General Instructions

1 . All question are compulsory .
2 . This question paper contains 29 questions.
3 . Question 1-4 in Section A are very short answer type questions carrying 1 mark each.
4 . Questions 5-12 in Section B are short answer type questions carrying 2 marks each.
5 . Question 13-23 in Section C are long answer I type questions carrying 4 marks each.
6 . Question 24-29 in Section D are long answer II type questiona carrying 6 marks each.

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Section A (1 Marks)

Question 1 .
If (A')' = and B = , Find [A + 2B]' .

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Question 2 .
If f:N → N , g:N → N and h:N → R defined as f(x) = 3x, g(y) = 4y + 3 and h(z) = sinz,∀ x,y,z ∈ N .Find the value of ho(gof).

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Question 3 .
Write the direction cosines of vector 3î - 2ĵ + 4k̂.

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Question 4 .
If P(A) =

2/7

, P(B)' =

4/7

and P(A∩B) =

1/7

evaluate P(

A/B

).

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Section B (2 Marks)

Question 5 .
If A is skew - symmetric matrix of order 3, then prove that det A = 0.

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Question 6 .
Find the value of k,if function , f(x) is continuous at x = 4 .
f(x) = kx + 1, if x ≤ 4 and 3x - 5, if x > 4 .

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Question 7 .
The side of an equilateral triangle is increasing at the rate of 4cm/s . At what rate is its area increasing , when the altitude of the triangle is 10√3 cm ?

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Question 8 .
Find ∫

dx/ 3- 6x - x²

.

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Question 9 .
Find x, if = î + (x-z)ĵ + 4k̂, = î - 3k̂ and = 3î + 3ĵ - 2k̂ are coplaner .

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Question 10 .
Find the vector and the cartesian equations of the line through the point (6,3,-2) and which is parallel to the vector 2î + 2ĵ - 5k̂ .

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Question 11 .
Formulate an L.P.P , use x for product A and y for product B.

	Machine (M1)	Machine (M2)	Profit
Product A	6	4	9
Product B	2	5	8
Machine Capacity	100	120

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Question 12 .
Find the mean number of tails in three tosses of a fair coin .

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Section C (4 Marks Each)

Question 13 .
Find the value of a for which the function f is defined as
f(x) = a sin

π/2

(x + 1), x ≤ 0

tanx - sinx/x³

, x > 0 .

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Question 14 .
Solve the equation for x ,
cos(tan^-1x) = sin (tan^-1

4/3

)

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Question 15 .
If f,g:R → R are two functions defined as f(x) = lxl + x , and g(x) = lxl - x, ∀ x ∈ R
then find fog and gof.

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Question 16 .
Prove that

π/2

- sin^-1(

1/3

) = sin^-1 (

2√2/3

)

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Question 17 .
Find the equation of line joining A(1,3) and B(0,0) using determinants and find the value of k if C(k,0) is a point such that area of triangle ABC is 3sq. units.

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Question 18 .
If a,b,c are positive and unequal show that the determinant is negative.

A =

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Question 19 .
If x^y = e^x-y then prove that

dy/dx

=

logx/{log(xe)}²

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Question 20 .
Solve the following differential equations.
(x loglxl)

dy/dx

+ y = 2loglxl

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Question 21 .
A company produces two products tables and chairs . Cost per unit for tables and a chairs is Rs. 20 and Rs. 30 respectively. One unit of table requires 4 labour hours and 3 machine hours . One unit of chair requires 5 labour hours and 4 machine hours. Atleast 200 labours hours and not more than 240 machine hours should be used. Atleast 30 table and atleast 20 chairs should be produced . Formulate as LPP and solve graphicaly to obtain optimal solution.

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Question 22 .
For 6 trails of an experiment , let X be a binomial random variate which satisfies the relation P(x = 4) = (1/4)P(x = 2). Find the probability of sucess.

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Question 23 .
Bag A contains 4 Red and 4 Black balls, while bag B contains 3 red and 5 black balls. Two balls are transferred randomly from bag A to bag B and then a ball is drawn from bag B at random. If the ball drawn from bag B is found to be red, find the probability that two red balls were transferred from A to B.

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Section D (6 Marks Each)

Question 24 .
Evaluate

(2 log lsin xl - loglsin 2xl)dx.

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Question 25 .
Show that the right circular cone of least curved surface and given volume has an altitude equal to √2 times tghe radius of the base.

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Question 26 .
Find the area of the region
{(x,y): x² + y² ≤ 16, x + y ≥ 4}

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Question 27 .
Show that the differential equation
x

dy/dx

sin(

y/x

) + x - y sin (

y/x

) = 0 is
homogeneous. Find the particular solution of this differential equation, given that x = 1, when y =

π/2

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Question 28 .
The scalar product of the vector a→ = î + ĵ +k̂ with a unit vector along the sum of vectors b→ = 2î + 4ĵ -5k̂ and c→ = μî + 2ĵ +3k̂ is equal to one. Find the value of μ.

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Question 29 .
Find the equation of the plane which contains the line of intersection of the planes
r→.(î - 2 ĵ + 3 k̂) - 4 = 0 and r→.(-2î - ĵ + k̂) + 5 = 0 and whose intercept on X- axis is equal to that of on Y-axis.

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