*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: For what values of does the quadratic equation have no real roots ?

Answer:

Given equation:

Comparing it with and

When , there are no real roots.

Therefore for , there will be no real roots.

Question 2: Find the distance between the points and .

Answer:

Given points and

Therefore the distance between the two points

Question 3: Find a rational number between and .

**Or**

Write the number of zeroes in the end of a number whose prime factorization is .

Answer:

and

Therefore is between and

is a rational number between and

**Or**

The given expression:

We know that zeros in an expression are a result of number of in it.

Hence the expression we can see that there will be zeros in the given expression.

Question 4: Let and their areas be respectively, and . If cm, find .

Answer:

Since

cm

Question 5: Evaluate:

**Or**

Express in terms of trigonometric ratios of the angle between and .

Answer:

**Or**

Question 6: Find the number of terms in the A.P. :

Answer:

Given AP:

Let be the term

Therefore is the term in the given AP

**Section – B**

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

Question 7: A bag contains balls, out of which some are white and the others are black. If the probability of drawing a black ball at random from the bag is , then find how many white balls are there in the bag.

Answer:

Total number of balls

Let the number of black balls

Hence the number of white balls

Question 8: A card is drawn at random from a pack of playing cards. Find the probability of drawing a card which is neither a spade nor a king.

Answer:

Total number of cards

Number of space cards (includes the king of space)

No of kings other than king of spade

Therefore Probability

Question 9: Find the solution of the pair of equations :

**Or**

Find the value(s) of for which the pair of equations has a unique solution.

Answer:

Given equation:

Let and

Therefore

… … … … … iii)

… … … … … iv)

Multiplying iv) by iii) and subtracting it from iii)

Therefore

Hence

Also

**Or**

Given equations:

If the system of equations are and and they have unique solution then it satisfy the following:

From first two terms

From First and Third term

or for the equations to have a unique solution.

Question 10: How many multiples of lie between and ?

**Or**

Determine the A.P. whose third term is and term exceeds the term by .

Answer:

Since the multiples of lie between and are

Therefore there are multiples of between and

**Or**

Let for the AP, first term and common difference

Given

… … … … … i)

Given

Therefore form i) we get

Therefore the AP is

Question 11: Use Euclid’s division algorithm to find the HCF of and .

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 and is

Question 12: The point divides the line segment , where and such that . Find the coordinates of .

Answer:

Given points and

Using sections formula,

**Section – C**

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

Question 13: Prove that :

**Or**

Answer:

LHS

. Hence proved.

**Or**

LHS

hence proved.

Question 14: In what ratio does the point divide the line segment joining the points and ? Hence find the value of .

**Or**

Find the value of for which the points and are collinear.

Answer:

Let divides in the ratio of

Therefore ratio is

**Or**

Given point and

If the points are collinear are of the

Question 15: is a right triangle in which . If cm and cm, find the diameter of the circle inscribed in the triangle.

Answer:

cm

Area of Area of Area of Area of

Question 16: In Figure 1, and are medians of a right-angled at . Prove that .

**Or**

Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

Answer:

Given and are medians

From

… … … … … i)

… … … … … ii)

… … … … … iii)

Adding ii) and iii)

Hence Proved.

**Or**

Given: is a rhombus

To prove:

We know that the diagonals of a rhombus bisect at right angles

Therefore from

Since

Hence proved.

Question 17: In Figure 2, two concentric circles with centre , have radii cm and cm. If , find the area of the shaded region.

Answer:

Area of shaded area

Question 18: Calculate the mode of the following distribution :

Class: | 10-15 | 15-20 | 20-25 | 25-30 | 30-35 |

Frequency: | 4 | 7 | 20 | 8 | 1 |

Answer:

Question 19: A cone of height cm and radius of base cm is made up of modelling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere and hence find the surface area of this sphere.

**Or**

A farmer connects a pipe of internal diameter cm from a canal into a cylindrical tank in his field which is m in diameter and m deep. If water flows through the pipe at the rate of km/hr, in how much time will the tank be filled ?

Answer:

Volume of cone

Let the radius of the sphere be

cm

Surface are of the sphere

**Or**

Rate of flow of water

Volume of tank

Volume of water flowing per minute

Therefore the time taken to fill the tank minutes

Question 20: Prove that is an irrational number when it is given that is an irrational number.

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.

Question 21: Sum of the areas of two squares is . If the sum of their perimeters is m, find the sides of the two squares.

Answer:

Let the sides of the two squares be and respectively

… … … … … i)

Also

… … … … … ii)

From i)

… … … … … iii)

From ii)

or

When

When

Hence the sides of the two squares are units and units

Question 22: Find the quadratic polynomial, sum and product of whose zeroes are and respectively. Also find the zeroes of the polynomial so obtained.

Answer:

Sum of zeros:

Product of zeros:

The quadratic polynomial is of the form:

(sum of zeros) (product of zeros)

or

When

When

Therefore zeros are

**Section – D**

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

Question 23: A plane left minutes later than the scheduled time and in order to reach its destination km away on time, it has to increase its speed by km/hr from its usual speed. Find the usual speed of the plane.

**Or**

Find the dimensions of a rectangular park whose perimeter is m and area .

Answer:

Let the speed of the plane

New speed of the plane

or ( not possible as speed cannot be negative)

**Or**

Let length and breadth

… … … … … i)

… … … … … ii)

From ii)

Substituting in i)

When m, m

When m, m

Hence the dimensions are m by m

Question 24: Find the value of , when in the A.P. given below .

Answer:

First term of AP

Common difference

Sum of terms

( total number of terms)

is the term

Question 25: If , show that

Answer:

… … … … … i)

Similarly,

… … … … … ii)

Dividing i) by ii) we get

Question 26: In (Figure 3), . Prove that

Answer:

Given

In … … … … … i)

In … … … … … ii)

Frim i) and ii)

Hence Proved.

Question 27: A moving boat is observed from the top of a m high cliff moving away from the cliff. The angle of depression of the boat changes from to in minutes. Find the speed of the boat in m/min.

**Or**

There are two poles, one each on either bank of a river just opposite to each other. One pole is m high. From the top of this pole, the angle of depression of the top and foot of the other pole are and respectively. Find the width of the river and height of the other pole.

Answer:

m

m

Therefore speed

**Or**

m (width of the river)

m

Hence the height of the pole is m

Question 28: Construct a triangle with sides cm, cm and cm and then another triangle whose sides are of the corresponding sides of the first triangle.

Answer:

Question 29: Calculate the mean of the following frequency distribution :

Class: | 10-30 | 30-50 | 50-70 | 70-90 | 90-110 | 110-130 |

Frequency: | 5 | 8 | 12 | 20 | 3 | 2 |

**Or**

The following table gives production yield in kg per hectare of wheat of 100 farms of a village :

Production Yield
(kg/hectare): |
40-45 | 45-50 | 50-55 | 55-60 | 60-65 | 65-70 |

Number of Farms: | 4 | 6 | 16 | 20 | 30 | 24 |

Change the distribution to a ‘more than type’ distribution, and draw its ogive.

Answer:

Class Interval | Frequency ( ) | ||

10-30 | 5 | 20 | 100 |

30-50 | 8 | 40 | 320 |

50-70 | 12 | 60 | 720 |

70-90 | 20 | 80 | 1600 |

90-110 | 3 | 100 | 300 |

110-130 | 2 | 120 | 2400 |

Mean

**Or**

Production Yield | Cumulative Frequency |

More than 40 | 100 |

More than 45 | 96 |

More than 50 | 90 |

More than 55 | 74 |

More than 60 | 54 |

More than 65 | 24 |

Question 30: A container opened at the top and made up of a metal sheet, is in the form of a frustum of a cone of height cm with radii of its lower and upper ends as cm and cm respectively. Find the cost of milk which can completely fill the container, at the rate of Rs. per liter. Also find the cost of metal sheet used to make the container, if it costs Rs. per . (Take )

Answer:

cm, cm, cm

cm

Volume of frustum

liters

Cost of milk Rs

Surface Area of the container

Cost of metal sheet

Cost of sheet metal Rs