*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: If HCF , find LCM .

Answer:

We know product of the two numbers

Question 2: Find the nature of roots of the quadratic equation .

Answer:

Given equation

Therefore Roots

Therefore the nature of the roots is imaginary

Question 3: Find the common difference of the Arithmetic Progression (A.P.)

Answer:

Given AP:

Common difference

To confirm, we also do

Hence

Question 4: Evaluate :

**Or**

If , calculate

Answer:

**Or**

If

Question 5: Write the coordinates of a point on which is equidistant from the points and .

Answer:

Let be equidistant from and

is

Question 6: In Figure 1, is an isosceles triangle right angled at with cm. Find the length of .

**Or**

In Figure 2, . Find the length of side , given that cm, cm and cm.

Answer:

Since is isosceles

(hypotenuse has to be the largest side, hence the other two sides are equal)

cm

**Or**

Since by Thales Theorem

cm

**Section – B**

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

Question 7: Write the smallest number which is divisible by both and .

Answer:

LCM of and

Hence the smallest number which is divisible by and is

Question 8: Find a relation between and if the points and are collinear.

**Or**

Find the area of a triangle whose vertices are given as and .

Answer:

If , and are collinear, then the area of the should be .

Area of

**Or**

Let the points be and

Area of triangle

Here

Substituting in the formula above

Area of

sq. units.

Question 9: The probability of selecting a blue marble at random from a jar that contains only blue, black and green marbles is . The probability of selecting a black marble at random from the same jar is . If the jar contains green marbles, find the total number of marbles in the jar.

Answer:

P(selecting Blue marble)

P(selecting Black marble)

P(selecting Green marble)

Therefore there are a total of marbles in the Jar.

Question 10: Find the value(s) of so that the pair of equations and has a unique solution.

Answer:

Pair of equations and

If the intersecting lines have a unique solution, then

From the first two terms,

From the second and the third terms,

Hence all values of except and

Question 11: The larger of two supplementary angles exceeds the smaller by . Find the angles.

**Or**

Sumit is times as old as his son. Five years later, he shall be two and a half times as old as his son. How old is Sumit at present ?

Answer:

Let and be the two supplementary angles

Also given (given one angle is greater than the other by )

Solving the two equations

Hence the two angles are ad

**Or**

Let the age of son be years

Therefore are of Sumit is

Age of son years later

Age of Sumit years later

Give:

years

Therefore age of Sumit years

Question 12: Find the mode of the following frequency distribution :

Class Interval: | 25-30 | 30-35 | 35-40 | 40-45 | 45-50 | 50-55 |

Frequency: | 25 | 34 | 50 | 42 | 38 | 14 |

Answer:

**Section – C**

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

Question 13: Prove that is an irrational number, given that is an irrational number.

**Or**

Using Euclid’s Algorithm, find the HCF of and .

Answer:

Let us assume that is a rational number.

The rational number is in the form of

is a rational no as is a rational number

But it is given that is an irrational number which contradicts our initial assumption.

Hence is an irrational number.

**Or**

Hence HCF

Question 14: Two right triangles and are drawn on the same hypotenuse and on the same side of . If and intersect at , prove that .

**Or**

Diagonals of a trapezium intersect each other at the point and . Find the ratio of the areas of triangles and .

Answer:

To prove:

Consider and

(given)

( vertically opposite angles)

(By AA criterion)

Hence proved.

**Or**

Given is a trapezium

and

Consider and

(vertically opposite angles)

(alternate angles)

(alternate angles)

(By AAA criterion)

By property of similar triangles

Question 15: In Figure 3, and are two parallel tangents to a circle with center and another tangent with point of contact intersecting at and at . Prove that .

Answer:

Given: , is a tangent

To prove:

Join through . is diameter of the circle

We know that tangents to a circle from an external point are equally inclined to the line segment joining this point to the center

and

Now and is transversal

From

Hence

Question 16: Find the ratio in which the line divides the line segment joining the points and . Find the coordinates of the point of intersection.

Answer:

Let divides point and in the ratio

Using section formula:

and in the ratio of

and

Since lies on the line

Therefore divided and in the ratio of

Hence which is the point of intersection.

Question 17: Evaluate:

Answer:

Question 18: In Figure 4, a square is inscribed in a quadrant . If cm, find the area of the shaded region. (Use )

**Or**

In Figure 5, is a square with side cm and inscribed in a circle. Find the area of the shaded region. (Use )

Answer:

cm (since is a square)

Radius

Area of square

Area of quadrant

Shaded area

**Or**

Area of square

Diameter

Therefore Radius of circle

Area of circle

Therefore shaded area

Question 19: A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is cm and the diameter of the cylinder is cm. Find the total volume of the solid. (Use )

Answer:

Radius of hemisphere

Therefore height of cylinder

Therefore total volume

Question 20: The marks obtained by students in an examination are given below :

Marks: | 30-35 | 35-40 | 40-45 | 45-50 | 50-55 | 55-60 | 60-65 |

Number of Students: | 14 | 16 | 28 | 23 | 18 | 8 | 3 |

Answer:

Marks ( Class interval) | No. of Students (Frequency) | Mean Value $latex (x) $ | |

30-35 | 14 | 32.5 | 455 |

35-40 | 16 | 37.5 | 600 |

40-45 | 28 | 42.5 | 1190 |

45-50 | 23 | 47.5 | 1092.5 |

50-55 | 18 | 52.5 | 945 |

55-60 | 8 | 57.5 | 460 |

60-65 | 3 | 62.5 | 187.5 |

Mean

Question 21: For what value of k, is the polynomial completely divisible by ?

**Or**

Find the zeroes of the quadratic polynomial and verify the relationship between the zeroes and the coefficients.

Answer:

divisible by the remainder of

Remainder should be

and are factors of

and are factors

or

Also are factors

Hence are zeros of

**Or**

Given polynomial

or

Comparing with we get and

Sum of zeros … … … … … i)

Sum of roots … … … … … ii)

Therefore i) = ii)

Products of zeros … … … … … iii)

Product of roots … … … … … iv)

Therefore iii) = iv)

Question 22: Write all the values of for which the quadratic equation has equal roots. Find the roots of the equation so obtained.

Answer:

Given quadratic equation is:

If the roots are equal then the discriminant is

i.e.

When

When

Therefore the roots of equation are

**Section – D**

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

Question 23: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.

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.

Question 24: Amit, standing on a horizontal plane, finds a bird flying at a distance of m from him at an elevation of . Deepak standing on the roof of a m high building, finds the angle of elevation of the same bird to be . Amit and Deepak are on opposite sides of the bird. Find the distance of the bird from Deepak.

Answer:

m

Question 25: A solid iron pole consists of a cylinder of height cm and base diameter cm, which is surmounted by another cylinder of height cm and radius cm. Find the mass of the pole, given that of iron has approximately gm mass. (Use )

Answer:

Volume of pole

Weight of the pole

Question 26: Construct an equilateral with each side cm. Then construct another triangle whose sides are times the corresponding sides of .

**Or**

Draw two concentric circles of radii cm and cm. Take a point on the outer circle and construct a pair of tangents and to the smaller circle. Measure .

Answer:

Question 27: Change the following data into ‘less than type’ distribution and draw its ogive :

Class Interval: | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 | 90-100 |

Frequency: | 7 | 5 | 8 | 10 | 6 | 6 | 8 |

Answer:

Question 28: Prove that:

**Or**

Prove that:

Answer:

LHS

**Or**

LHS

RHS

LHS

Hence Proved.

Question 29: Which term of the Arithmetic Progression will be ? Is any term of the A.P. ? Give reason for your answer.

**Or**

How many terms of the Arithmetic Progression must be taken so that their sum is ? Explain the double answer.

Answer:

Given AP:

Common difference

The term of AP

Hence is the term

Since the numbers in the given are not factors of will not be part of the AP.

**Or**

Given AP is

Common difference

or

We get double answer because the sum of the to term is .

Question 30: In a class test, the sum of Arun’s marks in Hindi and English is . Had he got marks more in Hindi and marks less in English, the product of the marks would have been . Find his marks in the two subjects.

Answer:

Let Arun’s marks in Hindi

Let Arun’s marks in English

… … … … … i)

Also … … … … … ii)

From i) … … … … … iii)

Substituting in ii)

or

Substituting in iii), when

When

Hence his marks in Hindi are or . When he gets in Hindi, he gets in English. When he gets in Hindi, he gets in English.