**VIKAS PRE UNIVERSITY COLLEGE, MANGALURU**

**CBSE CLASS – 10**

**MATHEMATICS – MODEL PAPER – II**

- If x = 2 and x = 0 are the zeroes of the polynomial p(x) = 2x
^{3}– 5x^{2}+ ax + b, find the values of a and b. - What is the common difference of an A.P in which a
_{18}– a_{14}= 32 - If the slope of a line is 1, then find the angle of inclination to x – axis.
- if tan x = sin 45° cos 45° + sin 30° , then find x.
- In the adjoining figure, DE||AB, AD= 7 cm, CD = 5 cm and BC = 18 cm. Find CE.

- Complete the missing entries in the following factor tree.

__SECTION – B__

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- Explain why 7 ´ 11 ´ 13 + 13 and 7 ´ 6 ´ 5 ´ 4 ´3 ´ 2 ´ 1 + 5 are composite numbers
**.** - How many three digit numbers are divisible by 7?
- Is the system of linear equations 2x + 3y – 9 = 0 and 4x + 6y – 18 = 0 consistent? Justify your answer.
- If (1, 2), (4,y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find the values of x and y.
- A box contains 90 discs which are numbered from 1 to 90.

If one disc is drawn at random from the box, find the probability that it bears

- a) a two – digit number
- b) a perfect square number
- A jar contains 24 marbles some are green and others are blue. If a marble is drawn at random from the jar, the probability that it is green is . Find the number of blue marbles in the jar.

__SECTION – C__

- Use Euclid’s division lemmnce to show that the cube of any positive integer is either of the form 9m, 9m + 1 or 9m + 8
- On dividing x
^{3}– 3x^{2}+ x + 2 by a polynomial g(x) the quotient and remainder were x – 2 and -2x + 4 respectively. Find g(x). - Ritu can row downstream 20 km in 2 hours and upstream 4 km in 2 hours. Find per speed of rowing in still water and the speed of the current.

Formulate this problem as a pair of linear equations and find their solution.

- The points A(4 -2), B(7, 2) C(0, 9) and D (-3, 5) form a parallelogram. Find the length of the altitude of the parallelogram on the base AB.

**OR**

** **Find the co-ordinates of the point s which divide the line segment joining A(-2, 2) and B(2, 8) into four equal parts.

- In DABC, DE||BC. If AD = x + 2, DB = 3x + 16, AE = x and EC = 3x + 5 then find x.

**OR**

** **In the given figure, D is point on side BC of DABC, such that . Prove that AD is the bisector of ÐBAC

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- Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
- Evaluate: –

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OR

Prove that

- A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle.

(use p = 3.14 and

- A drinking glass is in the shape of a frustum of a cone of height 14 cm. The diameters of its two circular end are 4 cm and 2 cm. Find the capacity of the glass.

OR

Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 min, if 8 cm of standing water is needed?

- The arithmetic mean of the following frequency distribution is 50 using the step deviation method” find the value of P.

Classes | 0-20 | 20-40 | 40-60 | 60-80 | 80-100 |

Frequency | 17 | P | 32 | 24 | 19 |

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__SECTION – D__

- An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11km/h more than that of passenger train find the average speed of two trains.

OR

Find the nature of the roots of the equation

** **x – 4, 7 ; and solve this equation by completing the square method.

- The ratio of the 11
^{th}term to the 18^{th}term of an AP is 2 : 3. Find the ratio of the 5^{th}term to the 21^{st}term and also the ratio of the sum of the first five terms to the sum of the first 21 terms.

\ S_{5} : S_{21} = 5 : 49.

- Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

OR

In the given figure, AD is a median of a triangle ABC and AM ^ BC. Prove that

- a) AC
^{2}= AD^{2}+ BC ´ DM + - b) AB
^{2}= AD^{2}– BC ´ DM + - Construct an isosceles triangle whose base is 8 cm and attitude 4 cm and then another triangle whose sides are times the corresponding sides of the isosceles triangle.

- Prove that

- A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite to the tower, the angle of elevation of the top of the tower is 60°. From another point 20 m away from this point on to the line joining this point to the foot of the tower, the angle of elevation of the top of the tower is 30°. Find the height of the tower and the wide of the canal.

Þ 20 = x =

X =

In right D ABC, tan60° = .

Þ …………………(2)

From (1) and (2) we get

.

Þ

Putting the value of h = in (2) we get

Hence the height of the tower is and the width of the canal is 10 m.

- Some people have made a group of volunteers to distribute milk to the children of a village suffering from the malnutrition. For this, they have arranged the 10 same sized buckets each in the form of a frustum of a cone to keep milk inside. The depth of each bucket is 24 cm and their diameter of top and bottom are 30 cm and 10 cm respectively. Find the total cost of the milk at the rate of Rs.40 per litre, if milk is kept full in each bucket. What values can be imbibed by the people?
- The mean of the following frequency distribution is 50, but the frequencies f
_{1}and f_{2}in classes 20 – 40 and 60 – 80, respectively are not known. Find these frequencies, if the sum of all the frequencies is 120.

Class | 0 – 20 | 20 – 40 | 40 – 60 | 60 – 80 | 80 – 100 |

Frequency | 17 | f_{1} |
32 | f_{2} |
19 |

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OR

The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them.

Monthly consumption
( in units) |
65 – 85 | 85 – 105 | 105 – 125 | 125 – 145 | 145 – 165 | 165-185 | 185 – 205 |

No. of consumers | 4 | 5 | 13 | 20 | 14 | 8 | 4 |

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VIKAS PRE UNIVERSITY COLLEGE, MANGALURU – CBSE CLASS – 10 – MATHEMATICS – MODEL PAPER – II