*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

Also,

Question 3: If a tower m high, casts a shadow m 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 and be the roots

Given

… … … … … i)

… … … … … ii)

Solving i) and ii)

Question 6: Which term of the progression is the first negative term?

Answer:

Given AP is

Confirming it once again:

and

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 and are tangents drawn at the end and on chord

To prove:

Construction: Join and

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 and at the points and respectively. If is the mid-point of , then find the coordinates of and .

Answer:

Let be and be

Also

and

Question 10: If the distances of from and 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 and are at distances of m and m from the foot respectively. If the angles of elevation from and of the top of the tower are complementary, then find the height of the tower.

Answer:

From … … … … … i)

From

… … … … … ii)

Substituting ii) in i) we get

m

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

and

Given

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 and ? Also find the value of .

Answer:

Given and

Let divides and in the ratio . Using sections formula,

Hence divides and in the ratio of

Similarly,

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:

Area of large semi circle

Area of large circle

Area of two small semi circles

Area of shaded small semi circle

Therefore shaded area

Question 17: In the given figure, two concentric circles with centre have radii cm and cm. If , find the area of the shaded region. [ Use ]

Answer:

Area of

Area of

Therefore area of

Shaded Area

Question 18: Water in a canal, m wide and m deep, is flowing with a speed of km/hour. How much area can it irrigate in minutes, if cm of standing water is required for irrigation ?

Answer:

Cross Section of canal

Speed of water

Therefore volume of water flow in 1 minute

Let the area irrigated in

Question 19: The slant height of a frustum of a cone is cm and the perimeters of its circular ends are cm and cm. Find the curved surface area of the frustum.

Answer:

We know that the curved surface area of frustum

Here

For

For

Now curved surface area

Hence the curved surface area

Question 20: The dimensions of a solid iron cuboid are m m m. It is melted and recast into a hollow cylindrical pipe of cm inner radius and thickness cm. 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.*

Question 21: Solve for

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

Portion of the tank filled by larger tap in 1 hour

Portion of the tank filled by Smaller tap in 1 hour

Given: Time taken when both taps are running to fill the tank hours

Portion of tank filled in 1 hour when both taps are running

(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:

Given:

… … … … … i)

We need to find:

… … … … … ii)

If we put … … … … … iii)

Substituting in i) we get

Hence

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 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 25: In Figure 3, and are two parallel tangents to a circle with center and another tangent with point of contact intersecting at and at . Prove that .

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

and

Now and is transversal

From

Hence

Question 26: Construct a triangle with side cm, . Then construct another triangle whose sides are times the corresponding sides of the .

Answer:

Question 27: An aeroplane is flying at a height of m 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 and respectively. Find the width of the river. [Use ]

Answer:

Let be the aeroplane & is the width of river

In

m

In

m

Therefore the width of the river m

Question 28: If the points and 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

Therefore Probability (Even sum outcome)

ii) Even products

Possible outcomes

Therefore Probability (Even product)

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

Area of semi circle

Therefore area of shaded region

Perimeter of shaded region

Question 31: In a rain-water harvesting system, the rain-water from a roof of m m drains into a cylindrical tank having diameter of base m and height m. If the tank is full, find the rainfall in cm. Write your views on water conservation.

Answer:

Let the rainfall be m

Therefore volume of water from roof volume of tank