*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: Given , if , then find

Answer:

Given

Question 2: What is the value of ?

Answer:

Question 3: If is one root of the quadratic equation , then find the value of .

Answer:

If is one root of the quadratic equation than it should satisfy the equation.

Question 4: Find the distance of a point from the origin.

Answer:

Distance of a point from the origin

Question 5: What is the HCF of smallest prime number and the smallest composite number ?

Answer:

Smallest composite number is

Smallest prime number is

Therefore HCF of and is

Question 6: In an AP, if the common difference , and the seventh term is , then find the first term.

Answer:

Given

**Section – B**

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

Question 7: An integer is chosen at random between and . Find the probability that it is :

(i) divisible by .

(ii) not divisible by .

Answer:

Number of integers between and

Total number of outcomes

(i) Numbers which are divisible by are:

Favorable Outcomes

Probability that integer is divisible by

(ii) Probability that integer is not divisible by

Question 8:Two different dice are tossed together. Find the probability :

(i) of getting a doublet

(ii) of getting a sum , of the numbers on the two dice.

Answer:

Given that two different dice are tossed.

Total number of outcomes

(i) Let be the event of getting a doublet.

Required probability

(ii) Let be the event of getting a sum of of the numbers of two dice.

Required probability

Question 9: Find the ratio in which divides the line segment joining the points and . Hence find .

Answer:

Let divides and in the ratio of

By section formula,

Also

Question 10: Given that is irrational, prove that is an irrational number.

Answer:

Given that, is irrational

Let us assume is rational.

As is rational. (Assumed) They must be in the form of where , and and are co prime.

We know is irrational but is a rational number

Therefore we contradict the statement that, is rational.

Hence proved that is irrational.

Question 11:

In Fig. 1, is a rectangle. Find the values of and .

Answer:

Given:

… … … … … i)

… … … … … ii)

Adding i) and ii) we get

Question 12: Find the sum of first multiples of .

Answer:

The first multiples of are :

Number of terms

The first term

Common difference

Hence the sum of the first multiples of is .

**Section – C**

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

Question 13: A plane left minutes late than its scheduled time and in order to reach the destination km away in time, it had to increase its speed by km/h from the usual speed. Find its usual speed.

Answer:

Distance to travel km

Let the usual speed km/hr

Increased speed km/hr

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

Therefore the usual speed km/hr

Question 14: Prove that the area of an equilateral triangle described on one side of the square is equal to half the area of the equilateral triangle described on one of its diagonal.

**Or**

If the area of two similar triangles are equal, prove that they are congruent.

Answer:

Let the side of the square

Therefore the sides of the equilateral as well

Now

Therefore the sides of the equilateral

Since the triangles are equilateral, each of the angles are . Hence by criterion, the two triangles are similar.

Therefore

Hence proven.

**Or**

Given: and

Also and

To Prove:

Since we know

Therefore we get

Therefore by SSS criterion.

Hence proved.

Question 15: Prove that the lengths of tangents drawn from an external point to a circle are equal.

Answer:

Given: Let the circle with center

Let be an external point from which tangents and are drawn as shown in the diagram

To prove:

Construction: Join and

Proof: As is tangent . Therefore

Similarly, As is tangent . Therefore

(Note: Tangents at any point on a circle is perpendicular to the radius through the point of contact)

In and

is common

(radius of the same circle)

Therefore both tangents are equal in length.

Question 16: A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 2. If the height of the cylinder is cm and its base is of radius cm. Find the total surface area of the article.

**Or**

A heap of rice is in the form of a cone of base diameter m and height m. Find the volume of the rice. How much canvas cloth is required to just cover the heap ?

Answer:

Radius of the cylinder cm Height cm

Curved surface area of cylinder

Surface area of two hemispheres

Therefore total surface area

**Or**

Diameter m Radius m Height m

Volume of Rice

Slant height m

Curved surface area of the heap

Hence Volume of Rice and Canvas required to cover the heap

Question 17: The table below shows the salaries of 280 persons:

Salary (in thousand Rs.) | No. of persons |

5-10 | 49 |

10-15 | 133 |

15-20 | 63 |

20-25 | 15 |

25-30 | 6 |

30-35 | 7 |

35-40 | 4 |

40-45 | 2 |

45-50 | 1 |

Answer:

Class Interval | Frequency | Cumulative Frequency |

5 – 10 | 49 | 49 |

10 – 15 | 133 | 182 |

15 – 20 | 63 | 245 |

20 – 25 | 15 | 260 |

25 – 30 | 6 | 266 |

30 – 35 | 7 | 273 |

35 – 40 | 4 | 277 |

40 – 45 | 2 | 279 |

45 – 50 | 1 | 280 |

Median

Therefore median salary Rs.

Question 18: If , evaluate

**Or**

If , where is an acute angle, find the value of .

Answer:

Given

Therefore

**Or**

Question 19: Find the area of the shaded region in Fig. 3, where arcs drawn with centres and intersect in pairs at mid-points and of the sides and respectively of a square of side cm. [Use ]

Answer:

Given square of side cm

Area of square

Area of

Therefore total area enclosed in arc’s

Therefore shaded area

Question 20: If and are the vertices of a parallelogram , find the values of a and b. Hence find the lengths of its sides.

**Or**

If and are the vertices of a quadrilateral, find the area of the quadrilateral .

Answer:

Mid point of

Mid point of

Since the mid point of and are the same

Similarly,

Therefore and

Hence is a Rhombus

**Or**

Join

Note: For a with vertices the area of the triangle is

sq. units

sq. units

sq. units

Question 21: Find HCF and LCM of and and verify that HCF LCM Product of the two given numbers.

Answer:

Therefore HCF of and

Also LCM of and

Product of two numbers

HCF LCM

Hence Product of two numbers HCF LCM

Question 22: Find all zeroes of the polynomial if two of its zeroes are and .

Answer:

Given and are zeros of the polynomial

Therefore is a factor of the polynomial Hence

Therefore is a factor of

and are factors of

Hence and are the other two zeros of the given polynomial

**Section – D**

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

Question 23: Draw a triangle with cm, cm and . Then construct a triangle whose sides are of the corresponding sides of the .

Answer:

Question 24: The sum of four consecutive numbers in an AP is and the ratio of the product of the first and the last term to the product of two middle terms is . Find the numbers.

Answer:

Let the four consecutive terms of the AP be and

Given:

… … … … … i)

Also

Substituting from i)

or

When

The the first four terms are

When

The the first four terms are

Question 25: In an equilateral , is a point on side such that . Prove that:

**Or**

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

Answer:

Given: is an equilateral triangle.

Also

To prove:

Construction: Draw

Consider and

is common

(equilateral triangle)

(By RHS criterion)

Using Pythagoras theorem,

In

… … … … … i)

In

… … … … … ii)

From i) and ii)

But

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 26: A motor boat whose speed is km/hr in still water takes hr more to go km upstream than to return downstream to the same spot. Find the speed of the stream.

**Or**

A train travels at a certain average speed for a distance of km and then travels at a distance of km at an average speed of km/hr more than its original speed. If it takes hours to complete total journey, what is the original average speed ?

Answer:

Distance traveled by the boat km

Speed of the boat km/hr

Let the speed of the stream km/hr

Therefore speed of boat upstream km/hr

Also speed of boat downstream km/hr

Hence:

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

Therefore the speed of the stream is km/hr

**Or**

Let the original speed is km/hr

Time taken to travel km hrs

New speed km/hr

Time taken to travel km

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

Hence original speed km/hr

Question 27: As observed from the top of a m high light house from the sea-level, the angles of depression of two ships are and . If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships. [Use ]

Answer:

In

m

In

m

m

Therefore the distance between the ships is m.

Question 28: The diameters of the lower and upper ends of a bucket in the form of a frustum of a cone are cm and cm respectively. If its height is cm, find :

(i) The area of the metal sheet used to make the bucket.

(ii) Why we should avoid the bucket made by ordinary plastic ? [Use ]

Answer:

Diameter of upper end of bucket cm

Radius of the upper end of the frustum of cone cm

Diameter of lower end of bucket cm

radius of the lower end of the frustum of cone cm

Height of the frustum of Cone cm

Volume of bucket

liters

cm

Therefore surface area

Hence, the Area of metal sheet used to make the bucket is

Question 29: The mean of the following distribution is . Find the frequency of the class .

Class: | 11-13 | 13-15 | 15-17 | 17-19 | 19-21 | 21-23 | 23-25 |

Frequency: | 3 | 6 | 9 | 13 | 5 | 4 |

**Or**

The following distribution gives the daily income of 50 workers of a factory:

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

Number of workers | 12 | 14 | 8 | 6 | 10 |

Convert the distribution above to a less than type cumulative frequency distribution and draw its ogive.

Answer:

Class Interval | Frequency | ||

11-13 | 3 | 12 | 36 |

13-15 | 6 | 14 | 84 |

15-17 | 9 | 16 | 144 |

17-19 | 13 | 18 | 234 |

19-21 | 20 | ||

21-23 | 5 | 22 | 110 |

23-25 | 4 | 24 | 96 |

Mean

**Or**

Daily wages in Rs (less than) | Frequency | Cumulative Frequency |

Less than 120 | 12 | 12 |

Less than 140 | 14 | 26 |

Less than 160 | 8 | 34 |

Less than 180 | 6 | 40 |

Less than 200 | 10 | 50 |

Question 30: Prove that:

Answer:

LHS

RHS. Hence Proved.