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General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Marks for each question are indicated against it.
(iv) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one mark each.
(v) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks each.
(vi) Questions 20 to 26 in Section-C are Long Answer II Type Questions carrying 6 marks each.
(vii) Please write down the serial number of the Question before attempting it.
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Question 1
  • Q1

    If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ. 

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  • Q2

    Write the element a23 of a 3 ✕ 3 matrix A = (aij) whose elements aij are given by aij=i-j2. 

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  • Q3

    Find the differential equation representing the family of curves v=Ar+ B, where A and B are arbitrary constants. 

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  • Q4

    Find the integrating factor of the differential equation
    e-2xx-yxdxdy=1. 

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  • Q5

    If a=7i^+j^-4 k^ and b=2 i^ + 6 j^ + 3k^, then find the projection of a onb. 

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  • Q6

    Find λ, if the vectors a=i^+3j^+k^, b=2i^-j^-k^ and c=λj^+3k^ are coplanar. 

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  • Q7

    A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B, If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.

    OR
    An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained. 

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  • Q8

    If r =xi^+yj^+zk^, find r ×i^·r ×j+xy 

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  • Q9

    Find the distance between the point (−1, −5, −10) and the point of intersection of the line x-23=y+14=z-212 and the plane xy + z = 5. 

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  • Q10

    If sin [cot−1 (x+1)] = cos(tan1x), then find x.


    OR

    If (tan1x)2 + (cot−1x)2 = 5π28, then find x. 

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  • Q11

    If y=tan-1 1+x2+1-x21+x2-1-x2, x21, then find dydx. 

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  • Q12

    If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that y2d2ydx2-xdydx+y=0. 

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  • Q13

    The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ? 

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  • Q14

    Find : x+33-4x-x2 dx 

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  • Q15

    Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The number of articles sold are given below:
     

    School      
    Article A B C
    Hand-fans 40 25 35
    Mats 50 40 50
    Plates 20 30 40

    Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose.

    Write one value generated by the above situation. 

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  • Q16

    If A=2012131-10 find A2-5A+4I and hence find a matrix X such that A2-5A+4I+X=O

    OR

    If A=1-230-14-221, find A'-1. 

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  • Q17

    If fx=a-10axa-1ax2axa, using properties of determinants find the value of f(2x) − f(x). 

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  • Q18

    Find : dxsin x+sin 2x
     

    OR

    Integrate the following w.r.t. x
    x2-3x+11-x2 

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  • Q19

    Evaluate : -xx cos ax-sin bx2 dx 

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  • Q20

    Solve the differential equation :

    tan-1y-xdy=1+y2dx.

    OR

    Find the particular solution of the differential equation dydx=xyx2+y2 given that y = 1, when x = 0. 

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  • Q21

    If lines x-12=y+13=z-14 and  x-31=y-k2=z1 intersect, then find the value of k and hence find the equation of the plane containing these lines. 

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  • Q22

    If A and B are two independent events such that PA¯  B =215 and PA  B¯ = 16, then find P(A) and P(B). 

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  • Q23

    Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π. 

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  • Q24

    Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below :

    2x + 4y  83x + y  6x + y  4x  0, y 0 

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  • Q25

    Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation. 

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  • Q26

    Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle x2+y2=4 at 1, 3

    OR

    Evaluate 13e2-3x+x2+1 dx as a limit of a sum.
     

     

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