Mathematics Model Question 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.