*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: Write the discriminant of the quadratic equation .

Answer:

Given Equation:

Comparing it with . and

Determinant

Question 2: Find after how many places of decimal the decimal form of the number will terminate.

**Or**

Express as a product of its prime factors.

Answer:

Simplifying

So the decimal form will terminate are digits.

**Or**

Question 3: Find the sum of first multiples of .

Answer:

is

Question 4: Find the value(s) of , if the distance between the points and is units.

Answer:

Given: and

units

Therefore could be or

Question 5: Two concentric circles of radii and are given. Find the length of the chord of the larger circle which touches the smaller circle.

Answer:

Length of the chord

Question 6: In Figure 1, cm, cm, and cm. Evaluate .

**Or**

If , find the value of .

Answer:

cm

**Or**

**Section – B**

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

Question 7: Points and are vertices of a parallelogram . Find the values of and .

**Or**

Points and trisect the line segment joining the points and such that is near to . Find the coordinates of points and .

Answer:

Let midpoint of be

is also the midpoint of

Therefore

**Or**

Using section formula [ divides in the ratio of ]

Similarly,

Using section formula [ divides in the ratio of ]

Question 8: Solve the following pair of linear equations :

Answer:

Given equations:

… … … … … i)

… … … … … ii)

Multiplying i) by and subtracting ii) from the resultant equation

Substituting into i)

Hence and

Question 9: If HCF of and is expressible in the form , then find the value of .

**Or**

On a morning walk, three persons step out together and their steps measure cm, cm and cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps ?

Answer:

Therefore HCF of and is

**Or**

Finding LCM of and

Therefore LCM

Therefore the minimum distance each should walk should be cm.

Question 10: A die is thrown once. Find the probability of getting (i) a composite number, (ii) a prime number.

Answer:

Total possible outcomes are

Therefore total number of events

i) Composite numbers are and

Therefore Probability (Composite Numbers)

ii) Prime numbers are

Therefore Probability (Prime Numbers)

Question 11: Using completing the square method, show that the equation has no solution.

Answer:

Given equation

A square of a number is always positive. Hence the given equation has no solution.

Question 12: Cards numbered to were put in a box. Poonam selects a card at random. What is the probability that Poonam selects a card which is a multiple of ?

Answer:

Total number of cards

Favorable outcomes

Therefore Probability

**Section – C**

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

Question 13: The perpendicular from on side of a meets at such that . Prove that .

**Or**

and are medians of triangles and respectively where . Prove that

Answer:

Given: with Also

To Prove: .

Proof: Let

Also

Using Pythagoras theorem, in right angled triangle

… … … … … i)

Using Pythagoras theorem, in right angled triangle

… … … … … ii)

Subtracting ii) from i)

(Putting back in the equation)

.

Hence Proved

**Or**

Given: and

is the median of and is the median of

Also

To Prove:

Proof: Since is a median

Similarly, is the median

Now,

(corresponding sides of similar triangles)

So,

(Since AD and PM are medians)

… … … … … i)

Also since

(corresponding angles of similar triangles)

Now in and

from i)

Hence (By SAS criterion)

Since corresponding sides of similar triangles are proportional

. Hence proved.

Question 14: Check whether is a factor of by dividing polynomial by polynomial , where ,

Answer:

If divisible by the remainder of

Therefore remainder is .

Therefore is not a factor of .

Question 15: Find the area of the triangle formed by joining the mid-points of the sides of the triangle , whose vertices are and .

Answer:

Mid-point of

Mid-point of

Mid-point of

Area of

sq. units.

Question 16: Draw the graph of the equations and . Using this graph, find the values of and which satisfy both the equations.

Answer:

For : When and when

For : When and when

From the graph the two lines intersect at . This is the point that satisfy both the equations.

Question 17: Prove that is an irrational number.

**Or**

Find the largest number which on dividing and leaves remainders and respectively.

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

If is even, then is also even in which case is not in simplest form. If is odd then is also odd. Therefore:

Since is an integer, the left hand side is even. Since is an integer, the right hand side is odd and we have found a contradiction, therefore our hypothesis is false.

Therefore is an irrational number

**Or**

Since, and are the remainders of and respectively. Thus, after subtracting these remainders from the numbers.

We have the numbers, and

Which is divisible by the required number.

Now, required number HCF of and

By Euclid’s division algorithm … … … … … (i)

dividend divisor quotient remainder

For largest number, put and

Now

HCF and

Now, we take and , then again using Euclid’s division algorithm,

[ Now ]

HCF and

Hence, is the largest number which divides and leaving remainder and respectively

Question 18: and are interior angles of a triangle . Show that

(i)

(ii) If , then find the value of

**Or**

If and , , then find the values of and .

Answer:

i) To prove:

In , sum of angles

… … … … … i)

Hence Proved.

ii) Similarly,

… … … … … ii)

From i) and ii) and

**Or**

Given

… … … … … i)

Similarly,

… … … … … ii)

Solving i) and ii)

From i)

Hence and

Question 19: In Figure 2, is a chord of length cm of a circle of radius cm. The tangents at and intersect at a point . Find the length .

**Or**

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle.

Answer:

(Length of the tangents from an external point to a circle are equal)

In

is the bisector of

Since

( perpendicular from center to a chord bisects the chord)

cm

In Right triangle

cm

Let

In Right triangle

… … … … … i)

Since is a tangent,

In Right triangle

From i)

Hence is cm

**Or**

Given: A circle with center touches the side and of a quadrilateral at points and .

To Prove: and

Construction: Join and

Proof: Since the two tangents from an external point to a circle subtend equal angles at the center;

and

Since the sum of all the angles subtended at the point is

Similarly

Question 20: Water in a canal, m wide and m deep, is flowing with a speed of km/h. How much area will it irrigate in minutes if cm of standing water is needed ?

Answer:

Speed of water

Rate of flow

Therefore volume of water in mins

Let Area of irrigation

Question 21: A class teacher has the following absentee record of students of a class for the whole term. Find the mean number of days a student was absent.

Number of Days: | 0-6 | 6-12 | 12-18 | 18-24 | 24-30 | 30-36 | 35-42 |

Number of Students: | 10 | 11 | 7 | 4 | 4 | 3 | 1 |

Answer:

Class Interval | |||

0-6 | 10 | 3 | 30 |

6-12 | 11 | 9 | 99 |

12-18 | 7 | 15 | 105 |

18-24 | 4 | 21 | 84 |

24-30 | 4 | 27 | 108 |

30-36 | 3 | 33 | 99 |

36-42 | 1 | 39 | 39 |

Mean value

Question 22: A car has two wipers which do not overlap. Each wiper has a blade of

length cm sweeping through an angle . Find the total area cleaned at each sweep of the blades. (Take )

Answer:

Total area

**Section – D**

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

Question 23: A pole has to be erected at a point on the boundary of a circular park of diameter m in such a way that the difference of its distances from two diametrically opposite fixed gates and on the boundary is m. Is it possible to do so ? If yes, at what distances from the two gates should the pole be erected ?

Answer:

Let and

We know that diameter will subtend a right angle on circumference.

(this is not possible as b cannot be negative)

m

m

Question 24: If times the term of an Arithmetic Progression is equal to times its term and , show that the term of the A.P. is zero.

**Or**

The sum of the first three numbers in an Arithmetic Progression is . If the product of the first and the third term is times the common difference, find the three numbers.

Answer:

Let the first term of AP

common difference

We have to show that term is zero or

term

term

Given that

Hence Proved

**Or**

Let first three numbers of A.P are , and

Sum of the first three numbers or

Product of first and third term or or

Hence and three numbers are

Question 25: Construct a triangle with side cm, cm and . Then construct another triangle whose sides are of the corresponding sides of the triangle .

Answer:

Question 26: In Figure 3, a decorative block is shown which is made of two solids, a cube and a hemisphere. The base of the block is a cube with edge cm and the hemisphere fixed on the top has a diameter of cm. Find

(a) the total surface area of the block.

(b) the volume of the block formed. (Take )

**Or**

A bucket open at the top is in the form of a frustum of a cone with a capacity of . The radii of the top and bottom circular ends are cm and cm respectively. Find the height of the bucket and the area of metal sheet used in making the bucket. (Use )

Answer:

a) Total surface area

b) Volume of the block

**Or**

Volume

cm cm

Volume of the bucket

cm

Therefore height of bucket cm

cm

Therefore surface are of the metal sheet used

curved surface area base area

Hence the height of the bucket is cm and surface are of the metal sheet is

Question 27: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio.

**Or**

Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Answer:

Given: In intersects and at and respectively.

To Prove:

Construct: and

Proof:

Area of

Area of

… … … … … i)

Now … … … … … ii)

(Because they have the same base and are between the same parallel)

Similarly, … … … … … iii)

From i) , ii) and iii) we get . Hence proved.

**Or**

Given: A right angled , right angled at

To prove:

Draw:

Proof: In

(common angle)

( by AA criterion)

Therefore (corresponding sides are proportional)

… … … … … i)

Similarly,

and … … … … … ii)

Adding i) and ii)

. Hence proved.

Question 28: If , then prove that or

Answer:

We have

Dividing both sides with

Let

or

Hence

Question 29: Change the following distribution to a ‘more than type’ distribution. Hence draw the ‘more than type’ ogive for this distribution.

Class Interval: | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 |

Frequency | 10 | 8 | 12 | 24 | 6 | 25 | 15 |

Answer:

Less Than table

Less Than | Cumulative Frequency |

Less than 30 | 10 |

Less than 40 | 18 |

Less than 50 | 30 |

Less than 60 | 54 |

Less than 70 | 60 |

Less than 80 | 85 |

Less than 90 | 100 |

More than table

More Than | Cumulative Frequency |

More than 30 | 10 |

More than 40 | 18 |

More than 50 | 30 |

More than 60 | 54 |

More than 70 | 60 |

More than 80 | 85 |

More than 90 | 100 |

Draw the chart. The point of intersection is :

Question 30: The shadow of a tower standing on a level ground is found to be m longer when the Sun’s altitude is than when it was . Find the height of the tower. (Given )

Answer:

Let be the height of the tower.

… … … … … i)

Also

… … … … … ii)

From i) and ii)

m