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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

    Write the distance of the point (3, –5, 12) from x-axis. 

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  • Q2

    Evaluate :

    02πcos5x dx 

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  • Q3

    For what value of 'k' is the function fx=sin 5x3x+cos x, if x  0k,if x = 0 is continuous at x = 0? 

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  • Q4

    If |A| = 3 and A-1=3-1-5323, then write the adj A. 

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  • Q5

    Find : dx3-2x-x2 

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  • Q6

    A company produces two types of goods A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of golds while that of type B requires 1 g of silver and 2 g of gold. The company can procure a maximum of 9 g of silver and 8 g of gold. If each unit of type A brings a profit of Rs 40 and that of type B Rs 50, formulate LPP to maximize profit. 

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  • Q7

    If P(A) = 0·4, P(B) = p, P(A ⋃ B) = 0·6 and A and B are given to be independent events, find the value of 'p'. 

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  • Q8

    A line passes through the point with position vector 2i^3j^+4k^ and is perpendicular to the plane r·3i^+4j^-5k^=7. Find the equation of the line in cartesian and vector forms. 

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  • Q9

    Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all xπ4,π2. 

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  • Q10

    Find dydx at t=2π3 when x = 10 (t – sin t) and y = 12 (1 – cos t). 

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  • Q11

    If A and B are square matrices of order 3 such that |A| = –1, |B| = 3, then find the value of |2AB|. 

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  • Q12

    The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. Use π=227 

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  • Q13

    There are 4 cards numbered 1 to 4, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X. 

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  • Q14

    If a=2i^+j^-k^, b=4i^-7j^+k^, find a vector c such that a×c=b and a·c=6. 

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  • Q15

    Evaluate : -21x3-xdx

    OR

    Find :  e2x sin 3x+1 dx 

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  • Q16

    In a shop X, 30 tins of pure ghee and 40 tins of adulterated ghee which look alike, are kept for sale while in shop Y, similar 50 tins of pure ghee and 60 tins of adulterated ghee are there. One tin of ghee is purchased from one of the randomly selected shops and is found to be adulterated. Find the probability that it is purchased from shop Y. What measures should be taken to stop adulteration? 

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  • Q17

    Find : ex2+ex4+e2xdx 

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  • Q18

    If xy = e(xy), then show that dydx=yx-1xy+1.
     

    OR

    If logy = tan–1 x, then show that 1+x2d2ydx2+2x-1dydx=0. 

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  • Q19

    Using properties of determinants show that

    111+x11+y11+z11=xyz+yz+zx+xy.

     

    OR

    Find matrix X so that X123456=-7-8-9246. 

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  • Q20

    Solve the following LPP graphically :
    Maximise Z = 105x + 90y
    subject to the constraints
    x + y ≤ 50
    2x + y ≤ 80
    x ≥ 0, y ≥ 0. 

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  • Q21

    Find the general solution of the differential equation x cos yxdydx=y cosyx+x. 

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  • Q22

    Prove that:

    tan-11+x2+1-x21+x2-1-x2=π4+12 cos-1x2;1<x<1 

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  • Q23

    Using vectors, find the area of triangle ABC, with vertices A (1, 2, 3), B (2, –1, 4) and C (4, 5, –1). 

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  • Q24

    Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices area A(1, 2), B (2, 0) and C (4, 3).


    OR

    Using integration, find the area of the region {(x, y) : x2 + y2 ≤ 1 ≤ x + y}. 

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  • Q25

    A wire of length 34 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a rectangle whose length is twice its breadth. What should be the lengths of the two pieces, so that the combined area of the square and the rectangle is minimum? 

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  • Q26

    Let A = ℝ − {3}, B = ℝ − {1}. Let f : A → B be defined by fx=x-2x-3,  x  A. Show that f is bijective. Also, find
    (i) x, if f−1(x) = 4
    (ii) f−1(7)


    OR

    Let A = ℝ × ℝ and let * be a binary operation on A defined by (a, b) * (c, d) = (ad + bc, bd) for all (a, b), (c, d) ∈ ℝ × ℝ.
    (i) Show that * is commutative on A.
    (ii) Show that * is associative on A.
    (iii) Find the identity element of * in A. 

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  • Q27

    Find the vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0. Hence find whether the plane thus obtained contains the line x+25=y-34=z5 or not.

    OR
     
    Find the image P' of the point P having position vector i^+3j^+4k^ in the plane r · 2i^-j^+k^+3=0. Hence find the length of PP'. 

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  • Q28

    If A=1-2  02   1  30-2  1, find A–1 and hence solve the system of equations x – 2y = 10, 2x + y + 3z = 8 and –2y + z = 7. 

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  • Q29

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

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