*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: The HCF of two numbers and is and their LCM is . Find the product .

Answer:

and are two numbers.

H.C.F. of and

L.C.M. of and

So, By Fundamental theorem of Arithmetic , we have

So, The value of product of and is

Question 2: Find the value of for which is a solution of the equation .

**Or**

Find the value(s) of for which the quadratic equation has real and equal roots.

Answer:

Given equation:

If is a solution, then it should satisfy the given equation.

**Or**

Given equation:

For roots to be equal,

Question 3: If in an A.P., and , then find the value of .

Answer:

Given: and

Since

Question 4: If and is an acute angle, find the value of .

**Or**

Find the value of .

Answer:

**Or**

Question 5: The area of two similar triangles are sq. cm and sq. cm. Find the ratio of their corresponding sides.

Answer:

Let and are corresponding sides of the similar triangles

Hence the ratio of the corresponding sides

Question 6: Find the value of so that the point lies on the line represented by .

Answer:

If lies on the equation , it must satisfy the equation.

**Section – B**

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

Question 7: If , the sum of the first terms of an A.P. is given by , then find its term.

**Or**

If the term of an A.P. exceeds its term by , find the common difference.

Answer:

Sum of first terms of AP,

Now choose and put in the above formula, First term

Now put to get the sum of first two terms

This means First term + Second term

But first term as calculated above Hence, second term

So common difference (d) becomes,

So the AP becomes,

**Or**

Let the first term and the common difference

Given

Common difference

Question 8: The mid-point of the line segment joining and is . Find the values of and .

Answer:

Given points and

Mid Point of

Also

Hence

Question 9: A child has a die whose 6 faces show the letters given below :

The die is thrown once. What is the probability of getting (i) A (ii) B ?

Answer:

Number of possible events

No of A’s on dice

No of B’s of dice

i) Probability

ii) Probability

Question 10: Find the HCF of and using prime factorisation.

**Or**

Show that any positive odd integer is of the form or or , where is some integer.

Answer:

First factorize each number

Therefore HCF of and

**Or**

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 or

Since is odd, cannot be or or

Therefore any odd integer is of the form or or .

Question 11: Cards marked with numbers to (one number on one card) are placed in a box and mixed thoroughly. One card is drawn at random from the box. Find the probability that the number on the card taken out is (i) a prime number less than , (ii) a number which is a perfect square.

Answer:

Total number of ways to select a card (Cards are marked to )

i) Prime numbers less than are and only

Therefore Probability (prime number less than )

ii) Numbers which are perfect squares are

Therefore Probability (a number which is a perfect square)

Question 12: For what value of , does the system of linear equations

, have an infinite number of solutions ?

Answer:

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

For and have an infinite number of solutions:

From first two terms

From second and third term

From first and third term

Therefore , the equation with have infinite solutions.

**Section – C**

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

Question 13: Prove that is an irrational number.

Answer:

Assume is a rational number i.e. it can be expressed as a rational fraction of the form where are relatively prime numbers.

Since

we have or

This would imply that is a multiple of . Since is prime, this implies is a multiple of . Thus for some integer , and

Dividing by , this means

So is a multiple of , and, just as it did for , this means is a multiple of . But and were presumed to lack a common factor other than , so this is a contradiction, and the fraction for must fail to exist.

Hence is irrational.

Question 14: Find all the zeroes of the polynomial , if two of its zeroes are and .

Answer:

Two zeros are and

Therefore and are factors of

is a factor of

Therefore the other two zeros are

Question 15: Point P divides the line segment joining the points and such that . If lies on the line , find the value of .

**Or**

For what value of , are the points and collinear ?

Answer:

(trisects)

Applying section formula

Coordinates of

Since lies on

**Or**

Given point and

If the points are collinear are of the

Question 16: Prove that:

**Or**

If show that

Answer:

LHS

RHS. Hence proved.

**Or**

Given:

Squaring both sides

Add on both sides

Hence proven.

Question 17: A part of monthly hostel charges in a college hostel are fixed and the remaining depends on the number of days one has taken food in the mess. When a student takes food for days, he has to pay Rs. , whereas a student who takes food for days, has to pay Rs. . Find the fixed charges per month and the cost of food per day.

Answer:

Let the fixed charges

Let the charges for food per day

Therefore for Student A:

… … … … … i)

Therefore for Student B:

… … … … … ii)

Solving i) and ii)

Rs.

Rs.

Therefore the fixed charges are Rs. and food cost per day is Rs.

Question 18: In and is the mid-point of . Prove that .

**Or**

In Figure 1, is a point on produced of an isosceles , with side . If and , prove that .

Answer:

Given:

From

… … … … … i)

From

… … … … … ii)

From i) and ii)

Hence proved.

**Or**

Given:

and

In and

(given)

( By AA criterion)

(corresponding sides)

Question 19: Prove that the parallelogram circumscribing a circle is a rhombus.

Answer:

Given: be a parallelogram circumscribing a circle with center .

To prove: is a rhombus.

We know that the tangents drawn to a circle from an exterior point are equal in length.

Therefore, and .

Adding the above equations,

(Since, is a parallelogram so and )

Therefore, .

Hence, is a rhombus.

Question 20: In Figure 2, three sectors of a circle of radius cm, making angles of , and at the centre are shaded. Find the the area of the shaded region.

Answer:

Area of the shaded region

Question 21: The following table gives the number of participants in a yoga camp :

Age (in years): | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |

No. of participants: | 8 | 40 | 58 | 90 | 83 |

Find the modal age of the participants.

Answer:

First we find the modal class of the given data which is the highest participant class i.e.

Modal class

Formula to find Mode is

Question 22: A juice seller was serving his customers using glasses as shown in Figure 3. The inner diameter of the cylindrical glass was cm but bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass. If the height of a glass was cm, find the apparent and actual capacity of the glass. (Use )

**Or**

A girl empties a cylindrical bucket full of sand, of base radius cm and height cm on the floor to form a conical heap of sand. If the height of this conical heap is cm, then find its slant height correct to one place of decimal.

Answer:

Apparent volume

Actual capacity

**Or**

Volume of sand remains the same

cm

Slant height cm

**Section – D**

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

Question 23: A train travels km at a uniform speed. If the speed had been km/hr more, it would have taken hr less for the same journey. Find the speed of the train.

**Or**

Solve for

Answer:

Distance traveled km

Let the original speed km/hr

km/hr or km/hr (this is not possible as speed cannot be negative)

Hence the original speed km/hr

**Or**

or

Question 24: If the sum of the first terms of an A.P. is and the sum of the first terms is ; then show that the sum of the first terms is .

Answer:

Let be the first term and be the common difference of the AP

Given

… … … … … i)

Also

… … … … … ii)

Subtracting ii) from i)

Now

Question 25: In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then prove that the angle opposite to the first side is a right angle.

Answer:

Given:

To prove:

Construction: is a right angled at such that and

Proof: From

(Pythagoras theorem)

(by construction) … … … … … i)

But (given) … … … … … ii)

from i) and ii)

… … … … … iii)

Now in and

(by construction)

(by construction)

(from iii)

(by SSS criterion)

But by construction

. Hence proved.

Question 26: Construct an isosceles triangle whose base is cm and altitude cm and then another triangle whose sides are times the corresponding sides of the isosceles triangle.

Answer:

Question 27: A boy standing on a horizontal plane finds a bird flying at a distance of m from him at an elevation of . A girl standing on the roof of a m high building, finds the elevation of the same bird to be . The boy and the girl are on the opposite sides of the bird. Find the distance of the bird from the girl. (Given )

**Or**

The angle of elevation of an aeroplane from a point on the ground is . After a flight of seconds, the angle of elevation changes to . If the plane is flying at a constant height of metres, find the speed of the aeroplane.

Answer:

In

m

m

In

m

**Or**

In

m

In

m

Hence m

Question 28: Find the values of frequencies and in the following frequency distribution table, if and median is .

Marks: | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | Total |

No. of Students | 10 | 25 | 30 | 10 | 100 |

**Or**

For the following frequency distribution, draw a cumulative frequency curve (ogive) of ‘more than type’ and hence obtain the median value.

Class: | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | Total |

Frequency: | 5 | 10 | 20 | 23 | 17 | 11 | 9 |

Answer:

Class Interval | Frequency ( ) | Cumulative Frequency ( ) |

0 – 10 | 10 | 10 |

10 – 20 | ||

20 – 20 | 25 | |

30 – 40 | 30 | |

40 – 40 | ||

50 – 60 | 10 |

… … … … … i)

For the above distribution, median class is

Median

From i)

**Or**

Class Interval (More than) | Frequency |

More than 0 | 100 |

More than 10 | 95 |

More than 20 | 80 |

More than 30 | 60 |

More than 40 | 37 |

More than 50 | 20 |

More than 60 | 9 |

Class Interval (Less than) | Frequency |

Less than 10 | 5 |

Less than 20 | 20 |

Less than 30 | 40 |

Less than 40 | 63 |

Less than 50 | 80 |

Less than 60 | 91 |

Less than 70 | 100 |

Question 29: Prove that:

Answer:

LHS

Since

RHS. Hence proved.

Question 30: An open metallic bucket is in the shape of a frustum of a cone. If the diameters of the two circular ends of the bucket are cm and cm and the vertical height of the bucket is cm, find the area of the metallic sheet used to make the bucket. Also find the volume of the water it can hold. (Use )

Answer:

cm

cm

cm

Volume of bucket

liters

cm

Therefore surface area