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: 90
General Instructions:
(i) All questions are compulsory
(ii) The question paper consists of 31 questions divided into four sections – A, B, C and D
(iii) Section A consists of 4 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 11 questions of 4 marks each.
(iv) Use of calculator is not permitted.
SECTION – A
Question number 1 to 6 carry 1 mark each.
Question 1: If the quadratic equation has two equal roots, then find the value of
.
Answer:
The given quadratic equation is
For real roots, Discriminant
is not possible as whole equation will become
then. Hence
.
Question 2: In Figure 1, a tower is
m high and
, its shadow on the ground, is
m long. Find the Sun’s altitude.

Answer:
Let be the tower and
be it’s shadow.
Therefore sunset is at an altitude of .
Question 3: Two different dice are tossed together. Find the probability that the product of the two numbers on the top of the dice is .
Answer:
Two dice are tossed. Therefore
Therefore the total number of outcomes when two dices are tossed
Favorable events of getting the product as are
Question 4: In Figure 2, is a chord of a circle with centre
and
is a tangent. If
, find
.

Answer:
Given: is a tangent so
(sum of the angles in a triangle is
)
Section – B
Question number 5 to 10 carry 2 mark each.
Question 5: In Figure 3, two tangents and
are drawn from an external point
to the circle with centre
. If
, then prove that
.

Answer:
… … … … … i)
And in and
(from equation i)
(Radii of same circle)
(common side)
( by RHS criterion)
So, … … … … … ii)
And … … … … … iii)
Substitute (given) And from equation iii) we get
So in we get
(substitute value from equation ii) we get)
Question 6: In Figure 4, a triangle is drawn to circumscribe a circle of radius
, such that the segments
and
are respectively of lengths
and
. If the area of
is
, then find the lengths of sides
and
.

Answer:
Given
Construction: Join and
Proof: Area of the area of
area of
area of
Let
,
(equal tangents)
,
(equal tangents)
Therefore
Therefore sides are ,
,
Question 7: Solve the following quadratic equation for
Answer:
Use formula:
Question 8: In an AP, if and
, then find the AP, where
denotes the sum of its first
terms.
Answer:
Let the first term be and the common difference be
… … … … … i)
… … … … … ii)
Solving i) and ii) we get
Substituting in ii) we get
Hence the AP is
Question 9: The points and
are the vertices of a right triangle, right-angled at
. Find the value of
.
Answer:
and
or
Question 10: Find the relation between and
if the points
and
are collinear.
Answer:
It is given that and
are collinear.
Therefore the area of
Section – C
Question number 11 to 20 carry 3 mark each.
Question 11: The term of an AP is twice its
term. If its
term is
, then find the sum of its first
terms
Answer:
Let be the first term and
be the common difference
Given,
… … … … … i)
Given
… … … … … ii)
Solving i) and ii) we get
Question 12: Solve for
Answer:
On comparing it with we get
and
Question 13: The angle of elevation of an aeroplane from a point A on the ground is . After a flight of
seconds, the angle of elevation changes to
. If the aeroplane is flying at a constant height of
, find the speed of the plane in km/hr.
Answer:
Time taken to travel sec
Question 14: If the coordinates of points and
are
and
respectively, find the coordinates of
such that
, where
lies on the line segment
.
Answer:
Let the coordinates of point
Hence divides
in the ratio of
Now using section’s formula
Question 15: The probability of selecting a red ball at random from a jar that contains only red, blue and orange balls is . The probability of selecting a blue ball at random from the same jar is
. If the jar contains
orange balls, find the total number of balls in the jar.
Answer:
Let the number of red balls be and number of Blue balls be
No of Orange balls
Total number of balls
… … … … … i)
… … … … … ii)
Solving i) and ii) we get
Hence the total number of balls
Question 16: Find the area of the minor segment of a circle of radius , when its central angle is
. Also find the area of the corresponding major segment. (Use
)
Answer:
Radius
Area of
We draw
In and
(by construction)
(both are radius of the same circle)
is common
(by RHS criterion)
Also, since
… … … … … i)
In right
Similarly, In right
Therefore area of segment
Area of major segment
Question 17: Due to sudden floods, some welfare associations jointly requested the government to get tents fixed immediately and offered to contribute
of the cost. If the lower part of each tent is of the form of a cylinder of diameter
and height
with the conical upper part of same diameter but of height
, and the canvas to be used costs Rs.
per sq. m, find the amount, the associations will have to pay. What values are shown by these associations ? (Use
)
Answer:
Surface Area of canvas CSA of Cone
CSA of cylinder
Rs.
Rs. per tent
Hence for tents, Association will pay
Rs.
This is an example of showing humanity and helpful nature of the association.
Question 18: A hemispherical bowl of internal diameter contains liquid. This liquid is filled into
cylindrical bottles of diameter
. Find the height of the each bottle, if
liquid is wasted in this transfer.
Answer:
Let the height of Bottle
Volume of Bottle
Total volume of bottles
Therefore height of bottle is
Question 19: A cubical block of side is surmounted by a hemisphere. What is the largest diameter that the hemisphere can have ? Find the cost of painting the total surface area of the solid so formed, at the rate of Rs.
per
sq. cm. [ Use
]
Answer:
The greatest diameter of the hemisphere can be
TSA of solid SA of cube
CSA of hemisphere
Area of base of hemisphere
Rs.
Question 20: cones, each of diameter
and height
, are melted and recast into a metallic sphere. Find the diameter of the sphere and hence find its surface area. (Use
)
Answer:
Diameter of cone
Height of cone
Volume of
cones
Let the radius of the sphere
Section – D
Question number 21 to 31 carry 4 mark each.
Question 21: The diagonal of a rectangular field is etres more than the shorter side. If the longer side is
etres more than the shorter side, then find the lengths of the sides of the field.
Answer:
Let the shorter side
or
(not possible)
Hence the shorter side , diagonal
and Longer side
Question 22: Find the term of the AP
, if it has a total of
terms and hence find the sum of its last
terms
Answer:
Given AP:
and
For sum of the last terms, lets look at the series backwards
Therefore sum of the last terms
Question 23: A train travels at a certain average speed for a distance of km and then travels a distance of
km at an average speed of
km/h more than the first speed. If it takes
hours to complete the total journey, what is its first speed ?
Answer:
Total time taken hr
or
(not possible)
Question 24: Prove that the lengths of the tangents drawn from an external point to a circle are equal.
Answer:
Given: Circle with center
and
are tangents touching the circle at
and
respectively.
To Prove:
Construction: Join and
Proof: Since is a tangent,
Similarly,
Consider and
(radius of the circle)
is common
(By RHS criterion)
. Hence proved.
Question 25: Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.
Answer:
Given: is the mid point of
,
is the center,
is the chord
To Prove:
(tangent)
Since is the mid point of
,
In and
(radius)
is common
(By SAS criterion)
Corresponding angles are equal, hence . Hence proved.
Question 26: Construct a in which
,
and
. Construct another
similar to
with base
.
Answer:
Question 27: At a point etres above the level of water in a lake, the angle of elevation of a cloud is
. The angle of depression of the reflection of the cloud in the lake, at
is
. Find the distance of the cloud from
.
Answer:
Question 28: A card is drawn at random from a well-shuffled deck of playing cards.
Find the probability that the card drawn is
(i) a card of spade or an ace.
(ii) a black king.
(iii) neither a jack nor a king.
(iv) either a king or a queen.
Answer:
Total number of cards
Question 29: Find the values of so that the area of the triangle with vertices
and
is
sq. units.
Answer:
Vertices:
or
Question 30: In Figure 5, is a square lawn with side
etres. Two circular flower beds are there on the sides
and
with center at
, the intersection of its diagonals. Find the total area of the two flower beds (shaded parts).

Answer:
Area of square lawn
Let
(diagonals of square are perpendicular to each other)
Area of one flower bed
Total area of flower bed
Question 31: From each end of a solid metal cylinder, metal was scooped out in hemispherical form of same diameter. The height of the cylinder is and its base is of radius
. The rest of the cylinder is melted and converted into a cylindrical wire of
thickness. Find the length of the wire. (Use
)
Answer:
Volume of cylinder
Volume melted
Diameter of cylindrical wire