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: The HCF of two numbers and
is
and their LCM is
. Find the product
.
Answer:
and
are two numbers.
H.C.F. of and
L.C.M. of and
So, By Fundamental theorem of Arithmetic , we have
So, The value of product of and
is
Question 2: Find the value of for which
is a solution of the equation
.
Or
Find the value(s) of for which the quadratic equation
has real and equal roots.
Answer:
Given equation:
If is a solution, then it should satisfy the given equation.
Or
Given equation:
For roots to be equal,
Question 3: If in an A.P., and
, then find the value of
.
Answer:
Given: and
Since
Question 4: If and
is an acute angle, find the value of
.
Or
Find the value of .
Answer:
Or
Question 5: The area of two similar triangles are sq. cm and
sq. cm. Find the ratio of their corresponding sides.
Answer:
Let and
are corresponding sides of the similar triangles
Hence the ratio of the corresponding sides
Question 6: Find the value of so that the point
lies on the line represented by
.
Answer:
If lies on the equation
, it must satisfy the equation.
Section – B
Question number 7 to 12 carry 2 mark each.
Question 7: If , the sum of the first
terms of an A.P. is given by
, then find its
term.
Or
If the term of an A.P. exceeds its
term by
, find the common difference.
Answer:
Sum of first terms of AP,
Now choose and put in the above formula, First term
Now put to get the sum of first two terms
This means First term + Second term
But first term as calculated above Hence, second term
So common difference (d) becomes,
So the AP becomes,
Or
Let the first term and the common difference
Given
Common difference
Question 8: The mid-point of the line segment joining and
is
. Find the values of
and
.
Answer:
Given points and
Hence
Question 9: A child has a die whose 6 faces show the letters given below :
The die is thrown once. What is the probability of getting (i) A (ii) B ?
Answer:
Number of possible events
No of A’s on dice
No of B’s of dice
Question 10: Find the HCF of and
using prime factorization.
Or
Show that any positive odd integer is of the form or
or
, where
is some integer.
Answer:
First factorize each number
Therefore HCF of and
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
or
Therefore any odd integer is of the form or
or
.
Question 11: Cards marked with numbers to
(one number on one card) are placed in a box and mixed thoroughly. One card is drawn at random from the box. Find the probability that the number on the card taken out is (i) a prime number less than
, (ii) a number which is a perfect square.
Answer:
Total number of ways to select a card (Cards are marked
to
)
i) Prime numbers less than are
and
only
ii) Numbers which are perfect squares are
Question 12: For what value of , does the system of linear equations
,
have an infinite number of solutions ?
Answer:
If the system of equations are and
and they have infinitely many solutions then it satisfy the following:
For and
have an infinite number of solutions:
From first two terms
From second and third term
From first and third term
Therefore , the equation with have infinite solutions.
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.
This would imply that is a multiple of
. Since
is prime, this implies
is a multiple of
. Thus
for some integer
, and
Dividing by , this means
So is a multiple of
, and, just as it did for
, this means
is a multiple of
. But
and
were presumed to lack a common factor other than
, so this is a contradiction, and the fraction
for
must fail to exist.
Hence is irrational.
Question 14: Find all the zeroes of the polynomial , if two of its zeroes are
and
.
Answer:
Two zeros are and
Therefore and
are factors of
is a factor of
Therefore the other two zeros are
Question 15: Point P divides the line segment joining the points and
such that
. If
lies on the line
, find the value of
.
Or
For what value of , are the points
and
collinear ?
Answer:
(trisects)
Applying section formula
Since lies on
Or
Given point and
If the points are collinear are of the
Question 16: Prove that:
Or
If show that
Answer:
LHS
RHS. Hence proved.
Or
Given:
Squaring both sides
Add on both sides
Hence proven.
Question 17: A part of monthly hostel charges in a college hostel are fixed and the remaining depends on the number of days one has taken food in the mess. When a student takes food for
days, he has to pay Rs.
, whereas a student
who takes food for
days, has to pay Rs.
. Find the fixed charges per month and the cost of food per day.
Answer:
Let the fixed charges
Let the charges for food per day
Therefore for Student A:
… … … … … i)
Therefore for Student B:
… … … … … ii)
Solving i) and ii)
Rs.
Rs.
Therefore the fixed charges are Rs. and food cost per day is
Rs.
Question 18: In and
is the mid-point of
. Prove that
.
Or
In Figure 1, is a point on
produced of an isosceles
, with side
. If
and
, prove that
.

Answer:
Given:
From
… … … … … i)
From
… … … … … ii)
From i) and ii)
Hence proved.
Or
Given:
and
In and
(given)
( By AA criterion)
Question 19: Prove that the parallelogram circumscribing a circle is a rhombus.
Answer:
Given: be a parallelogram circumscribing a circle with center
.
To prove: is a rhombus.
We know that the tangents drawn to a circle from an exterior point are equal in length.
Therefore, and
.
Adding the above equations,
(Since, is a parallelogram so
and
)
Therefore, .
Hence, is a rhombus.
Question 20: In Figure 2, three sectors of a circle of radius cm, making angles of
,
and
at the centre are shaded. Find the the area of the shaded region.

Answer:
Area of the shaded region
Question 21: The following table gives the number of participants in a yoga camp :
Age (in years): | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
No. of participants: | 8 | 40 | 58 | 90 | 83 |
Find the modal age of the participants.
Answer:
First we find the modal class of the given data which is the highest participant class i.e.
Modal class
Formula to find Mode is
Question 22: A juice seller was serving his customers using glasses as shown in Figure 3. The inner diameter of the cylindrical glass was cm but bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass. If the height of a glass was
cm, find the apparent and actual capacity of the glass. (Use
)

Or
A girl empties a cylindrical bucket full of sand, of base radius cm and height
cm on the floor to form a conical heap of sand. If the height of this conical heap is
cm, then find its slant height correct to one place of decimal.
Answer:
Apparent volume
Or
Volume of sand remains the same
cm
Slant height cm
Section – D
Question number 23 to 30 carry 4 mark each.
Question 23: A train travels km at a uniform speed. If the speed had been
km/hr more, it would have taken
hr less for the same journey. Find the speed of the train.
Or
Answer:
Distance traveled km
Let the original speed km/hr
km/hr or
km/hr (this is not possible as speed cannot be negative)
Hence the original speed km/hr
Or
or
Question 24: If the sum of the first terms of an A.P. is
and the sum of the first
terms is
; then show that the sum of the first
terms is
.
Answer:
Let be the first term and
be the common difference of the AP
Given
… … … … … i)
Also
… … … … … ii)
Subtracting ii) from i)
Question 25: In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then prove that the angle opposite to the first side is a right angle.
Answer:
Given:
To prove:
Construction: is a right angled at
such that
and
Proof: From
(Pythagoras theorem)
(by construction) … … … … … i)
But (given) … … … … … ii)
from i) and ii)
… … … … … iii)
Now in and
(by construction)
(by construction)
(from iii)
(by SSS criterion)
But by construction
. Hence proved.
Question 26: Construct an isosceles triangle whose base is and altitude
and then another triangle whose sides are
times the corresponding sides of the isosceles triangle.
Answer:
Question 27: A boy standing on a horizontal plane finds a bird flying at a distance of m from him at an elevation of
. A girl standing on the roof of a
m high building, finds the elevation of the same bird to be
. The boy and the girl are on the opposite sides of the bird. Find the distance of the bird from the girl. (Given
)
Or
The angle of elevation of an airplane from a point on the ground is
. After a flight of
seconds, the angle of elevation changes to
. If the plane is flying at a constant height of
meters, find the speed of the airplane.
Answer:
In
In
Or
In
m
In
Hence
Question 28: Find the values of frequencies and
in the following frequency distribution table, if
and median is
.
Marks: | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | Total |
No. of Students | 10 | 25 | 30 | 10 | 100 |
Or
For the following frequency distribution, draw a cumulative frequency curve (ogive) of ‘more than type’ and hence obtain the median value.
Class: | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | Total |
Frequency: | 5 | 10 | 20 | 23 | 17 | 11 | 9 |
Answer:
Class Interval | Frequency ( |
Cumulative Frequency ( |
0 – 10 | 10 | 10 |
10 – 20 | ||
20 – 20 | 25 | |
30 – 40 | 30 | |
40 – 40 | ||
50 – 60 | 10 |
… … … … … i)
For the above distribution, median class is
From i)
Or
Class Interval (More than) | Frequency |
More than 0 | 100 |
More than 10 | 95 |
More than 20 | 80 |
More than 30 | 60 |
More than 40 | 37 |
More than 50 | 20 |
More than 60 | 9 |
Class Interval (Less than) | Frequency |
Less than 10 | 5 |
Less than 20 | 20 |
Less than 30 | 40 |
Less than 40 | 63 |
Less than 50 | 80 |
Less than 60 | 91 |
Less than 70 | 100 |
Question 29: Prove that:
Answer:
Since
RHS. Hence proved.
Question 30: An open metallic bucket is in the shape of a frustum of a cone. If the diameters of the two circular ends of the bucket are cm and
cm and the vertical height of the bucket is
cm, find the area of the metallic sheet used to make the bucket. Also find the volume of the water it can hold. (Use
)
Answer:
liters
Therefore surface area