CBSE Class 10 - Maths - Polynomials - NCERT Ex 2.1

POLYNOMIALS

NCERT Exercise 2.1 and other Q & A

Q1: The graphs of y = p(x) are given in below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

Answer: 

(i) As shown in the graph, it does not intersect x-axis. Hence it has no real zeroes.

(ii) From the graph, it intersects x-axis at one point. ∴ it has one real zero.

(iii) As the graph shows, it intersects x-axis at three points. ∴ the graph polynomial has three real zeroes.

(iv) From the graph, it is clear it intersects x-axis at two points. Hence it has two zeroes.

(v) As the graph shows, it meets x-axis at four points. ∴ it has four real zeroes.

CBSE Class 12 - Computer Science (C++) Questions on Class Design

Computer Science (C++) - Class Design

Questions Asked in Board Papers on Class Design

Q1(CBSE 2013):  Define a class Tourist in C++ with the following specifications:
     Data Members:

• CarNo - to store bus number
• Origin - to store Place name
• Destination - to store Place name
• Type - to store Car Type such as 'E' for Economy
• Distance - to store the Distance in Kilometers
• Charge - to store the Car Fare

    Member Functions:

• A constructor function to initialize Type as 'E' and Freight as 250
• A function CalcCharge( ) to calculate Fare as per the following criteria:
  

  Type	Charge
  'E'	16*Distance
  'A'	22*Distance
  'L'	30*Distance

  
  
• A function Enter( ) to allow user to enter values for CarNo, Origin, Destination, Type and Distance. Also this function should call CalcCharge( ) to calculate Fare.
• A function Show( ) to display the content of all the data members on screen.

 Answer:

 const int BUFFSIZE = 40;
 class Tourist
 {
   // private data members
   int carNum;
   char origin[BUFFSIZE];
   char destination[BUFFSIZE];
   char carType;
   int distance;
   int charges;
  public:
    // member functions
    // constructor
   Tourist()
    {
     carType = 'E';
     charges = 250;
    }

   // calculate charges
   void CalcCharges()
    {
     if (Type == 'E')
     {
      charges = 16*distance;
     }
    else if (Type == 'A')
     {
      charges = 22*distance;
     }
    else if (Type == 'C')
     {
      charges = 30*distance;
     }
 };

 // enter values
 void Tourist::Enter()
 {
   cout << "Enter CarNo, Origin, Destination, Type, Distance:";
   cin >> carNum;
   gets(origin);
   gets(destination);
   cin >> carType >> Distance;
   CalcCharges();
 }
 
 // show/display details
 void Show()
 {
   cout << "Car No: " << carNum 
        << "\nOrigin: " << origing 
        << "\nDestination: " << destination
        << "\nType: " << carType
        << "\nDistance: " << distance
        << "\nCharges: " << charges << endl; 
 }
 };

Q2(CBSE 2012): Define a class Flat in C++ with the following description:
     Private Members:

CBSE Class 10 - Maths - CH2 - Polynomials - Study Points

Polynomials

Study Points

1. An expression of the form a₀ + a₁x + a₂x² + ----- + a_nxⁿ where a_n is called a polynomial in variable x of degree n. where; a₀ ,a₁, ----- a_n are real numbers and are called terms/co-efficients of the polynomial and each power of x is a non negative integer.

2. Polynomials in the variable x are denoted by f(x), g(x), h(x) etc.
    e.g. f(x) = a₀ + a₁x + a₂x² + ----- + a_nxⁿ

3. A polynomial p(x) = a (where a is constant) is of degree 0 and is called a constant polynomial.

4. A polynomial p(x) = ax + b is of degree 1 and is called a linear polynomial. e.g. 4x - 3, 5x...

5. A polynomial p(x) = ax² + bx + c of degree 2 and is called a quadratic polynomial.
    e.g. 3x² -x + 5, 1- x²...

6. A polynomial p(x) = ax³ + bx² + cx + d of degree 3 and is called a cubic polynomial.
    e.g. √5x³ - 2x² + 5x -1...

CBSE Class 12 - Computer Science - Sample Question Paper - (2014-15)

Class 12
Computer Science
Sample Q Paper (2014-15)

	CLASS-XII COMPUTER SCIENCE (Subject Code 083) SAMPLE PAPER 2014 - 15
Time allowed : 3 hours		Maximum Marks: 70
	Instructions: (i) All questions are compulsory. (ii) Programming Language: Section A C+ +. (iii) Programming Language : Section B Python. (iv) Answer either Section A or B, and Section C is compulsory.
Section A (C++)
Q1 (a)	Differentiate between ordinary function and member functions in C++. Explain with an example.	[2]
(b)	Write the related library function name based upon the given information in C++. (i) Get single character using keyboard. This function is available in stdio.h file. (ii) To check whether given character is alpha numeric character or not. This function is available in ctype.h file.	[1]
(c)	Rewrite the following C++ program after removing all the syntactical errors (if any), underlining each correction. : include <iostream.h> #define PI=3.14 void main( ) { float r;a; cout<<’enter any radius’; cin>>r; a=PI*pow(r,2); cout<< ”Area=” << a }	[2]
(d)	Write the output from the following C++ program code: #include <iostream.h> #include <ctype.h> void strcon(char s[]) { for(int i=0,l=0;s[i]!='\0';i++,l++); for(int j=0; j<l; j++) { if (isupper(s[j])) s[j]=tolower(s[j])+2; else if ( islower(s[j])) s[j]=toupper(s[j])-2; else s[j]='@'; } } void main() { char *c="Romeo Joliet"; strcon(c); cout<<"Text= "<<c<<endl; c=c+3; cout<<"New Text= "<<c<<endl; c=c+5-2; cout<<"last Text= "<<c }	[2]
(e)	Find the output of the following C++ program: #include<iostream.h> #include<conio.h> #include<ctype.h> class Class { int Cno,total; char section; public: Class(int no=1) { Cno=no; section='A'; total=30; } void addmission(int c=20) { section++; total+=c; } void ClassShow() { cout<<Cno<<":"<<section<<":"<<total<<endl; } }; void main() { Class C1(5),C2; C1.addmission(25); C1.ClassShow(); C2.addmission(); C1.addmission(30); C2.ClassShow(); C1.ClassShow(); }

CBSE Class 9 - Maths - Number Systems (Very Short Questions and Answers)

NUMBER SYSTEMS 

Very Short Q & A

Q1: Categorize the following numbers as
(i) Natural Numbers
(ii) Whole Numbers
(iii) Integers

(iv) Rational Numbers

7, 3/2, 0, 2, -9/5, 1/4, -8

Answer:
(i) Natural Numbers {2, 7}

(ii) Whole Numbers {0, 2, 7}

(iii) Integers {-8, 0, 2, 7} 

(iv) Rational Numbers {-8, -9/5, 0, 1/4, 3/2, 2, 7}

Q2: What is the rational number which does not have reciprocal?

Answer: 0 (zero)

Q3: Write the greatest negative integer.

Answer: -1

Q4: Name the smallest natural number.

CBSE Class 10 - Maths - CH1 - Real Numbers (Euclid's Division Lemma)

Euclid's Division Lemma

Important Questions asked in Examination...

Euclid (4th BC)
credits: wikipedia

Q1: State Euclid's Division Lemma.

Answer: Given positive integers a and b ( b ≠ 0 ), there exists unique integer numbers q and r satisfying a = bq + r, 0 ≤ r < |b|. where
    a is called dividend
    b is called divisor
    q is called quotient
    r is called remainder.

    e.g. 17 = 5 × 3 + 2

Q2: Prove Euclid's Division Lemma.

Answer: According to Euclid's Division lemma, for a positive pair of integers there exists unique integers q and r, such that
                      a = bq + r, where 0 ≤ r < b

Let us assume q and r are not unique i.e. let there exist another pair q0 and r0 i.e. a = bq0 + r0, where 0 ≤ r0 < b

⇒ bq + r = bq0 + r0
⇒ b(q - q0) = r - r0                                          ................ (I)

Since 0 ≤ r < b and 0 ≤ r0 < b, thus 0 ≤ r - r0 < b ......... (II)

The above eq (I) tells that b divides (r - r0) and (r - r0) is an integer less than b. This means (r - r0) must be 0.
⇒ r - r0 = 0 ⇒ r = r0

Eq (I) will be, b(q - q0) = 0
Since b ≠ 0, ⇒ (q - q0) = 0 ⇒ q = q0

Since r = r0 and q = q0, ∴ q and r are unique.

Q3: Prove that the product of two consecutive positive integers is divisible by 2?

CBSE Class 9 - CH 12 - Sound (Study Points)

Sound

Visual Representation of Sound Waveform
(image courtsey: openclipart)

Study Points

1. Sound travels from one point to another in the form of waves.

2. Sound is produced due to vibration of different objects.

3. Sound waves are longitudinal in nature.

4. Sound travels as a longitudinal wave through a material medium.

5. Sound travels as successive compressions and rarefactions in the medium.

6. In sound propagation, it is the energy of the sound that travels and not the particles of the medium.

7. Sound cannot travel in vacuum.

8. Sound waves cannot be polarised.

9. The change in density from one maximum value to the minimum value and again to the maximum value makes one complete oscillation.

10. The part or region of a longitudinal wave in which the density of the particles of the medium is higher than the normal density is known as compression.

11. The part or region of a longitudinal wave in which the density of the particles of the medium is lesser than the normal density is called a rarefaction.

12. The distance between two consecutive compressions or two consecutive rarefactions is called the wavelength, λ.