dx} = x^{tanx} [{tanx \over x} + sec^2x .logx] + (tanx)^x [{x\over sinxcosx} + log tanx]\)

29. 14. Using differentials find approximate value of (0.37)^1/2 

    Answer:

    \(\sqrt {0.37} = \sqrt {0.36 + 0.01}\)

    because \(\sqrt {0.36}\)  = 0.6

    let y = \(\sqrt x\)

    y + Ay = \(\sqrt {x + Ax}\)

    Ay = \(\sqrt {x + Ax}\)  - \(\sqrt x\)

    \(({dy \over dx})\)Ay = \(\sqrt {x + Ax}\)  - \(\sqrt x\)    ------------ 1

    let x = 0.36    Ax = 0.1

    from 1  =>  \({1 \over 2\sqrt x} . Ax = {\sqrt {0.36 + 0.01}} - {\sqrt {0.36}}\)

    \({0.1 \over 2 * 0.6} = \sqrt {0.37} - 0.6\)

    \({0.1 \over 1.2} + 0.6 = \sqrt {0.37}\)

    \((0.37)^{1\over 2} = {0.82 \over 1.2 } = 0.6833\)

     

31. 15. Evaluate \(\int {x^2 +1 \over x^4 +1}\) dx 

    Answer:

    I = \(\int {x^2 +1 \over x^4 +1}\) dx 

    I = \(\int {x^2 \over x^4 +1}\). dx + \(\int {dx \over 1 + (x^2)^2}\) 

    I = \(\int {1 + {1 \over x^2} \over ({x^2 + 1 \over x^2})} dx\) 

    put \(x - {1 \over x} = b\)

    \(({1 + {1 \over x^2}})dx = dt\)

    \(\int {dt \over t^2 +2} = \int {dt \over t^2 + (\sqrt2)^2} \)

    = \({1 \over \sqrt 2} tan^{-1} (t \sqrt 2) + c\)

    =\({1 \over \sqrt 2} tan^{-1} ({x^2 - 1 \over \sqrt 2 x }) + c\)

32. Or 
     
    Evaluate \(\int {dx \over x^2+1}\)

    Answer:

    I = \(\int {dx \over x^2+1}\)
    

    I = \(tan^{-1}x + c\)

33. 16. Find the area of region bounded by the ellipse  \({x^2 \over 9 } + {y^2 \over 4} = 1\)

    Answer:

    The equation and ellipse is  \({x^2 \over 9 } + {y^2 \over 4} = 1\)

    \( {y^2 \over 4} = 1 -{x^2 \over 9 } \)

    \(y^2 = {4 \over 9 } (9 - x^2)\)

    \(y = {2\over 3} \sqrt {9-x^2}\)

    The ellipse is symmetrical about both the axexs: 

    Req Area =  4 ( Area  AOB)

    =  4 \(\int_0^3 y.dx = 4 \int_0^3 {2 \over 3} \sqrt 9 - x^2 dx\)

    = \({8 \over 3} \int_0^3 \sqrt{(3)^2- (x)^2} dx\)

    = \({8 \over 3} [ {x \sqrt{(3)^2 - (x)^2}\over 2} + {(3)^2 \over 2} sin^{-1} {x \over 3} ]\)

    = \(6 \pi \) square units

34. 17. Find the particular solution of differential equation [x sin2 (y/x)-y] dx +xdy = 0;y(1)=𝜋/4

    Answer:

    [x sin2 (y/x)-y] dx +xdy = 0;y(1)=𝜋/4

    y(1) = 𝜋/4

    \(sin^2({y \over x}) - {y\over x} + {dy \over dx} = 0 \)

    \({dy \over dx} = {y \over x} - sin ^2 ({y \over x})\)-----------1

    Put y = vx 

    \({dy \over dx} = v + x{dv \over dx}\)

    1 becomes,

    \(v + {x {dv \over dx} } =v - sin^2 v\)

    \({x {dv \over dx} } = -sin^2 v\)

    \({dv \over sin ^2 v} = -{dx \over x}\)

    \(\int cosec^2v dv = - \int {dx \over x}\)

    -cotv = -log |x| + c

    log |x| - cotv = c

    log |x| + cot y/x = c

    Now , y(1) = 𝜋/4

    => log|1| - cot (𝜋/4) = c

    => c = -1

    log |x| - cot (y/x) = -1 

36. Or

    Find the particular solution of differential equation " given that tanx \(dy\over dx\) +y = 2x tan x + x² , x != 0 given that y=0 when x = 𝜋/2

    Answer:

    tanx \(dy\over dx\) +y = 2x tan x + x² 

    \(y ({\pi\over 2 }) = 0\)

    \(dy\over dx\) + y.cotx = 2x  + x² cot x

    comparing it with \(dy\over dx\) + Py = Q

    P = cot x ;  Q = x² cotx + 2x

    \(\int P.dx = \int cotx dx = log sinx\)

    I.F = \(e^{\int Pdx} e^{logsinx} = sinx\) 

    y sinx = \(\int ( { x^2 cot x + 2x}) sinx dx +c\)

    y sinx = \(\int ( { x^2 cos x + 2xsinx }) dx +c\)

    y sinx = \(\int { x^2 cos x dx + 2 \int} xsinx dx +c\)

    y sinx = \(x^2 sinx + c\)

     

37. 18. If à = 2í-3j+4k ਅਤੇ 6 = 5i +j-k represents sider parallelogram then find both diagonals and a unit vector perpendicular to both dingonals

    Answer:

38. 19. Two cards are drawn (without replacement from a well shulle  distribution table and mean  of number of kings. 

    Answer:

39. 20.  Solve the following system oflincar equations by matrix mehord: 
     
    x - 2y +3z = -5, 3 x +y +c= 8, 2x –y +2z = 1 

    Answer:

    x - 2y +3z = -5 

    3 x +y +c= 8

    2x –y +2z = 1 

    A = \( \begin{vmatrix} 1 & -2 & 3 \\ 3 & 1 & 1 \\ 2 & -1 & 2 \end{vmatrix}\)  ;     x = \( \begin{vmatrix} x \\ y \\ z & \end{vmatrix}\)  ;   B = \( \begin{vmatrix} -5 \\ 8 \\ 1 & \end{vmatrix}\)

    |A| = \( \begin{vmatrix} 1 & -2 & 3 \\ 3 & 1 & 1 \\ 2 & -1 & 2 \end{vmatrix}\)

    = 1(2 + 1) + 2(6 - 2) + 3(-3 - 2)

    = 3 + 2(4) + 3 (-5) 

    = 3 + 8 - 15      = 11 - 15     =  -4   != 0

    A₁₁  = \( \begin{vmatrix} 1 & 1 & \\ -1 & 2 \end{vmatrix}\)  =   2 + 1 = 3

    A₁₂  = -  \( \begin{vmatrix} 3 & 1 & \\ 2 & 2 \end{vmatrix}\)  =  - ( 6 - 2 ) = - 4

    A₁₃  =  \( \begin{vmatrix} 3 & 1 & \\ -2 & -1 \end{vmatrix}\)  = -3 - 2 = -5

    A21  =  \( \begin{vmatrix} -2 & 3 & \\ -1 & 2 \end{vmatrix}\)  = -(-4 + 3) = 1

    A22  =  \( \begin{vmatrix} 1 & 3 & \\ 2 & 2 \end{vmatrix}\)  = 2 - 6 = -4

    A23  =  -\( \begin{vmatrix} 1 & 2 & \\ 2 & -1 \end{vmatrix}\)  = -(-1 + 4) = -3

    A₃₁  =  \( \begin{vmatrix} -2 & 3 & \\ 1 & 1 \end{vmatrix}\)  = -2 - 3 = -5

    A₃₂  =  -\( \begin{vmatrix} 1 & 3 & \\ 3 & 1 \end{vmatrix}\)  = -(1 - 9) = 8

    A₃₃  =  \( \begin{vmatrix} 1 & -2 & \\ 3 & 1 \end{vmatrix}\)  = 1 + 6 = 7

    adj A  =   \( \begin{vmatrix} 3 & -4 & 5 \\ 1 & -4 & -3 \\ -5 & 8 & 7 \end{vmatrix}^t\)

    =  \( \begin{vmatrix} 3 & 1 & -5 \\ -4 &-4 & 8 \\ 5 & -3 & 7 \end{vmatrix}\)

    \(A^{-1} = {adj A \over |A|}\)

    \(A^{-1} = {1\over 4}\)\( \begin{vmatrix} 3 & 1 & -5 \\ -4 &-4 & 8 \\ 5 & -3 & 7 \end{vmatrix}\)

    x =   \(A^{-1}B = {1\over 4}\)  =     \( \begin{vmatrix} 3 & 1 & -5 \\ -4 &-4 & 8 \\ 5 & -3 & 7 \end{vmatrix}\)\( \begin{vmatrix} -5 \\ 8 \\ 1 & \end{vmatrix}\)

    X = \(- {1\over 4}\) \( \begin{vmatrix} -15 & 8 & -5 \\ -20 &-32 & 8 \\ 25 & -24 & 7 \end{vmatrix}\)

    X = \(- {1\over 4}\)\( \begin{vmatrix} -12 \\ -44 \\ -42 & \end{vmatrix}\)   =  \( \begin{vmatrix} 3 \\ 11 \\ {42 \over 4} & \end{vmatrix}\)

41.                                         Or 
     
     Using elementary transformations find inverse of \( \begin{vmatrix} 2 & 4 & 1 \\ 1 & 2 & 3 \\ 1 & -3 & 0 \end{vmatrix}\)

    Answer:

     A   =   \( \begin{vmatrix} 2 & 4 & 1 \\ 1 & 2 & 3 \\ 1 & -3 & 0 \end{vmatrix}\)

     

42. 21. A window is in the form of rectangle surmounted by a semi-circular opening. The perimeter of window is 30 m. Find the dimensions of window so that it can admit maximum light through the whole opening.                                      

    Answer:

43.                                                      Or 
     
    Prove that volume of largest cone, which can be inscribed in a sphere, is 8/27 part  of sphere. 

    Answer:

44. 22. Find the distance between the point (2, 3,-1) and foot of perpendicular drawn from (3, 1) to the plane X-y+3 z=10

    Answer:

46. 23. Find the equation of plane passing from the point A (2.-1, 1), B (4.3, 2) and C (6,5,-? Also prove that point (5. - 1, lies on the plane given by points A, B and C. 

    Answer:

47. 24. Maximise and minimise : Z=15x + 30y Subject to the constraints : x+y <= 8, 2x +y >= 28, x - 2y >=0, x, y >= 0

    Answer:

48.                                              Or 
     
    Maximise and minimize Z = 4x + 3y -7  Subject to the constraints : x+y <= 10, x +y >= 3, x<=8,  y <= 9 , x , y >-0 

    Answer:

Question paper 2

1. 1. (i) If y = sin (sin^-1 x + cos^-1 x),  x € [-1, 1] then  dy/dx  is 
        

          (a) 𝜋/2           (b) −𝜋/2           (c) 0          d) 1

   Answer:

2. (ii) \(\int e^x ( log x {1\over x}) \) dx is equal to 

   (a) e^x + c     (b) e^x logx + c    (c) e^x/x + c     (d) log x + c

   Answer:

3. (iv) If P(E) denotes probability of occurrence of event E then 
    
          (a) P (E)  € [-1, 1]         (b) P (E) € (1, 2)          (c) P (E) € (0, 1)          (d) P (E) €  [0, 1]  
    

   Answer:

4. (v) If is a binary operation such that a * b = a² +b² then 3 * 5 is  

   
       (a) 34           (b) 9             (c) 8              (d) 25

   Answer:

6. (vii) If  f(x) = {  \( {{{sinx\over x}\over k-1} , {x! = 0 \over x=0}}\)  } ,𝑥 = 0 is continuous at x=0 then 

   
         (a) 2        (b) 0            (c) -1              (d) 1 

   Answer:

7. (viii) Distance between plane 3x +4y-20 = 0 and point (0, 0,-7) is 
    
          (a) 4 units         (b) 3 units       (c) 2 units         (d) 1 unit 

   Answer:

8. (ix) If cos^-1  x = y  then 
    
   (a) −𝜋/2 <= y <=  𝜋/2               (b)  -π <=y <= π     

   (c) 0 <= y <= 𝜋/2                      (d)  0 <= y <= π 

   Answer:

9. (x) Integrating factor of differential equation dy/dx + 𝑦 = 3 is                                               
    
   (a) x       (b) e      (c) e^x     (d) logx 

   Answer:

11. 2. Evaluate  \(\int\limits_0^{x/2} sin^3X / sin^3X + cos^3X { sin^3x \over sin^3x + cos^3x }dx\)   

    Answer:

12. 3. Find particular solution of differential equation  \({dy \over dx} {1+y^2 \over 1+x^2}\)  given that  x=0 or y= 1 

    Answer:

13. 4. Find the angle between the plane 2x+3 y-5z= 10 and the line passing from the points (2, 3,-1)       Or            (1, 2, 1) 

    Answer:

14. 5. If matrix A = [aij]_{3 x 2} , and aij = (3i-2j)² or matrix A find them 

    Answer:

16. 6. Form differential equation representing the family of lines making equal intercepts on the co-ordinate axes.

    Answer:

17. 7. If P (A) = 7/13, P (B) = 9/13 and P (AUB) = 12/13 then find (A|B)

    Answer:

18. 8. Check whether Lagrange's mean value theorem is applicable on f(x) = sin x + cos x Interval [0, 𝜋/2]

    Answer:

19. 9. Evaluate \(\int {7dx \over x(x^7-1)}\)

    Answer:

21. 10. If y = (x)^tanx + (tanx)^x then find 𝑑𝑦/𝑑𝑥

    Answer:

22. 11. Using differentials find approximate value of (0.37)^1/2 

    Answer:

23. 12. Evaluate \(\int {x^2 +1 \over x^4 +1}\) dx 

    Or 
     
    Evaluate \(\int {dx \over x^2+1}\)

    Answer:

24. 13 If à = 2í-3j+4k ਅਤੇ 6 = 5i +j-k represents sider parallelogram then find both diagonals and a unit vector perpendicular to both dingonals.

    Answer:

26. 14. Two cards are drawn (without replacement from a well shulle  distribution table and mean  of number of kings. 

    Answer:

27. 15. Find the particular solution of differential equation [x sin2 (y/x)-y] dx +xdy = 0;y(1)=𝜋/4

    Or

    Find the particular solution of differential equation " given that tanx 𝑑𝑥/𝑑𝑥 +y = 2x tan x + x² , x != 0 given that y=0 when x = 𝜋/2

    Answer:

28. 16. Prove that function f : R --> R,  f(x) =  \(3−2𝑥 \over 7\)   in one-one and onto. Also find  f^-1

    Answer:

29. 17.  Express  \( \begin{matrix} 2 & 5 & -1 \\ 3 & 1 & 5 \\ 7 & 6 & 9 \end{matrix}\)  as sum of symmetric and skew-symmetric matrices. 

      Or 

    If x,y,z are different and  \( \begin{matrix} x & x^2 & 1 + x^3 \\ y & y^2 & 1 + y^3 \\ z & z^2 & 1 + z^3 \end{matrix}\)  =   0   then prove that  xyz =-1  

    Answer:

31. 18. Prove that : sin^-1 ( \(5\over13\) ) +cos^-1 (\(4\over 5\))  =  \(1\over 2\)  sin^-1 \(3696 \over 4225\)

    Answer:

32. 19. Find the area of region bounded by the ellipse  \({x^2 \over 9 } + {y^2 \over 4} = 1\)

    Answer:

33. 20. A window is in the form of rectangle surmounted by a semi-circular opening. The perimeter of window is 30 m. Find the dimensions of window so that it can admit maximum light through the whole opening.                                      

    Or 
     
    Prove that volume of largest cone, which can be inscribed in a sphere, is 8/27 part  of sphere.

    Answer:

34. 21. Maximise and minimise : Z=15x + 30y Subject to the constraints : x+y <= 8, 2x +y >= 28, x - 2y >=0, x, y >= 0   
                                         Or 
     
    Maximise and minimize Z = 4x + 3y -7  Subject to the constraints : x+y <= 10, x +y >= 3, x<=8,  y <= 9 , x , y >-0 

    Answer:

36. 22.  Solve the following system oflincar equations by matrix mehord: 
     
    x - 2y +3z = -5, 3 x +y +c= 8, 2x –y +2z = 1 
     
                                         Or 
     
     Using elementary transformations find inverse of \( \begin{matrix} 2 & 4 & 1 \\ 1 & 2 & 3 \\ 1 & -3 & 0 \end{matrix}\)

    Answer: