*Instructions:*

- Please check that this question paper consists of 11 pages.
- Code number given on the right hand side of the question paper should be written on the title page of the answer book by the candidate.
- Please check that this question paper consists of 30 questions.
**Please write down the serial number of the question before attempting it.**- 15 minutes times has been allotted to read this question paper. The question paper will be distributed at 10:15 am. From 10:15 am to 10:30 am, the students will read the question paper only and will not write any answer on the answer book during this period.

**SUMMATIVE ASSESSMENT – II**

**MATHEMATICS**

Time allowed: 3 hours Maximum Marks: 80

*General Instructions:*

*(i) All questions are compulsory*

*(ii) The question paper consists of 30 questions divided into four sections – A, B, C and D*

*(iii) Section A consists of 6 questions of 1 mark each. Section B consists of 6 questions of 2 marks each. Section C consists of 10 questions of 3 marks each. Section D consists of 8 questions of 4 marks each.*

*(iv) There is no overall choice. However, an internal choice has been provided in two questions of 1 mark, two questions of 2 marks, four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternative in all such questions. *

*(v) Use of calculator is not permitted.*

**SECTION – A**

*Question number 1 to 6 carry 1 mark each.*

Question 1: Find the value of for which the roots of the quadratic equation

are equal.

Answer:

Given equation:

For roots to be equal,

Question 2: Find the value of for which the distance between the points is units.

Answer:

Given distance between the points is units.

Answer:

, ,

Answer:

From the given AP

Question 5: If , find the value of

Answer:

Question 6: is drawn parallel to the base of a , meeting

Answer:

by AAA criterion

**Section – B**

*Question number 7 to 12 carry 2 mark each.*

Question 7: A bag contains red balls and some blue balls. If the probability of drawing a blue ball from the bag is three times that of a red ball, find the number of blue balls in the bag.

Answer:

Number of Red balls

Let the number of Blue balls

No of Blue balls in the bag

Question 8: The terms of an A.P. are respectively. Find the sum of first terms of the A.P.

Answer:

Let the first term and the common difference

… … … … … i)

… … … … … ii)

Subtracting ii) from i)

Therefore the sum of the first terms is

Question 9: Using Euclid’s Division Algorithm, find the HCF of .

Answer:

According to Eculid’s division theorem, any positive number can be expressed as where is the quotient, is the divisor and is the remainder and

So HCF of is

Question 10: If the point is equidistant from the points , find the value of .

Answer:

Given point is equidistant from the points

Applying distance formula

Question 11: Find the value of for which the pair of linear equations has infinitely many solutions.

Answer:

If the system of equations are and they have infinitely many solutions then it satisfy the following:

From first two terms

From Last two terms

Therefore for equations has infinitely many solutions.

Question 12: A card is drawn at random from a well shuffled pack of playing cards. Find the probability of getting (i) a red king (ii) a queen or a jack

Answer:

Total number of cards

Number of Red kings

Number of queens and jacks

**Section – C**

*Question number 13 to 22 carry 3 mark each.*

Question 13: Show that any positive odd integer is of the form for some integer .

Answer:

Let be any positive integer.

Eculid’s division theorem, any positive number can be expressed as where is the quotient, is the divisor and is the remainder and

Take

Since , the possible remainders are

That is can be

Since is odd, cannot be

Therefore any odd integer is of the form .

Question 14: The ten’s digit of a number is twice its unit’s digit. The number obtained by interchanging the digits is less than the original number. Find the original number.

Answer:

Let tens digit be and unit digit be

Therefore the number

Interchanged number

And (tens digit is twice the unit ) That is, … … … … … i)

Now according to the question

Then

Therefore the number is

Question 15: The line segment joining the points is trisected at the points , where is nearer to . If lies on the line , find the value of .

**Or**

The coordinate of a point is twice its coordinate. If is equidistant from the points , find the coordinates of .

Answer:

(trisects)

Applying section formula

lies on

**Or**

Let be the required point

Given: is equidistant from the points

Now

Hence is

Answer:

Comparing given cubic equation to we get

Now we verify the relations between zeros and their coefficients

Hence verified.

Question 17: Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

Answer:

Given: Circle with center . Let be tangents from external point P.

To Prove:

Proof: Since is a tangent,

Similarly, since is a tangent,

In Quadrilateral

Hence proved.

Question 18: are points on the sides of such that . Show that .

**Or**

Answer:

Given: and points on sides of

To Prove:

Proof: In

(given)

(common angle)

(by AA criterion)

Hence proved.

**Or**

Given: is an equilateral triangle.

To Prove:

Construction: Draw

Proof: Consider

is common

(equilateral triangle)

(By RHS criterion)

Using Pythagoras theorem,

In

… … … … … i)

In

… … … … … ii)

From i) and ii)

But

Hence proved.

**Or**

If , show that

Answer:

Hence proved.

**Or**

Given:

Hence proved.

Question 20: A chord of a circle, of radius cm, subtends an angle of at the center of the circle. Find the area of major and minor segments (Take )

Answer:

Radius cm

Therefore Area of

Area of

We draw

In

(by construction)

(both are radius of the same circle)

is common

(by RHS criterion)

Also, since

… … … … … i)

In right

Similarly, In right

Therefore area of segment

Area of major segment

Question 21: A sphere of diameter cm is dropped in a right circular cylindrical vessel, partly filled with water. If the sphere is completely submerged in water, the water level in the vessel rises by cm. Find the diameter of the cylindrical vessel.

**Or**

A cylinder whose height is two-third of its diameter, has the same volume as that of a sphere of radius cm. Find the radius of base of the cylinder.

Answer:

Volume of sphere Volume of water displaced

Therefore diameter

**Or**

Let the diameter of the cylinder

Question 22: The following table gives the daily income of laborer:

Daily Income (Rs.): | 100-120 | 120-140 | 140-160 | 160-180 | 180-200 |

Number of Laborer: | 12 | 14 | 8 | 6 | 10 |

Find the mean and mode of the above data.

Answer:

We have

Daily Income | Mid Value | Frequency | Cumulative Frequency | |

100-120 | 110 | 12 | 12 | 1320 |

120-140 | 130 | 14 | 26 | 1820 |

140-160 | 150 | 8 | 34 | 1200 |

160-180 | 170 | 6 | 40 | 1020 |

180-200 | 190 | 10 | 50 | 1900 |

Cumulative frequency just greater than is and corresponding class is

Thus, the Median class is

Mode

**Section – D**

*Question number 23 to 30 carry 4 mark each.*

Question 23: Two taps together can fill a tank in hours. The tap of larger diameter takes hours less than the smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately.

**Or**

Answer:

Let the time taken by the smaller tap hours

Therefore the time taken by the larger tap hours

When , time taken by the larger tap to fill the tank hours. is not possible because then the time taken by the larger tap would be negative which is not possible.

**Or**

Question 24: Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

**Or**

Prove that in a triangle, if the square of one side is equal to sum of the squares of the other two sides, the angle opposite the first side is a right angle.

Answer:

Given:

Construction: Draw

Proof:

… … … … … i)

Now in

(given by similarity)

(by construction)

(by AA criterion)

… … … … … ii)

From i) and ii)

Similarly, we can prove that

Hence proved

**Or**

Given:

To Prove:

Construction: is a right angled at such that

Proof: From

(Pythagoras theorem)

(by construction) … … … … … i)

But (given) … … … … … ii)

from i) and ii)

… … … … … iii)

Now in

(by construction)

(by construction)

(from iii)

(by SSS criterion)

But by construction

. Hence proved.

Question 25: Write the steps of construction for drawing a in which

. Now write the steps of construction for drawing a triangle whose sides are of the corresponding sides of .

Answer:

Question 26: The sum of the first n terms of an A.P. is . If its term is , find the value of . Also find the term of the A.P.

**Or**

The and the last terms of an A.P. are respectively. If there are terms in the A.P., find the A.P. and its term.

Answer:

Sum of the first n terms

For

For

common difference

Therefore

Hence

**Or**

The of an AP where the first term is and the common difference is

… … … … … i)

Also

… … … … … ii)

Subtracting i) from ii) we get

Therefore AP is

Hence the AP is

Question 27: Prove that:

Answer:

RHS

Hence Proved.

Question 28: A statue, m tall, stands on a pedestal. From a point on the ground the angle of elevation of the top of the statue is and from the same point angle of elevation of the top of the pedestal is . Find the height of the pedestal. (use )

Answer:

In

… … i)

From

… … … … … ii)

Substituting i) and ii) we get

Therefore the height of the pedestal

Question 29: Sudhakar donated cylindrical drums to store cereals to an orphanage. If radius of each drum is m and height m, find the volume of each drum. If each drum costs Rs. , find the amount spent by Sudhakar for orphanage. What value is exhibited in the question. (Use )

Answer:

Radius of drum

Height of the drum

Therefore amount spent

Question 30: The median of the following data is . If the total frequency is , find the values of and .

Classes | Frequency |

0-10 | 2 |

10-20 | 5 |

20-30 | |

30-40 | 12 |

40-50 | 17 |

50-60 | 20 |

60-70 | 7 |

70-80 | 9 |

80-90 | 7 |

90-100 | 4 |

Answer:

Classes | Frequency | Cumulative Frequency |

0-10 | 2 | 2 |

10-20 | 5 | 7 |

20-30 | 7 | |

30-40 | 12 | 19 |

40-50 | 17 | 36 |

50-60 | 20 | 56 |

60-70 | 56 | |

70-80 | 9 | 65 |

80-90 | 7 | 72 |

90-100 | 4 | 76 |

… … … … … i)

Median class

Hence