**Sample Paper -2008**

**Class – X**

**Subject-Mathematics**

Time-3hrs M.M-80

__SECTION – A__

- Write a quadratic equation whose roots are 3+Ö3 and 3-Ö3.
- Show that any positive odd integer is of the form 4q+1 or 4+3, where q is some integer.
- If tanB = ¾, and A+B = 90°, then find the value of cotA.
- If the zeros of the polynomial f(x) =x
^{3}-3x^{2}+x+1 are (a-b) , a and (a+b) find the value of and b. - A cylinder, a cone and a hemi-sphere have equal base and same height. What is the ratio of their volumes?
- Give an example of polynomial p(x), g(x), q(x) and r(x), satisfying

P(x) = g(x).q(x) + r(x), deg r(x) = 0

- A die is thrown once. What is the probability of getting an even prime number?
- Find the 20
^{th}term of the sequence -2, 0, 2, 4 ------------. - Find the perimeter of the sector whose base radius is 14 cm and central angle is 120°.
- For what value of ‘k’ the following pair of linear equations has infinitely many solutions?

10x + 5y – (k-5) = 0 and 20x + 10y – k = 0.

# SECTION – B

- Express sin 52° + cos 67° in terms of trigometric ratios of angles between 0° and 45°. ‘OR’

Sin (A+B) =1/2 and cos (A+B) =1/2, 0°°, A>B, find A and B.

- How many three digit numbers are divisible by 7?
- Find the values of y for which the distance between the points A (-3, 2) and B (4, y) is 7.
- ABC is a triangle right angled at A and AD^BC. Show that AC
^{2}= BC.CD. - Two dice are thrown once. What is the probability that the sum of the two numbers appearing on the top of the dice is less than or equal to 12?

__SECTION – C__

- While covering a distance of 30km Ajeet takes 2hrs more than Amit. If Ajeet doubles his pace, he would take1hour less than Amit. Find the ratio of their walking.

‘OR’

A number consists of two digits, the difference of digits is 3.If 4 times the number is equal to 7 times the number obtained on reversing digits. Find the number.

- Find the zeroes of the quadratic equation:

f(x)= abx^{2}+(b^{2}-ac)x-bc. - The angles of a quadrilateral in AP whose common difference is 10
^{0 }.Find the angles.

‘OR’

Which term of the AP: 114, 109, 104, --------- is the first negative term?

- Draw the graph of the following pair of linear equations: x + 3y = 6 and 2x – 3y = 12 and find the area of the region bounded by x = 0, y = 0 and 2x – 3y = 12.
- Prove that
__sinA + cosA__+__sinA – cosA__=__2__

SinA – cosA sinA + cosA sin^{2}A – cos^{2}A

OR

If 2 tan A = 1, find the value of __3 Cos A + 2 Sin A__

2 Cos A – Sin A

- For what value of ‘k’ the points A (1, 5), B (k, 1) and C (4, 11) are collinear?
- Draw a tr ABC in which BC=6cm , AB=5cm and ÐABC=60 then construct a tr similar to the given tr whose each corresponding side is 3/4
^{th}of that of tr ABC. - A train overtakes two persons who are walking at a speed of 2 km/hr and 4 km/hr respectively in he same direction in which the train is going , the train overtakes them in 9 seconds and 10 seconds respectively. Find the length and speed of the train.
- Find the ratio in which the line segment joining the points A (3, -6) and B (5, 3) is divided by x-axis.
- Water flows at the rate of 10 m/min through a circular pipe of 5 mm diameter. How long would it take to fill a conical vessel whose diameter at the base is 40 cm and depth 24 cm?
- Prove that in a right triangle the square of the hypotenuse is equal to the sum of square of the other two sides. Using the result of this theorem prove that the sum of squares on the sides of a rhombus is equal to the sum of squares on its diagonals.
- If the angle of elevation of a cloud from a point h metres above a lake is α and the angle of depression of its reflection in the lake is β. Prove that the distance of the cloud from the point of observation is 2h sec α/((tan β-tan α).

‘OR’

A balloon moving in a straight line passes vertically above points A and B on a horizontal plane 1000 m apart. When above A it has an altitude of 60^{0 }as seen from B and when at B it has an altitude of 45^{0} as seen from A. Find the distance from A of the point where the balloon will touch the ground.

- The height of a cone is 30 cm . A small cone is cut off at the top by the plane parallel to the base. If its volume be 1/27 of the volume of the given cone , at what height above the base, the section has been made.
- An employee finds that if he increases the wages of each worker by Rs50 and the employs one worker less he reduces his weekly wages bill by Rs230 from Rs6800 to Rs6570. taking the original weekly wages be x obtain eq in x and find the weekly wages of each worker.
- Find the median from the following table:

Marks | No. of Students |

Below 10 | 15 |

Below 20 | 35 |

Below 30 | 60 |

Below 40 | 84 |

Below 50 | 94 |

Below 60 | 127 |

Below 70 | 198 |

Below 80 | 249 |