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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 for any 2 × 2 square matrix A, A(adj A) = 8008, then write the value of |A|. 

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  • Q2

    Determine the value of 'k' for which the following function is continuous at x = 3:

    fx=x+32-36x-3,x3k             ,x=3 

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  • Q3

    Find :

    sin2x-cos2xsin x cos x dx 

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  • Q4

    Find the distance between the planes 2x – y + 2z = 5 and 5x – 2.5y + 5z = 20. 

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  • Q5

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

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  • Q6

    Find the value of c in Rolle's theorem for the function f(x) = x3 – 3x in  -3, 0. 

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  • Q7

    The volume of a cube is increasing at the rate of 9 cm3/s. How fast is its surface area increasing when the length of an edge is 10 cm? 

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  • Q8

    Show that the function f(x) = x3 – 3x2 + 6x – 100 is increasing on ℝ. 

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  • Q9

    The x-coordinate of a point on the line joining the points P(2, 2, 1) and Q(5, 1, –2) is 4. Find its z-coordinate. 

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  • Q10

    A die, whose faces are marked 1, 2, 3, in red and 4, 5, 6 in green, is tossed. Let A be the event "number obtained is even" and B be the event "number obtained is red". Find if A and B are independent events. 

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  • Q11

    Two tailors, A and B earn Rs 300 and Rs 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP. 

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  • Q12

    Find :

    dx5-8x-x2 

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  • Q13

    If tan-1 x-3x-4+tan-1 x+3x+4=π4, then find the value of x. 

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  • Q14

    Using properties of determinants, prove that

    a2+2a2a+112a+1a+21331=a-13

    OR

    Find matrix A such that

    2-110-34A=-1-81-2922 

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  • Q15

    If xy + yx = ab, then find dydx.
     

    OR

    If ey(x + 1) = 1, then show that d2ydx2=dydx2. 

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  • Q16

    Find:                             

    cos θ4+sin2θ5-4cos2θ 

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  • Q17

    Evaluate :

    0πx tan xsec x+tan xdx

    OR

    Evaluate :

    14x-1+x-2+x-4 dx 

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  • Q18

    Solve the differential equation (tan–1 x – y) dx = (1 + x2) dy.   

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  • Q19

    Show that the points A, B, C with position vectors 2i^j^+k^, i^3j^-5k^ and 3i^4j^-4k^ respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle. 

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  • Q20

    Find the value of λ, if four points with position vectors 3i^+6j^+9k^, i^+2j^+3k^, 2i^+3j^+k^ and 4i^+6j^+λk^ are coplanar. 

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  • Q21

    There are 4 cards numbered 1, 3, 5 and 7, 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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  • Q22

    Of the students in a school, it is known that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chosen at random from the school and he was found to have an A grade. What is the probability that the student has 100% attendance? Is regularity required only in school? Justify your answer. 

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  • Q23

    Maximise Z = x + 2y

    subject to the constraints

    x + 2y ≥ 100

    2x – y ≤ 0

    2x + y ≤ 200

    x, y ≥ 0

    Solve the above LPP graphically. 

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  • Q24

    Determine the product -4   4   4-7   1   3   5-3-1  1-1   11-2-22   1   3 and use it to solve the system of equations x – y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1. 

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  • Q25

    Consider f:--43R-43 given by f(x)=4x+33x+4. Show that f is bijective. Find the inverse of f and hence find f–1 (0) and x such that f–1 (x) = 2.

    OR
     
    Let A=× and let * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a, b), (c, d) ∊ A. Determine, whether * is commutative and associative. Then, with respect to * on A
    (i) Find the identity element in A.
    (ii) Find the invertible elements of A. 

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  • Q26

    Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube. 

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  • Q27

    Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices are A(4, 1), B(6, 6) and C(8, 4).


    OR
     
    Find the area enclosed between the parabola 4y = 3x2 and the straight line 3x – 2y + 12 = 0. 

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  • Q28

    Find the particular solution of the differential equation x-ydydx=x+2y, given that y = 0 when x = 1. 

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  • Q29

    Find the coordinates of the point where the line through the points (3, –4, –5) and (2, –3, 1), crosses the plane determined by the points (1, 2, 3), (4, 2, –3) and (0, 4, 3).

    OR
     
    A variable plane which remains at a constant distance 3p from the origin cuts the coordinate axes at A, B, C. Show that the locus of the centroid of triangle ABC is 1x2+1y2+1z2=1p2. 

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