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General Instructions:
(i) All questions are compulsory.
(ii) This question paper contains 29 questions.
(iii) Questions 1- 4 in Section A are very short-answer type questions carrying 1 mark each.
(iv) Questions 5-12 in Section B are short-answer type questions carrying 2 marks each.
(v) Questions 13-23 in Section C are long-answer I type questions carrying 4 marks each.
(vi) Questions 24-29 in Section D are long-answer II type questions carrying 6 marks each.
Question 1
  • Q1

    If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis. 

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  • Q2

    Evaluate : 233x dx. 

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  • Q3

    Determine the value of the constant 'k' so that function fx=kxx, if x < 03,if x  0 is continuous at x = 0. 

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  • Q4

    If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A–1) = (det A)k. 

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  • Q5

    Prove that if E and F are independent events, then the events E and F' are also independent. 

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  • Q6

    A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is Rs 100 and that on a bracelet is Rs 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit?
    It is being given that at least one of each must be produced. 

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  • Q7

    Find dxx2+4x+8 

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  • Q8

    Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7y = 35z. 

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  • Q9

    Show that the function fx=4x3-18x2+27x-7 is always increasing on . 

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  • Q10

    The volume of a sphere is increasing at the rate of 3 cubic centimeter per second. Find the rate of increase of its surface area, when the radius is 2 cm. 

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  • Q11

    Show that all the diagonal elements of a skew symmetric matrix are zero. 

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  • Q12

    If y=sin-16x1-9x2, -132<x<132, then find dydx. 

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  • Q13

    Let a=i^+j^+k^,b=i^ and c=c1i^+c2j^+c3k^, then

    (a) Let c1 = 1 and c2 = 2, find c3 which makes a,b and c coplanar.

    (b) If c2 = –1 and c3 = 1, show that no value of c1 can make a,b and c coplanar. 

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  • Q14

    If a, b, c are mutually perpendicular vectors of equal magnitudes, show that the vector a+ b+ c is equally inclined to a, b and  c.  Also, find the angle which a+ b+ c makes with a or  b or  c. 

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  • Q15

    The random variable X can take only the values 0, 1, 2, 3. Give that P(X = 0) = P(X = 1) = p and P(X = 2) = P(X = 3) such that Σpixi2=2Σpixi, find the value of p. 

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  • Q16

    Often it is taken that a truthful person commands, more respect in the society. A man is known to speak the truth 4 out of 5 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.
    Do you also agree that the value of truthfulness leads to more respect in the society? 

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  • Q17

    Using properties of determinants, prove that xx+yx+2yx+2yxx+yx+yx+2yx=9y2x+y.
     

    OR

    Let A=2-134, B=5274, C=2538, find a matrix D such that CD − AB = O. 

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  • Q18

    Differentiate the function sin xx+sin-1x with respect to x.

    OR

    If xmyn=x+ym+n, prove that d2ydx2=0. 

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  • Q19

    Evaluate : 0πx sin x1+cos2xdx
     

    OR

    Evaluate : 03/2x sin πxdx 

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  • Q20

    Solve the following L.P.P graphically:
     

    Maximise Z = 20x + 10y
    Subject to the following constraints
    x + 2y ≤ 28,
     
    3x + y ≤ 24,
     
     x ≥ 2,
     
     x, y ≥ 0
     

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  • Q21

    Show that the family of curves for which dydx=x2+y22xy, is given by x2y2 = cx. 

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  • Q22

    Find : 3 sin x-2 cos x13-cos2 x-7 sin xdx 

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  • Q23

    Solve the following equation for x:

    cos tan-1 x=sin cot-134 

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  • Q24

    Using integration, find the area of region bounded by the triangle whose vertices are (–2, 1), (0, 4) and (2, 3).
     

    OR

    Find the area bounded by the circle x2 + y2 = 16 and the line 3y=x in the first quadrant, using integration. 

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  • Q25

    Find the equation of the plane through the line of intersection of r·2i^-3j^+4k^=1 and r·i^-j^+4=0 and perpendicular to the plane r·2i^-j^+k^+8=0. Hence find whether the plane thus obtained contains the line x − 1 = 2y − 4 = 3z − 12.
     

    OR
     
    Find the vector and Cartesian equations of a line passing through (1, 2, –4) and perpendicular to the two lines x-83=y+19-16=z-107 and x-153=y-298=z-5-5. 

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  • Q26

    Consider f : R+ → [−5, ∞), given by f(x) = 9x2 + 6x − 5. Show that f is invertible with f-1yy+6-13.

    Hence Find
    (i) f−1(10)
    (ii) y if f-1y=43,

    where R+ is the set of all non-negative real numbers.


    OR

    Discuss the commutativity and associativity of binary operation '*' defined on A = Q − {1} by the rule a * b = ab + ab for all, a, b ∊ A. Also find the identity element of * in A and hence find the invertible elements of A. 

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  • Q27

    If the sum of lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum, when the angle between them is π3. 

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  • Q28

    If A=2311  223  1-1, find A–1 and hence solve the system of equations 2x + y – 3z = 13, 3x + 2y + z = 4, x + 2yz = 8. 

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  • Q29

    Find the particular solution of the differential equation

    tan x·dydx=2x tan x+x2-y;tan x 0 given that y = 0 when x=π2. 

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