SSC Mathematics Model Question-8

CLASS – X

SUB : MATHEMATICS MODEL TEST

Time:  3 hrs                                                                                                                               Full Marks : 100

[The figures in the right margin indicate full marks]

1.      If A = { 1, 2, 3} and B = {2, 4, 6}, C= { 1, 4, 7}. Show that (A È B) È C = (A È B) È C and              (A ÇB) Ç C = A Ç (B Ç C) .                                                                                                                                     4

Or,   If A = {a,  b}, B ={ 2, 3} and C = {3, 4}, then find A ×(B È C) and A × (B Ç C)

2.      Show that, if the square of an odd natural number is divided by 8 then in each case the remainder will be 1.                                                                                                                                                                        4

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

a)  If , then prove that

b)  If x = Ö3 + Ö2, then find the value of .

c)  Resolve into factors (any two) : 

         i)    ii)  (a – m) x² – (x – a) xy + (m – x)y².  iii)  x³ + 6x²y + 11xy² + 6y³.

d)  Find the H.C.F of  x² – y² – z²– 2yz, y² – z² –x² – 2xz and z² – x² – y²– 2xy

e)  A boat man can go x km against current in p hours. It takes q hours to cover the same distance infavour of current. Find the speed of current and boat.

4.      Simplify :                                                                                                  5

Or,   Simplify :

5.      Using the properties of proportion, show that if , then , a ¹ b.            5

Or,   Divide Tk. 674 in the ratio .

6.      Find the solution set :                                                                                   4

Or,   3 benches in a classroom remain vacant if 4 students sit in each bench. But 6 students are to remain standing if 3 students sit in each bench. What is the number of students in that class ?

7.      For , show that .                                                                                   4

Or,   Sketch the graph of the equation 4x + 5y = 20. Find the length of the segment of the graph cut by the axes.

8.      Solve : ax – cy = 0 ; ay – cx = a² – c².                                                                                                    4

Or,   The area of a rectangular field is 300 squares metres and its semi-perimeter is 10 metres more than a diagonal. Find the length and breadth of the rectangular field.

9.      If the sum of n-terms of the series 9 + 7 + 5 + . . . . . is – 144, then find the value of n.                      4

Or,   If the 5^th term of a geometric series is  and the 10^th term is  . Find the 3^rd term of the series.

10.    (a)   Prove that, the locus of a point equidistant from two intersecting straight lines is the bisectors of the internal angles between the two given straight line.                                                                              6

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

(b)    Prove that the sum of the two opposite angles of a quadrilateral inscribed in a circle is two right angles.                                                                                                                                                       6

Or,   Prove that the angle made by a tangent to a circle with any chord drawn from the point of contact is equal to any angle in the alternate segment of the circle.

11.    a)      In the DABC, AB > AC and the bisector AD of ÐA intersects BC at D. Prove that ÐADB is an obtuse angle.                                                                                                           4

Or,   O is a point inside a rectangular region ABCD. Prove that OA² + OC² = OB² + OD².                      

b)     If two chords AB and CD of a circle with centre O intersect in the interior point E of the circle, then prove that, ÐAEC = ½ (ÐBOD + ÐAOC)                                                                                           4

Or,   A and B are the centres of two circles and C is their point of contact. A straight line drawn through the point C intersects the circles at the points P and Q. Prove that AP || BQ.

12.     Construct a triangle when the base of triangle an acute angle adjacent to the base and difference of the other two sides are given. [Description and sign are essential]                                                                              5

Or,   The length of two line segments are a and b respectively. Find the line segment of length c, such that  ab = c². [ Description and sign are essential]

Or,   Draw a circle circumscribing a given triangle. [ Description and sign are essential]

13.    Construct a triangle when two angle adjacent to the base and the length of the perpendicular from the vertex to the base are given.[ Description and sign are essential]                                                                                 5

Or,   Draw inscribed and inscribed circles of a square. [ Description and sign are essential]

14.    a)      If sinA + cosA = a and secA + cosecA = b, then prove that b(a² – 1) = 2a.                                 4

Or,  , then find the value of

b)     Solve : (0⁰0)                                                                                  4

Or,  Solve : sinq + cosq = 1 when 0⁰ £ q £ 90⁰

c)  The length of the shadow of a minar is 240 metres when the angle of elevation of the sun is 60⁰. What is the height of the minar ?                                                                                                                              4

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

15.    Inside a square field, there is a road around it of width 4 metres. If the area of the road is 1 hectare, then what is the inner area excluding the road ? .                                                                                                  4

Or,   The diameter of a wheel is 4.2 metres. How many times will the wheel move to describe a distance of 330 metres. ?

16.    The length, breadth and height of a rectangular parallelopiped are in the ratio 21 : 16 : 12 and its diagonal is 87 cm. Find the area of the whole surface of the solid.                                                                             4

Or,   The area of the curved surface of cylinder is 100 sq cm and its volume is 150 cubic cm. Find the height and the radius of the base of the cylinder.

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