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 31 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 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 4 carry 1 mark each.
Question 1: What is the common difference of an A.P. in which
Answer:
Given:
Question 2: If the angle between two tangents drawn from an external point to a circle of radius
and center
, is
, then find the length of
.
Answer:
Radius
Question 3: If a tower high, casts a shadow
long on the ground, then what is the angle of elevation of the sun?
Answer:
Question 4: The probability of selecting a rotten apple randomly from a heap of apples is
. What is the number of rotten apples in the heap?
Answer:
Let the number of rotten apples
Hence the number of rotten apples
Section – B
Question number 5 to 10 carry 2 mark each.
Question 5: Find the value of , for which one root of the quadratic equation
is
times the other.
Answer:
Given equation:
Let be the roots
Given
… … … … … i)
… … … … … ii)
Solving i) and ii)
Answer:
Let term be zero
term where the value is very close is zero
term or
term will be negative
Also term
Question 7: Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord.
Answer:
Given: A circle with center are tangents drawn at the end
on chord
To prove:
Construction: Join
Proof: In , we have
(radius of the same circle)
(angles opposite to equal sides of a triangle)
(radius is
tangents)
Hence proved.
Question 8: A circle touches all the four sides of a quadrilateral . Prove that
Answer:
Given: A circle touches quadrilateral on all four sides.
To prove:
Proof: Since tangents to a circle from external points are equal.
Hence proved.
Question 9: A line intersects the at the points
respectively. If
is the mid-point of
, then find the coordinates of
.
Answer:
Let be
be
Question 10: If the distances of from
are equal, then prove that
.
Answer:
Given:
Hence proved.
Section – C
Question number 13 to 22 carry 3 mark each.
Question 11: If , then prove that the equation
has no real roots.
Answer:
The equation is of the form
Since , the equation has no real roots.
Question 12: The first term of an A.P. is , the last term is
and the sum of all its terms is
. Find the number of terms and the common difference of the A.P.
Answer:
We know
Let term be the last
… … … … … i)
Also the sum of the terms is
… … … … … ii)
Substituting i) in ii) we get
Question 13: On a straight line passing through the foot of a tower, two points are at distances of
and
from the foot respectively. If the angles of elevation from
of the top of the tower are complementary, then find the height of the tower.
Answer:
… … … … … i)
From
… … … … … ii)
Substituting ii) in i) we get
Question 14: A bag contains white and some black balls. If the probability of drawing a black ball from the bag is thrice that of drawing a white ball, find the number of black balls in the bag.
Answer:
Let the number of black balls
Therefore number of Black balls
Question 15: In what ratio does the point divide the line segment joining the points divide the line segment joining the points
? Also find the value of
.
Answer:
Let divides
in the ratio
. Using sections formula,
Hence divides
in the ratio of
Question 16: Three semicircles each of diameter 3 cm, a circle of diameter 4·5 cm and a semicircle of radius 4·5 cm are drawn in the given figure. Find the area of the shaded region.
Answer:
Question 17: In the given figure, two concentric circles with centre have radii
and
. If
, find the area of the shaded region. [ Use
]
Answer:
Therefore area of
Question 18: Water in a canal, wide and
deep, is flowing with a speed of
/hour. How much area can it irrigate in
inutes, if
of standing water is required for irrigation ?
Answer:
Cross Section of canal
Let the area irrigated in
Question 19: The slant height of a frustum of a cone is and the perimeters of its circular ends are
and
. Find the curved surface area of the frustum.
Answer:
We know that the curved surface area of frustum
Here
Hence the curved surface area
Question 20: The dimensions of a solid iron cuboid are
. It is melted and recast into a hollow cylindrical pipe of
inner radius and thickness
. Find the length of the pipe.
Answer:
Dimensions of cuboid:
Therefore volume of cuboid
Dimension of the pipe: Inner radius
Thickness
Therefore outer radius
Let l be the length of the pipe
Section – D
Question number 21 to 30 carry 4 mark each.
Answer:
Question 22: Two taps running together can fill a tank in hours. If one tap takes
hours more than the other to fill the tank, then how much time will each tap take to fill the tank ?
Answer:
Time taken by larger tap to fill the tank hours
Therefore Time taken by smaller tap to fill the tank hours
hours
(not possible as time cannot be negative)
Therefore the time taken by the larger tap to fill the tank hours
and the time taken by the smaller tap to fill the tank hours
Question 23: If the ratio of the sum of the first terms of two A.Ps is
, then find the ratio of their
terms.
Answer:
… … … … … i)
We need to find:
… … … … … ii)
… … … … … iii)
Substituting in i) we get
Question 24: Prove that the lengths of two 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
are drawn as shown in the diagram
To prove:
Construction: Join
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
is common
(radius of the same circle)
Therefore both tangents are equal in length.
Question 25: In Figure 3, are two parallel tangents to a circle with center
and another tangent
with point of contact
intersecting
at
at
. Prove that
.
Figure 3
Answer:
Given: ,
is a tangent
To prove:
Join through
.
is diameter of the circle
We know that tangents to a circle from an external point are equally inclined to the line segment joining this point to the center
Now is transversal
From
Hence
Question 26: Construct a triangle with side
,
. Then construct another triangle whose sides are
times the corresponding sides of the
.
Answer:
Question 27: An aeroplane is flying at a height of above the ground. Flying at this height, the angles of depression from the aeroplane of two points on both banks of a river in opposite directions are
respectively. Find the width of the river. [Use
]
Answer:
Let be the aeroplane &
is the width of river
In
Therefore the width of the river
Question 28: If the points are collinear, then find the value of
.
Answer:
Given: are collinear
Therefore the area of
Question 29: Two different dice are thrown together. Find the probability that the numbers obtained have
(i) even sum, and
(ii) even product.
Answer:
Number of possible outcomes:
Therefore total number of possible outcomes
i) Even sum outcome
Possible outcomes
ii) Even products
Possible outcomes
Question 30: In the given figure, is a rectangle of dimensions
cm
cm. A semicircle is drawn with
as diameter. Find the area and the perimeter of the shaded region in the figure.
Answer:
Area of
Therefore area of shaded region
Question 31: In a rain-water harvesting system, the rain-water from a roof of
drains into a cylindrical tank having diameter of base
and height
. If the tank is full, find the rainfall in cm. Write your views on water conservation.
Answer:
Let the rainfall be
Therefore volume of water from roof volume of tank