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Question Paper - Mathematics and Statistics 2013 - 2014-H.S.C-12th Board Exam Maharashtra State Board (MSBSHSE)

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SubjectMathematics and Statistics
Year2013 - 2014 (March)


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

Which of the following represents direction cosines of the line :

(a)`0,1/sqrt2,1/2`

(b)`0,-sqrt3/2,1/sqrt2`

(c)`0,sqrt3/2,1/2`

(d)`1/2,1/2,1/2`

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Q: 1.1.2[2]

`A=[[1,2],[3,4]]` ans A(Adj A)=KI, then the value of 'K' is

(a) 2  

(b) -2

(c) 10

(d) -10

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Q: 1.1.3[2]
 

The general solution of the trigonometric equation tan2 θ = 1 is..........................

(a)`theta =npi+-(pi/3),n in z`

(b)`theta =npi+-pi/6, n in z`

(c)`theta=npi+-pi/4, n in z`

(d) `0=npi, n in z`

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

If `bara, barb, bar c` are the position vectors of the points A, B, C respectively and ` 2bara + 3barb - 5barc = 0` , then find the ratio in which the point C divides line segment  AB.

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Q: 1.2.2[2]

The Cartestation equation of  line is `(x-6)/2=(y+4)/7=(z-5)/3` find its vector equation.

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Q: 1.2.3[2]

Equation of a plane is vecr (3hati-4hatj+12hatk)=8. Find the length of the perpendicular from the origin to the plane.

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Q: 1.2.4[2]

Find the acute angle between the lines whose direction ratios are 5, 12, -13 and 3, - 4, 5.

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Q: 1.2.5[2]

Write the dual of the following statements:

(l) (p ∨ q) ∧ T
(2) Madhuri has curly hair and brown eyes .

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Q: 2[14]
Q: 2.1 | Attempt any TWO of the following[6]
Q: 2.1.1[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: 2.1.2[3]

Prove that three vectors `bara, barb and barc ` are coplanar, if and only if, there exists a non-zero linear combination `xbara+ybarb +z barc=0`

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Q: 2.1.3[3]

Using truth table prove that :

`~p^^q-=(p vv q)^^~p`

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

In any ΔABC, with usual notations, prove that b^2=c^2+a^2-2ca cosB.

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Q: 2.2.2[4]

Show that the equation `x^2-6xy+5y^2+10x-14y+9=0 ` represents a pair of lines. Find the acute angle between them. Also find the point of intersection of the lines.

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Q: 2.2.3[4]

Express the following equations in the matrix form and solve them by method of reduction :

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

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

find the symbolic fom of the following switching circuit, construct its switching table and interpret it.

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Q: 3.1.3[3]

if A, B, C, D are (1, i, I), (2, l ,3), (3; 2, 2) and (3, 3, 4) respetivly., then find the volume of the parallepiped with AB, AC and AD as concurrent edges

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

Find the equation of the plane passing through the line of intersection of planes 2x - y + z = 3 and 4x- 3y + 5z + 9 = 0 and parallel to the line

` (x+1)/2=(y+3)/4=(z-3)/5`

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Q: 3.2.2[4]

Minimize :Z=6x+4y

Subject to : 3x+2y ≥12

x+y ≥5

0 ≤x ≤4

0 ≤ y ≤ 4 

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Q: 3.2.3[4]

Show that:

`cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)`

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Q: 4[12]
Q: 4.1 | Select an write the correct answer from the given alternatives in each of the following:[6]
Q: 4.1.1[2]

If y =1-cosθ , x = 1-sinθ , then ` dy/dx at " "0 =pi/4`  is ............

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Q: 4.1.2[2]

The integrating factor of linear differential equation `dy/dx+ysecx=tanx` is

(a)secx- tan x

(b) sec x · tan x

(c)sex+tanx

(d) secx.cotx

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Q: 4.1.3[2]

The equation of tangent to the curve y = 3x2 - x + 1 at the point (1, 3) is 

(a) y=5x+2

(b)y=5x-2

(c)y=1/5x+2

(d)y=1/5x-2

 

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

Examine the continuity of the function
f(x) =sin x- cos x, for x ≠ 0

      =- 1 ,forx=0

at the poinl x = 0

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Q: 4.2.2[2]

Verify Rolle's theorem for the function  

f(x)=x2-5x+9 on [1,4]

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Q: 4.2.3[2]

Evaluate : `intsec^nxtanxdx`

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Q: 4.2.4[2]

The probability mass function (p.m.f.) of X is given below:

X=x123
P (X= x)1/52/52/5

 find E(X2)

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Q: 4.2.5[2]

Given that X~ B(n = 10, p), if E(X) = 8. find the value of p.

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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]

Obtain the differential equation by eliminating arbitrary constants A, B from the equation -
y = A cos (log x) + B sin (log x)

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Q: 5.1.3[3]

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

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

An open box is to be made out of a piece of a square card board of sides 18 cms. by cutting off equal squares from  the comers and tumi11g up the sides. Find the maximum volume of the box.

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Q: 5.2.2[4]

Prove that :

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

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Q: 5.2.3[4]

If the function f (x) is continuous in the interval [-2, 2],find the values of a and b where

`f(x)=(sinax)/x-2, for-2<=x<=0`

`=2x+1, for 0<=x<=1`

`=2bsqrt(x^2+3)-1, for 1<x<=2`

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

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

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Q: 6.1.2[3]

A fair coin is tossed 8 times. Find the probability that it shows heads at least once

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Q: 6.1.3[3]

If xpyq=(x+y)p+q then Prove that ` dy/dx=y/x`

 

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

Find the area of the sector of a circle bounded by the circle x2 + y2 = 16 and the line y = x in the ftrst quadrant.

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Q: 6.2.2[4]

Prove that `int sqrt(x^2-a^2)dx=x/2sqrt(x^2-a^2)-a^2/2log|x+sqrt(x^2-a^2)|+c`


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Q: 6.2.3[4]

A random variable X has the following probability distribution :

X=x0123456
P[X=x]k3k5k7k9k11k13k

(a) Find k
(b) find P(O <X< 4)
(c) Obtain cumulative distribution function (c. d. f.) of X.

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