Mathematics Model Question for Class-X

Class X

Subject : Higher Mathematics

Time: 3  hrs.                                                                                                                      Full Marks:  75

1.         If  A = { c : d} B = { 4, 5} and C = { 6, 7 },  then show that  A ´(BUC) = ( A´B) U ( A´C).             4

Or,       For any sets of A, B, C  show that AÇ( BUC) = (AÇB) U (AÇC)

2.         Answer any two of the following questions:                                                                                3x2=6

            a)         Resolve into factors : 18x³ + 15x² - x -2

            b)         If  ¹0 then show that ( a+b+c) (x+y+z) = ax + by+cz:

            c)         Resolve into partial fractions: .

3.         Use the method of Mathematical Induction to show that for all n Є N, 1²+2²+3²+ ..............+n² =                                                                                                                                                       4

Or,       If S = { n: nÎN and 5ⁿ – 2ⁿ is divisible by 3 }, then show that S = N.

4.         If  y = (a+b) + (a –b) and a²-b²=c³, then show that y³ -3cy -2a =0                                      5

Or,       If    , then show that a^a b^b c^c = 1

5.         Find the domain of the function F (x) =  and determine whether the function is one one or not.   4

 Or,      Sketch the graph of the relation  S = { (x, y): x² +(y-1)² =16}and determine the graph whether the relation is a function.

6.         Solve:   6 +  5                                                                                                           4

Or,       Solve and show the solution set on the number line:  .

7.         Solve : x² – xy = 14,  y² + xy = 60                                                                                                        4

Or,       Solve : 8y^x – y^2x  = 16  

                                     2^x = y²

8.         Impose a condition on x under which the infinite series, +........... ( upto infinits) will have a sum and find the sum.                                                                                                                4

9.         Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares on

            corresponding sides.                                                                                                                             6

Or,       Prove that the circum-centre, the centroid and the orthocentre of any triangles are collinear.

10.       The medians of a triangle DABC meet at G. Prove that AB² + BC² +CA² =3 ( GA²+GB²+GC²)           4

Or,       ABC is an isosceles triangle and AD is perpendicular to BC. If R be the circumradius of the

            triangle, then prove that AB² = 2R. AD. 

11.       Construct a triangle having given the base, the vertical angle and the sum of the other two sides.

            ( sign of construction and description are essential)                                                                                    5

Or,       Construct a circle which touches a given straight line at a given point in it and passes through another

            given point outside that line. ( sign of construction and description are essential.)

12.       If  a , b,  c are the position vectors of A, B, C respectively and if the point C divides AB in the ratio

            m:n internally, then prove that  c =                                                                                                  4

Or,       If  a , b , c,  d are the position vectors respectively of the points A, B, C D. then show that ABCD

            will be a parallelogram if and only if b – a = c – d .

13.       A right circular cone, a semi- sphere and a cylinder of equal heights stand on equal bases. Show      4

            that their volumes are in the ratio 1: 2: 3:.

Or,       Find the length of the diagoanl and the volume of the cube of which the length of the diagonal

            of a face is 8cm.

14.       Answer any three of the following questions :                                                                 3x4=12

            a)         What is Radian? Prove that, Radian is a constant angle.

            b)         If a cos- b sin = c, show that a sin + b cos = ±

            c)         Solve : 5 cosec²x – 7 cotx cosecx – 2 = 0 When  0⁰≤x  ≤360⁰.

            d)         If tan  =  and cos is negative then find out the value of  

            e)         A boy running along a circular track at the rate of 5 km. per hour covers an arc in 36 seconds

            which subtends an angle 56⁰ at the centre. Find the diameter of the circle.

15.       Find the standard deviation from the following frequency distribution table .                                5

x	0	1	2	3	4	5	6	7
f	5	10	15	18	25	19	11	6

Or,

            Find the arithmetic mean from the frequency distribution table of the marks obtained in mathematics in    an examination of 50 marks.

Obtained marks	5	10	15	20	25	30	35	40	45	50
Number of students	5	15	20	25	30	35	45	15	6	4