Free Download CBSE Papers: Class 10 Maths Sample Papers, Important Questions, Worksheets

TIPS FOR CBSE CLASS 10TH MATHEMATICS BASIC BOARD EXAM 2023

As the question paper of the board examination is always based on NCERT, therefore first complete the entire syllabus from the NCERT Class 10th Mathematics textbook to understand the concepts of each and every chapter.
Write all the concepts formulae and statements of theorems on sheets for quick revision.
The unit which carries more marks in the question paper should be given more time to practice.
After completing the entire syllabus from the NCERT textbook, complete my MLL study materials for Basic based Exam. No need to go through reference books like RD Sharma, RS Aggarwal as it gives a broader view of questions which is not advisable for Basic opted students.
After completing the revision work, solve CBSE Sample paper for Basic Board Exam
Practicing the sample papers also teaches the concept of time management to the students. I am sure that this exercise will help you to complete your board exam in 21⁄2 hrs.
Read the question paper with full concentration. Use the first fifteen minutes carefully and effectively by going through the question paper at least twice and marking all the questions which seem difficult and leave them for last.
First; attempt those questions which you are sure about starting from Section D to A.
Understand the worth of each question. Never spend too much time (5-10 minutes) on 1 or 2 mark

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UNIT I : NUMBER SYSTEMS

1. REAL NUMBERS

Euclid's division lemma, Fundamental Theorem of Arithmetic - statements after reviewing work done earlier and after illustrating and motivating through examples, Proofs of results - irrationality of

2 , 3, 5 decimal expansions of rational recurring decimals.

UNIT II : ALGEBRA

1. POLYNOMIALS

Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials. Statement and simple problems on division algorithm for polynomials with real coefficients.

2. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES (15) Periods

Pair of linear equations in two variables and their graphical solution. Geometric representation of different possibilities of solutions/inconsistency.
Algebraic conditions for number of solutions. Solution of pair of linear equations in two variables algebraically - by substitution, by elimination and by cross multiplication. Simple situational problems must be included. Simple problems on equations reducible to linear equations may be included.

3. QUADRATIC EQUATIONS (15) Periods

Standard form of a quadratic equation ax2 + bx + c = 0, (a „j 0). Solution of the quadratic equations (only real roots) by factorisation and by using quadratic formula. Relationship between discriminant and nature of roots. Problems related to day to day activities to be incorporated. Situational problems based on quadratic equations related to day to day activities to be incorporated.

4. ARITHMETIC PROGRESSIONS (8) Periods

Motivation for studying AP. Derivation of standard results of finding the nth term and sum of first n terms and their application in solving daily life problems.

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UNIT III : COORDINATE GEOMETRY

1. LINES (In two-dimensions) (14) Periods

Review the concepts of coordinate geometry done earlier including graphs of linear equations. Distance between two points and section formula (internal). Area of a triangle.

UNIT IV : GEOMETRY

1. TRIANGLES (15) Periods

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.
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.

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.

7. (Prove) The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides.

8. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

9. (Prove) In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angles opposite to the first side is a right triangle.

2. CIRCLES (8) Periods

Tangents to a circle motivated by chords drawn from points coming closer and closer to the point.
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 circle are equal.

UNIT V : TRIGONOMETRY

1. INTRODUCTION TO TRIGONOMETRY (8) Periods (10) Periods

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 00 & 900. Values (with proofs) of the trigonometric ratios of 300, 450 & 600. Relationships between the ratios.

2. TRIGONOMETRIC IDENTITIES (15) Periods

Proof and applications of the identity sin2A + cos2A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles.

3. HEIGHTS AND DISTANCES (8) Periods

Simple and believable problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation / depression should be only 300, 450 & 600

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UNIT VI : MENSURATION

1. AREAS RELATED TO CIRCLES (12) Periods

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 60o, 90o & 120o only. Plane figures involving triangles, simple quadrilaterals and circle should be taken.)

2. SURFACE AREAS AND VOLUMES (12) Periods

(i) Problems on finding surface areas and volumes of combinations of any two of the following: cubes, cuboids, spheres, hemispheres and right circular cylinders/cones. Frustum of a cone.

(ii) 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 (18) Periods

Mean, median and mode of grouped data (bimodal situation to be avoided). Cumulative frequency graph.

2. PROBABILITY

Classical definition of probability. Connection with probability as given in problems on single events, not using set notation.

3. CONSTRUCTIONS

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

3. Construction of a triangle similar to a given triangle.

CHAPTER - 1 REAL NUMBERS IMPORTANT FORMULAS & CONCEPTS

EUCLID’S DIVISION LEMMA

Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, where 0rb.

Here we call ‘a’ as dividend, ‘b’ as divisor, ‘q’ as quotient and ‘r’ as remainder.  Dividend = (Divisor x Quotient) + Remainder
If in Euclid’s lemma r = 0 then b would be HCF of ‘a’ and ‘b’.

NATURAL NUMBERS

Counting numbers are called natural numbers i.e. 1, 2, 3, 4, 5, ................ are natural numbers.

WHOLE NUMBERS

All counting numbers/natural numbers along with 0 are called whole numbers i.e. 0, 1, 2, 3, 4, 5 ................ are whole numbers.

INTEGERS

All natural numbers, negative of natural numbers and 0, together are called integers. i.e. .......... – 3, – 2, – 1, 0, 1, 2, 3, 4, .............. are integers.

ALGORITHM
An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.

LEMMA
A lemma is a proven statement used for proving another statement.

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EUCLID’S DIVISION ALGORITHM

Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. Recall that the HCF of two positive integers a and b is the largest positive integer d that divides both a and b.

To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:
Step 1 : Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c =dq+r, 0rd.
Step 2 : If r = 0, d is the HCF of c and d. If r  0 apply the division lemma to d and r.
Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.
This algorithm works because HCF (c, d) = HCF (d, r) where the symbol HCF (c, d) denotes the HCF of c and d, etc.

The Fundamental Theorem of Arithmetic

Every composite number can be expressed ( factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

The prime factorisation of a natural number is unique, except for the order of its factors.

HCF is the highest common factor also known as GCD i.e. greatest common divisor.

LCM of two numbers is their least common multiple.

Property of HCF and LCM of two positive integers ‘a’ and ‘b’:

Practice Questions

Find the largest number which divides 245 and 1029 leaving remainder 5 in each case.
Find the largest number which divides 2053 and 967 and leaves a remainder of 5 and 7 respectively.
Two tankers contain 850 litres and 680 litres of kerosene oil respectively. Find the maximum capacity of a container which can measure the kerosene oil of both the tankers when used an exact number of times.
In a morning walk, three persons step off together. Their steps measure 80 cm, 85 cm and 90 cm respectively. What is the minimum distance each should walk so that all can cover the same distance in complete steps?
Find the least number which when divided by 12, 16, 24 and 36 leaves a remainder 7 in each case.
The length, breadth and height of a room are 825 cm, 675 cm and 450 cm respectively. Find the longest tape which can measure the three dimensions of the room exactly.
Determine the smallest 3-digit number which is exactly divisible by 6, 8 and 12.
Determine the greatest 3-digit number exactly divisible by 8, 10 and 12.
The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they change simultaneously again?

Three tankers contain 403 litres, 434 litres and 465 litres of diesel respectively. Find the maximum capacity of a container that can measure the diesel of the three containers exact number of times.
Find the least number which when divided by 6, 15 and 18 leave remainder 5 in each case.
Find the smallest 4-digit number which is divisible by 18, 24 and 32.
Renu purchases two bags of fertiliser of weights 75 kg and 69 kg. Find the maximum value of weight which can measure the weight of the fertiliser exact number of times.
In a seminar, the number, the number of participants in Hindi, English and Mathematics are 60, 84 and 108, respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject.
144 cartons of Coke cans and 90 cartons of Pepsi cans are to be stacked in a canteen. If each stack is of the same height and is to contain cartons of the same drink, what would be the greatest number of cartons each stack would have?
A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What would be the greatest capacity of such a tin?
Express each of the following positive integers as the product of its prime factors: (i) 3825 (ii) 5005 (iii) 7429
Express each of the following positive integers as the product of its prime factors: (i) 140 (ii) 156 (iii) 234
There is circular path around a sports field. Priya takes 18 minutes to drive one round of the field, while Ravish takes 12 minutes for the same. Suppose they both start at the same point and at the same time and go in the same direction. After how many minutes will they meet again at the starting point?
In a morning walk, three persons step off together and their steps measure 80 cm, 85 cm and 90 cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?
A circular field has a circumference of 360 km. Three cyclists start together and can cycle 48, 60 and 72 km a day, round the field. When will they meet again?
Find the smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively.
Find the smallest number which when increased by 17 is exactly divisible by 520 and 468.
Find the greatest numbers that will divide 445, 572 and 699 leaving remainders 4, 5 and 6 respectively.
Find the greatest number which divides 2011 and 2423 leaving remainders 9 and 5 respectively
Find the greatest number which divides 615 and 963 leaving remainder 6 in each case.
Find the greatest number which divides 285 and 1249 leaving remainders 9 and 7 respectively.

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CLASS X : CHAPTER - 2 POLYNOMIALS IMPORTANT FORMULAS & CONCEPTS

An algebraic expression of the form p(x) = a0 + a1x + a2x2 + a3x3 + ................anxn, where a ≠ 0, is called a polynomial in variable x of degree n.
Here, a0, a1, a2, a3, .........,an are real numbers and each power of x is a non-negative integer.
e.g. 3x2 – 5x + 2 is a polynomial of degree 2.

3 x  2 is not a polynomial.

If p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of the polynomial p(x). For example, 4x + 2 is a polynomial in the variable x of degree 1, 2y2 – 3y + 4 is a polynomial in the variable y of degree 2,

A polynomial of degree 0 is called a constant polynomial.
A polynomial p(x) = ax + b of degree 1 is called a linear polynomial.
A polynomial p(x) = ax2 + bx + c of degree 2 is called a quadratic polynomial.
A polynomial p(x) = ax3 + bx2 + cx + d of degree 3 is called a cubic polynomial.
A polynomial p(x) = ax4 + bx3 + cx2 + dx + e of degree 4 is called a bi-quadratic polynomial.

VALUE OF A POLYNOMIAL AT A GIVEN POINT x = k

If p(x) is a polynomial in x, and if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k, and is denoted by p(k).

ZERO OF A POLYNOMIAL
A real number k is said to be a zero of a polynomial p(x), if p(k) = 0.

Geometrically, the zeroes of a polynomial p(x) are precisely the x-coordinates of the points, where the graph of y = p(x) intersects the x -axis.

A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.

In general, a polynomial of degree ‘n’ has at the most ‘n’ zeroes.

PRACTICE QUESTIONS CLASS X : CHAPTER - 2 POLYNOMIALS

If p(x)=3x3 –2x2 +6x–5, find p(2).
Draw the graph of the polynomial f(x) = x2 – 2x – 8.
Draw the graph of the polynomial f(x)=3–2x–x2 .
Draw the graph of the polynomial f(x) = –3x2 + 2x – 1.
Draw the graph of the polynomial f(x)=x2 –6x+9.
Draw the graph of the polynomial f(x) = x3.
Draw the graph of the polynomial f(x) = x3 – 4x.
Draw the graph of the polynomial f(x) = x3 – 2x2.
Draw the graph of the polynomial f(x) = –4x2 + 4x – 1.
Draw the graph of the polynomial f(x) = 2x2 – 4x + 5.
Find the quadratic polynomial whose zeroes are 2 + 3 and 2 – 3 .

If the product of zeroes of the polynomial ax2 – 6x – 6 is 4, find the value of ‘a’.
If one zero of the polynomial (a2 + 9)x2 + 13x + 6a is reciprocal of the other. Find the value of a.
Write a quadratic polynomial, sum of whose zeroes is 2 3 and their product is 2.
Find a polynomial whose zeroes are 2 and –3.
Find the sum and product of zeroes of p(x) = 2(x2 – 3) + x.
Find a quadratic polynomial, the sum of whose zeroes is 4 and one zero is 5.
Find the zeroes of the polynomial p(x) = 2x2  3x  2 2 .

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CLASS X : CHAPTER - 3 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

IMPORTANT FORMULAS & CONCEPTS

An equation of the form ax + by + c = 0, where a, b and c are real numbers (a0,b0), is called a linear equation in two variables x and y.

The numbers a and b are called the coefficients of the equation ax + by + c = 0 and the number c is called the constant of the equation ax + by + c = 0.

Two linear equations in the same two variables are called a pair of linear equations in two variables. The most general form of a pair of linear equations is

a1x+b1y+c1 =0

a2x+b2y+c2 =0
where a1, a2, b1, b2, c1, c2 are real numbers, such that a12 + b12 ≠ 0, a22 + b22 ≠ 0.

CONSISTENT SYSTEM

A system of simultaneous linear equations is said to be consistent, if it has at least one solution.

INCONSISTENT SYSTEM

A system of simultaneous linear equations is said to be inconsistent, if it has no solution.

METHOD TO SOLVE A PAIR OF LINEAR EQUATION OF TWO VARIABLES

A pair of linear equations in two variables can be represented, and solved, by the: (i) graphical method (ii) algebraic method

GRAPHICAL METHOD OF SOLUTION OF A PAIR OF LINEAR EQUATIONS

The graph of a pair of linear equations in two variables is represented by two lines.

If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is consistent.

ALGEBRAIC METHODS OF SOLVING A PAIR OF LINEAR EQUATIONS Substitution Method
Following are the steps to solve the pair of linear equations by substitution method:

a1x+b1y+c1 =0...(i)and

a2x+b2y+c2 =0...(ii)
Step 1: We pick either of the equations and write one variable in terms of the other

11
Step 2: Substitute the value of x in equation (i) from equation (iii) obtained in step 1.
Step 3: Substituting this value of y in equation (iii) obtained in step 1, we get the values of x and y.

Elimination Method

Following are the steps to solve the pair of linear equations by elimination method:
Step 1: First multiply both the equations by some suitable non-zero constants to make the coefficients of one variable (either x or y) numerically equal.
Step 2: Then add or subtract one equation from the other so that one variable gets eliminated.

If you get an equation in one variable, go to Step 3.
If in Step 2, we obtain a true statement involving no variable, then the original pair of equations has infinitely many solutions.

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CLASS X : CHAPTER - 3 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

Sum of two numbers is 50 and their difference is 10, then the numbers are
(a) 30 and 20 (b) 24 and 14 (c) 12 and 2 (d) none of these
The sum of the digits of a two-digit number is 12. The number obtained by interchanging its digit exceeds the given number by 18, then the number is

(a) 72 (b) 75 (c) 57 (d) none of these
The sum of a two-digit number and the number obtained by interchanging its digit is 99. If the digits differ by 3, then the number is

(a) 36 (b) 33 (c) 66 (d) none of these
Seven times a two-digit number is equal to four times the number obtained by reversing the order of its digit. If the difference between the digits is 3, then the number is

(a) 36 (b) 33 (c) 66 (d) none of these
A two-digit number is 4 more than 6 times the sum of its digits. If 18 is subtracted from the number, the digits are reversed, then the number is

(a) 36 (b) 46 (c) 64 (d) none of these
The sum of two numbers is 1000 and the difference between their squares is 25600, then the numbers are

(a) 616 and 384 (b) 628 and 372 (c) 564 and 436 (d) none of these
Five years ago, A was thrice as old as B and ten years later A shall be twice as old as B, then the present age of A is

(a) 20 (b) 50 (c) 30 (d) none of these
The sum of thrice the first and the second is 142 and four times the first exceeds the second by 138, then the numbers are

(a) 40 and 20 (b) 40 and 22 (c) 12 and 22 (d) none of these
The sum of twice the first and thrice the second is 92 and four times the first exceeds seven times the second by 2, then the numbers are

(a) 25 and 20 (b) 25 and 14 (c) 14 and 22 (d) none of these
The difference between two numbers is 14 and the difference between their squares is 448, then the numbers are

CLASS X : CHAPTER - 4 QUADRATIC EQUATIONS IMPORTANT FORMULAS & CONCEPTS

An algebraic expression of the form p(x) = a0 + a1x + a2x2 + a3x3 + ................anxn, where a ≠ 0, is called a polynomial in variable x of degree n.

Here, a0, a1, a2, a3, .........,an are real numbers and each power of x is a non-negative integer.
e.g. 3x2 – 5x + 2 is a polynomial of degree 2.

3 x  2 is not a polynomial.

QUADRATIC EQUATION

A polynomial p(x) = ax2 + bx + c of degree 2 is called a quadratic polynomial, then p(x) = 0 is known as quadratic equation.
e.g.2x2 –3x+2=0,x2 +5x+6=0are quadratic equations.

METHODS TO FIND THE SOLUTION OF QUADRATIC EQUATIONS

Three methods to find the solution of quadratic equation:

Factorisation method
Method of completing the square
Quadratic formula method

FACTORISATION METHOD

Steps to find the solution of given quadratic equation by factorisation
Firstly, write the given quadratic equation in standard form ax2 + bx + c = 0. Findtwonumbers and  suchthatsumof and  isequaltobandproductof and  is equal to ac.

Write the middle term bx as  x   x and factorise it by splitting the middle term and let factors are(x+p)and(x+q)i.e.ax2 +bx+c=0  (x+p)(x+q)=0

Now equate reach factor to zero and find the values of x.
These values of x are the required roots/solutions of the given quadratic equation.

METHOD OF COMPLETING THE SQUARE

Steps to find the solution of given quadratic equation by Method of completing the square: Firstly, write the given quadratic equation in standard form ax2 + bx + c = 0.
Make coefficient of x2 unity by dividing all by a then we get

x2  b x  c  0 aa

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CLASS X : CHAPTER - 4 QUADRATIC EQUATIONS NATURE OF ROOTS

Find the value of k for which the quadratic equation 2x2 + kx + 3 = 0 has two real equal roots.
Find the value of k for which the quadratic equation kx(x – 3) + 9 = 0 has two real equal roots.
Find the value of k for which the quadratic equation 4x2 – 3kx + 1 = 0 has two real equal roots..
If–4 is a root of the equation x2 +px–4=0 and the equation x2 +px+q=0 has equal roots, find the value of p and q.
If–5 is a root of the equation 2x2 +px–15=0 and the equation p(x2 +x)+k=0 has equal roots, find the value of k.
Find the value of k for which the quadratic equation (k – 12)x2 + 2(k – 12)x + 2 = 0 has two real equal roots..
Find the value of k for which the quadratic equation k2x2 – 2(k – 1)x + 4 = 0 has two real equal roots..
If the roots of the equation(a–b)x2 +(b–c)x+(c–a)=0 are equal, prove that b+c=2a.
Prove that both the roots of the equation(x–a)(x–b)+(x–b)(x–c)+(x–c)(x–a)=0are

real but they are equal only when a = b = c.
Find the positive value of k for which the equation x2 + kx +64 = 0 and x2 – 8x +k = 0 will have real roots.

Find the value of k for which the quadratic equation kx2 – 6x – 2 = 0 has two real roots.
Find the value of k for which the quadratic equation 3x2 + 2x + k= 0 has two real roots.
Find the value of k for which the quadratic equation 2x2 + kx + 2 = 0 has two real roots.

14.Show that the equation3x2 +7x+8=0 is not true for any real value of x.

15.Show that the equation 2(a2 +b2)x2 +2(a+b)x+1=0 has no real roots, when a  b.

Find the value of k for which the quadratic equation kx2 + 2x + 1 = 0 has two real and distinct roots.
Find the value of p for which the quadratic equation 2x2 + px + 8 = 0 has two real and distinct roots.
If the equation (1 + m2)x2 + 2mcx + (c2 – a2) = 0 has equal roots, prove that c2 = a2(1 + m2).

SPEED, DISTANCE AND TIME RELATED QUESTIONS

A motor boat whose speed is 18 km/hr in still water takes 1 hour more to go 24 upstream than to return to the same point. Find the speed of the stream.
A motorboat whose speed is 9km/hr in still water, goes 15 km downstream and comes back in a total time of 3 hours 45 minutes. Find the speed of the stream.
A passenger train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km/hr from its usual speed. Find its usual speed.
In a flight for 3000 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 100 km/hr and consequently time of flight increased by one hour. Find the original duration of flight.
A plane left 30 minutes later than the schedule time and in order to reach its destination 1500 km away in time it has to increase its speed by 250 km/hr from its usual speed. Find its usual speed.
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/h more than that of the passenger train, find the average speed of the two trains.

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have

taken 1 hour less for the same journey. Find the speed of the train.
In a flight for 6000 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 400 km/hr and consequently time of flight increased by 30 minutes. Find the original duration of flight.
The time taken by a man to cover 300 km on a scooter was 1 1 hours more than the time taken by 2 him during the return journey. If the speed in returning be 10 km/hr more than the speed in going, find its speed in each direction.
A motorboat whose speed is 15 km/hr in still water, goes 30 km downstream and comes back in a total time of 4 hours 30 minutes. Find the speed of the stream.
The speed of a boat in still water is 8 km/hr. It can go 15 km upstream and 22 km downstream in 5 hours. Find the speed of the stream.
A motor boat goes 10 km upstream and returns back to the starting point in 55 minutes. If the speed of the motor boat in still water is 22 km/hr, find the speed of the current.
A sailor can row a boat 8 km downstream and return back to the starting point in 1 hour 40 minutes. If the speed of the stream is 2 km/hr, find the speed of the boat in still water.
A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hr more, it would have taken 30 minutes less for the journey. Find the original speed of the train.
The distance between Mumbai and Pune is 192 km. Travelling by the Deccan Queen, it takes 48 minutes less than another train. Calculate the speed of the Deccan Queen if the speeds of the two trains differ by 20 km/hr.
An aeroplane left 30 minutes later than it schedule time and in order to reach its destination 1500 km away in time, it had to increase its speed by 250 km/hr from its usual speed. Determine its usual speed.

Class 10 Maths MCQ on Circles

Quadratic Equations MCQ Questions for Class 10 Maths

Class 10 Maths MCQ on Pair of Linear Equations in two Variables

MCQ Questions on Triangles for Class 10 Maths

Class 10 MCQ Questions on Surface Areas and Volumes

CBSE Class 10 Maths MCQ on Statistics

Class 10 Maths MCQ on Some Applications to Trigonometry

MCQ Questions on Real Numbers for Class 10

MCQ Questions on Quadratic Equations for Class 10

Polynomials MCQ Questions Class 10 Maths

Introduction to Trigonometry MCQ for Class 10 Maths

Coordinate Geometry MCQ Questions Class 10 Maths

Circles MCQ Questions Class 10 Maths

MCQ Questions on Arithmetic Progression Class 10 Maths

Quadratic Equations MCQ Questions Class 10 Maths

Linear Equations MCQ Class 10 Maths

Triangles MCQ Questions Class 10 Maths

Surface Areas and Volumes MCQ Class 10 Maths

Statistics MCQ Questions Class 10 Maths

Some Applications to Trigonometry MCQ Questions Class 10

Real Numbers MCQ Questions Class 10 Maths

MCQ Questions on Quadratic Equations for Class 10 Maths

CBSE Class 10 Maths MCQ on Polynomials

Pair of Linear Equations in two Variables Class 10 MCQ

CBSE Class 10 Maths MCQ on Introduction to Trigonometry

CBSE Class 10 Maths MCQ on Coordinate Geometry

CBSE Class 10 Maths MCQ on Constructions

CBSE Class 10 Maths MCQ on Circles

Arithmetic Progression Class 10 MCQ Questions

CLASS X : CHAPTER - 6 TRIANGLES IMPORTANT FORMULAS & CONCEPTS

All those objects which have the same shape but different sizes are called similar objects. Two triangles are similar if
(i) their corresponding angles are equal (or)
(ii) their corresponding sides have lengths in the same ratio (or proportional)

Basic Proportionality theorem or Thales Theorem

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.

Converse of Basic Proportionality Theorem ( Converse of Thales Theorem)

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.

Angle Bisector Theorem

The internal (external) bisector of an angle of a triangle divides the opposite side internally (externally) in the ratio of the corresponding sides containing the angle.

Converse of Angle Bisector Theorem

If a straight line through one vertex of a triangle divides the opposite side internally (externally) in the ratio of the other two sides, then the line bisects the angle internally (externally) at the vertex.

Criteria for similarity of triangles

The following three criteria are sufficient to prove that two triangles are similar.

(i) AAA( Angle-Angle-Angle ) similarity criterion

If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar.
Remark: If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

(ii) SSS (Side-Side-Side) similarity criterion for Two Triangles

In two triangles, if the sides of one triangle are proportional (in the same ratio) to the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.

(iii) SAS (Side-Angle-Side) similarity criterion for Two Triangles

If one angle of a triangle is equal to one angle of the other triangle and if the corresponding sides including these angles are proportional, then the two triangles are similar.

Areas of Similar Triangles

The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

Pythagoras theorem (Baudhayan theorem)

In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Converse of Pythagoras theorem

In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the first side is a right angle.

CBSE Class 10 Maths MCQ on Areas Related to Circles

Probability Formulas For Class 10 Maths

Statistics Formulas For Class 10 Maths

Surface Areas and Volumes Formulas For Class 10 Maths

Constructions Formulas For Class 10 Maths

Circles Formulas For Class 10 Maths

Trigonometry Formulas For Class 10 Maths

Coordinate Geometry Formulas for Class 10 Maths

Triangles Formulas For Class 10 Maths

Arithmetic Progressions Formulas for Class 10 Maths

Quadratic Equations Formulas Class 10 Maths

Chapter 3 Maths Formulas for Class 10

Polynomials Formulas for Class 10 Maths

Real Numbers Formulas for CBSE Class 10

Pair of Linear Equation in Two variables Class 10 Maths Test Paper

Arithmetic Progression Class 10 Maths Test Paper

CLASS X : CHAPTER - 7 COORDINATE GEOMETRY IMPORTANT FORMULAS & CONCEPTS

Points to remember

The distance of a point from the y-axis is called its x-coordinate, or abscissa.

The distance of a point from the x-axis is called its y-coordinate, or ordinate.

The coordinates of a point on the x-axis are of the form (x, 0).

The coordinates of a point on the y-axis are of the form (0, y).

Problems based on geometrical figure

To show that a given figure is a
Parallelogram – prove that the opposite sides are equal
Rectangle – prove that the opposite sides are equal and the diagonals are equal.

Parallelogram but not rectangle – prove that the opposite sides are equal and the diagonals are not equal.

Rhombus – prove that the four sides are equal
Square – prove that the four sides are equal and the diagonals are equal.
Rhombus but not square – prove that the four sides are equal and the diagonals are not equal. Isosceles triangle – prove any two sides are equal.
Equilateral triangle – prove that all three sides are equal.
Right triangle – prove that sides of triangle satisfies Pythagoras theorem.

MCQ WORKSHEET-II CLASS X: CHAPTER – 7 COORDINATE GEOMETRY

Find the area of the triangle whose vertices are A(2, 4), B(–3, 7) and C(–4, 5) (a) 11sq. units (b) 22 sq. units (c) 7 sq. units (d) 6.5 sq. units
Find the area of the triangle whose vertices are A(10, –6), B(2, 5) and C(–1, 3) (a) 12.5 sq. units (b) 24.5 sq. units (c) 7 sq. units (d) 6.5 sq. units
Find the area of the triangle whose vertices are A(4, 4), B(3, –16) and C(3, –2) (a) 12.5 sq. units (b) 24.5 sq. units (c) 7 sq. units (d) 6.5 sq. units
For what value of x are the points A(–3, 12), B(7, 6) and C(x, 9) collinear? (a) 1 (b) –1 (c) 2 (d) –2
For what value of y are the points A(1, 4), B(3, y) and C(–3, 16) collinear? (a) 1 (b) –1 (c) 2 (d) –2
Find the value of p for which the points A(–1, 3), B(2, p) and C(5, –1) collinear? (a) 1 (b) –1 (c) 2 (d) –2
What is the midpoint of a line with endpoints (–3, 4) and (10, –5)?
(a) (–13, –9) (b) (–6.5, –4.5) (c) (3.5, –0.5) (d) none of these
A straight line is drawn joining the points (3, 4) and (5,6). If the line is extended, the ordinate of the point on the line, whose abscissa is –1 is
(a) 1 (b) –1 (c) 2 (d) 0
If the distance between the points (8, p) and (4, 3) is 5 then value of p is
(a) 6 (b) 0 (c) both (a) and (b) (d) none of these
The fourth vertex of the rectangle whose three vertices taken in order are (4,1), (7, 4), (13, –2) is (a) (10, –5) (b) (10, 5) (c) (8, 3) (d) (8, –3)
If four vertices of a parallelogram taken in order are (–3, –1), (a, b), (3, 3) and (4, 3). Then a : b = (a)1:4 (b)4:1 (c)1:2 (d)2: 1

Area of the triangle formed by (1, – 4), (3, – 2) and (– 3,16) is
(a) 40 sq. units (b) 48 sq. units (c) 24 sq. units (d) none of these
The points (2, 5), (4, - 1), (6, - 7) are vertices of an ___________ triangle (a) isosceles (b) equilateral (c) scalene (d) right angled
The area of triangle formed by the points (p, 2 - 2p), (l-p,2p) and (-4-p, 6- 2p) is 70 sq. units. How many integral value of p are possible ?
(a) 2 (b) 3 (c) 4 (d) none of these
If the origin is the mid-point of the line segment joined by the points (2,3) and (x,y), then the value of (x,y) is
(a) (2, –3) (b) (2, 3) (c) (–2, 3) (d) (–2, –3)

Circles Class 10 Maths Test Paper

Construction Class 10 Maths Test Paper

Coordinate Geometry Chapter Wise Test Papers For Class 10 Maths

Heights and Distance Chapter Wise Test Papers For Class 10 Maths

CBSE Class 10 Polynomials Test Paper

Probability Chapter Wise Test Papers For Class 10 Maths

Quadratic Equations Class 10 Maths Test Paper

Real Numbers Chapter Wise Test Papers For Class 10 Maths

Statistics Chapter Wise Test Papers For Class 10 Maths

Surface Areas and Volumes Class 10 Maths Test Paper

Trigonometry Class 10 Maths Test Paper

Constructions Class 10 Important Questions and Answers

Important Questions Arithmetic Progression Class 10 Solved

Important Questions On Circles for Class 10 with Solutions

Coordinate Geometry Class 10 Extra Questions & Solutions

Pair of Linear Equations in two variables Class 10 Important Questions & Solutions

Polynomials Class 10 Maths Extra Questions with Answers

Probability Class 10 Extra Questions with Answers PDF

Quadratic Equation Class 10 Important Questions & Solutions

CLASS X : CHAPTER - 10 CIRCLES IMPORTANT FORMULAS & CONCEPTS

The collection of all the points in a plane, which are at a fixed distance from a fixed point in the plane, is called a circle. The line segment joining the centre and any point on the circle is also called a radius of the circle.

A circle divides the plane on which it lies into three parts. They are: (i) inside the circle, which is also called the interior of the circle; (ii) the circle and (iii) outside the circle, which is also called the exterior of the circle. The circle and its interior make up the circular region.

The chord is the line segment having its two end points lying on the circumference of the circle.

The chord, which passes through the centre of the circle, is called a diameter of the circle.

A diameter is the longest chord and all diameters have the same length, which is equal to two times the radius.

A piece of a circle between two points is called an arc.
The longer one is called the major arc PQ and the shorter one is called the minor arc PQ.

The length of the complete circle is called its circumference.

The region between a chord and either of its arcs is called a segment of the circular region or simply a segment of the circle. There are two types of segments also, which are the major segment and the minor segment.

The region between an arc and the two radii, joining the centre to the end points of the arc is called a sector. The minor arc corresponds to the minor sector and the major arc corresponds to the major sector.

In the below figure, the region OPQ is the minor sector and remaining part of the circular region is the major sector. When two arcs are equal, that is, each is a semicircle, then both segments and both sectors become the same and each is known as a semicircular region. The line segment joining the centre and any point on the circle is also called a radius of the circle.

The chord is the line segment having its two end points lying on the circumference of the circle.

The chord, which passes through the centre of the circle, is called a diameter of the circle.

A diameter is the longest chord and all diameters have the same length, which is equal to two times the radius.

A piece of a circle between two points is called an arc.
The longer one is called the major arc PQ and the shorter one is called the minor arc PQ.

The length of the complete circle is called its circumference.

CLASS X: CHAPTER – 10 CIRCLES

Prove that “The tangent at any point of a circle is perpendicular to the radius through the point of contact”.
Prove that “The lengths of tangents drawn from an external point to a circle are equal.”
Prove that “The centre lies on the bisector of the angle between the two tangents drawn from an

external point to a circle.”
Find the length of the tangent drawn to a circle of radius 3 cm, from a point distant 5 cm from the centre.
A point P is at a distance 13 cm from the centre C of a circle and PT is a tangent to the given circle. If PT = 12 cm, find the radius of the circle.
From appoint Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre of the circle is 25 cm. Find the radius of the circle.
The tangent to a circle of radius 6 cm from an external point P, is of length 8 cm. Calculate the distance of P from the nearest point of the circle.
Prove that in two concentric circles, the chord of the bigger circle, which touches the smaller circle is bisected at the point of contact.
PQR circumscribes a circle of radius r such that angle Q = 900, PQ = 3 cm and QR = 4 cm. Find r.
Prove that the parallelogram circumscribing a circle is a rhombus.