Algebra

Algebra : Linear Quadratic Equations

 

Linear equations are fundamental in the CAT syllabus and often appear in both direct and indirect forms in the exam.

LINEAR EQUATION:-

Linear equations are algebraic equations of  degree = 1, meaning the highest power of the variable is 1. They generally take the form:

ax+b=0

Where:

  • a and b are constants.
  • x is the variable.

KEY CONCEPT:-

  • Solving Linear Equations:
    • Isolate the variable on one side of the equation by performing arithmetic operations (addition, subtraction, multiplication, division).
    • Example: 3x+4=7 can be solved as:

3x = 7-4

X=

X= 1

  • Systems of Linear Equations:
    • Two or more linear equations involving the same set of variables.
    • Solutions can be found using methods like substitution, elimination, or graphical methods.
    • Example: 2x + 3y = 5

x-y = 1

  • Solve one equation for one variable, substitute in the other, and solve.
  • Application in Word Problems:
    • Translate word problems into linear equations. Define variables, form equations, and solve.

NIMBUS MANTRA:

  • Practice different types of linear equation problems, including those involving fractions, decimals, and word problems.
  • Understand how to quickly manipulate and solve systems of linear equations.

Quadratic Equations

Quadratic equations are algebraic equations of the degree = 2, meaning the highest power of the variable is 2. They generally take the form:

ax2+bx+c=0

Where:

  • a, b, and c are constants.
  • x is the variable.

Key Concepts:

  1. Standard Form:
    • The general quadratic equation is written as ax2+bx+c=0
    • The solutions for x can be found using the quadratic formula:
      x =
  2. Factorization Method:
    • Express the quadratic equation as a product of two binomials.
    • Example: x2 – 5x + 6 = 0

(x-2)(x-3)=0

Hence x = 2 or x= 3

  1. Completing the Square:
    • Rewrite the quadratic equation in the form (x−p)2=q and solve for x.
    • Example: For x2+6x+5=0, completing the square gives (x+3)2−4=0.
  2. Discriminant Analysis:
    • The discriminant Δ=b2−4ac determines the nature of the roots:
      • Δ>0 : Two distinct real roots.
      • Δ=0 Exactly one real root (repeated).
      • Δ<0 No real roots, but two complex roots.
  1. Graphical Interpretation:
    • The graph of a quadratic equation is a parabola.
    • The roots are the x-intercepts of the parabola.

NIMBUS MANTRA:

  • Master different methods of solving quadratic equations: factorization, completing the square, and using the quadratic formula.
  • Be quick with discriminant analysis to determine the nature of roots.
  • Understand the graphical representation, as it helps in visualizing problems and solving inequality-related questions.

Practice and Strategy:

  • Time Management: Focus on solving problems quickly from NIMBUS study material . Linear and quadratic equations often appear in combination with other algebraic concepts, so efficiency is key.
  • Mock Tests: Regularly take NIMBUS mock tests and practice under timed conditions to build speed and accuracy.

 

Acollection of questions that typically appear from Linear (1st order equations – equation of straight lines) and Quadratic (parabolic functions) and cubic equations.

 

  1. If one of the roots of the quadratic equation x2+ mx + 24 = 0 is 1.5, then what is the value of m?
    1. -22.5
    2. 16
    3. -10.5
    4. -17.5

Concept: Co-efficient of quadratic equation

Correct answer 4 Level –  Easy

  1. Find the remainder when the polynomial x4 – 3x2 + 7x – 10 is divided by (x – 2).
    1. 8
    2. -20
    3. 18
    4. 0

Concept: Polynomial Division

Correct answer 1 Level – Easy

 

  1. If one of the roots of the quadratic equation 2x2 – 7x + q = 0 is 3, find the other root.
    1. -3
    2. -1/2
    3. 1/2
    4. 1/4

Concept: Quadratic equations – roots

Correct answer 1 Level – Easy

 

  1. A railway half ticket costs half the full fare and the reservation charge is the same on half ticket as on full ticket. One reserved first class ticket from Chennai to Trivandrum costs Rs. 216 and one full and one half reserved first class tickets cost Rs. 327. What is the basic first class full fare and what is the reservation charge?
    1. Rs. 105 and Rs. 6
    2. Rs. 216 and Rs. 12
    3. Rs. 210 and Rs. 12
    4. Rs. 210 and Rs. 6

Concept: Word Prolems – Linear Equations

Correct answer  4 Level – Medium

 

  1. If p and q are the roots of the equation x2 – bx + c = 0, what is the equation if the roots are (pq + p + q) and (pq – p – q)?
    1. x2 – 2cx + (c2 – b2) = 0
    2. x2 – 2bx + (b2 + c2) = 0
    3. bcx2 – 2(b + c)x + c2 = 0
    4. x2 + 2bx – (c2 – b2) = 0

Concept: Quadratic – roots to equation

Correct answer 1 Level – Medium

  1. If (x + 2)2= 9 and (y + 3) 2 = 25, then the maximum value of xyxy is ____.
    1. 40
    2. 52
    3. 2
    4. 10

Concept: Maximum – Minimum

Correct answer 1 Medium

  1. For what values of ‘m’ is y = 0, if y = x2 + (2m + 1)x + m2 – 1? x is a real number.
    1. m ≥ -2
    2. m < 0
    3. m = 0
    4. m ≥ -1.25

Concept: Nature of roots

Correct answer 4 Medium

  1. For what value of ‘m’ will the quadratic equation x2 + mx + 4 = 0 have real and equal roots?
    1. 4
    2. -4
    3. 4 or -4
    4. 16

Concept: Nature of roots – quadratic equation

Correct answer 3 Easy

  1. Rajesh is 10 years younger to Baskar. 10 years back, Rajesh’s age was two-thirds that of Baskar’s. How old is Baskar now?
    1. 30
    2. 40
    3. 20
    4. 16
    5. 28

Concept: Word Problems – Simultaneous Equations

Correct answer 2 Medium

  1. An owner of a pizza stand sold small slices of pizza for Rs. 150 each and large slices for Rs. 250 each. One night he sold 5000 slices, for a total of Rs. 10.50 lakh. How many small slices were sold?
    1. 3000
    2. 2000
    3. 4000
    4. 2500
    5. 3500

Concept: Word Problems – Linear Equation

Correct answer 2 Medium

  1. Jack has three more cards than Bill. Together they have 47 cards. If x represents the number of cards Bill has, then an equation that can be used to determine the number of cards each one has is
    1. x + 3 = 47
    2. 2x + 3 = 47
    3. x – 3 = 47
    4. 2x – 3 = 47
    5. 3x + 3 = 47

Concept: Framing Equation

Correct answer 2 Easy

 

  1. It costs Rs. x each to make the first thousand copies of a compact disk and Rs. y to make each subsequent copy. If z is greater than 1,000, how many Rupees will it cost to make z copies of the compact disk?
    1. 1000 x + yz
    2. zx – zy
    3. 1000 (z – x) + xy
    4. 1000 (z – y) + xz
    5. 1000 (x- y) + yz

Concept: Framing Expressions

Correct answer 5 Easy

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