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Question Paper - Mathematics and Statistics 2014 - 2015-H.S.C-12th Board Exam Maharashtra State Board (MSBSHSE)

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SubjectMathematics and Statistics
Year2014 - 2015 (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]

if  `A=[[2,0,0],[0,2,0],[0,0,2]]` then A6=  ......................

(a) 6A

(b) 12A

(c) 16A

(d) 32A

 

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Q: 1.1.2[2]

The principal solution of cos-1(1/2) is :

(a) π/3

(b) π/6

(c) 2π/3

(d) 3π/2

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Q: 1.1.3[2]

If an equation hxy + gx + fy + c = 0 represents a pair of lines, then.........................

(a) fg = ch                       (b) gh = cf

(c) Jh = cg                     (d) hf= - eg

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Q: 1.2 | Attempt any THREE of the following [6]
Q: 1.2.1[2]

Write the converse and contrapositive of the statement — “If two triangles are congruent, then their areas are equal.”

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Q: 1.2.2[2]
Q: 1.2.3[2]

Find the angle between the planes `bar r.(2bar i+barj-bark)=3 `

 

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Q: 1.2.4[2]

The Cartesian equations of line are 3x -1 = 6y + 2 = 1 - z. Find the vector equation of line.

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Q: 1.2.5[2]
Q: 2[14]
Q: 2.1 | Attempt any TWO of the following [6]
Q: 2.1.1[3]

If A, B, C, D are (1, 1, 1), (2, I, 3), (3, 2, 2), (3, 3, 4) respectively, then find the volume of parallelopiped with AB, AC and AD as the concurrent edges.

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Q: 2.1.2[3]

Discuss the statement pattern, using truth table : ~(~p ∧ ~q) v q

 

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Q: 2.1.3[3]
Q: 2.2 |  Attempt any TWO of the following[8]
Q: 2.2.1[4]

Find the direction cosines of the line perpendicular to the lines whose direction ratios are -2, 1,-1 and -3, - 4, 1 

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Q: 2.2.2[4]
Q: 2.2.3[4]

The sum of three numbers is 6. When second number is subtracted from thrice the sum of first and third number, we get number 10. Four times the sum of third number is subtracted from five times the sum of first and second number, the result is 3. Using above information, find these three numbers by matrix method.

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

If θ is the acute angle between the lines represented by equation ax2 + 2hxy + by2 = 0  then prove that `tantheta=|(2sqrt(h^2-ab))/(a+b)|, a+b!=0`

 

 

 

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Q: 3.1.2[3]

If the lines `(x-1)/2=(y+1)/3=(z-1)/4 ` and `(x-3)/1=(y-k)/2=z/1` intersect each other then find value of k

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Q: 3.1.3[3]

Construct the switching circuit for the following statement : [p v (~ p ∧ q)] v [(- q ∧ r) v ~ p]

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

Find the general solution of : cos x - sin x = 1.

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Q: 3.2.2[4]
Q: 3.2.3[4]
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]

A random variable X has the following probability distribution:

then E(X)=....................

(a) 0.8

(b) 0.9

(c) 0.7

(d) 1.1

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Q: 4.1.2[2]

If `int_0^alpha3x2dx=8` then the value of α is :

(a) 0

(b) -2

(c) 2 

(d) ±2

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Q: 4.1.3[2]

The differential equation of y=c/x+c2 is :

(a)`x^4(dy/dx)^2-xdy/dx=y`

(b)`(d^2y)/dx^2+xdy/dx+y=0`

(c)`x^3(dy/dx)^2+xdy/dx=y`

(d)`(d^2y)/dx^2+dy/dx-y=0`

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Q: 4.2 | Attempt any THREE of the following[6]
Q: 4.2.1[2]

Evaluate : `int e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]dx`

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Q: 4.2.2[2]

If   `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+..... oo))),` then show that `dy/dx=cosx/(2y-1)`

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Q: 4.2.3[2]

Evaluate :`int_0^(pi/2)1/(1+cosx)dx`

 

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Q: 4.2.4[2]
Q: 4.2.5[2]

A fair coin is tossed five times. Find the probability that it shows exactly three times head.

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Q: 5[14]
Q: 5.1 | Attempt any TWO of the following[6]
Q: 5.1.1[3]
Q: 5.1.2[3]
Q: 5.1.3[3]

If `f(x)=[tan(pi/4+x)]^(1/x), `

           = k                        ,for x=0

is continuous at x=0 , find k.

 

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Q: 5.2 | Attempt any TWO of the following :[8]
Q: 5.2.1[4]
Q: 5.2.2[4]
Q: 5.2.3[4]

Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`

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Q: 6[14]
Q: 6.1 | Attempt any two of the following [6]
Q: 6.1.1[3]
Q: 6.1.2[3]
Q: 6.1.3[3]

Let the p. m. f. (probability mass function) of random variable x be

`p(x)=(4/x)(5/9)^x(4/9)^(4-x), x=0, 1, 2, 3, 4`

                                                =0 otherwise

find E(x) and var (x)

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Q: 6.2 | Attempt any two of the following [8]
Q: 6.2.1[4]

Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x). 

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Q: 6.2.2[4]

Solve the differential equation (x2 + y2)dx- 2xydy = 0

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Q: 6.2.3[4]

Given the p. d. f. (probability density function) of a continuous random variable x as :

 `f(x)=x^2/3, -1`

         = 0 , otherwise

Determine the c. d. f. (cumulative distribution function) of x and hence find P(x < 1), P(x ≤ -2), P(x > 0), P(1 < x < 2)

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