**NOTE:**

(I) Please check that this question paper consists of 15 pages.

(II) 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.

(III) Please check that this question paper consists of 40 questions.

(IV)** Please write down the serial number of the question before attempting it.**

(V) 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.

**MATHEMATICS (STANDARD)**

—————————————————————————————————————————————–Time allowed: * 3 hours* Maximum Marks:

*—————————————————————————————————————————————–*

**80****General Instructions:**

**Read the following instructions very carefully and strictly follow them:**

(i) This question paper comprises four sections – A, B, C and D. This question paper carries 40 questions. All questions are compulsory.

(ii) **Section A** – Question numbers **1** to **20** comprises of **20** questions of **one** mark each.

(iii) **Section B** – Question numbers **21** to **26** comprises of **6** questions of **two** marks each.

(iv) **Section C** – Question numbers **27** to **34** comprises of **8** questions of **three** marks each.

(v) **Section D** – Question numbers **35** to **40** comprises of **6** questions of **four** marks each.

(vi) There is no overall choice in the question paper. However, an internal choice has been provided in 2 questions of one mark, 2 questions of two marks, 3 questions of three marks and 3 questions of four marks. You have to attempt only **one** of the choices in such questions.

(vii) In addition to this, separate instructions are given with each section and question, wherever necessary.

(viii) Use of calculators is not permitted.

**Section – A**

Question numbers **1** to **10** are multiple choice questions of **1** mark each. Select the correct option.

Question 1: The point on x-axis equidistant from the points and is

(a) (b) (c) (d)

Answer:

Since is on x-axis, the coordinate of would be

Since is equidistant from and we get

Therefore coordinate of is

Question 2: The coordinate of the point which is the reflection of the point in x -axis is

(a) (b) (c) (d)

Answer:

Because the reflection is on x-axis, the y coordinate would not change.

Question 3: If the point divides the line segment joining and in the ratio , then the value of is

(a) (b) (c) (d)

Answer:

Using section formula, we get

Question 4: The sum of exponents of prime factors in the prime factorisation of is

(a) (b) (c) (d)

Answer:

Therefore the sum of exponents of prime factors in the prime factorisation of is

Question 5: Euclid’s division Lemma states that for two positive integers and there exists unique integer and satisfying and

(a) (b) (c) (d)

Answer:

Euclid’s division Lemma states that for two positive integers and there exists unique integer and satisfying and

Question 6: The zeros of the polynomial are

(a) (b) (c) (d)

Answer:

Therefore zeros are

Question 7: The value of k for which the system of linear equations and is inconsistent is

(a) (b) (c) (d)

Answer:

For and

For and

For a unique solution,

Therefore for all values other than the given equations will have a unique solution. Hence for , the given equations will be inconsistent.

Question 8: The roots of the quadratic equation are

(a) (b) (c) (d)

Answer:

Question 9: The common difference of the AP is

(a) (b) (c) (d)

Answer:

First term

Common difference

Question 10: The term of the A.P. , is

(a) (b) (c) (d)

Answer:

First term

Common difference

Therefore term

In Question Nos. **11** to **15**, Fill in the blanks. Each question carries **1** mark.

Question 11: In Fig. 3, the angles of depressions from the observing positions and respectively of the object are

Answer:

Please refer to the adjoining diagram

Therefore the angle of depression are .

Question 12: In cm, cm and cm, then

**OR**

Two triangles are similar if their corresponding sides are

Answer:

We have to first check if the is a right angled triangle.

If it is a right angled triangle, then Pythagoras theorem should hold

. Therefore the is a right angled triangle.

**OR**

Two triangles are similar if there corresponding sides are .

If and are similar, then

Question 13: In given Fig. 2, the length cm.

Answer:

is a right angled triangle

cm

Since

Hence cm

Question 14: In Fig. 1, and , then

Answer:

Given

Let

Since

Using “Basic Proportionality Theorem” we get

Question 15: The value of is

**OR**

The value of is

Answer:

**OR**

Question Nos. **16** to **20** are short answer type questions of **1** mark each.

Question 16: A die is thrown once. What is the probability of getting a prime number.

Answer:

Total outcomes that are possible are

Therefore the number of possible outcomes

No of prime numbers on the dice are

Therefore the number of favorable outcomes

Hence the probability of getting a prime number

Question 17: If a number is chosen at random from the numbers then find the probability of .

**OR**

What is the probability that a randomly taken leap year has Sundays?

Answer:

Total number of cases

If , then can be either only.

Therefore number of favorable events

Therefore the probability

**OR**

There are weeks in a way which means there will be Sundays in a year.

There are days in a year but we have days in a leap year.

There are days in a week.

If we multiply the weeks by the days we have which equals to . This means that there are extra days in a leap year which will make it .

The probability of having Sundays in a leap year is thus: the remaining two days can be any of this formation:

Sunday-Monday, Monday-Tuesday, Tuesday-Wednesday, Wednesday-Thursday, Thursday-Friday, Friday-Saturday, Saturday-Sunday.

However, to get Sundays in a leap year, none of the remaining two days must be a Sunday. Therefore, out of the combinations above, that can be only realized out of times. The connection “Sunday-Monday and Saturday-Sunday” most be scraped off.

The probability of having Sundays in a leap year is therefore

Question 18: If , then find the value of the expression .

Answer:

Given

Now

Hence .

Question 19: Find the area of the sector of a circle of radius cm whose central angle is Take

Answer:

Given: radius cm

Central angle

We know, if the radius of the circle is and the length of the arc is , then

cm

Therefore perimeter of the sector cm

Question 20: Find the class – marks of the classes and .

Answer:

Given: classes and

We know, Class Marks

For Class interval lower limit , upper limit

Therefore Class Marks of

Similarly, For Class interval lower limit , upper limit

Therefore Class Marks of

**Section – B**

Question Nos. **21** to **26** carry **2** marks each.

Question 21: A teacher asked of his students to write a polynomial in one variable on a paper and then to hand over the paper. The following were the answers given by the students:

, , , , , , , , ,

Answer the following questions:

(i) How many of the above are not polynomials?

(ii) How many of the above are quadratic polynomials?

Answer:

Definitions:

- A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables
- A quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree.

: This is a polynomial.

: This is a polynomial. This is also a quadratic polynomial as the highest-degree term is of the second degree.

: This is a polynomial.

: This is NOT a polynomial as one of the exponent is not an integer.

: This is a polynomial.

: This is a polynomial.

: This is NOT a polynomial as one of the exponent is a negative integer.

: This is a polynomial.

: This is a polynomial.

: This is NOT a polynomial as one of the exponent is a negative integer.

Therefore:

(i) How many of the above are not polynomials? – Three

(ii) How many of the above are quadratic polynomials? – One

Question 22: A child has a die whose six faces show the letters shown below:

The Die is thrown once. What is the probability of getting (i) (ii) ?

Answer:

Probability

Number of faces in the dice

i) The probability of getting

Probability

ii) The proobability of getting

Probability

Question 23: In Fig. 4, and are two triangles on the same base , If intersects at , show that

**OR**

In Fig. 5, if , then prove that

Answer:

Given: and have common bases.

To prove:

Construction: Draw and

Proof: … … … … … i)

Similarly, … … … … … ii)

Therefore

Consider and

(vertically opposite angle)

(by AA similarity criterion)

… … … … … iv)

Therefore from iii) and iv) we get

**OR**

Given:

To prove:

Proof: Since

In … … … … … i)

Similarly, in … … … … … ii)

Subtracting ii) from i) we get

Hence proved.

Question 24: Prove that

**OR**

Show that

Answer:

LHS

RHS. Hence proved.

**OR**

LHS

Since

RHS. Hence proved.

Question 25: Find the mode for the following frequency distribution:

Class | 15-20 | 20-25 | 25-30 | 30-35 | 35-40 | 40-45 |

Frequency | 3 | 8 | 9 | 10 | 3 | 2 |

Answer:

Here the maximum frequency is and the corresponding class is

is the modal class

Mode

Question 26: From a solid right circular cylinder of height 14 cm and base radius 6 cm, a right circular code of the same height and same base radius is removed. Find the volume of the remaining solid.

Answer:

Volume of the cylinder

Let the radius . Therefore height

Therefore

cm

Therefore the height of the cylinder is cm

**Section – C**

Question Nos. **27** to **34** carry **3** marks each.

Question 27: If a circle touches the side of a triangle at and the extended sides and at and respectively, prove that

Answer:

Given: A circle touching the side of at and produced at and respectively.

We know that tangents drawn from an external point to the same circle are equal.

Hence and

Also

… … … … … i)

Also

… … … … … ii)

Adding i) and ii) we get

Question 28: The area of a circular playground is . Find the cost of fencing this ground at the rate of Rs. per meter.

Answer:

Area

Let the radius cm

cm

Circumference cm

Therefore cost of fencing Rs.

Question 29: If the mid point of the line segment joining the points and , is and , find the value of .

**OR**

Find the area of triangle with and the mid points of the sides through being and .

Answer:

If is the mid point then

and

Also

**OR**

Since and are mid points

Hence is

Hence is

Now the area of a triangle where we know all the three vertices

sq. units

Question 30: In Fig. 6, if and their sides of the lengths (in cm) are marked along them, then find the length of the sides of each of the triangles.

Answer:

Since

Therefore

Therefore

Similarly,

Question 31: If and find the value of

and

**OR**

Solve for

Answer:

Given equations: and

Adding the two equations we get

Substituting it back we get

Hence

**OR**

or

Question 32: Which term of the A.P. is the first negative term.

**OR**

Find the middle term of the A.P. .

Answer:

Given AP:

First term

Common difference

Let’s see which term be closest to

Therefore the terms would be the first negative term

**OR**

Given AP:

First term

Common difference

Let be the term

Therefore

Therefore the middle term is term

Question 33: Water in a canal m wide and m deep, is flowing with the speed of km/hr. How much area will it irrigate in minutes, if cm standing water is required.

Answer:

Canal is in the shape of cuboid: breadth m and height m

Speed of water km/hr m/min

Volume of water flow

Volume of water flow in minutes

Therefore the Area irrigated

Question 34: Show that:

Answer:

LHS

RHS.

Hence proved.

**Section – D**

Question Nos. **35** to **40** carry **3** marks each.

Question 35: The mean of the frequency distribution is . The frequency f in the class interval is missing. Determine .

Class Interval | 11-13 | 13-15 | 15-17 | 17-19 | 19-21 | 21-23 | 23-25 |

Frequency | 3 | 6 | 9 | 13 | f | 5 | 4 |

**OR**

The following table gives production yield per hectare of wheat of farms of a village

Production yield | 40-45 | 45-50 | 50-55 | 55-60 | 60-65 | 65-70 |

No. of farms | 4 | 6 | 16 | 20 | 30 | 24 |

Answer:

Class Interval | Frequency | Mean | |

11-13 | 3 | 12 | 36 |

13-15 | 6 | 14 | 84 |

15-17 | 9 | 16 | 144 |

17-19 | 13 | 18 | 234 |

19-21 | 20 | ||

21-23 | 5 | 22 | 110 |

23-25 | 4 | 24 | 96 |

Total |

Mean

**OR**

Production Yield | Number of Farms | Production Yield (x-axis) | Number of Farms (y-axis) |

40-45 | 4 | More than 40 | 100 |

45-50 | 6 | More than 45 | 96 |

50-55 | 16 | More than 50 | 90 |

55-60 | 20 | More than 55 | 74 |

60-65 | 30 | More than 60 | 54 |

65-70 | 24 | More than 65 | 24 |

Now plot

Question 36: From a point on the ground, the angles of elevation of the bottom and top of the tower fixed at the top of a m high building are and respectively. Find the height of the tower.

Answer:

Let the height of the tower be m

In

… … … … … i)

In

… … … … … ii)

Substituting in i) we get

m

Question 37: It takes 12 hours to fill a swimming pool using two pipes.If the pipe of the larger diameter is used for four hours and the pipe of the smaller diameter for 9 hours, only half of the pool can be filled. How long would it take for each of the pipes to fill the pool separately.

Answer:

Let the pipe with the larger diameter and the smaller diameter be pipe and respectively.

Let Pipe works at liters/hours and Pipe works at liters/hours.

Therefore the capacity of the pool liters

Given,

For larger pipe to fill the swimming pool

hours

For smaller pipe to fill the swimming pool

hours

Question 38: Prove that is an irrational number.

Answer:

Let’s prove this by the method of contradiction.

Say, is a rational number.

Therefore It can be expressed in the form where are co-prime integers.

{Squaring both the sides}

… … … … … i)

is a multiple of . {Euclid’s Division Lemma}

is also a multiple of . {Fundamental Theorm of arithmetic}

… … … … … ii)

From equations i) and ii), we get,

is a multiple of . {Euclid’s Division Lemma}

is a multiple of .{Fundamental Theorm of Arithmetic}

Hence, have a common factor . This contradicts that they are co-primes.

Therefore, is not a rational number. This proves that is an irrational number.

Question 39: Draw a circle of radius cm. From a point , cm from its center, draw two tangents to the circle.

**OR**

Construct a with cm, cm and . Now construct another triangle whose sides are times the corresponding sides of .

Answer:

Draw a circle of radius cm

Mark any point at a distance of cm from the center

Join and locate the midpoint

Taking as a center draw a circle of with as the radius.

The two circles intersect at and

Join and and they would be the tangents to the circle from the point .

**OR**

Step 1: First construct the . Draw latex A $ as a center, draw an angle of and draw a line through the point to make a line. Then taking an arc of cm draw an arc to intersect the line and mark point . Join to . This is your .

Step 2: Draw a ray making an acute angle with on the opposite site of vertex .

Step 3: Mark 3 points where

Step 4: Join and draw a line through parallel to to intersect extended at

Step 5: Draw a line through parallel to line to intersect at extended

is of

Question 40: A solid is in the shape of a hemisphere surmounted by a cone. If the radius of hemisphere and the base radius of cone is cm and height of the cone is cm, find the volume of the solid. Take

Answer:

We know: Volume of a sphere where is the radius of the sphere

Volume of a cone where radius of the base of the cone and is the height of the cone.

Volume of hemisphere

Volume of cone

Therefore total volume