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: Write the discriminant of the quadratic equation .
Answer:
Given Equation:
Comparing it with .
and
Determinant
Question 2: Find after how many places of decimal the decimal form of the number will terminate.
Or
Express as a product of its prime factors.
Answer:
Simplifying
So the decimal form will terminate are digits.
Or
Question 3: Find the sum of first multiples of
.
Answer:
is
Question 4: Find the value(s) of , if the distance between the points
and
is
units.
Answer:
Given: and
units
Therefore could be
or
Question 5: Two concentric circles of radii and
are given. Find the length of the chord of the larger circle which touches the smaller circle.
Answer:
Length of the chord
Question 6: In Figure 1, cm,
cm,
and
cm. Evaluate
.

Or
If
, find the value of
.
Answer:
cm
Or
Section – B
Question number 7 to 12 carry 2 mark each.
Question 7: Points and
are vertices of a parallelogram
. Find the values of
and
.
Or
Points and
trisect the line segment joining the points
and
such that
is near to
. Find the coordinates of points
and
.
Answer:
Let midpoint of be
is also the midpoint of
Therefore
Or
Using section formula [ divides
in the ratio of
]
Similarly,
Using section formula [ divides
in the ratio of
]
Question 8: Solve the following pair of linear equations :
Answer:
Given equations:
… … … … … i)
… … … … … ii)
Multiplying i) by and subtracting ii) from the resultant equation
Substituting into i)
Hence
and
Question 9: If HCF of and
is expressible in the form
, then find the value of
.
Or
On a morning walk, three persons step out together and their steps measure cm,
cm and
cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps ?
Answer:
Therefore HCF of and
is
Or
Finding LCM of and
Therefore LCM
Therefore the minimum distance each should walk should be cm.
Question 10: A die is thrown once. Find the probability of getting (i) a composite number, (ii) a prime number.
Answer:
Total possible outcomes are
Therefore total number of events
i) Composite numbers are and
Therefore Probability (Composite Numbers)
ii) Prime numbers are
Therefore Probability (Prime Numbers)
Question 11: Using completing the square method, show that the equation has no solution.
Answer:
Given equation
A square of a number is always positive. Hence the given equation has no solution.
Question 12: Cards numbered to
were put in a box. Poonam selects a card at random. What is the probability that Poonam selects a card which is a multiple of
?
Answer:
Total number of cards
Favorable outcomes
Therefore Probability
Section – C
Question number 13 to 22 carry 3 mark each.
Question 13: The perpendicular from on side
of a
meets
at
such that
. Prove that
.
Or
and
are medians of triangles
and
respectively where
. Prove that
Answer:
Given: with
Also
To Prove: .
Proof: Let
Also
Using Pythagoras theorem, in right angled triangle
… … … … … i)
Using Pythagoras theorem, in right angled triangle
… … … … … ii)
Subtracting ii) from i)
(Putting
back in the equation)
.
Hence Proved
Or
Given: and
is the median of
and
is the median of
Also
To Prove:
Proof: Since is a median
Similarly, is the median
Now,
(corresponding sides of similar triangles)
So,
(Since AD and PM are medians)
… … … … … i)
Also since
(corresponding angles of similar triangles)
Now in and
from i)
Hence (By SAS criterion)
Since corresponding sides of similar triangles are proportional
. Hence proved.
Question 14: Check whether is a factor of
by dividing polynomial
by polynomial
, where
,
Answer:
If divisible by
the remainder of
Therefore remainder is .
Therefore is not a factor of
.
Question 15: Find the area of the triangle formed by joining the mid-points of the sides of the triangle , whose vertices are
and
.
Answer:
Mid-point of
Mid-point of
Mid-point of
Area of
sq. units.
Question 16: Draw the graph of the equations and
. Using this graph, find the values of
and
which satisfy both the equations.
Answer:
For : When
and when
For : When
and when
From the graph the two lines intersect at . This is the point that satisfy both the equations.
Question 17: Prove that is an irrational number.
Or
Find the largest number which on dividing and
leaves remainders
and
respectively.
Answer:
Assume is a rational number
i.e. it can be expressed as a rational fraction of the form where
are relatively prime numbers.
Since
We have
or
If is even, then
is also even in which case
is not in simplest form. If
is odd then
is also odd. Therefore:
Since is an integer, the left hand side is even. Since
is an integer, the right hand side is odd and we have found a contradiction, therefore our hypothesis is false.
Therefore is an irrational number
Or
Since, and
are the remainders of
and
respectively. Thus, after subtracting these remainders from the numbers.
We have the numbers, and
Which is divisible by the required number.
Now, required number HCF of
and
By Euclid’s division algorithm … … … … … (i)
dividend
divisor
quotient
remainder
For largest number, put and
Now
HCF
and
Now, we take and
, then again using Euclid’s division algorithm,
[ Now
]
HCF
and
Hence, is the largest number which divides
and
leaving remainder
and
respectively
Question 18: and
are interior angles of a triangle
. Show that
(i)
(ii) If , then find the value of
Or
If and
,
, then find the values of
and
.
Answer:
i) To prove:
In , sum of angles
… … … … … i)
Hence Proved.
ii) Similarly,
… … … … … ii)
From i) and ii) and
Or
Given
… … … … … i)
Similarly,
… … … … … ii)
Solving i) and ii)
From i)
Hence and
Question 19: In Figure 2, is a chord of length
cm of a circle of radius
cm. The tangents at
and
intersect at a point
. Find the length
.

Or
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle.
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)
cm
In Right triangle
cm
Let
In Right triangle
… … … … … i)
Since is a tangent,
In Right triangle
From i)
Hence is
cm
Or
Given: A circle with center touches the side
and
of a quadrilateral
at points
and
.
To Prove: and
Construction: Join and
Proof: Since the two tangents from an external point to a circle subtend equal angles at the center;
and
Since the sum of all the angles subtended at the point is
Similarly
Question 20: Water in a canal, m wide and
m deep, is flowing with a speed of
km/h. How much area will it irrigate in
minutes if
cm of standing water is needed ?
Answer:
Speed of water
Rate of flow
Therefore volume of water in mins
Let Area of irrigation
Question 21: A class teacher has the following absentee record of students of a class for the whole term. Find the mean number of days a student was absent.
Number of Days: | 0-6 | 6-12 | 12-18 | 18-24 | 24-30 | 30-36 | 35-42 |
Number of Students: | 10 | 11 | 7 | 4 | 4 | 3 | 1 |
Answer:
Class Interval | |||
0-6 | 10 | 3 | 30 |
6-12 | 11 | 9 | 99 |
12-18 | 7 | 15 | 105 |
18-24 | 4 | 21 | 84 |
24-30 | 4 | 27 | 108 |
30-36 | 3 | 33 | 99 |
36-42 | 1 | 39 | 39 |
Mean value
Question 22: A car has two wipers which do not overlap. Each wiper has a blade of
length cm sweeping through an angle
. Find the total area cleaned at each sweep of the blades. (Take
)
Answer:
Total area
Section – D
Question number 23 to 30 carry 4 mark each.
Question 23: A pole has to be erected at a point on the boundary of a circular park of diameter m in such a way that the difference of its distances from two diametrically opposite fixed gates
and
on the boundary is
m. Is it possible to do so ? If yes, at what distances from the two gates should the pole be erected ?
Answer:
Let and
We know that diameter will subtend a right angle on circumference.
(this is not possible as b cannot be negative)
m
m
Question 24: If times the
term of an Arithmetic Progression is equal to
times its
term and
, show that the
term of the A.P. is zero.
Or
The sum of the first three numbers in an Arithmetic Progression is . If the product of the first and the third term is
times the common difference, find the three numbers.
Answer:
Let the first term of AP
common difference
We have to show that term is zero or
term
term
Given that
Hence Proved
Or
Let first three numbers of A.P are ,
and
Sum of the first three numbers or
Product of first and third term or
or
Hence and three numbers are
Question 25: Construct a triangle with side
cm,
cm and
. Then construct another triangle whose sides are
of the corresponding sides of the triangle
.
Answer:
Question 26: In Figure 3, a decorative block is shown which is made of two solids, a cube and a hemisphere. The base of the block is a cube with edge cm and the hemisphere fixed on the top has a diameter of
cm. Find
(a) the total surface area of the block.
(b) the volume of the block formed. (Take
)

Or
A bucket open at the top is in the form of a frustum of a cone with a capacity of . The radii of the top and bottom circular ends are
cm and
cm respectively. Find the height of the bucket and the area of metal sheet used in making the bucket. (Use
)
Answer:
a) Total surface area
b) Volume of the block
Or
Volume
cm
cm
Volume of the bucket
cm
Therefore height of bucket cm
cm
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 27: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio.
Or
Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
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.
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 28: If , then prove that
or
Answer:
We have
Dividing both sides with
Let
or
Hence
Question 29: Change the following distribution to a ‘more than type’ distribution. Hence draw the ‘more than type’ ogive for this distribution.
Class Interval: | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 |
Frequency | 10 | 8 | 12 | 24 | 6 | 25 | 15 |
Answer:
Less Than table
Less Than | Cumulative Frequency |
Less than 30 | 10 |
Less than 40 | 18 |
Less than 50 | 30 |
Less than 60 | 54 |
Less than 70 | 60 |
Less than 80 | 85 |
Less than 90 | 100 |
More than table
More Than | Cumulative Frequency |
More than 30 | 10 |
More than 40 | 18 |
More than 50 | 30 |
More than 60 | 54 |
More than 70 | 60 |
More than 80 | 85 |
More than 90 | 100 |
Draw the chart. The point of intersection is :
Question 30: The shadow of a tower standing on a level ground is found to be m longer when the Sun’s altitude is
than when it was
. Find the height of the tower. (Given
)
Answer:
Let be the height of the tower.
… … … … … i)
Also
… … … … … ii)
From i) and ii)
m