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: In Fig. 1, is a tangent at point
to a circle with center
. If
is the diameter and
, find
.

Answer:
Construction: Join
Now in
(radius of the same circle)
Therefore ,
Also,
Therefore,
Question 2: For what value of will
, and
are the consecutive terms of an A.P.?
Answer:
If three terms ,
and
are in A.P. then,
Since and
are in A.P.
Question 3: A ladder, leaning against a wall, makes an angle of with the horizontal. If the foot of the ladder is
away from the wall, find the length of the ladder.
Answer:
Let be a ladder,
is the wall and
is the ground. The situation given in the question is shown in the figure.
. Therefore, the length of ladder is 5 m.
Question 4: A card is drawn at random from a well shuffled pack of playing cards. Find the probability of getting neither a Red card or the queen.
Answer:
There are red cards in a deck of
cards
Also there are queens in a deck of
cards,
queens are black and
are red.
Now, numbers of card which are neither red nor queen are
Section – B
Question number 5 to 10 carry 2 mark each.
Question 5: If is the root of the quadratic equation
and the quadratic equation
has equal roots, find the value of
.
Answer:
If is root of
For to have equal roots,
(Determinant should be zero)
Question 6: Let and
be points of trisection of the line segment joining the points
and
such that
is nearer to
. Find the coordinates of
and
.
Answer:
Since and
be the points of trisection of the line segment joining the points
and
such that
is nearer to
. Therefore
divides line segment in the rail
and
divides in
By section formula Coordinates of are
By section formula Coordinates of Q are
Question 7: In Fig. 2, a quadrilateral is drawn to circumscribe a circle, with center
, in such a way that the sides
, and
touch the circle at the points
and
respectively. Prove that
.

Answer:
Since tangents drawn from the exterior point to a circle are equal in length.
As, and
are tangents from exterior point
so,
… … i)
Similarly,
and
are tangents from exterior point
so,
… … … ii)
and
are tangents from exterior point
so,
… … … iii)
and
are tangents from exterior point
so,
… … … iv)
Adding i) , ii), iii) and iv) , we get
Hence proved.
Question 8: Prove that the points and
are the vertices of a right angled isosceles triangle.
Answer:
If and
be the vertices of the triangle. From distance formula,
Therefore we see that
This implies that is a right angled isosceles triangle.
Question 9: The term of an AP is zero. Prove that the
term of the AP is three times its
term.
Answer:
Let be the first term and
be the common difference of an A.P. having
terms.
Its term is given by,
Therefore
… … … … … i)
… … … … … ii)
From equation i) and ii)
Hence proved.
Question 10: In Fig. 3, from an external point , two tangents
and
are drawn to a circle with center
and radius
. If
, show that
.

Answer:
In (tangent to a circle perpendicular to the radius through the point of contact)
We know
Consider and
(radius of the same circle)
(tangents from the same external point)
is common
(by SSS criterion)
is bisector of
Also (by SAS criterion)
Therefore
In ,
(angles of a triangle)
Similarly,
Section – C
Question number 11 to 20 carry 3 mark each.
Question 11: In Fig. 4, is the center of the circle such that the diameter
cm and
cm.
is joined. Find the area of shaded region. (Take
)

Answer:
In (angle in a semicircle)
Therefore
Area of shaded region Area of semicircle
Area of triangle
Question 12: In Fig. 5, a tent is an a shape of cylinder surmounted by a conical top of same diameter. If the height of the diameter of the cylindrical part are 1 m and
m respectively and the slant height of the conical part is
m, find the cost of the canvas needed to make the tent if the canvas is available at the rate of
Rs sq. meter. (Use
)

Answer:
Canvas needed to make the tent Curved surface area of the conical part
curved surface area of cylindrical part.
Slant height of the conical part is 8 m.
Height of cylindrical part is
Total surface area
Therefore cost of canvas used
Question 13: If the point is equidistant from the points
and
. Prove that
.
Answer:
Since the point is equidistant from the points
and
Therefore
Using distance formula
Question 14: In Fig. 6, find the area of the shaded region, enclosed between two concentric circles of radii cm and
cm where
. (Use
)

Answer:
Area of shaded region Area of circular ring
Area of region ABDC
Question 15: If the ratio of the sum first terms of two APs is
, find the ratio of their
term.
Answer:
For first AP, let the first term be and the common difference be
and Sum of the first
terms be
Similarly, for Second AP, let the first term be and the common difference be
and Sum of the first
terms be
For term, replace
by
Hence the ratio of the term is
Answer:
Question 17: A conical vessel with a base of radius and height
, is full of water. This water is emptied into a cylindrical vessel of base radius
. Find the height to which the water will rise in the cylindrical vessel. (Use
)
Answer:
Let be the radius, height and slant height of the cone.
Let and
be the radius and height of the cylinder.
Since volume will remain same
Volume of water in conical vessel Volume in cylindrical vessel
Question 18: A sphere of diameter , is dropped into a right circular cylindrical vessel, partly filled with water. If the sphere is completely submerged in water, the water level in the cylindrical vessel rises by
. Find the diameter of the cylindrical vessel.
Answer:
Diameter of sphere radius
Let the radius of cylinder is
Question 19: A man standing on the deck of the ship, which is eter above the water level, observes the angle of elevation of the top of the hill as
and the angle of depression of the base of the hill as
. Find the distance of the hill from the ship and the height of the ship.
Answer:
The man is standing on the deck of the ship at point .
Let be the hill with base at
.
It is given that the angle of elevation of point from
is
and the angle of depression of point
from
is
So,
(alternate angles)
Suppose and
In , we have
In , we have
Therefore distance of the hill from the ship is
And height of the hill is
Question 20: Three different coins are tossed together. Find the probability of getting (i) exactly two heads (ii) at least two heads (iii) at least two tails
Answer:
If three different coins tossed together,
Total possible outcomes are TTT, TTH , THT, THH, HTT, HTH, HHT, HHH
Number of total possible outcomes
i ) Exactly two heads (favorable outcomes) TTH , THT, HTT
Number of favorable outcomes =3.
ii ) At least two heads is (favorable outcomes) THH, HTH, HHT, HHH
Number of favorable outcomes =4.
iii ) At least two tails is (favorable outcomes) TTT, TTH , THT, HTT
Number of favorable outcomes =4.
Section – D
Question number 21 to 31 carry 4 mark each.
Question 21: Due to heavy flood in a state, thousands were rendered homeless. schools collectively offered to the state government to provide place and canvas for
tents to be fixed by government and decided to share the whole expenditure equally. The lower part of the tent is cylindrical of base radius
and height
, with conical upper part of the same base radius but of height
. If the canvas used to make the tent costs Rs.
sq. m, find the amount shared by each school to set up the tents. What value is generated by the above problem.(Use
)
Answer:
Let the radius of base of both cylinder and cone be
Let the height of cone be and that of cylinder be
.
Let the slant height of cone be , Then,
Area of tent CSA of cylindrical base
CSA of cone
Cost of canvas used in making one tent
Cost of canvas used in making tents
Question 22: Prove that the lengths of the tangents drawn from an external point to the circle are equal.
Answer:
Let us draw a circle with center and two tangents
and
are drawn from a external point
to the circle as shown in the figure given below,
Consider and
(Radius)
is common
Therefore (By SAS criterion)
Hence
Question 23: Draw a circle of radius cm. Draw two tangents to the circle inclined at an angle of
to each other.
Answer:
i) Draw a circle of radius with center
.
ii) Take a point on the circumference of circle and join
.
iii) Draw a perpendicular to at
iv) Draw a radius , making an angle of
with
v) Draw a perpendicular to at
. Let these perpendiculars intersect at
.
Question 24: In Fig. 7, two equal circles with center and
, touch each other at
.
produced meets the circle with center
at
.
is a tangent to the circle with center
, at the point
.
is perpendicular to
. Find the value of
.

Answer:
As, circles are equal, so their radius are also equal.
That means
Consider and
is common
(By AA criterion)
Answer:
or
Since
Question 26: The angle of elevation of the top of a vertical tower
from a point
on the ground is
. From a point
vertically above
, the angle of elevation of the top
of tower is
. Find the height of the tower
and the distance
.
Answer:
Let , therefore in
In
Therefore the height of the tower
Therefore horizontal distance is
Question 27: The houses in a row are numbered consecutively from to
. Show that there exists a value of
such that the sum of the numbers of houses preceding the house numbered
is equal to sum of the numbers of houses following
.
Answer:
The number of houses preceding is
Sum of the number of houses preceding the house numbered,
Here,
Given
Substituting
Question 28: In Fig. 8, the vertices of are
and
. A line segment
is drawn to intersect the sides
and
at
and
respectively such that
. Calculate the area of
and compare it with the area of
.

Answer:
sq. units
Now in and
is common
sq. units
Question 29: A number is selected randomly from the numbers
and
. Another number
is selected at random from the numbers
and
. Find the probability that the product of
and
is less than
.
Answer:
Number can be selected in four ways. Corresponding to each such way, there are four ways of selecting
.
Therefore two numbers can be selected in one of the following ways,
Therefore, the two numbers can be selected in ways
The number of combinations where are
Therefore the number of possible combinations are
Therefore probability that the product is less than
is
Question 30: In Fig. 9, is shown a sector of a circle with center
containing
.
is perpendicular to the radius
and meets
produced at
. Prove that the perimeter of shaded region is
.

Answer:
Answer:
Perimeter of shaded region Length of arc
Question 31: A motor boat whose speed is /h in still water takes
hour more to go
upstream than to return down stream to the same spot. Find the speed of the stream.
Answer:
Let the speed of stream be .
Therefore, Speed of boat in upstream is
In downstream, speed of boat is
Given, Time taken in the upstream journey ‒ Time taken in the downstream journey hour
. Since speed cannot be negative,
/hr.
Therefore, Speed of boat in upstream is /hr
In downstream, speed of boat is /hr