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Question Paper - Mathematics and Statistics 2015 - 2016-H.S.C-12th Board Exam Maharashtra State Board (MSBSHSE)

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SubjectMathematics and Statistics
Year2015 - 2016 (March)

 Topics
 Marks
 Topics
 Marks

Marks: 80
Q: 1[12]
Q: 1.1 | Select and write the most appropriate answer from the given alternatives in each of the following sub-questions[6]
Q: 1.1.1[2]

The negation of p ∧ (q → r) is

  1. p ∨ (~q ∨ r)
  2. ~p ∧ (q → r)
  3. ~p ∧ (~q → ~r)
  4. ~p ∨ (q ∧ ~r)
view question and solution
Q: 1.1.2[2]

If `sin^-1(1-x) -2sin^-1x = pi/2` then x is

  1. -1/2
  2. 1
  3. 0
  4. 1/2
 
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Q: 1.1.3[2]

The joint equation of the pair of lines passing through (2,3) and parallel to the coordinate axes is

  1.  xy -3x - 2y + 6 = 0
  2. xy +3x + 2y + 6 = 0
  3. xy = 0
  4. xy - 3x - 2y - 6 = 0
view question and solution
Q: 1.2 | Attempt any 3 of the following[6]
Q: 1.2.1[2]

Find (AB)-1 if

`A=[(1,2,3),(1,-2,-3)], B=[(1,-1),(1,2),(1,-2)]`

 
view question and solution
Q: 1.2.2[2]

Find the vector equation of the plane passing through a point having position vector `3 hat i- 2 hat j + hat k` and perpendicular to the vector `4 hat i + 3 hat j + 2 hat k`

 
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Q: 1.2.3[2]
Q: 1.2.4[2]

Find k, if one of the lines given by 6x2 + kxy + y2 = 0 is 2x + y = 0

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Q: 1.2.5[2]

If the lines

`(x-1)/-3=(y-2)/(2k)=(z-3)/2 and (x-1)/(3k)=(y-5)/1=(z-6)/-5`

are at right angle then find the value of k

 
view question and solution
Q: 2[14]
Q: 2.1 | Attempt any TWO of the following[6]
Q: 2.1.1[5]

Examine whether the following logical statement pattern is tautology, contradiction or contingency.

[(p → q) ∧ q] → p

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Q: 2.1.2[3]

By vector method prove that the medians of a triangle are concurrent.

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Q: 2.1.3[3]

Find the shortest distance between the lines

`bar r = (4 hat i - hat j) + lambda(hat i + 2 hat j - 3 hat k)`

and

`bar r = (hat i - hat j + 2 hat k) + mu(hat i + 4 hat j -5 hat k)`

where λ and μ are parameters

 
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Q: 2.2 | Attempt any TWO of the following :[8]
Q: 2.2.1[4]
Q: 2.2.2[4]

Minimize `z=4x+5y ` subject to `2x+y>=7, 2x+3y<=15, x<=3,x>=0, y>=0` solve using graphical method.

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Q: 2.2.3[4]

The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is Rs. 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is Rs. 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is Rs. 70. Find the cost of each item per dozen by using matrices.

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Q: 3[14]
Q: 3.1 | Attempt any TWO of the following:[6]
Q: 3.1.1[3]
Q: 3.1.2[3]
Q: 3.1.3[3]

how that every homogeneous equation of degree two in x and y, i.e., ax2 + 2hxy + by2 = 0 represents a pair of lines passing through origin if h2ab0.

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Q: 3.2 | Attempt any TWO of the following[8]
Q: 3.2.1[4]

If a line drawn from the point A( 1, 2, 1) is perpendicular to the line joining P(1, 4, 6) and Q(5, 4, 4) then find the co-ordinates of the foot of the perpendicular.

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Q: 3.2.2[4]

Find the vector equation of the plane passing through the points `hati +hatj-2hatk, hati+2hatj+hatk,2hati-hati+hatk`. Hence find the cartesian equation of the plane.

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Q: 3.2.3[4]

Find the general solution of `sin x+sin3x+sin5x=0`

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Q: 4[12]
Q: 4.1 | Select and write the most appropriate answer from the given alternatives in each of the following sub-questions :[6]
Q: 4.1.1[2]

if the function

`f(x)=k+x, for x<1`

       `=4x+3, for x>=1`

id continuous at x=1 then k=

(a) 7

(b) 8

(c) 6

(d) -6

view question and solution
Q: 4.1.2[2]

The equation of tangent to the curve y=`y=x^2+4x+1` at

(-1,-2) is...............

(a)  2x -y = 0                        (b)  2x+y-5 = 0

(c)  2x-y-1=0                        (d)  x+y-1=0

view question and solution
Q: 4.1.3[2]
Q: 4.2 | Attempt any THREE of the following:[6]
Q: 4.2.1[2]
Q: 4.2.2[2]

The displacement 's' of a moving particle at time 't' is given by s = 5 + 20t — 2t2. Find its acceleration when the velocity is zero.

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Q: 4.2.3[2]

Find the area bounded by the curve y2 = 4axx-axis and the lines x = 0 and x = a.

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Q: 4.2.4[2]

The probability distribution of a discrete random variable X is:

X=x12345
P(X=x)k2k3k4k5k

find P(X≤4)

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Q: 4.2.5[2]

Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`

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Q: 5[14]
Q: 5.1 | Attempt any TWO of the following[6]
Q: 5.1.1[3]

Ify y=f(u) is a differentiable function of u and u = g(x) is a differentiable function of x then prove that y = f (g(x)) is a  differentiable function of x and

`(dy)/(dx)=(dy)/(du)*(du)/(dx)`

 

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Q: 5.1.2[3]
Q: 5.1.3[3]

Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`

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Q: 5.2 | Attempt any TWO of the following[8]
Q: 5.2.1[4]

Discuss the continuity of the following functions. If the function have a removable discontinuity, redefine the function so as to remove the discontinuity

``

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Q: 5.2.2[4]

Prove that : `int sqrt(a^2-x^2)dx=x/2sqrt(a^2-x^2)=a^2/2sin^-1(x/a)+c`

 

 

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Q: 5.2.3[4]

A body is heated at 110°C and placed in air at 10°C. After 1 hour its temperature is 60°C. How much additional time is required for it to cool to 35°C?

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Q: 6[14]
Q: 6.1 | Attempt any TWO of the following :[6]
Q: 6.1.1[3]

Prove that: `int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`

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Q: 6.1.2[3]

Evaluate : `int (1+logx)/(x(2+logx)(3+logx))dx`

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Q: 6.1.3[3]
Q: 6.2 | Attempt any TWO of the following :[8]
Q: 6.2.1[4]

Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.

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Q: 6.2.2[4]

A wire of length l is cut into two parts. One part is bent into a circle and other into a square. Show that the sum of areas of the circle and square is the least, if the radius of circle is half the side of the square.

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Q: 6.2.3[4]

The following is the p.d.f. (ProbabiIity Density Function) of a continuous random variable X :

`f(x)=x/32,0

        = 0 otherwise

(a) Find the expression for c.d.f. (Cumulative Distribution Function) of X.

(b) Also find its value at x = 0.5 and 9.

 

 

 

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