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    Half-Yearly Examinations - Maths Paper-II

    Model Paper-I

    Part-A

    Time: 2 Hours Maximum Marks: 35

    Section-I

    Group-A

    (Geometry, Analytical Geometry, Statistics)

    1. Prove that the tangents at the ends of a diameter of a circle are parallel.

    2. Find the point on xaxis which is equidistant from (2,3) and (4,2)

    3. Find the equation of straight line passing through the points (4,7) and (1,5)

    4. Write the merits of Arithmetic Mean?

    Group-B

    (Matrices, Computing)

    5. Show that AB 0, BA=0, If..

    6. A matrix D has an inverse. D1= Find D.

    7. Write the characteristics of a computer.

    8. Define the i) Algorithem ii) Flow chart

    Section-II

    9. Find the distance between the centres of two cicles whose radii are 5 cm and 7 cm having

    three common tangents?

    3 41 2

    1 0B =

    0 a

    0 0A =

    1 0

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    11. Find the median of the following observations 1.8, 4.0, 2.7, 1.2 , 4.5, 2.3, 3.1 and 3.7 .

    12. Maximise the objective function at (0,120) and (80, 40).

    13. One end of the diameter of a circle is (2,3) and the centgre is (2,5). Find the co-ordinate

    of the other end of the diameter.

    14. Define programming language.

    Section-III

    Group-A

    15. State and prove Pythagoras Theorem.

    16. The point G(0,6) is the centroid of the triangle, two of whose vertices are A(4,4), B(6,12)

    Find the co-ordinates of the third vertex. Show that area of ABC = 3(area of AGB).

    17. Find the area of the triangle enclosed between the coordinate axes and the line passing

    through (8,3) and (4, 12).

    18. Marks scored by 100 students in a 25 marks unit test of mathematics is given below. Find

    the median.

    Marks 0-5 5-10 10-15 15-20 20-25

    Students 10 18 42 23 7

    Group-B

    (Matrices, Computing)

    19. Given that and (A+B)2 = A2+B2. Find a,b.

    20. Solve the following equations using matrix inversion method

    1 1 a 1 A = ,B =

    2 1 b 1

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    21. What are the different boxes used in a flow chart? Describe their functions in details?

    22. Write an algorithm and draw a flow chart to pick largest number of the three given number?

    Section-IV

    (Polynomials, Linear Programming)

    23. Draw a circumcircle to a ABC with measures AB= 4 cm, BC= 4 cm, and AC= 6 cm.

    24. Construct a triangle ABC in which BC= 5cm, A= 70 and median AD through A= 3.5cm.

    PART-B

    Marks : 30=5

    1. ABC DEF, m A + mB = 130 then f = . ( )

    A) 130 B) 140 C) 50 D) 40

    2. The line y= mx+c intersect the xaxis at the point . ( )

    A) (0,C) B) (C,0) C) (c/m, 0) D) (0, c/m)

    3. The slope of the line joining (4,6) and (2,5) is . ( )

    A) 6/5 B) 2/4 C) 5/6 D) 11/2

    4. The histogram consists of . ( )

    A) sectors B) triangles C) Squares D) rectangles

    5. The median of the scores 13, 23, 12, 18, 26, 19 and . ( )

    A)14 B) 26 C) 13 D) 18

    6. The arthmetic mean of a+2, a, a2 is . ( )

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    7. If and A= B then p and x are .( )

    A) p= 6, x= 2 B) p=2, x=6 C) p=3, x=4 D) p=4, x=3

    8. If then the order of AT is = . ( )

    A) 32 B) 22 C) 23 D) 33

    9. The father of computer . ( )

    A) Pascal B) Bill gates C) Charles Babbage D) Newin

    10. Vaccum tubes are used in generation of computers. ( )

    A) fourth B) First C) Second D) Third

    Answers : 1. C 2. C 3. D 4. D 5. D

    6. B 7. A 8. A 9. C 10. B

    II. Fill in the blanks with suitable words. Marks : 10=5

    11. Angle in a semi circle is .

    12. Basic proportionality theorem is also known as theorem.

    13. The slope of a line perpendicular to 2x+3y=4 is .

    14. If A.M. of 3, 5, 9, x, 11 is 7 then x = .

    15. Formula for calculation the median of frequency distribution is .

    16. The angle between the lines x2=0 and y+3=0 is .

    1 2 3A =

    4 5 6

    3 4 3 4 A = , B =

    6 p 2 x

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    18. , If A has not multiplicative inverse then x= .

    19., then AB = .

    20. Expand C.P.U. .

    Answers

    11. Right angle 12. Thales

    13. 3/2 14. 7

    15. x2+2x15=0 16. 90

    17. 45 18. a + b

    19. 1/13 20. 4

    III. Match the following. Marks : 10=5

    Group-A Group-B

    21. The height of the equilateral ( ) A) 2

    triangle of side 23 is

    22. If C= 90 in ABC and

    a =3, b=4, then C= ( ) B) yy1 = m(xx1)

    23. Slope and point form of ( ) C) a

    a line.

    24. The equation of y-axis is ( ) D) 3a

    25. A.M. of ad, a, a+d is ( ) E) 3

    [ ]5A , B x y2

    = =

    4 xA

    x 9

    =

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    G)

    H) 5

    Answers : 21. E 22. F 23. B 24. A 25. C

    Group-A Group-B

    26. ( ) A) 1

    27. ( ) B) 5

    28. If = 0 then a = ( ) C) A1.B1

    29. (AB)1 ( ) D)

    30. Computer ( ) E) cos

    F) B1.A1

    G) sin

    H) An electronic machine

    Answers : 26. G 27. A 28. B 29. F 30. H

    2a 56 3

    2Tan

    2sec 1sec

    x y 1a b

    + =

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    Key

    Section-I

    Group-A

    1. Prove that the tangents at the ends of a diameter of a circle are parallel.

    A. Given: Let 'O' be the centre of the circle and be a diameter. Let and be the two

    tangents drawn at A and B to the circle with centre 'O'

    R.T.P. //

    Proof: A = B = 90 1) (Tangent is perpendicular to the diameter at the point of contact)

    Let and be two lines and be a transversal then A and B = 90 + 90 = 180

    (since From 1)

    If two lines are cut off a transversal and a pair of interior

    angles So formed are supplementary then the two lines are

    parallel.

    //

    2. Find the point on x-axis which is equidistant from (2,3) and (4,2)?

    A. Let the required point be (x,0)

    Distance between (x,0) and (2,3) = Distance between (x,0) and (4,2)

    2x 8x 16 4= + +2x 4x 4 9 + +

    2(x 4) 4= +2(x 2) 9 +

    2(x 4) 4= +2(x 2) 9 +

    2 2(x 4) 0 ( 2)= + 2 2(x 2) (03) +

    BDHJJG

    ACHJJG

    ABHJJG

    BDHJJG

    ACHJJG

    BDHJJG

    ACHJJG

    BDHJJG

    ACHJJG

    AB

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    B D

    CA

    O

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    x24x+13 = x28x+20

    4x+13+8x20 = 0 4x7 = 0

    x = 7/4 The required point is (7/4, 0)

    3. Find the equation of straight line passing through the points (4,7) and (1,5)

    A. Equation of a line passing through two points is

    4. Write the merits of Arithmetic mean?

    A. Merits of Arithmetic mean:

    i) It is uniquely defined

    ii) It is based on all observations

    iii) It is easily understood

    iv) It is easy to compute

    Group-B

    5. Show that AB 0, BA=0, If..

    A.

    AB 0, BA=0

    1 0 0 0 0 0 0 0 0 0BA 0

    0 0 1 0 0 0 0 0 0 0+ +

    = = = = + + 0 0 1 0 0 0 0 0

    AB1 0 0 0 1 0 0 0

    + + = = + +

    0 0 1 0A , B

    1 0 0 0

    = =

    1 0B =

    0 a

    0 0A =

    1 0

    4x y 9 0 + y 7 4 x 16 + =

    y 7 4(x 4) + = 12y 7 + =

    4

    3(x 4)

    5 ( 7)y 7 (x 4)1 4

    + =

    121 1

    2 1

    y yy y (x x )

    x x

    =

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    6. A matrix D has an inverse. D1= Find D?

    A. Let D= D1= D. D1 = I

    2(3a+b=1) 1) 2(3c+d= 0) 3)

    4a+2b=0 2) 4c+2d = 1 4)

    substitute a=1 in eq-1) substitute c= 1/2 in eq (3)

    s(1) + b= 1 3(1/2) +d=0

    b= 13 = 2 d= 3/2

    D= =

    7. Write the characteristics of a computer?

    A. Characteristics of a computer-

    i) A computer can perform only those operation which are identified and concieved by a

    human being.

    1 -21/ 2 -3 / 2

    a bc d

    6c + 2d = 0

    4c + 2d = 1- - -

    2c = 1 c = 1/ 2

    6a + 2b = 2

    4a + 2b = 0- - -

    2a = 2 a = 1

    1 0

    0 1

    =

    3a + b 4a 2b

    3c d 4c 2d+

    + + 1 0a b 3 4 0 1c d 1 3

    =

    3 41 2

    a bc d

    3 41 2

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    iii) It can perform various logical operations.

    iv) It can perform million of computations and compile the result in a desired form in a few

    minutes

    8. Define the i) Algorithem ii) Flow chart

    A. Algotithm: An algorithm is a disign or a plan of obtaining a solution to a problem. It forms

    the central concept of the branch of computer science or informatics.

    Flow chart: A diagramatic or a pictoral representation of the sequence of steps for solving

    a problem is called flow chart.

    Section-II

    9. Find the distance between the centres of two cicles whose radii are 5 cm and 7 cm

    having three common tangents?

    A. If two circles have 3 common tangents, then they touch externally

    d= r1+r2 = 5+7 = 12 cms.

    10. Find the equation of the line passing through the point (5,7) and slope is 4.

    A. The given point (x1, y1) = (5,7)

    The given slope m = 4

    Equation of the line having slope m and passing through the point (x1, y1) is