Sample General Mathematics Question 8 for Class-X

Cadet College

Subject Code : 126

2nd Term end Examination-2008

Class-IX

Subject- Higher Mathematics

Time:   2-30 hrs.                                                                                                                                  Marks: 75

Algebra- 28

Answer any seven of the following questions.                                                                        4x 7 = 28

1.         Demonstrate with the help of Venn diagrams that ( AUB)(BUA) = (AUB) \ AÇB).

2.         If  A = { 1, 2 } and B = { 2, 5 }, then show that P(A) Ç P (B) = P ( A ÇB).

3.         Resolve into factors:     a( b - c)³ + b ( c - a )³ + c ( a - b)³.

4.         If    +    +   =   , then show that  bc+ ca + ab =0  or a = b = c .

5.         Simplify :         +   +  +

6.         Resolve into partial fractions:    

7.         If  S = { x: x ÎR and x² +1 =0 }, then find S² = R/S. What can you say about S ?

8.         If   ¹ 0, then show that ( a+b+c )( x + y + z) = ax + by + cz.

Geometry- 30

Answer any five of the following questions:                                                                                      6x5 =30

9.         The diagonals of the rectangle ABCD intersect at 0. It p is any point inside the rectangle, then prove        that PA² + PB² +PC² +PD² = AC² + 4PO²

10.       In the triangle ABC, AB = AC and AC is produced to D such that AC = AD, prove that

            BD² = 2BC²+AC².

11.       The madians of a DABC meet at G. Prove that AB² + BC² +CA² = 3 ( GA²+GB²+ GC²).

12.       To construct a triangle having given the base, the vertical angle and the difference of the other two

            two side. [ Description and tracing are required.]

 

13.       To divide a given straight line internally such that the rectangle contained by the whole and the one         part may be equal to the square on the other part. [ Tracing and description are required]

14.       To construct a circle which touches a given, straight line at a given point in it and passes through another given point outside that line. [ Tracing and description are required.]

Trigonometry - 12

15.       Define radian. Prove that  radian is a constant anlge.

16.       An angle is denoted by D⁰ and R^c in the sexagesimal and circular systems respectively. Prove that

           

17.       The angles of a triangle are in the ratio 2:5:3:  find the circular and sexagesimal measure of the largest             angle.

18.       A wheel revolues 20 times to travel a distance of 88 km, what is the radius of the wheel?

Sample General Mathematics Question 7 for Class-X

Mymensingh Girls' Cadet College

Subject Code : 126

Pre Test Examination- 2008

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

Sample General Mathematics Question 6 for Class-X

Class – X

Subject: General Mathematics

Time – 3 hours                                                                                                                        Full Marks – 100

Algebra

1. If A, B and C are the sets of the prime factors of 60, 70 and 80 respectively, determine A, B and C and show their relationship in a Venn diagram.                                                                                                              4

or

If x is any integer, A = {x: x² + x – 6 = 0} and B = {x: x² – 5x + 6 = 0}, then find AB and BA.

2. Find two rational and two irrational numbers between 0.10 and 0.11.                                                      4

3. Answer any three (03) of the following:                                                                                       35=15

(a) Simplify: (a + b)⁶– (a – b)⁶ – 12ab(a² – b²)².

(b) If p + = 1, find the value of (p + 2)³ + .

(c) If x = + 2, find the value of x² – .

(d) Resolve into factors: (i) (a + 1)x² + a²xy + (a – 1)y²         (ii) a⁸b⁸ + a⁴b⁴ + 1

(e) If ax² + bx + c is divisible by (x – p); find the remainder.

4. Simplify: x – {x^–1 + (y^–1 – x)^–1}^–1                                                                                                    5

or,       Simplify: 7log10 + 3log81 + 2log24 – 7log9 – 2log25 – 3log80.

5.         Three numbers are in continued proportion. The sum of the numbers is 21 and product of the numbers      is 64. Find the numbers.                                                                                                                                                 5

or, If a : b = b : c; establish a² b² c²= a³ + b³ + c³.

6. Solve:                                                                                         4

or, The digit in the tens place of a number consisting of two digits is twice the digit in the unit place. Show that the number is seven times the sum of the digits.

7. If f(x) =; what is the value of ?                                                                                             4

or, Sketch the graph of the equation (x – 3)² + (y + 5)² – 81 = 0.

8. Solve:and                                                                                                                        4

or, Eight years ago, the age of the father was eight times the age of the son. After ten years, the age of the father will be twice the age of the son. What are their present ages?

9. Find the 9^th term of the series 16 + 8 + 4 + … ….                                                                                     5

or, In a certain arithmetic series, if first three terms are 2x + 1, 3x and 4x – 1, what is the 9^th term?

Geometry

10. Answer any two (02) of the following:                                                                                     26=12

(a) Prove that, the sum of three angles of a triangle is equal to two right angles.

(b) Prove that, the locus of a point equidistant from the two fixed points is the perpendicular bisector of the line joining those points.

(c) ABC is a circle and O is a point outside of it. Two tangents OA and OB are drawn from O to ABC. Prove that, OA = OB.

11. Answer any two (02) of the following:                                                                                      24=08

(a) Determine the locus of a point equidistant from two parallel straight lines.

(b) Prove that, the diagonals of a parallelogram divide the parallelogram region into four equal triangular regions.

(c) Prove that, if two chords of a circle bisect each other, their point of intersection is the centre of the circle.

12. Answer any one (01) of the following:                                                                                                  5

(a) The diagonal of a square is x cm. Construct it. (Sign of drawing and construction is must)

(b) The lengths of three line segments are x, y and z cm. Determine a line segment p such that, x : y = z : p (Sign of drawing and construction is must)

13. Answer any one (01) of the following:                                                                                                  5

(a) Construct a triangle when two angles adjacent to the base and the length of the perpendicular from the vertex to the base are given. (Sign of drawing and construction is must)

(b) Determine the centre of a circle. (Sign of drawing and construction is must)

Trigonometry

14.       Prove geometrically that, sin²θ + cos² θ = 1, where θ is an acute angle.                                            4

or,       prove that, .

15.       If tanA = ; find the value of .                                                                             4

or,       A pole of 48 meters long breaks such that the two parts are not completely separated and the upper part             makes an angle 30⁰ with the ground. At what height did the pole break?

16.       The angle of elevation of the top of a tree to a point on the ground 60 meters form the foot of a tall tree   is 45^o, find the height of the tree.                                                                                                           4

or,        The angle of elevation of a point of the roof of a building is 45⁰ to a point on the ground on moving 40             meters towards the building, the angle of elevation becomes 60⁰ find the height of building.

Mensuration

17.       The perimeter of a rhombus is 360 cm and one of its diagonals is 27 cm. Find its other diagonal and             area.                                                                                                                                                     4

or,       An arc of the circle subtends an angle 30^o at the centre. If the diameter of the circle is 64 cm, find the             length of the arc.

18.         The volume of a rectangular parallelepiped is 220 cubic meters. If its diagonal is 15 meters and length      is 11 meters, find its breadth and height.                                                                                               4

or,       A metallic solid sphere of diameter 6 cm is melted and formed into a solid cylinder rod of radius 6 cm.             Find the length of the rod.