**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 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 2: 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 3: The zeros of the polynomial are

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

Answer:

Therefore zeros are

Question 4: 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 5: The roots of the quadratic equation are

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

Answer:

Question 6: The common difference of the AP is

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

Answer:

First term

Common difference

Question 7: The term of the A.P. , is

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

Answer:

First term

Common difference

Therefore term

Question 8: 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 9: 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 10: 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

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

Question 11: In Fig. 1, and , then

Answer:

Given

Let

Since

Using “Basic Proportionality Theorem” we get

Question 12: In given Fig. 2, the length cm.

Answer:

is a right angled triangle

cm

Since

Hence cm

Question 13: 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 14: The value of is equal to

Answer:

Question 15: 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 Nos. **16** to **20** are short answer type questions of **1** mark each.

Question 16: If , then find the value of the expression .

Answer:

Given

Now

Hence .

Question 17: In Fig. 4 is a sector of a circle of radius cm. Find the perimeter of the sector. 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 18: 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 19: 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

Question 20: 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

**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: In Fig. 5, and are two triangles on the same base , If intersects at , show that

**OR**

In Fig. 6, 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 23: Prove that

**OR**

Show that

Answer:

LHS

RHS. Hence proved.

**OR**

LHS

Since

RHS. Hence proved.

Question 24: The volume of the right circular cylinder with its height equal to the radius is . Find the height of the cylinder. Use

Answer:

Volume of the cylinder

Let the radius . Therefore height

Therefore

cm

Therefore the height of the cylinder is cm

Question 25: 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 26: Compute the mode for the following frequency distribution:

Size of the Items
(in cm) |
0-4 | 4-8 | 8-12 | 12-16 | 16-20 | 20-24 | 24-28 |

Frequency | 5 | 7 | 9 | 17 | 12 | 10 | 6 |

Answer:

Here the maximum frequency is and the corresponding class is

is the modal class

Mode

**Section – C**

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

Question 27: 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 28: Show that the sum of all terms of an A.P. whose first term is and the second term is and the last term is is equal to

**OR**

Solve the equation:

Answer:

First term

Second term

Therefore Common difference

We know term

We know

We know that sum of terms

Hence proved.

**OR**

Given

First term

Common difference

[this is not possible]

Hence

We know is the term

Question 29: In a flight of km, an aircraft was slowed down due to bad weather. The average speed of the trip was reduced by km/hr and the time of the flight increased by minutes. Find the duration of the flight.

Answer:

Distance traveled km

Let the average speed km/hr

Therefore

km/hr or km/hr [this is not possible as speed cannot be negative]

Therefore normal duration of flight hr

Question 30: 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 31: In Fig. 7, 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 32: 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 33: If , prove that

Answer:

Given

Squaring both sides

… … … … … i)

Now, LHS

RHS. Hence proved.

Question 34: 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.

**Section – D**

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

Question 35: 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 36: It takes hours to fill a swimming pool using two pipes. If the pipe of the larger diameter is used for hours and the pipe of the smaller diameter for 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 37: Draw a circle of cm radius with center and take a point outside the circle such that is cm. From , draw two tangents to the circle.

**OR**

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

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 cm. The taking as a center, draw an arc with cm radius. Similarly draw an arc of cm radius with as a center. Join and to point of intersection . This is the .

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

Step 3: Mark 4 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

Question 38: 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 39: Find the area of the shaded region in Fig. 8, if cm, cm, and is the center of the circle.

**OR**

Find the curved surface area of the frustum of a cone, the diameters of whose circular ends are m and m and its height is m.

Answer:

(Diameter or a circle subtends a right angle on any point of the circumference of the circle)

Radius

Area of semi circle

Area of

Therefore the shaded area

**OR**

Curved Surface Area where slant height, is the larger radius, is the smaller radius

Question 40: 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 |

Change the distribution to a ‘more than’ type distribution and draw an ogive.

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