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केन्द्रीय विद्यालय सगंठन

क्षेत्रीय कायाालय, जयपरु सभंाग

सत्र 2020-21

श्री बी. एल. मोरोविया

सभंागीय आयुक्त, के.वि.स.ं क्षते्रीय कायाालय, जयपरु सभंाग

के नतेतृ्ि एि ंमागादर्ान में वनर्मात

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MATHEMATICS (X) (CODE NO. 041)

Session 2020-21

Units Unit Name Marks

I NUMBER SYSTEMS 06

II ALGEBRA 20

III COORDINATE GEOMETRY 06

IV GEOMETRY 15

V TRIGONOMETRY 12

VI MENSURATION 10

VII STATISTICS & PROBABILTY 11

Total 80

UNIT I: NUMBER SYSTEMS

1. REAL NUMBER

Fundamental Theorem of Arithmetic - statements after reviewing work done earlier and after

illustrating and motivating through examples, Proofs of irrationality of Decimal

representation of rational numbers interms of terminating/non-terminating recurring decimals.

UNIT II: ALGEBRA

1. POLYNOMIALS

Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials.

2. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

Pair of linear equations in two

variables

and

graphical

method

of their

solution, consistency/inconsistency.

Algebraic conditions for number of solutions. Solution of a pair of linear equations in two

variables algebraically - by substitution, by elimination. Simple situational problems. Simple

problems on equations reducible to linear equations.

3. QUADRATIC EQUATIONS

Standard form of a quadratic equation ax2 + bx + c = 0, (a ≠ 0). Solutions of quadratic equations

(only real roots) by factorization, and by using quadratic formula. Relationship between discriminant and nature of roots.

4. ARITHMETIC PROGRESSIONS

Motivation for studying Arithmetic Progression Derivation of the nth

term and sum of the first n terms of A.P.

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UNIT III: COORDINATE GEOMETRY

1. LINES (In two-dimensions)

Review: Concepts of coordinate geometry, graphs of linear equations. Distance formula. Section formula (internal division).

UNIT IV: GEOMETRY

1. TRIANGLES

Definitions, examples, counter examples of similar triangles.

1. (Prove) If a line is drawn parallel to one side of a triangle to intersect the other two sides in

distinct points, the other two sides are divided in the same ratio.

2. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the

third side.

3. (Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are

proportional and the triangles are similar.

4. (Motivate) If the corresponding sides of two triangles are proportional, their corresponding

angles are equal and the two triangles are similar.

5. (Motivate) If one angle of a triangle is equal to one angle of another triangle and the sides

including these angles are proportional, the two triangles are similar.

6. (Motivate) If a perpendicular is drawn from the vertex of the right angle of a right triangle to the

hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and

to each other.

7. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on

the other two sides.

2. CIRCLES

Tangent to a circle at, point of contact

1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of

contact.

2. (Prove) The lengths of tangents drawn from an external point to a circle are equal.

3. CONSTRUCTIONS

1. Division of a line segment in a given ratio (internally). 2. Tangents to a circle from a point outside it.

UNIT V: TRIGONOMETRY

1. INTRODUCTION TO TRIGONOMETRY

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their

existence (well defined). Values of the trigonometric ratios of 300, 45

0 and 60

0.

Relationships between the ratios.

2. TRIGONOMETRIC IDENTITIES

Proof and applications of the identity sin2A + cos

2A = 1. Only simple identities to be given.

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3. HEIGHTS AND DISTANCES: Angle of elevation, Angle of Depression. (8) Periods

Simple problems on heights and distances. Problems should not involve more than two right

triangles. Angles of elevation / depression should be only 30°, 45°, 60°.

UNIT VI: MENSURATION 1. AREAS RELATED TO CIRCLES

Motivate the area of a circle; area of sectors and segments of a circle. Problems based on

areas and perimeter / circumference of the above said plane figures. (In calculating area of

segment of a circle, problems should be restricted to central angle of 60°and 90° only. Plane

figures involving triangles, simple quadrilaterals and circle should be taken.)

2. SURFACE AREAS AND VOLUMES

1. Surface areas and volumes of combinations of any two of the following: cubes, cuboids,

spheres, hemispheres and right circular cylinders/cones.

2. Problems involving converting one type of metallic solid into another and other mixed

problems. (Problems with combination of not more than two different solids be taken).

UNIT VII: STATISTICS AND PROBABILITY

1. STATISTICS

Mean, median and mode of grouped data (bimodal situation and step deviation method for

finding the mean to be avoided).

2. PROBABILITY

Classical definition of probability. Simple problems on finding the probability of an event.

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Quick Revision Module JAIPUR REGION

CLASS :- X

Mathematics

Chapter.1 REAL NUMBER Key Points:-

1. Fundamental Theorem of Arithmetic: Every composite number can be

expressed as a product of primes, and this factorisation is unique, apart from the

order in which the prime factors occur.

2. If ‘a’ and ‘b’ are any two positive integers then

HCF(a,b) × LCM(a,b) = a× b

3. If 𝑝

𝑞 be a rational number, such that the prime factorization of q is of the

form 2𝑛5𝑚, where n, m are non-negative integers. Then It has a decimal

expansion which terminates.

Questions for Revision

1. Express each number as a product of its prime factors:

(i) 5005

(ii) 720

2. Given that HCF (306,657) = 9, Find LCM (306,657)

3. Prove that √3 is irrational number.

4. Prove that 3 + 2√ 5 is irrational number.

5. Write two rational number between 2 3and .

6. Explain why 5 × 7×11 + 7 is a composite number.

7. Without actually performing the long division, state whether the following

rational numbers will have a terminating decimal expansion or a non-

terminating repeating decimal expansion .

(i) 13

3125 ( ii)

15

1600 (iii)

29

22×32

Solutions Chapter 1:-

1.(i) 5005 = 5 ×7 × 11 × 13 (ii) 720= 24 × 32 × 5

2. Given ,a=306 , b=657 and HCF(306,657)= 9

We know that , HCF(a,b) ×LCM(a,b) = a×b

⇒ 9 ×LCM(306,657) = 306× 657

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⇒ LCM(306,657) = 306×657

9 = 22338

3.Let us assume, to the contrary, that √3 is rational number of form p/q where p

and q are co- primes and q ≠ 0 .

So, p/q = √3

Squaring on both sides, and rearranging, we get , 𝑝2 = 3𝑞2

Therefore, 𝑝2 is divisible by 3, It means that p is also divisible by 3.

So, we can write p = 3c for some integer c.

Substituting for p, we get 9𝑐2= 3𝑞2, that is, 3𝑐2= 𝑞2.

This means that 𝑞2 is divisible by 3, and so q is also divisible by 3

Therefore, p and q have at least 3 as a common factor.

But this contradicts the fact that p and q are coprime.

This contradiction has arisen because of our incorrect assumption that √3 is

rational.

Hence √3 is an irrational Number.

4. Suppose that 3 + 2√ 5 is a rational number say “a”

a =3 + 2√ 5

⇒𝑎−3

2 = √ 5

As a is a rational number therefore a-3 is a rational and so 𝑎−3

2 is a rational

number but we know that √ 5 is an irrational .

This contradicts that our supposition is wrong.

Hence 3 + 2√ 5 is an irrational number.

5. Let N= 5 × 7×11 + 7

= 7(5 ×11 + 1 )

= 7 m where m = 5 ×11 + 1 for some integer m

⇒ N has 1,7 ,m,7m as its factors and hence it is a composite number.

6.(i) 13

3125 , q= 3125 = 55 = 20 × 55 , it is in the form 2𝑛 × 5𝑚.Therefore it is

terminating decimal

(ii) 15

1600 =

3

320 , q=320= 26 × 5 ,it is in the form 2𝑛 × 5𝑚.Therefore

it is terminating decimal

(iii) 29

22×32 ,q= 22 × 32, it is not in the form 2𝑛 × 5𝑚.Therefore it is

non-terminating decimal

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Chapter.2 POLYNOMIALS Key Points

# Degree of a polynomial:

The highest power of the variable in a polynomial is called its degree.

# Linear Polynomial (Straight line on graph)

A polynomial of degree 1 is called a linear polynomial

Exp : 2𝑥 + 5 , 𝑥 + 4

# Quadratic Polynomial: (parabola shape on the graph)

A polynomial of degree 2 is called a quadratic polynomial

Ex : 𝑥2 + 2𝑥 + 1

# Cubic Polynomial:

A polynomial of degree 3 is called a cubic polynomial.

Ex : 2𝑥3 + 4𝑥2 + 5𝑥 + 6

Bi-Quadratic Polynomial :

A polynomial of degree 4 is called a Bi-Quadratic polynomial.

Ex : 4𝑥4 + 3𝑥2 + 𝑥2 + 𝑥 + 5

Zeroes of Polynomial:

K is said to be zero of a polynomial 𝑝(𝑥) 𝑖𝑓 𝑝(𝑘) = 0

Geometrical definition of zeroes:

The zeroes of a polynomial 𝑝(𝑥) all the x - coordinates of the points where

the graph of 𝑦 = 𝑝(𝑥) intersects the 𝑥 − 𝑎𝑥𝑖𝑠.

(iii) (iv)

(i) The number of zeroes is 1 as the graph intersects the 𝑥 − 𝑎𝑥𝑖𝑠 at one point.

(ii) The number of zeroes is 2 as the graph intersects the 𝑥 − 𝑎𝑥𝑖𝑠 at two points.

(iii) The number of zeroes is 3 as the graph intersects the 𝑥 − 𝑎𝑥𝑖𝑠 at three points.

(iv) The number of zeroes is 4 as the graph intersects the 𝑥 − 𝑎𝑥𝑖𝑠 at four points.

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# Relationship between the zeroes and the coefficient of a polynomial:

(i) If 𝛼, 𝛽 are zeroes of 𝑝(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, 𝑡ℎ𝑎𝑛

Sum of zeroes 𝛼 + 𝛽 = −𝑏

𝑎−

−(𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑒𝑛𝑡 𝑜𝑓 𝑥)

𝐶𝑜𝑒𝑓𝑓𝑖𝑒𝑛𝑡 𝑜𝑓 𝑥2

Product of zeroes 𝛼𝛽 =𝑐

𝑎=

𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑒𝑛𝑡 𝑜𝑓 𝑥2

# Formation of a quadratic polynomial:

If 𝛼 , 𝛽 are zeroes or Roots of a quadratic polynomial 𝑝(𝑥) then

𝑝(𝑥) = 𝑥2 − (𝛼 + 𝛽)𝑥 + 𝛼𝛽

⟹ 𝑝(𝑥) = 𝑥2 − (𝑠𝑢𝑚 𝑜𝑓 𝑧𝑒𝑟𝑜𝑒𝑠)𝑥 + 𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑧𝑒𝑟𝑜𝑒𝑠.

Questions for Revision

Q.1 The graphs of y = p(x) are given below, for some polynomials p(x). Find

the number of zeroes of p(x),

( i )

(ii)

Which Polynomial is shown in the graph. Write zeroes of the polynomial

Q.2 Find the zeroes of the quadratic polynomial 𝑥2 + 11𝑥 + 30 and verify

relation between zeroes and the coefficients of the polynomial.

Q.3 If 𝑥 + 𝑎 is a factor of 2𝑥2 + 2𝑎𝑥 + 5𝑥 + 10 , 𝑓𝑖𝑛𝑑 𝑎 .

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Q.4 If the product of zeroes of the polynomial 𝑎𝑥2 − 6𝑥 − 6 𝑖𝑠 4 find the

value of a.

Q.5 If a polynomial p(x) has zeroes as 2 and 3, Write a polynomial whose zeros

are reciprocal of zeroes of p(x).

Solutions Chapter 2. Q1.( i) Since the curve of p(x) intersect the X-axis at three point therefore there

are Three zeroes of p(x).

(ii) Quadratic Polynomial is shown .Its zeroes are -2 and 3

Q.2 Let p(x)= 𝑥2 + 11𝑥 + 30

= 𝑥2 + 6𝑥 + 5𝑥 + 30

= x(x+6)+5(x+6)

=(x+6)(x+5) If x+6=0 then x=-6 and if x+5=0 then x=-5

Therefore, 𝛼 = −6 𝑎𝑛𝑑 𝛽 = −5

𝛼 + 𝛽 = (−6) + (−5) = −11 and 𝛼 × 𝛽 = (−6) × (−5)=30

a=1 ,b=11 and c=30 −𝑏

𝑎=

−11

1= −11 and

𝑐

𝑎=

30

1= 30 hence , 𝛼 + 𝛽 =

−𝑏

𝑎 and 𝛼 × 𝛽 =

𝑐

𝑎 is

verified.

Q3. Given that 𝑥 + 𝑎 is a factor of p(x) = 2𝑥2 + 2𝑎𝑥 + 5𝑥 + 10

Putting x = −𝑎 𝑖𝑛 𝑝(𝑥), 𝑤𝑒 ℎ𝑎𝑣𝑒, P(-a) = 2(𝑎)2 + 2𝑎(−𝑎) + 5(−𝑎) + 10

= 2𝑎2 − 2𝑎2 − 5𝑎 + 10 = −5𝑎 + 10

Since,𝑥 + 𝑎 is a factor of p(x) ⇒ p(-a) =0 ⇒ −5𝑎 + 10 = 0 ⇒ a=−10

−5 = 2 Ans.

Q4.𝑝(𝑥) = 𝑎𝑥2 − 6𝑥 − 6 Here a=a , b= -6 and c= -6

We know that, 𝛼 × 𝛽 = 𝑐

𝑎

Given that , 𝛼 × 𝛽 = 4 ⇒ 4 = 𝑐

𝑎 ⇒ 4a = 𝑐 ⇒ a =

𝑐

4 =

−6

4 =

−3

2 Ans.

Q5. Given, 𝛼 = 2 𝑎𝑛𝑑 𝛽 = 3

Let the given polynomial be P’(x) then its zeroes will be 1

𝛼 and

1

𝛽

P’(x)= 𝑥2 − (1

𝛼 +

1

𝛽)𝑥 +

1

𝛼 ×

1

𝛽

=𝑥2 − (1

2 +

1

3)𝑥 +

1

2 ×

1

3 = 𝑥2 −

5

6 𝑥 +

1

6 =

1

6(6𝑥2 − 5𝑥 + 1)

Ans. 6𝑥2 − 5𝑥 + 1

Chapter.3 PAIR OF LINEAR EQUATONS IN TWO VARIABLE

KEY POINTS

● The general form for a pair of linear equations in two variables x and y .

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

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Where a1, b1, c1, a2, b2, c2 are all real numbers and a1≠ 0, b1 ≠0, a2≠ 0, b2≠ 0.

● Graphical representation of a pair of linear equations in two variables:

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

● Will represent intersecting lines if -

1 1

2 2

a b

a b

unique solution. And these types of equations are called

consistent pair of linear equations.

Ex: x – 2y = 0; 3x + 4y – 20 = 0

● will represent overlapping or coincident lines if

1 1 1

2 2 2

a b c

a b c

i.e. Infinitely many solutions, consistent or dependent pair of linear

equations

Ex: 2x + 3y – 9 = 0; 4x + 6y – 18 = 0

The graph is Coincident lines,

● will represent parallel lines if

1 1 1

2 2 2

a b c

a b c

i.e. no solution and called inconsistent pair of linear

equations.

Ex: x + 2y – 4 = 0; 2x + 4y – 12 = 0

Parallel lines, no solution.

● Algebraic methods of solving a pair of linear equations:

o Substitution method

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o Elimination Method

LEVEL- I

1. Check whether given pair of lines is consistent or not , 3x+8y +7 =0 ,7x-

6y -9 =0

2. Write the relationship between the coefficients for which the pair of

linear equations has unique solution.

3. Write any one equation of the line which is parallel to 2x + 3y -8=0

LEVEL-II

1. Solve the equations: 3x – y = 3; 7x + 2y = 20 Ans. x = 2 y = 3

2. Find the fraction which becomes 2/3 when the numerator is increased

by 2 and equal 4/7 when the denominator is increased by 4.

Ans. x= 16 y= 27 Required Fraction=16/27

3. If the system of equation 4x+y = 3 and (2k-1)x +( k-1) y = 2k+1 is

inconsistent ,then find k. Ans. k = 3/2

LEVEL – III

1. Draw the graph of the equations 4x – y = 4; 4x + y = 12. Determine the

vertices of the triangle formed by the lines representing these equations and the

x- axis. Shade the triangular region so formed.

2. For what value of k will the equation x +5y-7=0 and 4x +20y +k=0

represent coincident

lines? Ans. k = -28

3. Solve 2x +3y =11 and 2x -4y = -24 and hence find the value of m for

which y = mx +3 .

Ans x= -2 y = 5 m = -1

4. 5 pencils and 7 pens together cost Rs. 50 whereas 7 pencils and 5 pens

together cost Rs. 46. Find the cost of one pencil and that of one pen.

Ans. pencil-Rs 3 pen- Rs 5

Chapter 4. QUADRATIC EQUATIONS

KEY POINTS

1. The general form : ax2+bx+c=0 , a≠0 , a,b,c are real numbers.

2. Discriminant: D= b2-4ac

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3. Quadratic formula or Sreedharacharya Method to find roots of the

Quadratic equation:

𝑥 =−𝑏±√𝑏2−4𝑎𝑐

2𝑎

4. Sum of the roots , 𝛼+β= −𝑏

𝑎 and product of the roots , αβ=

𝑐

𝑎

5. Forming of Quadratic equation when the roots α,β are given:

x2 – (α+β)x + αβ= 0

6. Nature of roots of ax2 + bx +c =0

i) If D>0 then real and unequal ( Distinct) roots

ii) If D=0 then real and equal roots

iii) If D<0 then no real roots

LEVEL – I

1. If -2 is root of the equation x2 + kx -6 =0 , then find value of k.

Ans. -1

2. The product of roots of a quadratic equation 2x2 +7x -4=0 is ? Ans.

-2

3. If the equation 9x2 +6x -k =0 has equal roots , then find value of k. Ans.

-1

LEVEL- II

1. Find the discriminant of the quadratic equation 2x2 -4x +3 =0 and

hence find nature of its roots. Ans. D=-8 <0 so

no real roots

2. Find the roots of x2-3x -10 =0 Ans. -2,5

3. Find two numbers whose sum is 27 and product is 182. Ans. 13, 14

LEVEL -III

1 Solve the equation: 𝑥

𝑥+1 +

𝑥+1

𝑥 =

34

15 , x≠0 , x≠ -1 Ans. x=3/2 or x=-5/2

2. Find the values of k for each of the following quadratic equations,

so that they have two equal roots

(i) 2x2 + kx + 3 = 0 (ii) kx (x – 2) + 6 = 0

Ans. i) k= ±2√6 ii)k=6

LEVEL IV

1. An express train takes 1 hour less than a passenger train to

travel 132 km between Mysore and Bangalore (without taking into

consideration the time they stop at intermediate stations). If the average

speed of the express train is 11km/hr more than that of the passenger

train, find the average speed of the two Trains.

Ans. passenger-33km/hr express-44km/hr

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2. Sum of the areas of two squares is 468 m2 . If the difference of their

perimeter is 24m. Find the sides of two squares.

Ans. 18m ,12m

Chapter.5 ARITHMETIC PROGRESSION

KEY POINTS

1. an = a + (n-1)d where a is first term of AP , d is common difference

2. d = a2 – a1

3. Sn = 𝑛

2 {2a + (n – 1 ) d }

4. Sn = 𝑛

2 ( a + l ) where l is last term of given AP

LEVEL – I

1. Find nth term of -15, -18, -21, ...... Ans. -3 (n+4 )

2. Find the common difference of AP 1, -2, -5, ….. Ans. -3

3. If 2p, p +10, 3p +2 are in AP then find p. Ans. p= 6

LEVEL – II

1. If a= 5, d= 3 and an = 50, then find n. Ans. 16

2. How many three digit numbers are divisible by 7 ? Ans. 128

3. Given a= 2, d= 8, sn = 90 find n and an . Ans. n= 5 , an = 34

4. Find sum of all odd numbers between 0 and 50. Ans. 625

5. Write the next term of an AP √8, √18, √32, --- Ans. 5√2

LEVEL – III

1. If sum of n terms of an AP is 2n2 +5n , then find its nth term. Ans. 4n

+3

2. If the nth term of an AP is (2n +1 ) , find the sum of first n terms of the

AP. Ans. n (n +2 )

3. The 7th term of an AP is 32 and its 13th term is 62. Find a and d . Ans.

2,5

4. Find the 20th term from the last term of the AP : 3, 8, 13, . . ., 253.

Ans158

Chapter.6.Triangles KEY POINTS

BASIC PROPORTIONALITY THEOREM OR THALES THEOREM

Statement

If a straight line is drawn parallel to one side of a triangle intersecting the

other two sides, then it divides the two sides in the same ratio.

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CONVERSE OF BASIC PROPORTIONALITY THEOREM

(CONVERSE OF THALES THEOREM)

Statement

If a straight line divides any two sides of a triangle in the same ratio, then

the line must be parallel to the third side.

PYTHAGORAS THEOREM-

Statement

In a right angled triangle, the square of the hypotenuse is equal to the sum

of the squares of the other two sides.

15

Questions

1. In given figure, DE ∥ BC. Find the length of side AD if AE = 1.8 cm, BD

= 7.2 cm and CE = 5.4 cm.

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2. In ∆ABC, DE || BC, find the value of x.

3. If ∆ ABC ~ ∆ 𝐷𝐸𝐹 such that area of ∆ ABC is 9 cm2 and area of ∆ 𝐷𝐸𝐹

is 16 cm2 and BC = 2.1 cm. Find the length of EF.

4. ABC is an isosceles triangle right angled at C. Prove that AB2 = 2AC2

5. In the figure ABD is a triangle right angled at A and AC is perpendicular

to BD then show that AC2= BC x DC

6. Prove that the area of the equilateral triangle described on the side of a

square is half the area of the equilateral triangle described on its diagonal.

7. In an equilateral triangle ABC, D is a point on side BC such that BD = 1

3

BC. Prove that 9AD2= 7AB2

8. Prove that if a line is drawn parallel to one side of a triangle intersecting

the other two sides, then it divides the two sides in the same ratio.

(Basic Proportionality Theorem OR Thales Theorem)

9. Prove that in a right angled triangle, the square of the hypotenuse is equal

to the sum of the squares of the other two sides. (Pythagoras Theorem)

Ch.7 COORDINATE GEOMETRY

Key points:

1. Distance formula

Distance between two points P (𝑥1 , 𝑦1) and Q (𝑥2 , 𝑦2) is given by

PQ = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2

2. Section formula

The Co-ordinates of a point A (x, y) which divides the join of two points P

(x , y) and Q (𝑥2 , 𝑦2) Internally in the ratio 𝑚 ∶ 𝑛 is given by

17

𝑥 = 𝑚𝑥2+ 𝑛𝑥1

𝑚+𝑛 𝑦 =

𝑚𝑦2+ 𝑛𝑦1

𝑚+𝑛

⋆The Co - ordinate of a midpoint of PQ is given by

𝑥 = 𝑥1+ 𝑥2

2 , 𝑦 =

𝑦1+ 𝑦2

2

Questions for revision: -

1. Find the distance of point (-4, 3) from the origin.

2. Find a point on x – axis, which is equidistant from the point (7,6) and (-3, 4).

3. If A = (2, 7) and B = (x, - 5). Find the value of x so that AB = 13 units.

4. The Co-ordinates of points A and B are (1 , 2) and (2 , 3) respectively. Find

the Co-ordinates of point P, in AB so that AP : PB = 4 : 3..

5. In what ratio does the point (3 ,-2) divide the line segment joining the points

(5 , -3) and (- 9 , 4).

6. Three vertices of the parallelogram are given in the form (1, 2) (2, 4) & (3, 7).

Finds its fourth vertex .

7. Find the coordinates of a point A, where AB is a diameter of a circle whose center

is (2,-3) and B is (1,4) .

Solutions Chapter7.

1.Coordinates of origin are (0,0)

Dist =√{0 − (−4)}2 + {0 − 3}2 = √{0 + 4}2 + {−3}2=√16 + 9 = √25 =5 ans.

2. Let the point(x,0) on x- axis be P and A=(7,6) , B =(-3,4)

Given, PA=PB ⇒ 𝑃𝐴2 = 𝑃𝐵2 ⇒ {𝑥 − 7}2 + {0 − 6}2 = {𝑥 − (−3)}2 +{0 − 4}2

⇒ 𝑥2 − 14𝑥 + 49 + 36 = {𝑥 + 3}2 + {−4}2

⇒ 𝑥2 − 14𝑥 + 49 + 36 = 𝑥2 + 6𝑥 + 9 + 16

⇒ −14𝑥 + 85 = 6𝑥 + 25

⇒ −14𝑥 − 6𝑥 = 25 − 85

⇒ −20𝑥 = −60 ⇒x = −60

−20 = 3

Ans. (3,0)

18

3. Given AB =13 ⇒ 𝐴𝐵2 =169 ⇒ {2 − 𝑥}2 + {7 + 5}2=169

⇒ 4 − 4𝑥 + 𝑥2 + 144 =169 ⇒ 𝑥2 − 4𝑥 + 148 − 169 =0

⇒ 𝑥2 − 4𝑥 − 21 =0 ⇒ 𝑥2 − 7𝑥 + 3𝑥 − 21 =0 ⇒ 𝑥(𝑥 − 7) + 3(𝑥 − 7) = 0

⇒ (𝑥 − 7)(𝑥 + 3) = 0 ⇒ x = 7 or x=-3

4. Given ,A (1 , 2) and B (2 , 3)

Point P on AB such that that AP : PB = 4 : 3. ⇒ m:n =4:3

Using section formula,

𝑥 = 𝑚𝑥2+ 𝑛𝑥1

𝑚+𝑛 =

4×2+3×1

4+3 =

8+3

4+3 =

11

7 , 𝑦 =

𝑚𝑦2+ 𝑛𝑦1

𝑚+𝑛=

4×3+3×2

4+3 =

12+6

4+3 =

18

7

⇒ 𝑃(11

7,

18

7) Ans.

5. Let the required ratio be m:n , Using section formula,

𝑥 = 𝑚𝑥2+ 𝑛𝑥1

𝑚+𝑛 ⇒ 3 =

𝑚×(−9)+𝑛×5

𝑚+𝑛 ⇒ 3 =

−9𝑚+5𝑛

𝑚+𝑛

⇒ 3(𝑚 + 𝑛) = −9𝑚 + 5𝑛 ⇒ 3𝑚 + 3𝑛 = −9𝑚 + 5𝑛 ⇒ 3𝑚 + 9𝑚 = 5𝑛 − 3𝑛

⇒ 12𝑚 = 2𝑛 ⇒𝑚

𝑛=

2

12 ⇒

𝑚

𝑛=

1

6 Ans.

6. Three vertices of the parallelogram are given in the form (1, 2) (2, 4) & (3, 7).

Finds its fourth vertex

Let A(1,2) B(2,4) , C(3,7) and D (x,y)

Since diagonal of parallelogram bisect each other

Mid point of AC = Mid point of BD

(1+ 3

2 ,

2+ 7

2)= (

2+ 𝑥

2 ,

4+ 𝑦

2) ⇒

1+ 3

2=

2+ 𝑥

2 𝑎𝑛𝑑

2+ 7

2=

4+ 𝑦

2 ⇒ 𝑥 = 2 𝑎𝑛𝑑 𝑦 = 5

7. Let the coordinates of A(x,y)

As C is the centre of of the circle so it the mid pont of Daimeter AB

Using Mid point formula, 𝑥 = 𝑥1+ 𝑥2

2 , 𝑦 =

𝑦1+ 𝑦2

2

2 = 𝑥+ 1

2 ,−3 =

𝑦+ 4

2 ⇒ x=3 and y= -10

Ans.(3,-10)

Chapter.8 INTRODUCTION TO TRIGONOMETRY

Key Points 2 2sin cos 1 1

tan , cot , sin cos 1, cos , seccos sin sin cos

ec

19

0° 30° 45° 60° 90°

sin 𝜃 0 1

2

1

√2

√3

2

1

cos 𝜃 1 √3

2

1

√2

1

2

0

tan 𝜃 0 1

√3

1 √3 Not defined

cot 𝜃 Not defined √3 1 1

√3

0

sec 𝜃 1 2

√3 √2 2 Not defined

cosec 𝜃 Not defined 2 √2 2

√3

1

1. 7 (1 sin )(1 sin )

cot , :8 (1 cos )(1 cos )

If evaluate

.

2. Find the value of 0

2 0

2 tan 30

1 tan 30.

3. If 1

( ) 3 ( ) ;0 90 ; ,3

Tan A B and Tan A B A B A B Find A and B.

4. Prove the following identities, where the angles involved are acute angles

for which the expressions are defined.

1 sinsec tan

1 sin

AA A

A

5. Prove that

sec (1 sin )(sec tan ) 1A A A A

6. Prove that

tan cot1 sec cos

1 cot 1 tanec

2 4

2

4 2

sin 2sin7.Pr sec 1

2cos cosovethat

Practice Questions

1. 2

tan 2 1, sin cos4

If showthat

2. sin tan (1 cot ) 1

3 sin cos , ( )sin cos 3

If find the value of Ans

3. 3 7 3

sin , cos tan . (cos , tan )4 4 7

If A calculate A and A Ans A A

20

4. .Evaluatethe following 2 2 2

2 2 2

2 2

) 2 tan 45 cos 30 sin 60 ( .2)

5cos 60 4sec 30 tan 45 67) ( . )

sin 30 cos 30 12

i Ans

ii Ans

2 2

2 2

2 2

2 2

5. sin , cos tan .

2( .cos , tan )

2

a bIf find thevalueof and

a b

ab a bAns

a b ab

6. cos sin 2 sin , cos sin 2 cosIf provethat

7. Prove That 2cos sin

sin cos1 tan sin cos

A AA A

A A A

2 2 2

4 4

2 2

8. tan sin tan sin , ( ) 16

sin cos9.Pr 1.

1 2sin cos

If m and n than showthat m n mn

ovethat

SOLUTIONS CHAPTER 8 1.

2 2 2 2 2 2

2

2

7cot ,cot , 7 , 8

8

, 49 64

8 7113 , 113 ,sin ,cos

113 113

8 8(1 sin )(1 sin ) (1 )(1 )

113 113

8 64 491 1

113 113113

(1 cos )(1 cos )

7 7 49 64(1 )(1 ) 1

113 113113 113

(1

BCBC k AC k

AC

AB BC AC AB k k

AB k AB k

49

sin )(1 sin ) 4911364(1 cos )(1 cos ) 64

113

2.

21

2

1 22( )

2 33 33 tan 60

1 1 231 ( ) 133

X

3.

1( ) 3 ( )

3

60 30

45, 15

Tan A B and Tan A B

A B and A B

After Solving we get A B

4.

1 sinsec tan

1 sin

AA A

A

LHS =1 sin

1 sin

A

A

2 2

2 2 2

1 sinsec tan

1 sin

(1 sin )(1 sin ) (1 sin ) (1 sin ) (1 sin )

(1 sin )(1 sin ) 1 sin cos cos

1 sinsec tan

cos cos

AA A

A

A A A A A

A A A A A

AA A RHS

A A

5.

2 2

2 2 2

sec (1 sin )(sec tan ) 1

sec (1 sin )(sec tan )

1 1 sin (1 sin )(1 sin ) 1 sin cos( )(1 sin )( ) 1cos cos cos cos cos cos

A A A A

LHS A A A A

A A A A AA RHS

A A A A A A

6.

2

tan cot1 sec cos

1 cot 1 tan

tan cot

1 cot 1 tan

sin cos sin cos

cos sin cos sincos sin sin cos cos sin

1 1sin cos sin cos

sin sin cos cos sin

cos (sin cos ) sin (cos sin ) cos (

ec

LHS

X X

2

3 3 2 2

cos

sin cos ) sin (cos sin )

sin cos (sin cos )(sin cos sin cos )

sin cos (sin cos ) sin cos (sin cos )

1 sin cos 1 sin cos

cos sin cos sin cos sin

sec cos 1ec RHS

22

2 42

4 2

2 42

4 2

2 4

4 42

4 2

4 4

2 2 42

2

2 2 22

2

2

sin 2sin7.Pr sec 1

2cos cos

sin 2sinsec

2cos cos

sin 2sin

cos cossec2cos cos

cos cos

tan sec 2 tansec

2 sec

tan (sec 2 tan )sec

2 sec

tasec

ovethat

LHS

2 2

2 2 2 2 2

2

2 2

n (2 sec ){sec 2 tan sec 2(sec 1) 2 sec }

2 sec

sec tan 1

Chapter9. SOME APPLICATIONS OF TRIGONOMETRY

1. A man on a cliff observes a boat at an angle of depression of 30° which is

approaching the shore to the point immediately beneath the observer with

a uniform speed. Six minutes later, the angle of depression of the boat is

found to be 60°. Find the time taken by the boat to reach the shore.

2. A TV tower stands vertically on a bank of a canal. From a point on the

other bank directly opposite the tower, the angle of elevation of the top of

the tower is 60°. From another point 20 m away from this point on the

line joining this point to the foot of the tower, angle of elevation of the

top of the tower is 30°. Find the height of the tower and the width of the

canal.

23

3. The angle of elevation of the top of a tower 30m high from the foot of

another tower in the same plane is 60° and the angle of elevation of the

top of the second tower from the foot of the first tower is 30°. Find the

distance between the two towers and also height of the other tower.

4. The aviation technology has evolved many upgradations in the last few

years. It has taken in account speed, direction and distance as well as

other features of the flight. The wind plays a vital role, when a flight will

travel

Angle of elevation: The angle of elevation of an object viewed is the

angle formed by the line of sight with the horizontal, when it is above the

horizontal level.

(a) If the point C moves towards the point B, then angle of elevation :

(i) Increase

(ii) Decrease

(iii)Remain same

(iv)None of these

24

(b) The of distance BC is equal to ?

(i) 500√3 m (ii) 500 m

(iii) 300√3 m (iv) 400√3 m

(c) If angle of elevation changes from 60° to 45°. What will be the new distance of

BC?

(i) 1500 m (ii) 1400 m

(iii) 1300 m (iv) 1200 m

(d) Find the area of ∆𝐴𝐵𝐶.

(i) 30000√3 𝑚2 (ii) 375000 𝑚2

(iii) 37500√3 𝑚2 (iv) 375000√3 𝑚2

5. There is a small island in the middle of a 100m wide river and a tall tree stands on the

island. P and Q are points directly opposite to each other on two banks and in line

with the tree. If the angles of elevation of the top of the tree from P and Q are

respectively 30° and 45° , then find the height of the tree.

Practice questions 1. Point A is the position of jet fighter flying in the sky. The angle of elevation of point

A from ground is shown. After 15 seconds , the jet fighter moves in the direction AP

and reaches at point P. The angle of elevation of point P on the ground is

shown(Assume that fighter is flying at the constant height above the ground.)

a. The angle of elevation of point A(Jet fighter plane) from the ground is

(i) 30° (ii) 60° (iii) 90° (iv) none of these

b. The angle of elevation of a point P from the ground is

(i) 30° (ii) 60° (iii) 90° (iv) none of these

c. What is the distance AP, if jet is flying with speed 720 km/hr in 15 seconds?

(i) 300m (ii) 3000m (iii) 30m (iv) none of these

d. If the jet is flying at the speed of 720 km/hr then the constant height at which it

is flying

(i) 1500m (ii) 1500

√3 (iii) 1500√3 (iv)

150

√3

25

e. AB is called

(i) line of sight (ii) angle of elevation (iii) angle of depression (iv) none

of these

Ans. (a). (ii) 60° (b) (i) 30° (c) (ii) (d) (iii) 1500√3 (e) (i) line of sight

2. A building is made by keeping the lower window of a building at a particular height

above the ground and upper window is constructed at some vertically above the lower

window. Position of both windows are shown in diagram.

Both windows are designed and constructed in order to have proper sun light.

At certain instant, the angle of elevation of balloon from these windows are shown.

Balloon is flying at centre height H above the ground.

a) Length BQ (in terms of H)

i. (H-2)m ii. 𝐻−2

√3m iii. (H-2) √3 m iv. H√3

b) Length AR(in terms of H)

i. (H-2)m ii. 𝐻−2

√3m iii. (H-8) √3 m iv. H√3

c) Height H

i. 10m ii. 9m iii. 8m iv. 11m

d) Distance PC

i. 3m ii. 3√3m iii. √3m iv. 4√3m

e) SR=

i. 6m ii. 2m iii. 11m iv. 3m

Ans . (a) ii (b) iii (c) iv (d) ii (e) iv

3. A contractor constructs a vertical pillar at a horizontal distance of 300m from a fixed

point. It was decided that angle of elevation of the top of the complete pillar from that

point to be 60°. He finished the job by making a pillar such that the angle of elevation

of its top was 30°. Find the height of the pillar to be increased as per the terms of the

contract.

(Ans. 200√3 m)

26

4. Two pillars of equal height are on either side of a road, which 100m wide. The angles

of the top of the pillars are 60° and 30° at a point on the road between the pillars. Find

the position of the point between the pillars. Also find the height of each pillar.

(Ans. 25m, 43.3m)

5. In a storm, a tree got bent by the wind. The top of the tree meets the ground at an

angle of 30°, whose distance of 30m from the root. At what height from the bottom

did the tree get bent? What was the original height of the tree?

(Ans. 17.3 m, 51.9 m)

6. An army pilot is flying an aeroplane at an altitude of 1800 m observes some

suspicious activity of two ships which are sailing towards it in the same directions and

immediately report it to the navy chief. The angle of depression of the ships as

observed from the aeroplane are 60° and 30° respectively. Find the distance between

two ships.

(Ans. 1200√3 m)

SOLUTIONS CHAPTER 9 1. Let the height of the cliff be h m., according to the figure it is AB and the observer is

standing at point A. A boat reaches the point C from Point D in 6 minutes.

Let DC =𝑥1, CB=𝑥2

Now from triangle ABC and ABD

2

2 1 2

1 2 1 22

12 1 2 2 1 2

1tan 60 , . 3 , tan 30

3

, . 33 3

3 , 2 ,2

h hh x

x x x

x x x xh x

xx x x x x x

But it is given that in covering the distance 𝑥1, 6 minutes are taken.

Time taken in covering the distance 𝑥2 =6

2= 3minutes

Time taken in reaching B from point D=6+3=9 minutes.

2. Let BC =X m be the width of the canal and AB= h m be the height of the TV tower.

From given figure

tan 60 , 3

13 ...........( ), tan 30 ,

203

20.............( )

3

AB h

BC x

AB hh x i

BD x

xh ii

From (i) and (ii) 20

3 , 3 20 , 2 203

10

xx x x x

x

27

Substituting the value of x in (i)

h= 10(√3)=10X1.732

h=17.32 m

3. Let BC= 30m and AD=h m be the height of the first and second towers and x be the

distance between the towers.

AB= x m

From the given diagram

1

tan 30 , , 33

30 30tan 60 , 3 ,

3

30 3 30 310 3

33 3

AD hx h

AB x

BCx

AB x

x X

On putting the value of x we get

10√3 =h√3

h=10 m

4. Ans a. (i) increase

b. In ∆𝐴𝐵𝐶

3 1500 1500 2sin 60 , , 1000 3

2 3

AB XAC m

AC AC

c. In ∆𝐴𝐵𝐶 1500 1500

an 60 , 3 , 500 33

ABBC m

AC BC

d. If angle of elevation changes from 60° to 45° then in ∆𝐴𝐵𝐶 1500

tan 45 , 1 , 1500AB

BC mBC BC

e. Area of ∆𝐴𝐵𝐶 =1

2xBCxAB

1500 3 1500 375000 3

2x x m

5. Let OA be the tree of height h m.

Given PQ=100 m and angles of elevation are 30° and 45°

In right angle ∆𝑃𝑂𝐴

1

tan30 , , 3 ..........( )3

OA hOP h i

OP OP

Now in right angle ∆𝑄𝑂𝐴

28

tan 45 , 1 , ........( )

( ) ( )

3 , ( 3 1) , 100 ( 3 1)

100 ( 3 1) 100( 3 1)50( 3 1) 36.6

23 1 ( 3 1)

OA hh OQ ii

OQ OQ

i ii

OP OQ h h PQ h h

h X m

Chapter10.Circle KEY CONCEPTS

● Circle: A circle is a collection of all points in a plane which are at a constant

distance (radius) from a fixed point (centre).

● Tangent to a Circle :

It is a line that intersects the circle at only one point. There is only one tangent at a

point of the circle. The tangent to a circle is a special case of the secant, when the

two end points of its corresponding chord coincide.

Theorems:

(1)The tangent at any point of a circle is perpendicular to the radius through the

point of contact.

(2)The lengths of tangents drawn from an external point to a circle are equal.

● Number of tangents from a point on a circle-

(1)There is no tangent to a circle passing through a point lying inside the circle.

(2) There are exactly two tangents to a circle through a point lying outside the

circle.

(3)There is one and only one tangent to a circle passing through a point lying on

the

circle.

●

1. How many tangents can a circle have?

2. Find the length of the tangent drawn from a point whose distance from the center of a

circle is 25 cm and the radius of the circle is 7 cm.

3. If the tangent at a point P to a circle with centre O cuts a line through O at Q such that

PQ = 24 cm and OQ = 25 cm. Find the radius of circle?

29

4. A point P is 13 cm from the centre of the circle. The length of the tangent drawn from

P to the circle is 12 cm. Find the radius of the circle.

5. In the given fig. P, Q and R are the points of contact. If AB = 4 cm, BP = 2 cm then

the perimeter of ∆ ABC is

6. Prove that a tangent to a circle is perpendicular to the radius through the point of

contact.

7. Prove that the lengths of two tangents drawn from an external point to a circle are

equal.

8. A circle touches all the four sides of a quadrilateral ABCD.

Prove that AB + CD = BC + DA

9. A circle is touching the side BC of ∆ ABC at P and touching AB and AC produced at

Q and R respectively.

Prove that AQ = 1

2 (𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 ∆ 𝐴𝐵𝐶)

10. In the given figure, XY and X’Y’ are two parallel tangents to a circle with centre O

and another tangent AB with point of contact C intersects XY at A and X’Y’ at B.

Prove that AOB = 90.

11. Prove that opposite sides of a quadrilateral circumscribing a circle subtend

supplementary angles at the centre of circle.

Chapter11.Construction

1. Draw a line segment of length 8 cm and divide it in the ratio 4:5.

30

2. Draw a line segment of length 7.6 cm and divide it in the ratio 5:8. Measure the two

parts.

3. Draw a circle of radius 6 cm. From a point 10 cm away from its centre. Construct the

pair of tangents to the circle and measure their lengths.

4. Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an

angle of 600.

5. Draw a pair of tangents to a circle of radius 4.5 cm which are inclined to each other at

an angle of 450.

6. Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of

radius 6 cm and measure its lengths. Also verify the measurements by actual calculation.

7. Draw two concentric circles of radii 3 cm and 5 cm. Construct a tangent to the smaller

circle from a point on the longer circle. Also measure its length.

Chapter 12- AREA RELATED TO CIRCLE

Key Concepts: Area of a square = side x side

Area of rectangle = length x breadth

Circumference of a circle = 2πr

Area of a circle = πr2 [where r is the radius of a circle]

Area of a semi-circle = 𝜋

2 𝑟2

Area of a quadrant = 𝜋

4 𝑟2

Area of a circular path or ring:

Then area of shaded part = πR2 – πr2 = π(R2 – r2) = π(R + r)(R – r)

Let ‘R’ and ‘r’ he radii of two circles

Minor arc and Major Arc: An arc length is called a

major arc if the arc length enclosed by the two radii is

greater than a semi-circle.

If the arc subtends angle ‘θ’ at the centre, then the

Length of minor arc = θ

360×2𝜋𝑟=

θ

180×𝜋𝑟

Length of major arc = 360−θ

360 x 2𝜋𝑟

Sector of a Circle and its Area:

31

A region of a circle enclosed by any two radii and

corresponding arc is called the sector of a circle.

(i) A sector is called a minor sector if the minor arc of the circle is part

of its boundary.

is minor sector and OACB is major sector.

Area of minor sector = θ

360×𝜋𝑟2 , Perimeter of minor sector =

2𝑟+θ

360×2𝜋𝑟

Area of major sector = 360−θ

360×𝜋𝑟2

Perimeter of major sector = 2r + 360−θ

360 x 2𝜋𝑟

Minor Segment and Major Segment: The region enclosed by an arc and a chord is called a

segment of the circle.

The region enclosed by the chord PQ & minor arc PRQ is called the minor segment.

The region enclosed by the chord PQ & major arc PSQ is called the major segment.

Area of Minor segment = Area of the corresponding sector – Area of the corresponding

triangle

= θ

360×𝜋𝑟2 -

1

2r2sinθ

Area of major segment = Area of a circle – Area of the minor segment

= 𝜋𝑟2 - ( θ

360×𝜋𝑟2 -

1

2r2sinθ )

Q1. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute

hand in 5 minutes.

Q2. Find the area of a quadrant of a circle whose radius is 56 cm .

Q3. Find the area of the sector of a circle with radius 6 cm and of angle 60° .

Q4. In a circle of diameter 42 cm, if an arc subtends an angle of 60°at the centre, then

what will be the length of minor and major arc. (Take π= 22/7)

Q5. The radii of two circles are 8 cm and 6 cm, respectively. Find the radius of the

circle having area equal to the sum of the areas of the two circles.

(ANS: 10 cm)

Q6. In Fig. 12.30,OACB is a quadrant of a circle with centre O and

radius 3.5cm. If OD = 2cm , find the area of the quadrant

OACB and shaded region.

( Ans. 77

8 cm2 ,

49

8 cm2 )

32

Q7. In Fig. 12.31, a square OABC is inscribed in a quadrant

OPBQ. If OA = 20 cm, find the area of the shaded region.

(Use π = 3.14)

( Ans. 228 cm2)

Q8. A car has two wipers which do not overlap. Each wiper has a

blade of length 21 cm sweeping through an angle of 120°. Find

the total area cleaned at each sweep of the blades

( Ans. 924 cm2)

Q9. The wheel of a car are of diameter 80cm each. How many complete revolutions does

each wheel make in 10 minutes when the car is travelling at a speed of 66km per hour ?

(Ans. 4375)

Q10. The radii of two circles are 19cm and 9cm respectively .find the radius of the circle

which has circumference equal to the sum of the circumferences of the two circles.

(Ans. 28cm)

Chapter 13- SURFACE AREAS AND VOLUMES

KEY CONCEPTS

● CUBOID :

Total surface area of a cuboid = 2( lb + bh + hl )

sq units

Lateral Surface area = 2 h (l+ b) sq units

Volume of a cuboid = lxbxh cubic units

CUBE :

Total Surface Area of a Cube = 6a2 sq units

LSA of cube = 4a2 sq units

Volume of the Cube = a3 cubic units

(length of each side is a)

33

● Cylinder / Right Circular Cylinder :

Total Surface Area of cylinder = 2πr ( r + h) sq units

Curved Surface Area of cylinder = 2πrh sq units

Volume of cylinder= πr2h cubic

units

● Right Circular Hollow Cylinder :

Area of each end = π (R2 – r2) [ R and r be the external

radius and internal radius ] sq units

Curved Surface Area of Hollow Cylinder = 2πrh +2πRh

sq units

Total Surface Area = 2πrh + 2πRh + 2πR2 - 2πr2

sq units

Volume of material = πh ( R2- r2 )

cubic units

● Sphere :

Surface Area of sphere= 4πr2 sq units

Volume of sphere = 4

3 πr3 cubic units

● Hemisphere

Curved Surface Area of hemisphere = 2πr2 sq

units

Total Surface Area of hemisphere =3πr2 sq

units

Volume of hemisphere= 2

3 πr3

cubic units

34

Cone/ Right Circular Cone :

Curved Surface Area of cone= πrl sq units

[Where l = Slant height and l2 = r2 + h2 ]

Total Surface Area of cone =πrl + π r2

= π r(l+r) sq units

Volume = 1

3 πr2h cubic units

Q1. 2 cubes each of volume 64 cm3 are joined end to end.

Find the surface area of the resulting cuboid. (Ans. 160cm2)

Q2. A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius.

The total height of the toy is 15.5 cm. Find the total surface area of the toy. (Ans. 572 cm2 )

Q3. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest

diameter the hemisphere can have? Find the surface area of the

solid (Ans. 332.5cm2)

Q4. A pen stand made of wood is in the shape of a cuboid with

four conical depressions to hold pens. The dimensions of the

cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the

depressions is 0.5 cm and the depth is 4.2 cm. Find the volume

of wood in the entire stand (see Fig.).

(Ans. 520.60cm3 )

Q5. A metallic sphere of radius 4.2 cm is melted and recast into

the shape of a cylinder of radius 6 cm. Find the height of the cylinder. (Ans. 2.74cm)

Q6. A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread

out to form a platform 22 m by 14 m. Find the height of the platform. (Ans. 2.5 cm )

Q7. How many silver coins, 1.75 cm in diameter and of thickness 2 mm, must be melted to

form a cuboids of dimensions 5.5 cm × 10 cm × 3.5 cm? ( Ans. 400)

Q8. Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How

much area will it irrigate in 30 minutes, if 8 cm of standing water is needed?

(Ans. 562500 m2 = 56.25 hectares )

Q9. A tent is in the shape of a cylinder surmounted by a conical top. If the height and

diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the

top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of the

canvas of the tent at the rate of Rs 500 per m2. (Note that the base of the tent will not be

covered with canvas.)

( Ans. required surface area of tent=44 m2 and Rs. 22000 )

Q10. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The

diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the

inner surface area of the vessel.

35

(The inner surface area = CSA of the cylindrical part + CSA of hemispherical part=572

cm2 )

Chapter.14.STATISTICS

Key Points

Name Formula:

Mean of grouped data

Median of grouped data

Mode of grouped data

Q1. Find the mode of the following frequency distribution.

Class Frequency

0-10 8

10-20 10

20-30 10

30-40 16

40-50 12

50-60 6

60-70 7

Q2. Find the mean of the following distribution:

Class: 3-5 5-7 7-9 9-11 11-13

Frequency 5 10 10 7 8

Q3. Find the median of the following data?

Class

Interval

30-40 40-50 50-60 60-70 70-80 80-90 90-100

Frequency 7 5 8 10 6 6 8

Q4. If the median of the following frequency distribution is 32.5, find the values

of f1 and f2.

Class Frequency

0-10 f1

36

10-20 5

20-30 9

30-40 12

40-50 f2

50-60 3

60-70 2

Total 40

Q5.Find the median of the following data:

Daily income Cumulative frequency

Less than 120 12

Less than 140 26

Less than 160 34

Less than 180 40

Less than 200 50

SOLUTIONS CHAPTER 14

Ans.1

Here, the maximum class frequency is 16

Modal class = 30-40

lower limit (l) of modal class = 30

Class size (h) =10

Frequency (f1) of the modal class = 16

Frequency (f0) of preceding class = 10

37

Frequency (f2) of succeeding class = 12

Ans.2

Class interval Frequency(fi) Mid value (xi ) fixi

3-5 5 4 20

5-7 10 6 60

7-9 10 8 80

9-11 7 10 70

11-13 8 12 96

Total: 40 326

Mean = 326

40

= 8.15

Ans. 3

Class interval Frequency Cumulative frequency

30-40 7 7

40-50 5 12

50-60 8 20

60-70 10 30

70-80 6 36

38

80-90 6 42

90-100 8 50

Total 50

Here N=50, so 𝑁

2 = 50/2 = 25

Now we see which number in the last column is nearest and greater than 25. We find

it as 30 , so the median class will be the class corresponding to this i.e 60-70

Where l is the lower limit of the median class.- here it is 60-70

C.f. is the cumulative frequency of the preceding class –here it is 20

f is the frequency of the median class – here it is 10

h is the class size i.e upper limit – lower limit, here it is 70-60 = 10

Median = 60+ 25−20

10 × 10

= 60 + 5

=65

Ans. 4 Given : Median = 32.5

39

Ans.5

Daily income Cumulative

frequency

Class Interval: Frequency

Less than 120 12 Less than 120 12

Less than 140 26 120-140 26-12 =14

Less than 160 34 140-160 34-26 = 8

Less than 180 40 160-180 40-34 = 6

Less than 200 50 180-200 50-40 = 10

Median:

40

N=50 , so N/2 =25 , so median class =120-140

Median = 120 + 25−12

14 × 20

= 120 + 13

14 ×20

= 120 +18.57 = 138.57

Chapter 15. Probability Key Points:

COIN

If a coin is tossed one time:

Sample space: { H,T}

Q1.A coin is tossed once , find the probability of getting :

A. a head B. a Tail C. No head

If a coin is tossed two times/ Two coins are tossed simultaneously

{HH} ,{HT}, {TT}, {TH} Total=4

Q.2 What is the probability of getting

A. head on both coins B. at least one head C. at

most one tail D. no tail

If a coin is tossed thrice/ Three coins tossed simultaneously

SS: HHH, HHT, HTH, THH, TTT, TTH, THT, HTT

Q3. If three coins are tossed simultaneously then find the probability of getting :

A exactly two heads B. at most two heads C.

Atleast one head and one tail D. no tails

DICE

41

When one dice is rolled

SS: {1,2,3,4,5,6}

Q4.A dice is rolled once, find the probability of getting :

A. A prime number

B. an even number when a dice is rolled

C. a multiple of 3

D. getting a number less than 5?

When Dice is rolled twice/ two dice are rolled simultaneously

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Total :36

Q5.Two dice are rolled simultaneously, find the probability of getting :

A. An even number as the sum B. The sum as a prime number

C. a total of atleast 10 D. a doublet of odd numbers

E. even number on both the dice F 5 as the sum

Playing Cards

42

Q.6 A card is drawn at random from a pack of 52 cards. Find the probability that the card

drawn is :

A. A black king B. spade or an ace C. a black card

D. The ace of spades E.A queen F. neither a red nor a

Queen

Q7. A bag contains 8 red, 6 white, and 4 black balls. A ball is drawn at random from the bag.

What is the probability of that the ball drawn is :

A. Red or white B. not black C. neither white nor black

Q8.Find the probability that a number selected at random from the numbers

1,2,3,4,5……35 is a :

A. Prime number B. multiple of 7 C. a multiple of 3 or 5

43

Q9 Cards numbered 1 to 30 are put in a bag. A card is drawn at random from this

bag. Find the probability that the number on the drawn card is:

A. Not divisible by 3 B. a prime number greater than 7

C.not a perfect square number.

SOLUTIONS CHAPTER 15

Ans.1 A. P(H) = 1

2

B.P(T) = 1

2

C. Probability of getting no head = 1 – probability of getting a head

P(�̅�) = 1- P(H)

= 1- 1

2

= 1

2

Ans.2.

A. Sample Space:{HH} total =1

P(getting heads on both coins) = 1

4

B. SS: {HT}, {HH}, {TH} total :3

P( at least one head) = 3

4

C. SS : {HH},{HT},{TH} total :3

P( at most one tail ) = 3

4

D. SS: {HH}

P(No tail)= 1

4

Ans. 3

A. P(Exactly two heads)= 3

8

B. P(atmost two heads)= 7

8

C. P(Atleast one head and one tail) = 3

4

D. P( no tails) = 1

8

Ans. 4

A. SS of Prime numbers= {2,3,5}

P(A prime number)= 3

6 =

1

2

B. SS of Even numbers= {2,3,5}

P(an even number)= 3

6 =

1

2

44

C. SS of Prime numbers= {3,6}

P(a multiple of 3)= 2

6 =

1

3

D.SS: {1,2,3,4}

P(getting a number less than 5)= 4

6 =

2

3

Ans 5.

A. SS: (1,1) (1,3), (1,5)

(2,2),(2,4),(2,6)

(3,1),(3,3), (3,5)

(4,2),(4,4),(4,6)

(5.1),(5,3),(5,5)

(6,2),(6,4),(6,6)

P(An even number as the sum)= 18

36 =

1

2

B. SS (1,1),(1,2),(1,4),(1,6)

(2,1),(2,3),(2,5),

(3,2),(3,4)

(4,1),(4,3),

(5,2),(5,6)

(6,1),(6,5)

P(The sum as a prime number) = 15

36 =

5

12

C. SS:(4,6),(5,5),(5,6),(6,4),(6,5),(6,6)

P(a total of atleast 10) = 6

36 =

1

6

D. SS:(1,1),(3,3),(5,5)

P( a doublet of odd numbers )= 3

36 =

1

12

E. SS: (2,2),(2,4),(2,6)

(4,2),(4,4),(4,6)

(6,2),(6,4),(6,6)

P(even number on both the dice)= 9

36 =

1

4

F. (1,4)

(2,3)

(3,2)

(4,1)

P(5 as the sum) = 4

36 =

1

9

45

Ans.6

A. P(A black king) = 2

52 =

1

26

B. P( spade or an ace) = P(spades)+P(ace without spade)= 13

52 +

3

52 =

16

52 =

1

4

C. P( a black card) = 26

52 =

1

2

D. P(The ace of spades ) = 1

52

E. P(A queen) = 4

52 =

1

13

F. P(neither a red nor a Queen)= 1- P(getting a red or a queen)

= 1-[P(red)+P(queen without red)]

=1-[26

52 +

2

52]

= 1-28

52 =

24

52 =

6

13

Ans.7

A. P(Red or white ) = 7

9

B. P( not black) = 7

9

C. P(neither white nor black)= 4

9

Ans.8

A. P(Prime number)= 11

35

B. P( multiple of 7) = 1

7

C. P( a multiple of 3 or 5)= 16

35

Ans.9 .

A. P(Not divisible by 3) = 2

3

B. P( a prime number greater than 7) = 1

5

C. P(not a perfect square number)= 5

6.

Class- X

Mathematics-Basic (241)

Sample Question Paper 2020-21

Max. Marks: 80 Duration:3 hours

46

General Instructions:

1. This question paper contains two parts A and B. 2. Both Part A and Part B have internal choices.

Part – A:

1. It consists of two sections- I and II 2. Section I has 16 questions. Internal choice is provided in 5 questions. 3. Section II has four case study-based questions. Each case study has 5 case-based sub-parts.

An examinee is to attempt any 4 out of 5 sub-parts.

Part – B:

1. Question No 21 to 26 are Very short answer Type questions of 2 mark each, 2. Question No 27 to 33 are Short Answer Type questions of 3 marks each 3. Question No 34 to 36 are Long Answer Type questions of 5 marks each. 4. Internal choice is provided in 2 questions of 2 marks, 2 questions of 3 marks and 1 question of

5 marks.

Questi Part-A Marks

on No.

Section-I

1

1. Express 156 as the product of primes.

1

2. Write a quadratic polynomial, sum of whose zeroes is 2 and product is -8.

Given that HCF (96,404) is 4, find the LCM ( 96,404).

1

3.

OR

State the fundamental Theorem of Arithmetic.

Page 1 of 11

47

4 On comparing the ratios of the coefficients, find out whether the pair of 1

equations x – 2y =0 and 3x + 4y -20 =0 is consistent or inconsistent.

5 If a and b are co-prime numbers, then find the HCF (a, b). 1

6

Find the area of a sector of a circle with radius 6cm if angle of the sector is 1

60°. (Take = 22/7)

OR

A horse tied to a pole with 28m long rope. Find the perimeter of the field where the horse can graze. (Take = 22/7)

7 In the given fig. DE || BC, ∟ADE =70° and ∟BAC=50°, then angle 1

∟BCA = ______

OR

In the given figure, AD = 2cm, BD = 3 cm, AE = 3.5 cm and AC = 7 cm. Is DE parallel to BC ?

Page 2 of 11

48

8 The cost of fencing a circular field at the rate of Rs.24 per metre is Rs. 5280. 1 Find the radius of the field.

9 A tree breaks due to storm and the broken part bends so that the top of the 1 tree touches the ground where it makes an angle 30° . The distance between

the foot of the tree to the point where the top touches the ground is 8m. Find

the height of the tree from where it is broken.

10 If the perimeter and the area of a circle are numerically equal, then find the 1

radius of the circle

11 Write the empirical relationship among mean, median and mode. 1

12 To divide a line segment BC internally in the ratio 3 : 5, we draw a ray BX 1

such that ∠CBX is an acute angle. What will be the minimum number of points

to be located at equal distances, on ray BX?

For what values of p does the pair of equations 4x + p y +8 =0 and 2x +2y +2 1

13

=0 has unique solution?

OR

What type of straight lines will be represented by the system of equations 2x + 3y =5 and 4x + 6y = 7 ?

14 A bag contains 3 red balls and 5 black balls. A ball is drawn at random from 1

the bag. What is the probability that the ball drawn is red?

OR

A die is thrown once. What is the probability of getting a prime number?

15 A tower stands vertically on the ground. From a point on the ground, which is 1

15m away from the foot of the tower, the angle of elevation of the top of the

tower is found to be 60°.Find the height of the tower.

16 ̅

1 Probability of an event E + Probability of the event E ( not E) is,_______

Page 3 of 11

49

Section-II

Case study-based questions are compulsory. Attempt any 4 sub parts

from each question. Each question carries 1 mark

17

Mathematics teacher of a school took her 10th standard students to show Red fort. It was a part of their Educational trip. The teacher had interest in history

as well. She narrated the facts of Red fort to students. Then the teacher said

in this monument one can find combination of solid figures. There are 2 pillars

which are cylindrical in shape. Also 2 domes at the corners which are

hemispherical.7 smaller domes at the centre. Flag hoisting ceremony on

Independence Day takes place near these domes.

i) How much cloth material will be required to cover 2 big domes each of radius 1

2.5 metres? (Take = 22/7)

a) 75m2 b) 78.57m2 c) 87.47m2 d) 25.8m2 b)

ii) Write the formula to find the volume of a cylindrical pillar. 1

a) Πr2h b) Πrl c) Πr(l + r) d) 2Πr

iii) Find the lateral surface area of two pillars if height of the pillar is 7m and 1

radius of the base is 1.4m.

a) 112.3cm2 b) 123.2m2 c) 90m2 d) 345.2cm2

iv) How much is the volume of a hemisphere if the radius of the base is 3.5m? 1

a) 85.9 m3 b) 80 m3 c) 98 m3 d) 89.83 m3

Page 4 of 11

50

v) What is the ratio of sum of volumes of two hemispheres of radius 1cm each to 1

the volume of a sphere of radius 2 cm?

a) 1:1 b) 1:8 c) 8 :1 d) 1:16

18 Class X students of a secondary school in Krishnagar have been allotted a

rectangular plot of a land for gardening activity. Saplings of Gulmohar are

planted on the boundary at a distance of 1m from each other. There is a

triangular grassy lawn in the plot as shown in the fig. The students are to sow

seeds of flowering plants on the remaining area of the plot.

Considering A as origin, answer question (i) to (v)

i) Considering A as the origin, what are the coordinates of A? 1

a) (0,1) b) (1,0) c) (0,0) d)(-1,-1)

ii) What are the coordinates of P? 1

a) (4,6) b)( 6,4) c) (4,5) d) (5,4)

iii) What are the coordinates of R? 1

a) (6,5) b) (5,6) c) ( 6,0) d) (7,4)

iv) What are the coordinates of D? 1

a) (16,0) b) (0,0) c) (0,16) d) (16,1)

v) What are the coordinate of P if D is taken as the origin? 1

a) ( 12,2) b ) (-12,6) c) (12,3) d) (6,10)

Page 5 of 11

51

19

Rahul is studying in X Standard. He is making a kite to fly it on a Sunday. Few

questions came to his mind while making the kite. Give answers to his

questions by looking at the figure.

i) Rahul tied the sticks at what angles to each other? 1

a) 30° b) 60° c) 90° d) 60°

ii) Which is the correct similarity criteria applicable for smaller triangles at the 1

upper part of this kite?

a) RHS b) SAS c) SSA d) AAS

iii) Sides of two similar triangles are in the ratio 4:9. Corresponding medians of 1

these triangles are in the ratio,

a) 2:3 b) 4:9 c) 81:16 d) 16:81

iv) In a triangle, if square of one side is equal to the sum of the squares of the 1

other two sides, then the angle opposite the first side is a right angle. This

theorem is called as,

a) Pythagoras theorem b) Thales theorem

c) Converse of Thales theorem d) Converse of Pythagoras theorem

v) What is the area of the kite, formed by two perpendicular sticks of length 6 cm 1

and 8 cm?

a) 48 cm2 b) 14 cm2 c) 24 cm2 d) 96 cm2

Page 6 of 11

52

20 Due to heavy storm an electric wire got bent as shown in the figure. It followed

a mathematical shape. Answer the following questions below.

i) Name the shape in which the wire is bent 1

a) Spiral b) ellipse c) linear d) Parabola

ii) How many zeroes are there for the polynomial (shape of the wire) 1

a) 2 b) 3 d) 1 d) 0

iii) The zeroes of the polynomial are 1

a) -1, 5 b) -1, 3 c) 3, 5 d) -4, 2

iv) What will be the expression of the polynomial? 1

a) x2+2x -3 b) x2 -2x +3 c) x2 - 2x -3d) x2 +2x+3

v) What is the value of the polynomial if x = -1? 1

a) 6 b) -18 c)) 18 d) 0

Part –B

All questions are compulsory. In case of internal choices, attempt

anyone.

21 Find the coordinates of the point which divides the line segment joining the 2

points (4, -3) and (8,5) in the ratio 3:1 internally.

Page 7 of 11

53

OR

Find a relation between x and y such that the point (x,y) is equidistant from the

points (7,1) and (3,5)

22 2

In the fig. if LM II CB and LN II CD, prove that

AM

= AN

MB ND

23 A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = 2

AD + BC.

24 Draw a line segment of length 7.8 cm and divide it in the ratio 5:8. Measure 2

the two parts.

25 Given 15 cot A =8, find sin A and sec A. 2

OR

Find tan P – cot R

Page 8 of 11

54

26 How many terms of the A. P : 9,17,25, .......must be taken to give a sum 636? 2

Part –B

All questions are compulsory. In case of internal choices, attempt

anyone.

27 Prove that √3 is an irrational number. 3

28 Two tangents TP and TQ are drawn to a circle with centre O from an external 3

point T. Prove that ∟PTQ = 2∟OPQ.

29 Meena went to a bank to withdraw Rs.2,000. She asked the cashier to give 3

her Rs.50 and Rs.100 notes only. Meena got 25 notes in all. Find how many

notes of Rs.50 and Rs.100 she received.

30 A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn 3

at random from the box, find the probability that it bears

(i) a two-digit number

(ii) a perfect square number.

(iii) a number divisible by 5.

OR

One card is drawn from a well shuffled deck of 52 cards. Find the probability of

getting

(i) A king of red colour.

(ii) A spade

(iii) The queen of diamonds

31 Metallic spheres of radii 6cm, 8cm and 10cm respectively are melted to form a 3

solid sphere. Find the radius of the resulting sphere.

Page 9 of 11

55

32 sin A−cosA+1 1 3

Prove that

=

sinA+cosA−1 secA−tanA

33 A motor boat whose speed in still water is 18 km/h, takes 1 hour more to go 24 3

km upstream than to return downstream to the same spot. Find the speed of

the stream.

OR

Find two consecutive odd positive integers, sum of whose squares is 290.

Part –B

All questions are compulsory. In case of internal choices, attempt

anyone.

34 The angles of depression of the top and bottom of a 8m tall building from the 5

top of a multi storied building are 30° and 45°, respectively. Find the height of

the multi storied building and the distance between the two buildings.

OR

A 1.2m tall girl spots a balloon moving with the wind in a horizontal line at a

height 88.2 m from the ground. The angle of elevation of the balloon from the

eyes of the girl at any instant is 60°.After sometime, the angle of elevation

reduces 30°.Find the distance travelled by the balloon during the interval.

35 The pth, qth and rth terms of an A.P. are a, b and c respectively. 5 Show that a(q – r) + b(r-p) + c(p – q) = 0

Page 10 of 11

56

36 A survey regarding the heights in (cm) of 51 girls of class X of a school was 5

conducted and the following data was obtained. Find the median height and

the mean using the formulae.

Height (in cm) Number of Girls

Less than 140 4

Less than 145 11

Less than 150 29

Less than 155 40

Less than 160 46

Less than 165 51

Class- X Session- 2020-21

Subject- Mathematics -Standard

Sample Question Paper Time Allowed: 3 Hours Maximum Marks: 80

General Instructions:

3. This question paper contains two parts A and B. 4. Both Part A and Part B have internal choices.

Part – A:

4. It consists three sections- I and II. 5. Section I has 16 questions of 1 mark each. Internal choice is provided in 5 questions. 6. Section II has 4 questions on case study. Each case study has 5 case-based sub-parts.

An examinee is to attempt any 4 out of 5 sub-parts.

5. Question No 21 to 26 are Very short answer Type questions of 2 mark each, 6. Question No 27 to 33 are Short Answer Type questions of 3 marks each 7. Question No 34 to 36 are Long Answer Type questions of 5 marks each. 8. Internal choice is provided in 2 questions of 2 marks, 2 questions of 3 marks and 1 question of 5

marks.

Question Part-A Marks

No. allocated

Section-I

Section I has 16 questions of 1 mark each. Internal choice is provided

in 5 questions.

1 If xy=180 and HCF(x,y)=3, then find the LCM(x,y). 1

57

OR

The decimal representation of 14587

will terminate after how many decimal

14

2 × 5

places?

2 If the sum of the zeroes of the quadratic polynomial 3x2-kx+6 is 3, then find 1 the value of k.

Page 1 of

14

58

3. For what value of k, the pair of linear equations 3x+y=3 and 6x+ky=8 does 1

not have a solution.

4. If 3 chairs and 1 table costs Rs. 1500 and 6 chairs and 1 table costs Rs.2400. Form 1

linear equations to represent this situation.

5. Which term of the A.P. 27, 24, 21,…..is zero? 1

OR

In an Arithmetic Progression, if d= - 4, n=7,an=4, then find a.

6. For what values of k, the equation 9x2+6kx+4=0 has equal roots?

7. Find the roots of the equation x2+7x+10=0 1

OR

For what value(s) of ‘a’ quadratic equation 30 2 − 6 + 1 = 0 has no real roots?

8. If PQ=28cm, then find the perimeter of ∆PLM 1

9. If two tangents are inclined at 60˚ are drawn to a circle of radius 3cm then 1

find length of each tangent.

OR

PQ is a tangent to a circle with centre O at point P. If ∆OPQ is an isosceles

triangle, then find ∠OQP.

Page 2 of 14

59

10. In the ∆ABC, D and E are points on side AB and AC respectively such that 1

DE II BC. If AE=2cm, AD=3cm and BD=4.5cm, then find CE.

11. In the figure, if B1, B2, B3,…... and A1,A2, A3,….. have been marked at 1

equal distances. In what ratio C divides AB?

12. += 1, = 30° and B is an acute angle, then find the value of B. 1

13. If x=2sin2Ɵ and y=2cos2Ɵ+1, then find x+y 1

14. In a circle of diameter 42cm,if an arc subtends an angle of 60˚ at the centre 1

where ∏=22/7, then what will be the length of arc.

15. 12 solid spheres of the same radii are made by melting a solid metallic 1

cylinder of base diameter 2cm and height 16cm. Find the diameter of the

each sphere.

16. Find the probability of getting a doublet in a throw of a pair of dice. 1

OR

Page 3 of 14

60

Find the probability of getting a black queen when a card is drawn at

random from a well-shuffled pack of 52 cards.

Section-II

Case study based questions are compulsory. Attempt any four

sub parts of each question. Each subpart carries 1 mark

17. Case Study based-1 SUN ROOM

The diagrams show the plans for a sun room. It will be built onto the wall of a

house. The four walls of the sunroom are square clear glass panels. The roof

is made using • Four clear glass panels, trapezium in shape, all the same size • One tinted glass panel, half a regular octagon in shape

(a) Refer to Top View 1

Find the mid-point of the segment joining the points J (6, 17) and I (9, 16).

(i) (33/2,15/2)

(ii) (3/2,1/2)

(iii)(15/2,33/2)

(iv) (1/2,3/2)

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(b) Refer to Top View 1

The distance of the point P from the y-axis is

(i) 4

(ii) 15

(iii) 19

(iv) 25

(c) Refer to Front View 1

The distance between the points A and S is

(i) 4

(ii) 8

(iii)16

(iv)20

(d) Refer to Front View 1

Find the co-ordinates of the point which divides the line segment joining the

points A and B in the ratio 1:3 internally.

(i) (8.5,2.0)

(ii) (2.0,9.5)

(iii) (3.0,7.5)

(iv) (2.0,8.5)

(e) Refer to Front View 1

If a point (x,y) is equidistant from the Q(9,8) and S(17,8),then

(i) x+y=13

(ii) x-13=0

(iii) y-13=0

(iv)x-y=13

18. Case Study Based- 2

SCALE FACTOR AND SIMILARITY

SCALE FACTOR

A scale drawing of an object is the same shape as the object but a different

size.

The scale of a drawing is a comparison of the length used on a drawing to

the length it represents. The scale is written as a ratio.

SIMILAR FIGURES

The ratio of two corresponding sides in similar figures is called the scale

factor.

Scale factor = ℎ

ℎ

If one shape can become another using Resizing then the

shapes are Similar

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62

Rotation or Turn

Reflection or Flip

Translation or Slide Hence, two shapes are Similar when one can become the other after

a resize, flip, slide or turn.

(a) A model of a boat is made on the scale of 1:4. The model is 120cm long. The 1

full size of the boat has a width of 60cm. What is the width of the scale

model?

(i) 20 cm (ii) 25 cm (iii) 15 cm

(iv)240 cm

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63

(b) What will effect the similarity of any two polygons? 1

(i) They are flipped horizontally

(ii)They are dilated by a scale factor

(iii)They are translated down

(iv)They are not the mirror image of one another

(c) If two similar triangles have a scale factor of a: b. Which statement regarding 1

the two triangles is true?

(i)The ratio of their perimeters is 3a : b

(ii)Their altitudes have a ratio a:b (iii)Their medians have a ratio : b

2

(iv)Their angle bisectors have a ratio a2 : b2

(d) The shadow of a stick 5m long is 2m. At the same time the shadow of a tree 1

12.5m high is

(i)3m

(ii)3.5m

(iii)4.5m

(iv)5m

(e) Below you see a student's mathematical model of a farmhouse roof with 1

measurements. The attic floor, ABCD in the model, is a square. The beams

that support the roof are the edges of a rectangular prism, EFGHKLMN. E is

the middle of AT, F is the middle of BT, G is the middle of CT, and H is the

middle of DT. All the edges of the pyramid in the model have length of 12 m.

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64

What is the length of EF, where EF is one of the horizontal edges of the block? (i)24m (ii)3m (iii)6m (iv)10m

19. Case Study Based- 3 Applications of Parabolas-Highway Overpasses/Underpasses A

highway underpass is parabolic in shape.

Parabola A parabola is the graph that

results from p(x)=ax2+bx+c

Parabolas are symmetric

about a vertical line known

as the Axis of Symmetry.

The Axis of Symmetry runs

through the maximum or

minimum point of the

parabola which is called the

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65

Vertex

(a) If the highway overpass is represented by x2–2x –8. Then its zeroes are (i) (2,-4)

(ii) (4,-2)

(iii) (-2,-2)

(iv) (-4,-4)

(b) The highway overpass is represented graphically.

Zeroes of a polynomial can be expressed graphically. Number of zeroes of

polynomial is equal to number of points where the graph of polynomial

(i) Intersects x-axis

(ii) Intersects y-axis

(iii) Intersects y-axis or x-axis

(iv)None of the above

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14

66

(c) Graph of a quadratic polynomial is a

(i) straight line

(ii) circle

(iii)parabola

(iv)ellipse

(d) The representation of Highway Underpass whose one zero is 6 and sum of

the zeroes is 0, is

(i)x2 – 6x + 2

(ii) x2 – 36

(iii)x2 – 6

(iv)x2 – 3

(e) The number of zeroes that polynomial f(x) = (x – 2)2 + 4 can have is: (i)1

(ii) 2

(iii) 0

(iv) 3

20. Case Study Based- 4

100m RACE

A stopwatch was used to

find the time that it took a

group of students to run 100

m.

Time 0-20 20-40 40-60 60-80 80-100

(in sec)

No. of 8 10 13 6 3

students

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67

(a) Estimate the mean time taken by a student to finish the race.

(i)54

(ii)63

(iii)43

(iv)50

(b) What wiil be the upper limit of the modal class ?

(i)20

(ii)40

(iii)60

(iv)80

(c) The construction of cummulative frequency table is useful in determining the

(i)Mean

(ii)Median

(iii)Mode

(iv)All of the above

(d) The sum of lower limits of median class and modal class is

(i)60

(ii)100

(iii)80

(iv)140

(e) How many students finished the race within 1 minute?

(i)18

(ii)37

(iii)31

(iv)8

Part –B

All questions are compulsory. In case of internal choices, attempt any

one.

21. 3 bells ring at an interval of 4,7 and 14 minutes. All three bell rang at 6 am, 2

when the three balls will the ring together next?

22. Find the point on x-axis which is equidistant from the points (2,-2) and (-4,2) 2

OR

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68

P (-2, 5) and Q (3, 2) are two points. Find the co-ordinates of the point R on

PQ such that PR=2QR

23. Find a quadratic polynomial whose zeroes are 5-3√2 and 5+3√2. 2

24. Draw a line segment AB of length 9cm. With A and B as centres, draw 2

circles of radius 5cm and 3cm respectively. Construct tangents to each circle

from the centre of the other circle.

25. If tanA=3/4, find the value of 1/sinA+1/cosA 2

OR

If √3 sinƟ-cosƟ=0 and 0˚<Ɵ <90˚, find the value of Ɵ

26. In the figure, quadrilateral ABCD is circumscribing a circle with centre O 2

and AD⊥AB. If radius of incircle is 10cm, then the value of x is

27.. Prove that 2-√3 is irrational, given that √3 is irrational. 3

28. If one root of the quadratic equation 3x2+px+4=0 is 2/3, then find the value 3 of p and the other root of the equation.

OR

The roots α and β of the quadratic equation x2-5x+3(k-1)=0 are such that α- β=1. Find the value k.

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69

29. In the figure, ABCD is a square of side 14 cm. Semi-circles are drawn with 3

each side of square as diameter. Find the area of the shaded region.

30. The perimeters of two similar triangles are 25cm and 15cm respectively. If 3

one side of the first triangle is 9cm, find the length of the corresponding side

of the second triangle.

OR

In an equilateral triangle ABC, D is a point on side BC such that BD = 1/3

BC. Prove that 9 AD2 = 7 AB2

31. The median of the following data is 16. Find the missing frequencies a and b, 3

if the total of the frequencies is 70.

Class 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40

Frequency 12 a 12 15 b 6 6 4

32. 3

If the angles of elevation of the top of the candle from two coins distant ‘a’

cm and ‘b’ cm (a>b) from its base and in the same straight line from it are

30˚ and 60˚, then find the height of the candle.

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Section V

33. The mode of the following data is 67. Find the missing frequency x. 3

Class 40-50 50-60 60-70 70-80 80-90

Frequency 5 x 15 12 7

34. The two palm trees are of equal heights and are standing opposite each 5

other on either side of the river, which is 80 m wide. From a point O

between them on the river the angles of elevation of the top of the trees

are 60° and 30°, respectively. Find the height of the trees and the

distances of the point O from the trees.

OR

The angles of depression of the top and bottom of a building 50 meters

high as observed from the top of a tower are 30˚ and 60˚ respectively.

Find the height of the tower, and also the horizontal distance between the

building and the tower.

35. Water is flowing through a cylindrical pipe of internal diameter 2cm, into a 5

cylindrical tank of base radius 40 cm at the rate of 0.7m/sec. By how

much will the water rise in the tank in half an hour?

36. A motorboat covers a distance of 16km upstream and 24km downstream 5

in 6 hours. In the same time it covers a distance of 12 km upstream and

36km downstream. Find the speed of the boat in still water and that of the

stream.

71

CBSE Class 10th 2021 Mathematics (041)

Deleted Portion

UNIT I-NUMBER SYSTEMS

Chapter Topics

REAL NUMBERS Euclid’s division lemma

UNIT II-ALGEBRA

Chapter Topics

POLYNOMIALS

Statement and simple problems on division algorithm for polynomials with real coefficients.

PAIR OF LINEAR

EQUATIONS IN TWO cross multiplication method

VARIABLES

QUADRATIC Situational problems based on equations reducible to

EQUATIONS quadratic equations

ARITHMETIC Application in solving daily life problems based on sum

PROGRESSIONS to n terms

UNIT III-COORDINATE GEOMETRY

Chapter Topics

COORDINATE Area of a triangle

GEOMETRY

UNIT IV-GEOMETRY

Chapter Topics

Proof of the following theorems are deleted · The ratio

of the areas of two similar triangles is equal to the ratio

TRIANGLES

of the squares of their corresponding sides. · In a

triangle, if the square on one side is equal to sumof the

squares on the other two sides, the angle opposite to the

first side is a right angle.

CIRCLES No Deletion

CONSTRUCTIONS Construction of a triangle similar to a given triangle.

UNIT V-

TRIGONOMETRY

72

Chapter Topics

INTRODUCTION TO Motivate the ratios whichever are defined at 0o and 90o

TRIGONOMETRY

TRIGONOMETRIC Trigonometric ratios of complementary angles

IDENTITIES

HEIGHTS AND No deletion

DISTANCES

UNIT VI-

MENSURATION

AREAS RELATED TO Problems on central angle of 120°

CIRCLES

SURFACE AREAS AND Frustum of a cone

VOLUMES

UNIT VI-STATISTICS & PROBABILITY

Chapter Topics

STATISTICS · Step deviation Method for finding the mean · Cumulative Frequency graph

PROBABILITY No deletion