Sample General Mathematics Question 4 for Class-X

Sylhet Cadet College

Pre-test Examination 2010

Mathematics (Compulsory)

Class X

Time: 3 hours

Full Marks - 75

 

1.      Determine the set of natural numbers which divides 346 and 556 with 31 as remainder in each case.         4

Or,    If   (x - a, y + 2a) = (y - 2a, 2x + a)  then find  (x, y).

2.      Show that, if the square of an od natural number is divided by 8, then each case the remainder will be 1.

3.      Answer any three of the followings:                                                                                   5×3=15

         i)       If   then find the value of  

         ii)      If x + y = a,  x² + y² = b² and  x³ + y³ =  c³, then prove that, a³ + 2c³ = 3ab².

         iii)     Resolve into factors:  a)   x³ -7xy² -6y³ ,   b)  (a-m)x² – (x-a)xy + (m-x)y² .

         iv)     Find the H. C. F.  of  x³ – x -24   and  x³ - 6x² + 18x – 27

         v)      Salary of Motin is x% higher than that of Jalil. As a result Jalil’s salary is y% less than that of Motin. Express y in terms of x.

4.      Evaluate:                                             5

Or,    Simplify:        

5.      Using the properties of proportion, show that, if , then show that,       where,  .                                                                                                                                5

Or,    If the length of each side of a square is increased by 20% then, what is the percentage of increase in the area enclosed by the square?

6.      Solve:              .                                                                                          4

Or,    The hypotenuse of a right-angled triangle is  13 cm. and its perimeter is 30cm. Find its area.

7.      If a α b and  b α c, then prove that,                                                                4

Or,    Sketch the graph of the equation    x² + y² - 8x -10y – 103 = 0

8.      Solve by determinant method:         ;   and    .                                  4

Or,    Twice the breadth of a rectangle is 10 meter than its length. If the area of the region enclosed by the rectangle is 600sq. meter, then find its length.

9.      If 15 terms of an arithmetic series is 60, then what is the sum of its first 29 terms.                     5

Or,    For all n€N, prove that, 1² + 2² + 3² +------------- + n² =   

10.    State and prove that the Pythagoras theorem by Euclid Methods.                                               6

Or,    Prove that, equal chords of a circle are equidistant from the centre.

11.    From an external point of a circle, if the two tangents are drawn, then prove that the distances of the points of contact from that point are equal.                                                                                   6

Or,    If the line segment joining the two points subtends an equal angles at two other points on the same side of it, then prove that the four points are concyclic.

                                                                                                                                                                           P.T.O

::  2  ::

12.    Prove that, the diagonals of a rhombus bisect each other at right-angles.

Or,    If X and Y are the middle points respectively of the sides AB and AC of the triangle ABC, then prove that, =.                                                                           4

13.    In the triangle ABC, if the bisectors of  and  meet at the point P and their exterior bisectors at Q, then show that the four points B, P, C, Q are concyclic.                                                         4

Or,    Two circles touches externally at the pint A. The tangent T, T´ touches the circles at P and Q. Prove that, .

14.    Answer any one of the following:                                                                                         5×1=5

         a)      Draw a direct common tangent to the two circles of unequal radii. (Sign of the drawing and description are required.)

         b)      Construct a trapezium when two parallel sides and the adjacent angles to the greater side are given. (Sign of the drawing and description are required.)

         c)      The base of triangle an acute angle adjacent to the base and the difference of the other two sides are given. (Sign of the drawing and description are required.) 

15.    Draw a parallelogram such that its one angle is given and the region bounded by it is equal to the given quadrilateral region. (sign of the drawing and description are required.)                            5

Or,    Draw an inscribed and an escribed circles of asquare. (sign of the drawing and description are required.)

16.(a)         If tanA + sinA = m and tanA – sinB = n, then prove that  m² – n²  =                      4

Or,    If tanA =, then  find the value of   .

b)      Find the value of tan30º by geometrical method.                                                                          4

Or,    Solve:          when

c)      From a Helicopter above a point between the two kilometer posts, the angle of depression of the two posts are 30° and 60° respectively. Find the height of the Helicopter.

Or,    A man standing on the bank of a river observes that the angle of elevation of a tower exactly opposite to him on the other bank is 60º. On moving 25 meters in the backward direction he observed that, the angle of elevation of the tower is 30º. Find the height of the tower.

17.    The perimeter of a square region is equal to the perimeter of a rectangular region. The length of the rectangular region is three times its breadth and its area is 768sq.meter. How many stones each of 40cm. square is required to cover the square.  

Or,    The length of the sides of a parallelogram are 12 meters and 8 meters respectively. If one diagonal is 10 meters, then find the other diagonal.

18.    The outer measurements of a rectangular box are 8cm., 6cm. and 4cm. respectively. If the area of the whole inner surface is 88sq cm. then find the thickness of the box.

Or,    The inner and outer diameters of an iron pipe are 12cm and 14cm and its height is 5meters. What is the weight of the iron contained in the pipe, if the weight of  1c.c. of iron is 7.2gm. 

* * *