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: Find the coordinates of a point , where
is the diameter of a circle whose center is
and
is the point
.
Answer:
Let the coordinates of be
Since is the center, it is also the midpoint of
Question 2: For what value of , the roots of the equation
are real?
Or
Find the value of , for which the roots of the equation
are reciprocal of each other.
Answer:
Given
For roots to be real,
Or
Given
Question 3: Find if
Or
Find the value of
Answer:
Or
Question 4: How many two digits number are divisible by ?
Answer:
Numbers divisible by are
Lowest digit number divisible by
is
The highest digit number divisible by
is
So the series starts with and ends with
Difference between the numbers is
So the AP will be
We need to find
Now
Therefore there are 30 two digit numbers divisible by 3
Question 5: In Fig. 1, cm and
cm. What is the ratio of the
to the
?

Answer:
Given
Consider and
(alternate angles)
is common
( alternate angles)
( AAA criterion)
Since , by Thales Theorem
Since triangles are similar,
Question 6: Find a rational number between and
.
Answer:
We know and
Therefore is between
and
Hence is between
and
Section – B
Question number 7 to 12 carry 2 mark each.
Question 7: Find the HCF of and
using Euclid’s algorithm.
Or
Show that every positive odd integer is of the form or
, where
is some integer.
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
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
Therefore any odd integer is of the form or
.
Question 8: Which term of
will be
more than its
term?
Or
If , the sum of the first
terms of an
is given by
, find the
term.
Answer:
Given is
Here
Therefore term is given by
Required term
Let it be the term, then
Hence term is the required term.
Or
Given
Put
Hence the common difference is
Therefore term
Question 9: Find the ratio in which the segment joining the points and
is divided by
. Also find the coordinates of this point on
.
Answer:
Given Points and
Let the point be the point which divides the segment joining
and
in the ratio
The using the section formula, we get
Therefore divides
and
in the ratio
Question 10: A game consists of tossing a coin three times and noting the outcome each time. If getting the same result in all the tosses is success, find the probability of losing the game.
Answer:
Possible outcomes of tossing a coin three times is
i.e. Total number of events
The favorable events are and
Question 11: A die is thrown once. Find the probability of getting a number which i) is a prime number ii) lies between and
.
Answer:
The possible outcomes when a dice is thrown one are
Therefore Total No. of events
No of prime events are prime numbers
No of events when the number lies between and
Therefore
Note: 1 only has one positive divisor itself
. So it is not a prime number. A prime number is a positive integer whose positive divisor are exactly
and the number itself.
Question 12: Find if the system of equations
has infinitely many solutions.
Answer:
Given: and
If the system of equations are and
and they have infinitely many solutions then it satisfy the following:
From the given system
Therefore
Taking the first two
… … … … … (i)
Taking the last two
… … … … … (ii)
From (i) and (ii) we get
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.
must be even and since
is even,
should be even
Let
We have
Since is even,
is even and since
is even,
must be even
However two even numbers cannot be relatively prime so cannot be expresses as a rational fraction.
Hence is irrational.
Question 14: Find the value of such that the polynomial
has sum of its zeros equal to half of their products.
Answer:
Polynomial is
Let and
be the zeros of the equation
On comparing with we get
Therefore
… … … … … i)
… … … … … ii)
Given
Question 15: A father’s age is three times the sum of the ages of his two children. After years his age will be two times the sum of their ages. Find the present age of the father.
Or
A fraction becomes when
is subtracted from the numerator and it becomes
when
is subtracted from the denominator. Find the fraction.
Answer:
Let the ages of sons be and
Given that father’s age is times the ages of his two sons
Therefore present age of father … … … … … i)
Given years hence, father’s are will be twice the sum of the ages of his sons
Father’s age after years
Ages of his sons year hence
and
… … … … … ii)
Using i) and ii) we get that the age of father is years.
Or
Let the fraction be
… … … … … i)
Also
… … … … … ii)
Subtracting ii) from i) we get
From ii)
Hence the fraction is
Question 16: Find the point on which is equidistant from the point
and
.
Or
The line segment joining the points and
is trisected at points
and
such that
is nearer to
. If
also lies on the line given by
, find the value of
.
Answer:
Let be equidistant from
and
Since
Hence the point is
Or
(trisects)
Applying section formula
Coordinates of
Since lies on
Question 17: Prove that .
Or
Answer:
To Prove:
LHS
RHS. Hence proved.
Or
To Prove:
LHS
RHS. Hence proved.
Question 18: In Fig. 2, is a chord of length
cm of a circle of radius
cm and center
. The tangent at
and
intersect at point
. Find the length of
.

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)
In Right triangle
cm
Let
In Right triangle
… … … … … i)
Since is a tangent,
In Right triangle
From i)
Hence is
Question 19: In Fig. 3, and
, prove that

Or
If and
are point of sides
and
respectively, of
, right angled at
, prove that
Answer:
Given:
In … … … … … i)
In … … … … … ii)
Adding i) and ii)
. Hence proved.
Or
… … … … … i)
… … … … … ii)
Adding i) and ii)
. Hence proved.
Question 20: Find the area of the shaded region in Fig. 4, if is a rectangle with sides
cm and
cm and
is the center of the circle. (Take
)

Answer:
(angle subtended by the diameter on the circumference is )
is a rectangle
cm
Therefore the radius of the circle cm
Area of the circle
Area of
Hence the shaded area
Question 21: Water in a canal, m wide and
m deep, is flowing at the speed of
km/hour. How much area will it irrigate in
minutes; if $latex 8 cm standing water is needed?
Answer:
Let Area of irrigation
Question 22: Find the mode of the following frequency distribution
Class | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
Frequency | 8 | 10 | 10 | 16 | 12 | 6 | 7 |
Answer:
Section – D
Question number 23 to 30 carry 4 mark each.
Question 23: Two water taps can together can fill in a tank in
hours. The tap with longer diameter takes
hours less than the tap with the smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately.
Or
A boat goes km upstream and
km downstream in
hours. In
hours it can go
km upstream and
km downstream. Determine the speed of the stream and that of the boat in still water.
Answer:
Let the time taken by the smaller tap to fill up the tank completely hours
Also it is given that the time taken by the larger tap is 2 hours less
hours
cannot be
hours as the larger tap takes
hours less.
Hence hours
Therefore small tap will fill the tank in hours and the larger tap will fill the tank in
hours.
Or
Let the speed of the boat in still water km/hr
Let the speed of the stream km/hr
Speed of the boat downstream km/hr
Sped of the boat upstream km/hr
Boat goes km upstream and
km downstream in
hours
… … … … … … i)
Boat goes km upstream and
km downstream in
hours
… … … … … … ii)
Substituting in i) and ii)
… … … … … … iii)
… … … … … … iv)
From iii) we get
Substituting in iv) we get
Substituting in iii) we get
Now solving for and
… … … … … … v)
… … … … … … vi)
Adding v) and vi) we get
From vi)
Therefore speed o boat in still water is km/hr and speed of stream is
km/hr
Question 24: If the sum of the first four terms of an is
and that of the first
terms is
. Find the sum of the first
terms.
Answer:
… … … … … i)
… … … … … ii)
solving i) and ii)
gives and
so
Question 25: Prove that
Answer:
RHS. Hence proved.
Question 26: A man in a boat is rowing away from the light house at m high takes
minutes to change the angle of elevation of the top of the light house from
to
. Find the speed of the boat in meters per minute. [ Use
]
Or
Two poles of equal height are standing opposite each other on either side of the road, which is m wide. From a point between them on the road, the angle of elevation of the top of the poles are
to
respectively. Find the height of the poles and the distance of the point from the poles.
Answer:
Or
Given
Question 27: Construct a in which
is
cm,
cm and
. The construct a triangle whose sides are
of the corresponding sides of
.
Answer:
Question 28: A bucket open at the top is in the form of a frustum of a cone with a capacity . The radii of the top and bottom of the circular ends of the bucket and
cm and
cm respectively. Find the height of the bucket, and also the area of the metal sheet used in making it. ( Use
)
Answer:
Volume
cm
cm
Therefore height of bucket
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 29: Prove that in a right angle triangle, the square of the hypotenuse is equal to the sum of square of the other two sides.
Answer:
Given: A right angled , right angled at
To prove:
Draw:
Proof: In
(common angle)
( by AA criterion)
… … … … … i)
Similarly,
and … … … … … ii)
Adding i) and ii)
. Hence proved.
Question 30: If the median of the following frequency distribution is , find the values of
and
.
Class | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | Total |
Frequency | 5 | 9 | 12 | 3 | 2 | 40 |
Or
The marks obtained by students in a class in an examination are given below.
Marks | No. of Students |
0-5 | 2 |
5-10 | 5 |
10-15 | 6 |
15-20 | 8 |
20-25 | 10 |
25-30 | 25 |
30-35 | 20 |
35-40 | 18 |
40-45 | 4 |
45-50 | 2 |
Answer:
Class Interval | Frequency | Cumulative Frequency |
0-10 | ||
10-20 | 5 | |
20-30 | 9 | |
30-40 | 12 | |
40-50 | ||
50-60 | 3 | |
60-70 | 2 | |
Given, Median
The median class
Given sum of frequencies
Hence and
Or
i)
Marks | Number of Students |
Less than 5 | 2 |
Less than 10 | 7 |
Less than 15 | 13 |
Less than 20 | 21 |
Less than 25 | 31 |
Less than 30 | 56 |
Less than 35 | 76 |
Less than 40 | 94 |
Less than 45 | 98 |
Less than 50 | 100 |
ii)
Hence Median