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Mathematics · Class 11 Science

Ch 2Basic Algebra — Class 11 Mathematics, concept-first.

Algebra lets us state a relationship once, using variables (symbols standing for real numbers), and then read off its truth for every particular number by substitution -- this is what makes it so much more powerful than working with one numerical example at a time. This chapter's variables always denote real numbers.

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Key concepts

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Real Number System

The real numbers are built up in stages, each solving a problem the previous set could not: (counting) (adds zero) (adds negatives, for debts) (adds ratios, for even division) (fills in every remaining point of the numbe…

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

Introduction

Algebra lets us state a relationship once, using variables (symbols standing for real numbers), and then read off its truth for every particular number by substitution -- this is what makes it so much…

2.2

Real Number System

Before working with real numbers formally, it helps to see how the number system was actually built up, one enlargement at a time, starting from counting.

2.2.1

Rational Numbers

Natural numbers are enough for counting, but not for representing a loss or a debt. Enlarging by including zero and the negatives of every natural number gives the integers The set (natural numbers to…

2.2.2

The Number Line

The number line gives rational numbers a geometric home: mark an origin for and a point one unit to its right for ; every other rational number sits units to the right of , and every rational (for ) s…

2.2.3

Irrational Numbers

Theorem. is not a rational number.

2.2.4

Properties of Real Numbers

The real numbers obey a standard list of algebraic and order properties -- these are the rules that justify every later algebraic manipulation in the chapter. For all :

+Exercise 2.1i5 questions
  1. Q1Classify each element of $\left\{\sqrt7,\ -\dfrac14,\ 0,\ 3.14,\ 4,\ \dfrac{22}{7}\right\}$ as a member of $N,\ Q,\ R-Q$ or $Z$.Free
  2. Q2Prove that $\sqrt3$ is an irrational number. (Hint: Follow the method used to prove $\sqrt2\notin Q$.)Free
  3. Q3Are there two distinct irrational numbers such that their difference is a rational number? Justify.Preview
  4. Q4Find two irrational numbers such that their sum is a rational number. Can you find two irrational numbers whose product is a rational number…Preview
  5. Q5Find a positive number smaller than $\dfrac{1}{2^{1000}}$. Justify.Preview
2.3

Absolute Value

For any real number , the numbers and sit the same distance from on the number line. That common distance is called the absolute value of , written : Absolute value defines a function from onto -- it…

2.3.1

Definition and Properties

Symmetry. For any , (both measure the same distance from ). Consequently exactly when or .

2.3.2

Equations Involving Absolute Value

To solve an equation containing , isolate the absolute value on one side, then split into the two cases from §2.3.1: (with ) becomes or .

2.3.3

Some Results For Absolute Value

A handful of algebraic identities about absolute value are used throughout the rest of the chapter:

2.3.4

Inequalities Involving Absolute Value

Two master rules handle every absolute-value inequality:

+Exercise 2.2i6 questions
  1. Q1Solve for $x$: (i) $|3-x|<7$. (ii) $|4x-5|\ge-2$. (iii) $\left|3-\dfrac34x\right|\le\dfrac14$. (iv) $|x|-10<-3$.Free
  2. Q2Solve $\dfrac{1}{|2x-1|}<6$ and express the solution using the interval notation.Free
  3. Q3Solve $-3|x|+5\le-2$ and graph the solution set in a number line.Preview
  4. Q4Solve $2|x+1|-6\le7$ and graph the solution set in a number line.Preview
  5. Q5Solve $\dfrac15|10x-2|<1$.Preview
  6. Q6Solve $|5x-12|<-2$.Preview
2.4

Linear Inequalities

A function of the form ( constants) is a linear function -- its graph is a straight line, with the slope and the -intercept; if , its -intercept is (solving ).

+Exercise 2.3i10 questions
  1. Q1Represent the following inequalities in the interval notation: (i) $x\ge-1$ and $x<4$ (ii) $x\le5$ and $x\ge-3$ (iii) $x<-1$ or $x<3$ (iv) $…Free
  2. Q2Solve $23x<100$ when (i) $x$ is a natural number, (ii) $x$ is an integer.Free
  3. Q3Solve $-2x\ge9$ when (i) $x$ is a real number, (ii) $x$ is an integer, (iii) $x$ is a natural number.Free
  4. Q4Solve: (i) $\dfrac{3(x-2)}{5}\le\dfrac{5(2-x)}{3}$. (ii) $\dfrac{5-x}{3}<\dfrac{x}{2}-4$.Preview
  5. Q5To secure A grade one must obtain an average of 90 marks or more in 5 subjects each of maximum 100 marks. If one scored 84, 87, 95, 91 in fi…Preview
  6. Q6A manufacturer has 600 litres of a 12 percent solution of acid. How many litres of a 30 percent acid solution must be added to it so that th…Preview
  7. Q7Find all pairs of consecutive odd natural numbers both of which are larger than 10 and their sum is less than 40.Preview
  8. Q8A model rocket is launched from the ground. The height $h$ reached by the rocket after $t$ seconds from lift off is given by $h(t)=-5t^2+100…Preview
  9. Q9A plumber can be paid according to the following schemes: In the first scheme he will be paid rupees 500 plus rupees 70 per hour, and in the…Preview
  10. Q10A and B are working on similar jobs but their annual salaries differ by more than Rs 6000. If B earns rupees 27000 per month, then what are…Preview
2.5

Quadratic Functions

Just as ( times) for , we now generalise linear functions to quadratic functions: , where are constants and . If for some , is called a zero of .

2.5.1

Quadratic Formula

Completing the square. Any quadratic can be rewritten as -- verified by expanding the bracket and simplifying.

+Exercise 2.4i10 questions
  1. Q1Construct a quadratic equation with roots $7$ and $-3$.Free
  2. Q2A quadratic polynomial has one of its zeros $1+\sqrt5$ and it satisfies $p(1)=2$. Find the quadratic polynomial.Free
  3. Q3If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2+\sqrt2x+3=0$, form a quadratic polynomial with zeroes $\dfrac1\alpha,\d…Free
  4. Q4If one root of $k(x-1)^2=5x-7$ is double the other root, show that $k=2$ or $-25$.Preview
  5. Q5If the difference of the roots of the equation $2x^2-(a+1)x+a-1=0$ is equal to their product, then prove that $a=2$.Preview
  6. Q6Find the condition that one of the roots of $ax^2+bx+c=0$ may be (i) negative of the other, (ii) thrice the other, (iii) reciprocal of the o…Preview
  7. Q7If the equations $x^2-ax+b=0$ and $x^2-ex+f=0$ have one root in common and if the second equation has equal roots, then prove that $ae=2(b+f…Preview
  8. Q8Discuss the nature of roots of (i) $-x^2+3x+1=0$, (ii) $4x^2-x-2=0$, (iii) $9x^2+5x=0$.Preview
  9. Q9Without sketching the graphs, find whether the graphs of the following functions will intersect the x-axis and if so in how many points. (i)…Preview
  10. Q10Write $f(x)=x^2+5x+4$ in completed square form.Preview
2.5.2

Quadratic Inequalities

Steps to solve or : (1) solve the equation ; (2) if there are no real solutions, the inequality holds for every (or for no , according to the sign of ) since the expression never changes sign; (3) if…

+Exercise 2.5i2 questions
  1. Q1Solve $2x^2+x-15\le0$.Free
  2. Q2Solve $-x^2+3x-2\ge0$.Preview
2.6

Polynomial Functions

An expression (with and a non-negative integer) is a polynomial in . When , the polynomial has degree ; is its leading coefficient and its constant term.

2.6.1

Division Algorithm

Division algorithm. Given polynomials and a nonzero , there exist unique polynomials (the quotient) and (the remainder) with If , then and are factors of .

+Exercise 2.6i4 questions
  1. Q1Find the zeros of the polynomial function $f(x)=4x^2-25$.Free
  2. Q2If $x=-2$ is one root of $x^3-x^2-17x=22$, then find the other roots of equation.Free
  3. Q3Find the real roots of $x^4=16$.Preview
  4. Q4Solve $(2x+1)^2-(3x+2)^2=0$.Preview
2.6.2

Important Identities

An equation that holds for every value in its domain is called an identity (as opposed to a conditional equation, true only for some values).

2.6.3

Method of Undetermined Coefficients

Given information about a polynomial's zeros and/or its value at specific points, we can construct it by writing it with unknown ('undetermined') coefficients and using the equality test of §2.6 -- ma…

+Exercise 2.7i2 questions
  1. Q1Factorize: $x^4+1$. (Hint: Try completing the square.)Free
  2. Q2If $x^2+x+1$ is a factor of the polynomial $3x^3+8x^2+8x+a$, then find the value of $a$.Preview
2.7

Rational Functions

A rational expression in is the ratio of two polynomials with , defined for every where . If , dividing (§2.6.1's division algorithm) gives turning an 'improper' rational expression into a polynomial…

2.7.1

Rational Inequalities

To solve a rational inequality such as : move every term to one side, , then combine into a single fraction, (dividing by flips the inequality).

+Exercise 2.8i3 questions
  1. Q1Find all values of $x$ for which $\dfrac{x^3(x-1)}{(x-2)}>0$.Free
  2. Q2Find all values of $x$ that satisfy the inequality $\dfrac{2x-3}{(x-2)(x-4)}<0$.Preview
  3. Q3Solve $\dfrac{x^2-4}{x^2-2x-15}\le0$.Preview
2.7.2

Partial Fractions

A rational expression is a proper fraction if ; every proper fraction whose denominator factors into linear and irreducible-quadratic pieces can be written uniquely as a sum of simpler pieces -- its p…

+Exercise 2.9i12 questions
  1. Q1$\dfrac{1}{x^2-a^2}$Free
  2. Q2$\dfrac{3x+1}{(x-2)(x+1)}$Free
  3. Q3$\dfrac{x}{(x^2+1)(x-1)(x+2)}$Free
  4. Q4$\dfrac{x}{(x-1)^3}$Preview
  5. Q5$\dfrac{1}{x^4-1}$Preview
  6. Q6$\dfrac{(x-1)^2}{x^3+x}$Preview
  7. Q7$\dfrac{x^2+x+1}{x^2-5x+6}$Preview
  8. Q8$\dfrac{x^3+2x+1}{x^2+5x+6}$Preview
  9. Q9$\dfrac{x+12}{(x+1)^2(x-2)}$Preview
  10. Q10$\dfrac{6x^2-x+1}{x^3+x^2+x+1}$Preview
  11. Q11$\dfrac{2x^2+5x-11}{x^2+2x-3}$Preview
  12. Q12$\dfrac{7+x}{(1+x)(1+x^2)}$Preview
2.7.3

Graphical Representation of Linear Inequalities

A straight line splits the Cartesian plane into two half-planes; a vertical line gives left/right half-planes and a horizontal line gives upper/lower half-planes.

+Exercise 2.10i7 questions
  1. Q1$x\le3y,\ x\ge y$.Free
  2. Q2$y\ge2x,\ -2x+3y\le6$.Free
  3. Q3$3x+5y\ge45,\ x\ge0,\ y\ge0$.Free
  4. Q4$2x+3y\le35,\ y\ge2,\ x\ge5$.Preview
  5. Q5$2x+3y\le6,\ x+4y\le4,\ x\ge0,\ y\ge0$.Preview
  6. Q6$x-2y\ge0,\ 2x-y\le-2,\ x\ge0,\ y\ge0$.Preview
  7. Q7$2x+y\ge8,\ x+2y\ge8,\ x+y\le6$.Preview
2.8

Exponents and Radicals

Having built polynomial and rational functions from whole-number powers, we now extend the exponent itself to any rational number -- and beyond.

2.8.1

Exponents

For and , ( times, ordinary repeated multiplication). For a negative integer exponent and , (so negative flips to a positive power in the denominator). Note for any .

2.8.2

Radicals

Motivating question. For and (), can be defined so that satisfies ? This is exactly asking to invert .

2.8.3

Exponential Function

For any and , is now fully defined (via §2.8.2's rational powers, extended by continuity to every real exponent); always.

+Exercise 2.11i8 questions
  1. Q1Simplify: (i) $(125)^{2/3}$, (ii) $16^{-3/4}$, (iii) $(-1000)^{-2/3}$, (iv) $\left(3^{-6}\right)^{1/3}$, (v) $\dfrac{27^{-2/3}}{27^{-1/3}}$.Free
  2. Q2Evaluate $\left[(256)^{-1/2}-\dfrac14\right]^{3}$.Free
  3. Q3If $(x^{1/2}+x^{-1/2})^2=\dfrac92$, then find the value of $(x^{1/2}-x^{-1/2})$ for $x>1$.Free
  4. Q4Simplify and hence find the value of $n$: $\dfrac{3^{2n}\cdot 9^{2}\cdot 3^{-n}}{3^{3n}} = 27$.Preview
  5. Q5Find the radius of the spherical tank whose volume is $\dfrac{32\pi}{3}$ cubic units.Preview
  6. Q6Simplify by rationalising the denominator: $\dfrac{7+\sqrt6}{3-\sqrt2}$.Preview
  7. Q7Simplify $\dfrac{1}{3-\sqrt8}-\dfrac{1}{\sqrt8-\sqrt7}+\dfrac{1}{\sqrt7-\sqrt6}-\dfrac{1}{\sqrt6-\sqrt5}+\dfrac{1}{\sqrt5-2}$.Preview
  8. Q8If $x=\sqrt2+\sqrt3$, find $x^2+\dfrac1{x^2}-2$.Preview
2.8.3.1

Compound Interest

Because is exactly the compound-amount formula (principal , rate , compounding periods per year, years), it is natural to ask what happens as the number of compounding periods per year grows without b…

2.9

Logarithm

Since () is a bijection from onto (§2.8.3), it has an inverse function, called the logarithmic function with base and written : if sends , then sends .

2.9.1

Properties of Logarithm

Properties of Logarithm (all with , unless stated):

+Exercise 2.12i12 questions
  1. Q1Let $b>0$ and $b\ne1$. Express $y=b^x$ in logarithmic form. Also state the domain and range of the logarithmic function.Free
  2. Q2Compute $\log_9 27-\log_{27}9$.Free
  3. Q3Solve $\log_8 x+\log_4 x+\log_2 x=11$.Free
  4. Q4Solve $\log_4\left(2^{8x}\right)=2\log_2 8$.Preview
  5. Q5If $a^2+b^2=7ab$, show that $\log\left(\dfrac{a+b}3\right)=\dfrac12(\log a+\log b)$.Preview
  6. Q6Prove $\log\dfrac{a^2}{bc}+\log\dfrac{b^2}{ca}+\log\dfrac{c^2}{ab}=0$.Preview
  7. Q7Prove that $\log 2+16\log\dfrac{16}{15}+12\log\dfrac{25}{24}+7\log\dfrac{81}{80}=1$.Preview
  8. Q8Prove $\log_{a^2}a\cdot\log_{b^2}b\cdot\log_{c^2}c=\dfrac18$.Preview
  9. Q9Prove $\log a+\log a^2+\log a^3+\cdots+\log a^n=\dfrac{n(n+1)}2\log a$.Preview
  10. Q10If $\dfrac{\log x}{y-z}=\dfrac{\log y}{z-x}=\dfrac{\log z}{x-y}$, then prove that $xyz=1$.Preview
  11. Q11Solve $\log_2x-3\log_{1/2}x=6$.Preview
  12. Q12Solve $\log_{5-x}(x^2-6x+65)=2$.Preview
2.10

Application of Algebra in Real Life

Algebra is not confined to the classroom -- it is the working language behind a wide range of everyday and professional calculations.

+Exercise 2.13i20 questions
  1. Q1If $|x+2|\le9$, then $x$ belongs to (1) $(-\infty,-7)$ (2) $[-11,7]$ (3) $(-\infty,-7)\cup[11,\infty)$ (4) $(-11,7)$Free
  2. Q2Given that $x,y$ and $b$ are real numbers, $x<y$, $b>0$, then (1) $xb<yb$ (2) $xb>yb$ (3) $xb\le yb$ (4) $\dfrac xb\ge\dfrac yb$Free
  3. Q3If $\dfrac{|x-2|}{x-2}\ge0$, then $x$ belongs to (1) $[2,\infty)$ (2) $(2,\infty)$ (3) $(-\infty,2)$ (4) $(-2,\infty)$Free
  4. Q4The solution of $5x-1<24$ and $5x+1>-24$ is (1) $(4,5)$ (2) $(-5,-4)$ (3) $(-5,5)$ (4) $(-5,4)$Preview
  5. Q5The solution set of the following inequality $|x-1|\ge|x-3|$ is (1) $[0,2]$ (2) $[2,\infty)$ (3) $(0,2)$ (4) $(-\infty,2)$Preview
  6. Q6The value of $\log_{\sqrt2}512$ is (1) $16$ (2) $18$ (3) $9$ (4) $12$Preview
  7. Q7The value of $\log_3\dfrac1{81}$ is (1) $-2$ (2) $-8$ (3) $-4$ (4) $-9$Preview
  8. Q8If $\log_{\sqrt x}0.25=4$, then the value of $x$ is (1) $0.5$ (2) $2.5$ (3) $1.5$ (4) $1.25$Preview
  9. Q9The value of $\log_ab\cdot\log_bc\cdot\log_ca$ is (1) $2$ (2) $1$ (3) $3$ (4) $4$Preview
  10. Q10If 3 is the logarithm of 343, then the base is (1) $5$ (2) $7$ (3) $6$ (4) $9$Preview
  11. Q11Find $a$ so that the sum and product of the roots of the equation $2x^2+(a-3)x+3a-5=0$ are equal is (1) $1$ (2) $2$ (3) $0$ (4) $4$Preview
  12. Q12If $a$ and $b$ are the roots of the equation $x^2-kx+16=0$ and satisfy $a^2+b^2=32$, then the value of $k$ is (1) $10$ (2) $-8$ (3) $-8,8$ (…Preview
  13. Q13The number of solutions of $x^2+|x-1|=1$ is (1) $1$ (2) $0$ (3) $2$ (4) $3$Preview
  14. Q14The equation whose roots are numerically equal but opposite in sign to the roots of $3x^2-5x-7=0$ is (1) $3x^2-5x-7=0$ (2) $3x^2+5x-7=0$ (3)…Preview
  15. Q15If 8 and 2 are the roots of $x^2+ax+c=0$ and 3, 3 are the roots of $x^2+dx+b=0$, then the roots of the equation $x^2+ax+b=0$ are (1) $1,2$ (…Preview
  16. Q16If $a$ and $b$ are the real roots of the equation $x^2-kx+c=0$, then the distance between the points $(a,0)$ and $(b,0)$ is (1) $\sqrt{k^2-4…Preview
  17. Q17If $\dfrac{kx}{(x+2)(x-1)}=\dfrac2{x+2}+\dfrac1{x-1}$, then the value of $k$ is (1) $1$ (2) $2$ (3) $3$ (4) $4$Preview
  18. Q18If $\dfrac{1-2x}{3+2x-x^2}=\dfrac A{3-x}+\dfrac B{x+1}$, then the value of $A+B$ is (1) $-\dfrac12$ (2) $-\dfrac23$ (3) $\dfrac12$ (4) $\dfr…Preview
  19. Q19The number of roots of $(x+3)^4+(x+5)^4=16$ is (1) $4$ (2) $2$ (3) $3$ (4) $0$Preview
  20. Q20The value of $\log_3 11\cdot\log_{11}13\cdot\log_{13}15\cdot\log_{15}27\cdot\log_{27}81$ is (1) $1$ (2) $2$ (3) $3$ (4) $4$Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 27 questions27 questions
  1. Q1If $\dfrac{ax}{(x+2)(2x-3)} = \dfrac{2}{x+2} + \dfrac{3}{2x-3}$ then $a =$ (a) $7$ (b) $4$ (c) $8$ (d) $5$Preview
  2. Q2If $|x+2|\le 8$, then $x$ belongs to: (a) $(6, 10)$ (b) $(-10, 6)$ (c) $[6, 10]$ (d) $[-10, 6]$Preview
  3. Q3(a) Solve the equation $\sqrt{6-4x-x^2}=x+4$. **OR** (b) Prove that in any $\triangle ABC$, $\Delta=\sqrt{s(s-a)(s-b)(s-c)}$, where $s$ is t…Preview
  4. Q4The solution of $5x - 1 < 24$ and $5x + 1 > -24$ is: (a) (4, 5) (b) (-5, -4) (c) (-5, 5) (d) (-5, 4)Preview
  5. Q5The number of solutions of $x^2 + |x - 1| = 1$ is: (a) 1 (b) 0 (c) 2 (d) 3Preview
  6. Q6Find the complete set of values of 'a' for which the quadratic $x^2 - ax + a + 2 = 0$ has equal roots.Preview
  7. Q7The value of $\log_{\sqrt{2}} 512$ is: (a) 9 (b) 16 (c) 12 (d) 18Preview
  8. Q8Solve $|2x-17|=3$ for $x$.Preview
  9. Q9Resolve into partial fractions: $\dfrac{x}{(x+3)(x-4)}$.Preview
  10. Q10The number of real solutions of the equation $x^2 - 3|x| + 2 = 0$ are: (a) 4 (b) 2 (c) 1 (d) 3Preview
  11. Q11Prove that $\log a + \log a^2 + \log a^3 + \ldots + \log a^n = \dfrac{n(n+1)}{2}\log a$.Preview
  12. Q12Solve the equation $\sqrt{6-4x-x^2} = x+4$.Preview
  13. Q13If one root of $k(x-1)^2 = 5x-7$ is double the other root, show that $k=2$ or $-25$. **OR** Express the matrix $A = \begin{bmatrix}1 & 3 & 5…Preview
  14. Q14The value of $\log_{\sqrt2}512$ is: (a) $9$ (b) $16$ (c) $12$ (d) $18$Preview
  15. Q15Simplify: $\dfrac{1}{3-\sqrt8}-\dfrac{1}{\sqrt8-\sqrt7}+\dfrac{1}{\sqrt7-\sqrt6}-\dfrac{1}{\sqrt6-\sqrt5}+\dfrac{1}{\sqrt5-2}$Preview
  16. Q16(a) Resolve into partial fractions $\dfrac{2x}{(x^2+1)(x-1)}$. **OR** (b) If $y=e^{\tan^{-1}x}$, show that $(1+x^2)y''+(2x-1)y'=0$.Preview
  17. Q17(a) Prove that $\log_{10}2+16\log_{10}\dfrac{16}{15}+12\log_{10}\dfrac{25}{24}+7\log_{10}\dfrac{81}{80}=1$ **OR** (b) There are two identica…Preview
  18. Q18The solution set of the following inequality $|x-1| \geq |x-3|$ is: (a) $(0, 2)$ (b) $[0, 2]$ (c) $(-\infty, 2)$ (d) $[2, \infty)$Preview
  19. Q19If $3$ is the logarithm of $343$, then the base is: (a) $6$ (b) $5$ (c) $9$ (d) $7$Preview
  20. Q20Solve $23x < 100$ when (i) $x$ is a natural number, (ii) $x$ is an integer.Preview
  21. Q21Solve : $\sqrt{x^2 - x - 2} = x + 1$Preview
  22. Q22(a) Resolve into partial fractions : $\dfrac{x^2+x+1}{x^2-5x+6}$ **OR** (b) Express the equation $\sqrt{3}x - y + 4 = 0$ in the following eq…Preview
  23. Q23If $|x + 2| \le 9$, then $x$ belongs to: (a) $(-\infty, -7) \cup [11, \infty)$ (b) $(-\infty, -7)$ (c) $(-11, 7)$ (d) $[-11, 7]$Preview
  24. Q24The value of $\log_3 11 \cdot \log_{11} 13 \cdot \log_{13} 15 \cdot \log_{15} 27$ is: (a) 3 (b) 1 (c) 4 (d) 2Preview
  25. Q25If $x = -2$ is one root of $x^3 - x^2 - 17x = 22$, then find the other roots of the equation.Preview
  26. Q26Resolve into Partial fractions: $\dfrac{1}{x^2-7^2}$Preview
  27. Q27Prove that $\log\dfrac{75}{16}-2\log\dfrac{5}{9}+\log\dfrac{32}{243}=\log 2$ **OR** By the principle of mathematical induction, prove that,…Preview