# 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 (October)

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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[bar a barb barc]!=0  then bar a . barp +bar b . bar q+bar c. bar r  is equal to

(a) 0

(b) 1

(c) 2

(d) 3

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Q: 1.1.2[2]

The inverse of the matrix [[2,0,0],[0,1,0],[0,0,-1]]is --------

(a) [[1/2,0,0],[0,1,0],[0,0,-1]]

(b) [[-1/2,0,0],[0,-1,0],[0,0,1]]

(c) [[-1,0,0],[0,-1/2,0],[0,0,1/2]]

(d) 1/2[[-1/2,0,0],[0,-1,0],[0,0,-1]]

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Q: 1.1.3[2]

Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........

(a) +-1/sqrt51,+-5/sqrt51,+-1/sqrt51

(b) +-5/sqrt51,+-5/sqrt51,+-5/sqrt51

(c) +-sqrt5,+-1,+-5

(d) +-sqrt51,+-sqrt51+-sqrt51

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

Write truth values of the following statements :sqrt5 is an irrational number but 3 +sqrt 5 is a complex number.

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Write truth values of the following statements : ∃ n ∈ N such that n + 5 > 10.

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Q: 1.2.2[2]

If bar c = 3bara- 2bar b

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Q: 1.2.3[2]

Find the vector equation of the plane which is at a distance of 5 units from the origin and which is normal to the vector  2hati + hatj + 2hatk.

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Q: 1.2.4[2]

The Cartesian equations of line are 3x+1=6y-2=1-z find its equation in vector form.

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Q: 1.2.5[2]

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are -2, 1, -1, and -3, -4, 1.

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Q: 2[12]
Q: 2.1 | Attempt any TWO of the following:[6]
Q: 2.1.1[3]

Using truth table, prove the following logical equivalence :

(p ∧ q)→r = p → (q→r)

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Q: 2.1.2[3]

Find the joint equation of the pair of lines through the origin each of which is making an angle of 30° with the line 3x + 2y - 11 = 0

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Q: 2.1.3[3]

Show that: 2sin^-1(3/5)=tan^-1(24/7)

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

Solve the following equations by the method of reduction :

2x-y + z=1,  x + 2y +3z = 8, 3x + y-4z=1.

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Q: 2.2.2[4]

Prove that the volume of a parallelopiped with coterminal edges as   bara ,bar b , barc

Hence find the volume of the parallelopiped with coterminal edges  bar i+barj, barj+bark

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Q: 2.2.3[4]

Solve the following LPP by using graphical method.

Maximize : Z = 6x + 4y

Subject to x ≤ 2, x + y ≤  3, -2x + y ≤  1, x ≥  0, y ≥ 0.

Also find maximum value of Z.

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

In ΔABC with usual notations, prove that 2a {sin^2(C/2)+csin^2 (A/2)} = (a +   c - b)

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Q: 3.1.2[3]

If p : It is a day time, q : It is warm, write the compound statements in verbal form

denoted by -

(a) p ∧ ~ q   (b)  ~ p  → q    (c)  q  ↔  p

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Q: 3.1.3[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: 3.2 | Attempt any TWO of the following:[8]
Q: 3.2.1[4]

Parametric form of the equation of the plane is bar r=(2hati+hatk)+lambdahati+mu(hat i+2hatj+hatk) λ and μ are parameters. Find normal to the plane and hence equation of the plane in normal form. Write its Cartesian form.

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Q: 3.2.2[4]

If the angle between the lines represented by ax2 + 2hxy + by2 = 0 is equal to the angle between the lines 2x2 - 5xy + 3y2 =0,

then show that 100(h2 - ab) = (a + b)2

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Q: 3.2.3[4]

Find the general solution of: sinx · tanx = tanx- sinx+ 1

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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]
Q: 4.1.2[2]

If X is a random variable with probability mass function

P(x) = kx ,  x=1,2,3

= 0 ,     otherwise

then , k=..............

(a) 1/5

(b) 1/4

(c) 1/6

(d) 2/3

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Q: 4.1.3[2]
Q: 4.2 | Attempt any THREE of the following:[6]
Q: 4.2.1[2]

If y=sin^-1(3x)+sec^-1(1/(3x)),

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Q: 4.2.2[2]
Q: 4.2.3[2]
Q: 4.2.4[2]

The probability that a certain kind of component will survive a check test is 0.5. Find the probability that exactly two of the next four components tested will survive.

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Q: 4.2.5[2]

Find the area of the region bounded by the curve y = sinx, the lines x=-π/2 , x=π/2 and X-axis

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

Examine the continuity of the following function at given point:

f(x)=(logx-log8)/(x-8) ,

 =8,

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

Solve : 3ex tanydx + (1 +ex) sec2 ydy = 0

Also, find the particular solution when x = 0 and y = π.

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

Evaluate : intlogx/(1+logx)^2dx

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Q: 5.2.3[4]
Q: 6[14]
Q: 6.1 | Attempt any TWO of the following:[6]
Q: 6.1.1[3]

Show that the function defined by f(x) =|cosx| is continuous function.

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Q: 6.1.2[3]

Solve the differential equation dy/dx=(y+sqrt(x^2+y^2))/x

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Q: 6.1.3[3]

Given X ~ B(n, p). If n = 20, E(X) = 10, find p, Var. (X) and   S.D. (X).

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Q: 6.2 | Attempt any TWO of the following:[8]
Q: 6.2.1[4]
Q: 6.2.2[4]

Verify Lagrange’s mean value theorem for the function f(x)=x+1/x, x ∈ [1, 3]

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Q: 6.2.3[4]
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