10th Sample Paper

MATHEMATICS

Time allowed: 2 hours                                                                            

SECTION – A

                                                    2      3     7       6      9      11

1.     Solve for x and y       : ¾ + ¾ = ¾ ;  ¾ + ¾ = ¾ , (x ¹ 0, y ¹ 0)

                                                 y      x     xy     y       x     xy

2.     If x² - 1 is HCF of the polynomials p(x) = (x² + 3x + 2) (x² - ax + 1) and q(x) = (x² - 3x + 2)(x² + bx + 1), find the values of a and b.

                                                                        3                    -              3           .   

3.           Simplify as a rational expression:     x³ +  x² + x + 1            x³ -  x² + x - 1

4.            Using factorisation, solve the following quadratic equation:

                                 x + 1     +             x -2      = 3 (x ¹ 1, -2)

                                 x -1                     x + 2              

5.     Determine the A.P. whose third term is 16 and the difference of 5 th term from 7^th term is 12.

6.     Find K if the given value of x is the Kth term of the given A.P.:

                    1              1

               5 ¾, 11, 16 ¾  , 22, …..; x = 550

                    2              2

7.       Find a₃₀ - a₂₀ for A.P. -9, -14, -19, -24, ……

8.       A loan of Rs. 12,750 is to be paid back in two equal half-yearly installments. If the interest is compounded half-yearly at 8% p.a., find the amount of each instalment.

9.     Two right-angled triangle ABC and DBC are on the same hypotenuse BC. Side AC and BD intersect at P. Prove that AP ´ PC = BP ´ PD.

SECTION B:

10.  Draw the graphs of the equations 2x - y = -8, 8x + 3y = 24. Determine the vertices of the triangle by the lines representing these equations and the x-axis. Shade the triangular region formed. Also find the area.

11.    A motor boat, whose speed is 9 km/hr in still water, goes 12 km downstream and comes back in a total time of 3 hours. Find the speed of the stream.

12  Draw a quadrilateral ABCD, in which AB = 3.2 cm, Ð B = 120°, Bc = 4 cm, CD = 3.8 cm and DA = 3 cm. Draw another quadrilateral AB¢C¢D¢ similar to quadrilateral ABCD such that AB¢ = 2.5 cm.

13 Find  the value of p which will make the product of 2p- 5 and p - 4 equal in value of p.

14.  If cos q - sin q = 1 , show that cos q + sin q = 1 or -1.

15 The line segment joining the points (-6, 8) and (8, -6) is divided into four equal parts. Find the coordinates of the points of section.

16  In D ABC, ÐC is a right angle. A semicircle is drawn on AB as diameter. P is any point on AC produced when joined, BP meets the semicircle in point D. Prove that AB² = AC.AP + BD . BP.

17A well with 10 m inside diameter is dug 14 m deep. Earth taken out of it and spread all around to a width of 5 m to form an embankment. Find the height of embankment.

18 Cards marked with numbers 1, 2, 3 …. , 50 are placed in a box and mixed thoroughly. One card is drawn from the box. Find the probability that number on the card is  i) divisible by 7   (ii) a number which is perfect square.

SECTION C:

19    The annual income of Mr Sanjay (excluding  HRA) is Rs. 1, 90,000. He  contributes Rs 4,500 per month in his provident fund and pays an annual premium  Rs 12,000 towards his insurance policy. Calculate the income tax including surcharge paid by him in the last month of the year if his earlier deduction as income tax for the first 11 months were at the rate of Rs 1, 000 per month. Assume the following for calculating income tax:

20   If a chord is drawn through the point of contact of a tangent to a circle, then the angles which this chord makes with the given tangent are equal respectively to the angles formed in the corresponding alternate segments. Prove it. Also use the above result to find Ð ABC when PQ is a tangent to the circle at A, ÐBAQ = 50° and ÐBAC = 35°.

21  If PAB is a secant to a circle intersecting the circle at A and B and PT is a tangent segment then PA ´ PB  = PT². Prove it.

22.  A bucket has top and bottom diameter of 40 cm and 20 cm respectively. Find the volume of the bucket if its depth is 12 cm. Also find the cost of tin sheet used for making the bucket at the rate of Rs 1.20 per dm².

23 iron pillar has some part in the form of a right circular cylinder and remaining in the form of a right circular cone. The radius of the base of each of the cone and cylinder is 8 cm. The cylinder part is 240 cm high and conical part is 36 cm high. Find the weight of the pillar if 1 cu cm of iron weights 7.8 gram. 

24.    Prove that the degree measure of an arc of a circle is twice the angle subtended by it at any point of the alternate segment of the circle with respect to the arc.