Samacheer Kalvi solutions for Mathematics - Volume 1 and 2 [English] Class 11 TN Board chapter 2 - Basic Algebra [Latest edition]

Classify each element of `{sqrt(7), (-1)/4, 0, 3, 1, 4, 4, 22/7}` as a member of N, Q, R − Q or Z

Prove that `sqrt(3)` is an irrational number.
(Hint: Follow the method that we have used to prove `sqrt(2)` ∉ Q)

Are there two distinct irrational numbers such that their difference is a rational number? Justify

Find two irrational numbers such that their sum is a rational number. Can you find two irrational numbers whose product is a rational number

Find a positive number smaller than `2^(1/1000)`. Justify

Solve for x: |3 − x| < 7

Solve for x: |4x − 5| ≥ −2

Solve for x: `|3 - 3/4x| ≤ 1/4`

Solve for x: |x| − 10 < −3

Solve `1/(|2x - 1|) < 6` and express the solution using the interval notation

Solve −3|x| + 5 ≤ −2 and graph the solution set in a number line

Solve 2|x + 1| − 6 ≤ 7 and graph the solution set in a number line

Solve `1/5|10x - 2| < 1`

Solve |5x − 12| < −2

Represent the following inequalities in the interval notation:

x ≥ −1 and x < 4

Represent the following inequalities in the interval notation:

x ≤ 5 and x ≥ − 3

Represent the following inequalities in the interval notation:

x < −1 or x < 3

Represent the following inequalities in the interval notation:

−2x > 0 or 3x − 4 < 11

Solve 23x < 100 when x is a natural number

Solve 23x < 100 when x is an integer

Solve −2x ≥ 9 when x is a real number

Solve −2x ≥ 9 when x is an integer

Solve −2x ≥ 9 when x is a natural number

Solve: `(3(x - 2))/5 ≤ (5(2 - x))/3`

Solve: `(5 - x)/3 < x/2 - 4`

To 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 first four subjects, what is the minimum mark one scored in the fifth subject to get A grade in the course?

A 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 the acid content in the resulting mixture will be more than 15 percent but less than 18 percent?

Find all pairs of consecutive odd natural numbers both of which are larger than 10 and their sum is less than 40

A 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) = −5t² +100t, 0 ≤ t ≤ 20. At what time the rocket is 495 feet above the ground?

A 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 second scheme he will be paid rupees 120 per hour. If he works x hours, then for what value of x does the first scheme give better wages?

A and B are working on similar jobs but their monthly salaries differ by more than Rs 6000. If B earns rupees 27000 per month, then what are the possibilities of A’s salary per month?

Construct a quadratic equation with roots 7 and −3

A quadratic polynomial has one of its zeros `1 + sqrt(5)` and it satisfies p(1) = 2. Find the quadratic polynomial

If α and β are the roots of the quadratic equation `x^2 + sqrt(2)x + 3` = 0, form a quadratic polynomial with zeroes `1/α, 1/β`

If one root of k(x − 1)² = 5x − 7 is double the other root, show that k = 2 or −25

If the difference of the roots of the equation 2x² − (a + 1)x + a − 1 = 0 is equal to their product, then prove that a = 2

Find the condition that one of the roots of ax² + bx + c may be negative of the other

Find the condition that one of the roots of ax² + bx + c may be thrice the other

Find the condition that one of the roots of ax² + bx + c may be reciprocal of the other

If the equations x² − ax + b = 0 and x² − ex + f = 0 have one root in common and if the second equation has equal roots, then prove that ae = 2(b + f)

Discuss the nature of roots of − x² + 3x + 1 = 0

Discuss the nature of roots of 4x² − x − 2 = 0

Discuss the nature of roots of 9x² + 5x = 0

Without sketching the graph, find whether the graph of the following function will intersect the x-axis and if so in how many points
y = x² + x + 2

Without sketching the graph, find whether the graph of the following function will intersect the x-axis and if so in how many points
y = x² − 3x − 7

Without sketching the graph, find whether the graph of the following function will intersect the x-axis and if so in how many points
y = x² + 6x + 9

Write f(x) = x² + 5x + 4 in completed square form

Solve 2x² + x – 15 ≤ 0

Solve – x² + 3x – 2 ≥ 0

Find the zeros of the polynomial function f(x) = 4x² − 25

If x = −2 is one root of x³ − x² − 17x = 22, then find the other roots of equation

Find the real roots of x⁴ = 16

Solve (2x + 1)² − (3x + 2)² = 0

Factorize: x⁴ + 1. (Hint: Try completing the square)

If x² + x + 1 is a factor of the polynomial 3x³ + 8x² + 8x + a, then find the value of a.

Find all values of x for which `(x^3(x - 1))/((x - 2)) > 0`

Find all values of x that satisfies the inequality `(2x - 3)/((x - 2)(x - 4)) < 0`

Solve `(x^2 - 4)/(x^2 - 2x - 15) ≤ 0`

Resolve the following rational expressions into partial fractions

`1/(x^2 - "a"^2)`

Resolve the following rational expressions into partial fractions

`(3x + 1)/((x - 2)(x + 1))`

Resolve the following rational expressions into partial fractions

`x/((x^2 + 1)(x - 1)(x + 2))`

Resolve the following rational expressions into partial fractions

`x/((x - 1)^3`

Resolve the following rational expressions into partial fractions

`1/(x^4 - 1)`

Resolve the following rational expressions into partial fractions

`(x - 1)^2/(x^3 + x)`

Resolve the following rational expressions into partial fractions

`(x^2 + x + 1)/(x^2 - 5x + 6)`

Resolve the following rational expressions into partial fractions

`(x^3 + 2x + 1)/(x^2 + 5x + 6)`

Resolve the following rational expressions into partial fractions

`(x + 12)/((x + 1)^2 (x - 2))`

Resolve the following rational expressions into partial fractions

`(6x^2 - x + 1)/(x^3 + x^2 + x + 1)`

Resolve the following rational expressions into partial fractions

`(2x^2 + 5x - 11)/(x^2 + 2x - 3)`

Resolve the following rational expressions into partial fractions

`(7 + x)/((1 + x)(1 + x^2))`

Determine the region in the plane determined by the inequalities:

x ≤ 3y, x ≥ y

Determine the region in the plane determined by the inequalities:

y ≥ 2x, −2x + 3y ≤ 6

Determine the region in the plane determined by the inequalities:

3x + 5y ≥ 45, x ≥ 0, y ≥ 0

Determine the region in the plane determined by the inequalities:

2x + 3y ≤ 35, y ≥ 2, x ≥ 5.

Determine the region in the plane determined by the inequalities:

2x + 3y ≤ 6, x + 4y ≤ 4, x ≥ 0, y ≥ 0

Determine the region in the plane determined by the inequalities:

x − 2y ≥ 0, 2x − y ≤ −2, x ≥ 0, y ≥ 0

Determine the region in the plane determined by the inequalities:

2x + y ≥ 8, x + 2y ≥ 8, x + y ≤ 6

Simplify: `(125)^(2/3)`

Simplify: `16^((-3)/4)`

Simplify: `(- 1000)^((-2)/3)`

Simplify: `(3^-6)^(1/3)`

Simplify: `(27^((-2)/3))/(27^((-1)/3))`

Evaluate `[((256)^(-1/2))^((-1)/4)]^3`

If `(x^(1/2) + x^(- 1/2))^2 = 9/2` then find the value of `(x^(1/2) - x^(-1/2))` for x >1

Simplify and hence find the value of n:

`(3^(2"n")*9^2*3^(-"n"))/(3^(3"n"))` = 27

Find the radius of the spherical tank whose volume is `(32pi)/3` units

. Simplify by rationalising the denominator `(7 + sqrt(6))/(3 - sqrt(2))`

Simplify `1/(3 - sqrt(8)) - 1/(sqrt(8) - sqrt(7)) + 1/(sqrt(7) - sqrt(6)) - 1/(sqrt(6) - sqrt(5)) + 1/(sqrt(5) - 2)`

If x = `sqrt(2) + sqrt(3)` find `(x^2 + 1)/(x^2 - 2)`

Let b > 0 and b ≠ 1. Express y = b^x in logarithmic form. Also state the domain and range of the logarithmic function

Compute log₉ 27 – log₂₇ 9

Solve log₈x + log₄x + log₂x = 11

Solve log₄ 2^8x = 2^log₂8 

If a² + b² = 7ab, show that `log  ("a" + "b")/3 = 1/2(log"a" + log "b")`

Prove `log  "a"^2/"bc" + log  "b"^2/"ca" + log  "c"^2/"ab"` = 0

Prove that `log 2 + 16log  16/15 + 12log  25/24 + 7log  81/80` = 1

Prove that log_a² a + log_b² b + log_c² c = `1/8`

Prove log a + log a² + log a³ + · · · + log aⁿ = `("n"("n" + 1))/2 log "a"`

If `log x/(y - z) = logy/(z - x) = logz/(x - y)`, then prove that xyz = 1

Solve `log_2 x − 3 log_(1/2) x` = 6

Solve log_{5 – x} (x² – 6x + 65) = 2

Choose the correct alternative:
If |x + 2| ≤ 9, then x belongs to

Choose the correct alternative:
Given that x, y and b are real numbers x < y, b > 0, then

Choose the correct alternative:
If  `|x - 2|/(x - 2) ≥ 0`, then x belongs to

Choose the correct alternative:
The solution of 5x − 1 < 24 and 5x + 1 > −24 is

Choose the correct alternative:
The solution set of the following inequality |x − 1| ≥ |x − 3| is

Choose the correct alternative:
The value of `log_(sqrt(5))` 512 is

Choose the correct alternative:
The value of `log_3  1/81` is 

Choose the correct alternative:
If `log_(sqrt(x)` 0.25 = 4, then the value of x is

Choose the correct alternative:
The value of log_ab  log_bc  log_ca is

Choose the correct alternative:
If 3 is the logarithm of 343, then the base is

Choose the correct alternative:
Find a so that the sum and product of the roots of the equation 2x² + (a − 3)x + 3a − 5 = 0 are equal is

Choose the correct alternative:
If a and b are the roots of the equation x² − kx + 16 = 0 and satisfy a² + b² = 32, then the value of k is

Choose the correct alternative:
The number of solutions of x² + |x − 1| = 1 is

Choose the correct alternative:
The equation whose roots are numerically equal but opposite in sign to the roots of 3x² − 5x − 7 = 0 is

Choose the correct alternative:
If 8 and 2 are the roots of x² + ax + c = 0 and 3, 3 are the roots of x² + dx + b = 0, then the roots of the equation x² + ax + b = 0 are

Choose the correct alternative:
If a and b are the real roots of the equation x² − kx + c = 0, then the distance between the points (a, 0) and (b, 0) is

Choose the correct alternative:
If `("k"x)/((x + 2)(x - 1)) = 2/(x + 2) + 1/(x - 1)`, then the value of k is

Choose the correct alternative:
If `(1 - 2x)/(3 + 2x - x^2) = "A"/(3 - x) + "B"/(x + 1)`, then the value of A + B is

Choose the correct alternative:
The number of roots of (x + 3)⁴ + (x + 5)⁴ = 16 is

Choose the correct alternative:
The value of log₃ 11 . log₁₁ 13 . log₁₃ 15 . log₁₅ 27 . log₂₇ 81 is