*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