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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

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

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

    VIEW SOLUTION

  • Q2

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

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  • Q3

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

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  • Q4

    Find :

    sin2x-cos2xsin x cos x dx 

    VIEW SOLUTION

  • Q5

    Find :

    dx5-8x-x2 

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  • Q6

    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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  • Q7

    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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  • Q8

    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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  • Q9

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

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  • Q10

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

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  • Q11

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

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  • Q12

    The volume of a sphere is increasing at the rate of 8 cm3/s. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm. 

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  • Q13

    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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  • Q14

    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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  • Q15

    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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  • Q16

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

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  • Q17

    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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  • Q18

    If xy + yx = ab, then find dydx.
     

    OR

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

    VIEW SOLUTION

  • Q19

    Evaluate :

    0πx tan xsec x+tan xdx

    OR

    Evaluate :

    14x-1+x-2+x-4 dx 

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  • Q20

    Solve the following linear programming problem graphically :
    Maximise Z = 7x + 10y
    subject to the constraints
    4x + 6y ≤ 240
    6x + 3y ≤ 240
    x ≥ 10
    x ≥ 0, y ≥ 0 

    VIEW SOLUTION

  • Q21

    Find :

    exdxex-12ex+2 

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  • Q22

    If a=2i^-j^-2k^ and b=7i^+2j^-3k^ , then express b in the form of b=b1+b2, where b1 is parallel to a and b2 is perpendicular to a. 

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  • Q23

    Find the general solution of the differential equation dydx-y=sin x. 

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  • Q24

    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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  • Q25

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

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  • Q26

    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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  • Q27

    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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  • Q28

    If A=2-3    53   2-41  1-2, then find A–1 and hence solve the system of linear equations 2x-3y+5z=11, 3x+2y-4z=-5 and x+y-2z=-3. 

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  • Q29

    A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening. 

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