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

Solved Question Paper

Math Mar-2018

PSEB • Class 12th • 1st • Mar-2018

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Solved Questions

Solved

  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 

    Very Short Answer Mar-2018 • PSEB Class 12th

  2. If cos-1  x = y  then 
     
    (a) −𝜋/2 <= y <=  ðœ‹/2               (b)  -π <=y <= π     

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

    Very Short Answer Mar-2018 • PSEB Class 12th

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

    Very Short Answer Mar-2018 • PSEB Class 12th

  4. 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

    Very Short Answer Mar-2018 • PSEB Class 12th

  5. If 
    f(x) = {  ( {{{sinxover 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 

    Very Short Answer Mar-2018 • PSEB Class 12th

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

    (a) ex + c     (b) ex logx + c    (c) ex/x + c     (d) log x + c

    Very Short Answer Mar-2018 • PSEB Class 12th

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

    Very Short Answer Mar-2018 • PSEB Class 12th

  8. 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 

    Very Short Answer Mar-2018 • PSEB Class 12th

  9. 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 

    Very Short Answer Mar-2018 • PSEB Class 12th

  10. 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] 

    Very Short Answer Mar-2018 • PSEB Class 12th

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

    Short Answer Mar-2018 • PSEB Class 12th

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

    Short Answer Mar-2018 • PSEB Class 12th

  13. Evaluate  (intlimits_0^{x/2} sin^3X / sin^3X + cos^3X { sin^3x over sin^3x + cos^3x }dx)   

    Short Answer Mar-2018 • PSEB Class 12th

  14. Evaluate (int {7dx over x(x^7-1)})

    Short Answer Mar-2018 • PSEB Class 12th

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

    Short Answer Mar-2018 • PSEB Class 12th

    ({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})

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

    Short Answer Mar-2018 • PSEB Class 12th

  17. 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)

    Short Answer Mar-2018 • PSEB Class 12th

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

    Short Answer Mar-2018 • PSEB Class 12th

    P (A) = 7/13, P (B) = 9/13 

    P (AUB) = 12/13  P(A|B) = ? 

    P(A|B) = (P(A igcap B) over P(B))

    (P (A igcup B) = P(A) + P(B) - P (A igcap B))

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

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

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

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

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

  19. 10  Prove that function f : R --> R,  f(x) =  (3−2𝑥 over 7)   in one-one and onto. Also find  f-1

    Short Answer Mar-2018 • PSEB Class 12th

     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 )

     

  20. Prove that : sin-1 ( (5over13) ) +cos-1 ((4over 5))  =  (1over 2)  sin-1 (3696 over 4225)

    Short Answer Mar-2018 • PSEB Class 12th

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

    Short Answer Mar-2018 • PSEB Class 12th

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

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

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

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

  22.   Or 

    If x,y,z are different and  ( egin{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  
     

    Short Answer Mar-2018 • PSEB Class 12th

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

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

    => A1  + A2 = 0

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

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

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

    R2 +  R2 -  R1   &  R3 + R3 - R1

    A1  = xyz ( egin{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) =   ( egin{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

  23. If y = (x)tanx + (tanx)x then find 𝑑𝑦/𝑑𝑥

     

    Long Answer Mar-2018 • PSEB Class 12th

    y = (x)tanx + (tanx)x

    Let u = xtanx ,  v =  tanxx

    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 [ {tanxover 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})

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

  24. Using differentials find approximate value of (0.37)1/2 

    Long Answer Mar-2018 • PSEB Class 12th

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

     

  25. Evaluate (int {x^2 +1 over x^4 +1}) dx 

    Long Answer Mar-2018 • PSEB Class 12th

    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)

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

    Long Answer Mar-2018 • PSEB Class 12th

    I = (int {dx over x^2+1})

    I = (tan^{-1}x + c)

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

    Long Answer Mar-2018 • PSEB Class 12th

    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 = {2over 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

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

    Long Answer Mar-2018 • PSEB Class 12th

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

    y(1) = ðœ‹/4

    (sin^2({y over x}) - {yover 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) = ðœ‹/4

    => log|1| - cot (𝜋/4) = c

    => c = -1

    log |x| - cot (y/x) = -1 

  29. Or

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

    Long Answer Mar-2018 • PSEB Class 12th

    tanx (dyover dx) +y = 2x tan x + x2 

    (y ({piover 2 }) = 0)

    (dyover dx) + y.cotx = 2x  + x2 cot x

    comparing it with (dyover 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)

     

  30. 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

    Long Answer Mar-2018 • PSEB Class 12th

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

    Long Answer Mar-2018 • PSEB Class 12th

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

    Long Answer Mar-2018 • PSEB Class 12th

    x - 2y +3z = -5 

    3 x +y +c= 8

    2x –y +2z = 1 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    Long Answer Mar-2018 • PSEB Class 12th

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

     

  34. 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.                                      

    Short Answer Mar-2018 • PSEB Class 12th

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

    Long Answer Mar-2018 • PSEB Class 12th

  36. 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

    Long Answer Mar-2018 • PSEB Class 12th

  37. 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. 

    Long Answer Mar-2018 • PSEB Class 12th

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

    Long Answer Mar-2018 • PSEB Class 12th

  39.                                              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 

    Long Answer Mar-2018 • PSEB Class 12th

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01 Where can I find PSEB 12th Class Math previous year question papers for 1st semester on BRpaper?

This Math subject page is meant to help students find PSEB Class 12 previous year Math question papers in one place. You can use it to access subject-wise paper links for revision and to review how board-level Math questions are presented.

02 Are the PSEB 12th Math papers on BRpaper arranged subject-wise?

Yes, this page is focused on Math for PSEB 12th Class Question Papers, so it helps students navigate previous year papers by subject. That makes it easier to revise Maths separately instead of searching across all subjects.

03 How should I use previous year Math papers for PSEB board exam revision?

Use old Math papers to practice question selection, step-based solving, and time management. They can also help you spot repeated topic areas, understand common question styles, and check which chapters need more revision before the exam.

04 What is the difference between previous year Math papers and sample or model papers?

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05 Can PSEB 12th Math previous year papers help me understand the exam pattern and marks distribution?

They can help you understand the general paper pattern, question types, and level of presentation commonly seen in Math exams. For exact sections or marks distribution for your session, it is better to also check the latest official sample or model paper because patterns can change.

06 How many previous year PSEB 12th Math papers should I practice?

Students often try to solve multiple recent papers so they can compare question style, difficulty, and recurring topics. A practical approach is to start with the latest available papers first, then use older ones for extra revision and self-assessment.

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