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 (October)

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 p ˄ q = F, p → q = F, then the truth value of p and q is :

(A) T, T

(B) T, F

(C) F, T

(D) F, F

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Q: 1.1.2[2]

If `A^-1=1/3[[1,4,-2],[-2,-5,4],[1,-2,1]]` and | A | = 3, then (adj. A) = _______





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Q: 1.1.3[2]

The slopes of the lines given by 12x2 + bxy + y2 = 0 differ by 7. Then the value of b is :

(A) 2

(B) ± 2

(C) ± 1

(D) 1

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

In a Δ ABC, with usual notations prove that:` (a -bcos C) /(b -a cos C )= cos B/ cos A`

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Q: 1.2.2[2]

Find ‘k’, if the equation kxy + 10x + 6y + 4 = 0 represents a pair of straight lines.

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Q: 1.2.3[2]

If A, B, C, D are four non-collinear points in the plane such that `bar(AD)+bar( BD)+bar( CD)=bar O` then prove that point D is the centroid of the ΔABC.

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Q: 1.2.4[2]

Find the direction cosines of the line 

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

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Q: 1.2.5[2]

Show that the points (1, 1, 1) and (-3, 0, 1) are equidistant from the plane `bar r (3bari+4barj-12bark)+13=0`

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

Using truth table prove that p ↔ q = (p ∧ q) ∨ (~p ∧ ~q).

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Q: 2.1.2[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: 2.1.3[3]

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

Find the inverse of the matrix,  `A=[[1,3,3],[1,4,3],[1,3,4]]`

by using column transformations.

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Q: 2.2.2[4]

In ΔABC, prove that : `tan((a-b)/2)=(a-b)/(a+b)cotC/2`



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Q: 2.2.3[4]

Show that the lines ` (x+1)/-3=(y-3)/2=(z+2)/1; ` are coplanar. Find the equation of the plane containing them.

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

Construct the simplified circuit for the following circuit:

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Q: 3.1.2[3]

Express `-bari-3barj+4bark `  as a linear combination of vectors  `2bari+barj-4bark,2bari-barj+3bark `



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Q: 3.1.3[3]

Find the length of the perpendicular from the point (3, 2, 1) to the line `(x-7)/2=(y-7)/2=(z-6)/3=lambda (say)`

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

Show that the angle between any two diagonals of a cube is `cos^-1(1/3)`

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Q: 3.2.2[4]

Minimize : Z = 6x + 4y

Subject to the conditions:

3x + 2y ≥ 12,

x + y ≥ 5,

0 ≤ x ≤ 4,

0 ≤ y ≤ 4

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Q: 3.2.3[4]

If `tan^-1((x-1)/(x-2))+cot^-1((x+2)/(x+1))=pi/4; `

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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 `y=sec^-1((sqrtx-1)/(x+sqrtx))+sin_1((x+sqrtx)/(sqrtx-1)), `

(A) x

(B) 1/x

(C) 1

(D) 0

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Q: 4.1.2[2]

If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:

(A) 0

(B) π

(C) π/2

(D) π/4

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Q: 4.1.3[2]

The solution of the differential equation dy/dx = sec x – y tan x is:

(A) y sec x = tan x + c

(B) y sec x + tan x = c

(C) sec x = y tan x + c

(D) sec x + y tan x = c

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

Evaluate: `int1/(xlogxlog(logx))dx`

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Q: 4.2.2[2]

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

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Q: 4.2.3[2]

Find k, such that the function  P(x)=k(4/x) ;x=0,1,2,3,4 k>0

                                                 =0 ,otherwise

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Q: 4.2.4[2]

Given is X ~ B (n, p). If E(X) = 6, and Var(X) = 4.2, find the value of n.

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Q: 4.2.5[2]

Solve the differential equation `y-xdy/dx=0`

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

Discuss the continuity of the function

`f(x)=(1-sinx)/(pi/2-x)^2, `

       = 3,                  for x=π/2


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Q: 5.1.2[3]

If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f'(x).

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Q: 5.1.3[3]

Differentiate `cos^-1((3cosx-2sinx)/sqrt13)` w. r. t. x.

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

Show that:  `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`

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Q: 5.2.2[4]

A rectangle has area 50 cm2 . Find its dimensions when its perimeter is the least

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Q: 5.2.3[4]

Prove that : `int_-a^af(x)dx=2int_0^af(x)dx` , if f (x) is an even function.

                                    = 0, if f (x) is an odd function.

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

If y = f (u) is a differential function of u and u = g(x) is a differential function of x, then prove that y = f [g(x)] is a differential function of x and `dy/dx=dy/(du) xx (du)/dx`

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Q: 6.1.2[3]

Each of the total five questions in a multiple choice examination has four choices, only one of which is correct. A student is attempting to guess the answer. The random variable x is the number of questions answered correctly. What is the probability that the student will give atleast one correct answer?

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Q: 6.1.3[3]

If f (x) = x 2 + a, for x ≥ 0 ` =2sqrt(x^2+1)+b, ` is continuous at x = 0, find a and b.

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

Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]

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Q: 6.2.2[4]

Solve the differential equation:  `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.

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Q: 6.2.3[4]

Find the expected value, variance and standard deviation of random variable X whose probability mass function (p.m.f.) is given below: 



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