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.
Answer:
Given
Question 2: What is the value of ?
Answer:
Question 3: If is one root of the quadratic equation
, then find the value of
.
Answer:
If is one root of the quadratic equation
than it should satisfy the equation.
Question 4: Find the distance of a point from the origin.
Answer:
Distance of a point from the origin
Question 5: What is the HCF of smallest prime number and the smallest composite number ?
Answer:
Smallest composite number is
Smallest prime number is
Therefore HCF of is
Question 6: In an AP, if the common difference , and the seventh term
is
, then find the first term.
Answer:
Given
Section – B
Question number 7 to 12 carry 2 mark each.
Question 7: An integer is chosen at random between . Find the probability that it is :
(i) divisible by .
(ii) not divisible by .
Answer:
Number of integers between
Total number of outcomes
(i) Numbers which are divisible by are:
Favorable Outcomes
(ii) Probability that integer is not divisible by
Question 8: Two different dice are tossed together. Find the probability:
(i) of getting a doublet
(ii) of getting a sum , of the numbers on the two dice.
Answer:
Given that two different dice are tossed.
Total number of outcomes
(i) Let be the event of getting a doublet.
(ii) Let be the event of getting a sum of
of the numbers of two dice.
Question 9: Find the ratio in which divides the line segment joining the points
. Hence find
.
Answer:
Let divides
in the ratio of
By section formula,
Question 10: Given that is irrational, prove that
is an irrational number.
Answer:
Given that, is irrational
Let us assume is rational.
As is rational. (Assumed) They must be in the form of
where
, and
are co prime.
Therefore we contradict the statement that, is rational.
Hence proved that is irrational.
Question 11:
In Fig. 1, is a rectangle. Find the values of
and
.

Answer:
Given:
… … … … … i)
… … … … … ii)
Adding i) and ii) we get
Question 12: Find the sum of first multiples of
.
Answer:
The first multiples of
are :
Number of terms
The first term
Common difference
Hence the sum of the first multiples of
is
.
Section – C
Question number 13 to 22 carry 3 mark each.
Question 13: A plane left minutes late than its scheduled time and in order to reach the destination
km away in time, it had to increase its speed by
km/h from the usual speed. Find its usual speed.
Answer:
Distance to travel km
Let the usual speed
Increased speed
km/hr or
km/hr (this is not possible as speed cannot be negative)
Therefore the usual speed
Question 14: Prove that the area of an equilateral triangle described on one side of the square is equal to half the area of the equilateral triangle described on one of its diagonal.
Or
If the area of two similar triangles are equal, prove that they are congruent.
Answer:
Let the side of the square
Therefore the sides of the equilateral as well
Now
Therefore the sides of the equilateral
Since the triangles are equilateral, each of the angles are . Hence by
criterion, the two triangles are similar.
Therefore
Hence proven.
Or
Given: and
Also and
To Prove:
Since we know
Therefore we get
Therefore by SSS criterion.
Hence proved.
Question 15: Prove that the lengths of tangents drawn from an external point to a circle are equal.
Answer:
Given: Let the circle with center
Let be an external point from which tangents
and
are drawn as shown in the diagram
To prove:
Construction: Join and
Proof: As is tangent
. Therefore
Similarly, As is tangent
. Therefore
(Note: Tangents at any point on a circle is perpendicular to the radius through the point of contact)
In and
is common
(radius of the same circle)
Therefore both tangents are equal in length.
Question 16: A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 2. If the height of the cylinder is cm and its base is of radius
cm. Find the total surface area of the article.

Or
A heap of rice is in the form of a cone of base diameter m and height
m. Find the volume of the rice. How much canvas cloth is required to just cover the heap ?
Answer:
cm Height
cm
Or
Diameter m
Radius
m Height
m
m
Hence Volume of Rice and Canvas required to cover the heap
Question 17: The table below shows the salaries of 280 persons:
Salary (in thousand Rs.) | No. of persons |
5-10 | 49 |
10-15 | 133 |
15-20 | 63 |
20-25 | 15 |
25-30 | 6 |
30-35 | 7 |
35-40 | 4 |
40-45 | 2 |
45-50 | 1 |
Answer:
Class Interval | Frequency |
Cumulative Frequency |
5 – 10 | 49 | 49 |
10 – 15 | 133 | 182 |
15 – 20 | 63 | 245 |
20 – 25 | 15 | 260 |
25 – 30 | 6 | 266 |
30 – 35 | 7 | 273 |
35 – 40 | 4 | 277 |
40 – 45 | 2 | 279 |
45 – 50 | 1 | 280 |
Median
Therefore median salary Rs.
Or
If , where
is an acute angle, find the value of
.
Answer:
Therefore
Or
Question 19: Find the area of the shaded region in Fig. 3, where arcs drawn with centres and
intersect in pairs at mid-points
and
of the sides
and
respectively of a square
of side
cm. [Use
]

Answer:
Given square of side
cm
Area of square
Area of
Therefore total area enclosed in arc’s
Therefore shaded area
Question 20: If and
are the vertices of a parallelogram
, find the values of a and b. Hence find the lengths of its sides.
Or
If and
are the vertices of a quadrilateral, find the area of the quadrilateral
.
Answer:
Since the mid point of and
are the same
Similarly,
Therefore and
Hence is a Rhombus
Or
Join
Note: For a with vertices
the area of the triangle is
sq. units
sq. units
Question 21: Find HCF and LCM of and
and verify that HCF
LCM
Product of the two given numbers.
Answer:
Therefore HCF of and
Also LCM of and
Product of two numbers
HCF LCM
Hence Product of two numbers HCF
LCM
Question 22: Find all zeroes of the polynomial if two of its zeroes are
and
.
Answer:
Given and
are zeros of the polynomial
Therefore is a factor of the polynomial
Hence
Therefore is a factor of
and
are factors of
Section – D
Question number 23 to 30 carry 4 mark each.
Question 23: Draw a triangle with
cm,
cm and
. Then construct a triangle whose sides are
of the corresponding sides of the
.
Answer:
Question 24: The sum of four consecutive numbers in an AP is and the ratio of the product of the first and the last term to the product of two middle terms is
. Find the numbers.
Answer:
Let the four consecutive terms of the AP be and
Given:
… … … … … i)
Substituting from i)
or
The the first four terms are
The the first four terms are
Question 25: In an equilateral ,
is a point on side
such that
. Prove that:
Or
Prove that, in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
Answer:
Given: is an equilateral triangle.
To prove:
Construction: Draw
Consider and
is common
(equilateral triangle)
(By RHS criterion)
Using Pythagoras theorem,
In
… … … … … i)
In
… … … … … ii)
From i) and ii)
But
Hence proved.
Or
To prove:
Draw:
Proof: In
(common angle)
( by AA criterion)
… … … … … i)
Similarly,
and … … … … … ii)
Adding i) and ii)
. Hence proved.
Question 26: A motor boat whose speed is km/hr in still water takes
hr more to go
km upstream than to return downstream to the same spot. Find the speed of the stream.
Or
A train travels at a certain average speed for a distance of km and then travels at a distance of
km at an average speed of
km/hr more than its original speed. If it takes
hours to complete total journey, what is the original average speed ?
Answer:
Distance traveled by the boat km
Speed of the boat
Let the speed of the stream
Therefore speed of boat upstream
Also speed of boat downstream
km/hr or
(not possible as speed cannot be negative)
Therefore the speed of the stream is
Or
Let the original speed is
New speed
or
(not possible as speed cannot be negative)
Hence original speed
Question 27: As observed from the top of a m high light house from the sea-level, the angles of depression of two ships are
and
. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships. [Use
]
Answer:
In
In
Therefore the distance between the ships is
Question 28: The diameters of the lower and upper ends of a bucket in the form of a frustum of a cone are cm and
cm respectively. If its height is
cm, find :
(i) The area of the metal sheet used to make the bucket.
(ii) Why we should avoid the bucket made by ordinary plastic ? [Use ]
Answer:
Diameter of upper end of bucket
Radius of the upper end of the frustum of cone
Diameter of lower end of bucket
radius of the lower end of the frustum of cone
Height of the frustum of Cone
Therefore surface area
Hence, the Area of metal sheet used to make the bucket is
Question 29: The mean of the following distribution is . Find the frequency
of the class
.
Class: | 11-13 | 13-15 | 15-17 | 17-19 | 19-21 | 21-23 | 23-25 |
Frequency: | 3 | 6 | 9 | 13 | 5 | 4 |
Or
The following distribution gives the daily income of 50 workers of a factory:
Daily Income (in Rs) | 100-120 | 120-140 | 140-160 | 160-180 | 180-200 |
Number of workers | 12 | 14 | 8 | 6 | 10 |
Convert the distribution above to a less than type cumulative frequency distribution and draw its ogive.
Answer:
Class Interval | Frequency |
||
11-13 | 3 | 12 | 36 |
13-15 | 6 | 14 | 84 |
15-17 | 9 | 16 | 144 |
17-19 | 13 | 18 | 234 |
19-21 | 20 | ||
21-23 | 5 | 22 | 110 |
23-25 | 4 | 24 | 96 |
Or
Daily wages in Rs (less than) | Frequency |
Cumulative Frequency |
Less than 120 | 12 | 12 |
Less than 140 | 14 | 26 |
Less than 160 | 8 | 34 |
Less than 180 | 6 | 40 |
Less than 200 | 10 | 50 |
Question 30: Prove that:
Answer: