Samacheer Kalvi solutions for Mathematics [English] Class 10 SSLC TN Board chapter 3 - Algebra [Latest edition]

Solve the following system of linear equations in three variables

x + y + z = 5; 2x – y + z = 9; x – 2y + 3z = 16

Solve the following system of linear equations in three variables

`1/x - 2/y + 4 = 0; 1/y - 1/z + 1 = 0; 2/z + 3/x = 14`

Solve the following system of linear equations in three variables

x + 20 = `(3y)/2 + 10` = 2z + 5 = 110 – (y + z)

Discuss the nature of solution of the following system of equation.

x + 2y – z = 6; –3x – 2y + 5z = –12; x – 2z = 3

Discuss the nature of solutions of the following system of equations

2y + z = 3(– x + 1); – x + 3y – z = – 4; 3x + 2y + z = `-1/2`

Discuss the nature of solutions of the following system of equations

`(y + z)/4 = (z + x)/3 = (x + y)/2`; x + y + z = 27

Vani, her father and her grandfather have an average age of 53. One-half of her grandfather’s age plus one-third of her father’s age plus one-fourth of Vani’s age is 65. Four years ago if Vani’s grandfather was four times as old as Vani then how old are they all now?

The sum of the digits of a three-digit number is 11. If the digits are reversed, the new number is 46 more than five times the former number. If the hundreds digit plus twice the tens digit is equal to the units digit, then find the original three-digit number?

There are 12 pieces of five, ten and twenty rupee currencies whose total value is ₹ 105. When first 2 sorts are interchanged in their numbers its value will be increased by ₹ 20. Find the number of currencies in each sort

Find the G.C.D. of the given polynomials

x⁴ + 3x³ – x – 3, x³ + x² – 5x + 3

Find the G.C.D. of the given polynomials

x⁴ – 1, x³ – 11x² + x – 11

Find the G.C.D. of the given polynomials

3x⁴ + 6x³ – 12x² – 24x, 4x⁴ + 14x³ + 8x² – 8x

Find the G.C.D. of the given polynomials

3x³ + 3x² + 3x + 3, 6x³ + 12x² + 6x + 12

Find the L.C.M. of the given expressions

4x²y, 8x³y²

Find the L.C.M. of the given expressions

– 9a³b², 12a²b²c

Find the L.C.M. of the given expressions

16m, – 12m²n², 8n²

Find the L.C.M. of the given expressions

p² – 3p + 2, p² – 4

Find the L.C.M. of the given expressions

2x² – 5x – 3, 4x² – 36

Find the L.C.M. of the given expressions

(2x² – 3xy)², (4x – 6y)³, (8x³ – 27y³)

Find the LCM and GCD for the following and verify that f(x) × g(x) = LCM × GCD

21x²y, 35xy²

Find the LCM and GCD for the following and verify that f(x) × g(x) = LCM × GCD

(x³ – 1) (x + 1), (x³ + 1)

Find the LCM and GCD for the following and verify that f(x) × g(x) = LCM × GCD

(x²y + xy²), (x² + xy)

Find the LCM pair of the following polynomials

 a² + 4a – 12, a² – 5a + 6 whose GCD is a – 2

Find the LCM pair of the following polynomials

x⁴ – 27a³x, (x – 3a)² whose GCD is (x – 3a)

Find the GCD pair of the following polynomials

12(x⁴ – x³), 8(x⁴ – 3x³ + 2x²) whose LCM is 24x³ (x – 1)(x – 2)

Find the GCD pair of the following polynomials

(x³ + y³), (x⁴ + x²y² + y⁴) whose LCM is (x³ + y³) (x² + xy + y²)

Given the LCM and GCD of the two polynomials p(x) and q(x) find the unknown polynomial in the following table

Given the LCM and GCD of the two polynomials p(x) and q(x) find the unknown polynomial in the following table

Reduce the following rational expression to its lowest form

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

Reduce the following rational expression to its lowest form

`(x^2 - 11x + 18)/(x^2 - 4x + 4)`

Reduce the following rational expression to its lowest form

`(9x^2 + 81x)/(x^3 + 8x^2 - 9x)`

Reduce the following rational expression to its lowest form

`("p"^2 - 3"p" - 40)/(2"p"^3 - 24"p"^2 + 64"p")`

Find the excluded values, of the following expression

`y/(y^2 - 25)`

Find the excluded values, of the following expression

`"t"/("t"^2 - 5"t" + 6)`

Find the excluded values, of the following expression

`(x^2 + 6x + 8)/(x^2 + x - 2)`

Find the excluded values, of the following expression

`(x^3 - 27)/(x^3 + x^2 - 6x)`

Simplify `(4x^2y)/(2z^2) xx (6xz^3)/(20y^4)`

Simplify `("p"^2 - 10"p" + 21)/("p" - 7) xx ("p"^2 + "p" - 12)/("p" - 3)^2`

Simplify `(5"t"^3)/(4"t" - 8) xx (6"t" - 12)/(10"t")`

Simplify `(x + 4)/(3x + 4y) xx (9x^2 - 16y^2)/(2x^2 + 3x - 20)`

Simplify `(x^3 - y^3)/(3x^2 + 9xy + 6y^2) xx (x^2 + 2xy + y^2)/(x^2 - y^2)`

Simplify `(2"a"^2 + 5"a" + 3)/(2"a"^2 + 7"a" + 6) ÷ ("a"^2 + 6"a" + 5)/(-5"a"^2 - 35"a" - 50)`

Simplify `("b"^2 + 3"b" - 28)/("b"^2 + 4"b" + 4) ÷ ("b"^2 - 49)/("b"^2 - 5"b" - 14)`

Simplify `(x + 2)/(4"y") ÷ (x^2 - x - 6)/(12y^2)`

Simplify `(12"t"^2 - 22"t" + 8)/(3"t") ÷ (3"t"^2 + 2"t" - 8)/(2"t"^2 + 4"t")`

If x = `("a"^2 + 3"a" - 4)/(3"a"^2 - 3)` and y = `("a"^2 + 2"a" - 8)/(2"a"^2 - 2"a" - 4)` find the value of x²y^–2  

If a polynomial p(x) = x² – 5x – 14 is divided by another polynomial q(x) we get `(x - 7)/(x + 2)`, find q(x)

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

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

Simplify `(x^3)/(x - y) + (y^3)/(y - x)`

Simplify `((2x + 1)(x - 2))/(x - 4) - ((2x^2 - 5x + 2))/(x - 4)`

Simplify `(4x)/(x^2 - 1) - (x + 1)/(x - 1)`

Subtract `1/(x^2 + 2)` from `(2x^3 + x^2 + 3)/(x^2 + 2)^2`

Identify rational expression should be subtracted from `(x^2 + 6x + 8)/(x^3 + 8)` to get `3/(x^2 - 2x + 4)`

If A = `(2x + 1)/(2x - 1)`, B = `(2x - 1)/(2x + 1)` find `1/("A" - "B") - (2"B")/("A"^2 - "B"^2)`

If A = `x/(x + 1)` B = `1/(x + 1)` prove that `(("A" + "B")^2 + ("A" - "B")^2)/("A" + "B") = (2(x^2 + 1))/(x(x + 1)^2`

Pari needs 4 hours to complete the work. His friend Yuvan needs 6 hours to complete the same work. How long will it take to complete if they work together?

Iniya bought 50 kg of fruits consisting of apples and bananas. She paid twice as much per kg for the apple as she did for the banana. If Iniya bought ₹ 1800 worth of apples and ₹ 600 worth bananas, then how many kgs of each fruit did she buy?

Find the square root of the following rational expression

`(400x^4y^12z^16)/(100x^8y^4z^4)`

Find the square root of the following rational expression

`(7x^2 + 2sqrt(14)x + 2)/(x^2 - 1/2 x + 1/16)`

Find the square root of the following rational expression

`(121("a" + "b")^8 (x + y)^8 ("b" - "c")^8)/(81("b" - "c")^4 ("a" - "b")^12 ("b" - "c")^4)`

Find the square root of the following

4x² + 20x + 25

Find the square root of the following

9x² – 24xy + 30xz – 40yz + 25z² + 16y² 

Find the square root of the following

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

Find the square root of the following

(4x² – 9x + 2)(7x² – 13x – 2)(28x² – 3x – 1)

Find the square root of the following

`(2x^2 + 17/6 x + 1) (3/2 x^2 + 4x + 2) (4/3 x^2 + 11/3 x + 2)`

Find the square root of the following polynomials by division method

x⁴ – 12x³ + 42x² – 36x + 9

Find the square root of the following polynomials by division method

37x² – 28x³ + 4x⁴ + 42x + 9

Find the square root of the following polynomials by division method

16x⁴ + 8x² + 1

Find the square root of the following polynomials by division method

121x⁴ – 198x³ – 183x² + 216x + 144

Find the values of a and b if the following polynomial is a perfect square

4x⁴ – 12x³ + 37x² + bx + a

Find the values of a and b if the following polynomial is a perfect square

ax⁴ + bx³ + 361x² + 220x + 100

Find the values of m and n if the following polynomial is a perfect square

`1/(x^4) - 6/(x^3) + 13/(x^2) + "m"/x + "n"`

Find the values of m and n if the following polynomial is a perfect square

 x⁴ – 8x³ + mx² + nx + 16

Determine the quadratic equation, whose sum and product of roots are – 9, 20

Determine the quadratic equation, whose sum and product of roots are `5/3, 4`

Determine the quadratic equation, whose sum and product of roots are `(-3)/2`, – 1

Determine the quadratic equation, whose sum and product of roots are – (2 – a)², (a + 5)²

Find the sum and product of the roots for the following quadratic equation

x² + 3x – 28 = 0

Find the sum and product of the roots for the following quadratic equation

x² + 3x = 0

Find the sum and product of the roots for the following quadratic equation

`3 + 1/"a" = 10/"a"^2`

Find the sum and product of the roots for the following quadratic equation

3y² – y – 4 = 0

Solve the following quadratic equation by factorization method

4x² – 7x – 2 = 0

Solve the following quadratic equation by factorization method

3(p² – 6) = p(p + 5)

Solve the following quadratic equation by factorization method

`sqrt("a"("a" - 7)) = 3sqrt(2)`

Solve the following quadratic equation by factorization method

`sqrt(2)x^2 + 7x + 5sqrt(2)` = 0

Solve the following quadratic equation by factorization method

`2x^2 - x + 1/8` = 0

The number of volleyball games that must be scheduled in a league with n teams is given by G(n) = `("n"^2 - "n")/2` where each team plays with every other team exactly once. A league schedules 15 games. How many teams are in the league?

Solve the following quadratic equation by completing the square method

9x² – 12x + 4 = 0

Solve the following quadratic equation by completing the square method

`(5x + 7)/(x - 1)` = 3x + 2

Solve the following quadratic equation by formula method

2x² – 5x + 2 = 0

Solve the following quadratic equation by formula method

`sqrt(2)"f"^2 - 6"f" + 3sqrt(2)` = 0

Solve the following quadratic equation by formula method

3y² – 20y – 23 = 0

Solve the following quadratic equation by formula method

36y² – 12ay + (a² – b²) = 0

A ball rolls down a slope and travels a distance d = t² – 0.75t feet in t seconds. Find the time when the distance travelled by the ball is 11.25 feet

If the difference between a number and its reciprocal is `24/5`, find the number

A garden measuring 12m by 16m is to have a pedestrian pathway that is ‘ω’ meters wide installed all the way around so that it increases the total area to 285 m². What is the width of the pathway?

A bus covers a distance of 90 km at a uniform speed. Had the speed been `(15"km")/"hour"` more it would have taken 30 minutes less for the journey. Find the original speed of the bus

A girl is twice as old as her sister. Five years hence, the product of their ages (in years) will be 375. Find their present ages.

A pole has to be erected at a point on the boundary of a circular ground of diameter 20 m in such a way that the difference of its distances from two diametrically opposite fixed gates P and Q on the boundary is 4 m. Is it possible to do so? If answer is yes at what distance from the two gates should the pole be erected?

From a group of 2x² black bees, square root of half of the group went to a tree. Again eight-ninth of the bees went to the same tree. The remaining two got caught up in a fragrant lotus. How many bees were there in total?

Music is been played in two opposite galleries with certain group of people. In the first gallery a group of 4 singers were singing and in the second gallery 9 singers were singing. The two galleries are separated by the distance of 70 m. Where should a person stand for hearing the same intensity of the singers voice?

(Hint: The ratio of the sound intensity is equal to the square of the ratio of their corresponding distances)

There is a square field whose side is 10 m. A square flower bed is prepared in its centre leaving a gravel path all round the flower bed. The total cost of laying the flower bed and gravelling the path at ₹ 3 and ₹ 4 per square metre respectively is ₹ 364. Find the width of the gravel path

The hypotenuse of a right angled triangle is 25 cm and its perimeter 56 cm. Find the length of the smallest side

Determine the nature of the roots for the following quadratic equation

15x² + 11x + 2 = 0

Determine the nature of the roots for the following quadratic equation

x² – x – 1 = 0

Determine the nature of the roots for the following quadratic equation

`sqrt(2)"t"^2 - 3"t" + 3sqrt(2)` = 0

Determine the nature of the roots for the following quadratic equation

`9y^2 - 6sqrt(2)y + 2` = 0

Determine the nature of the roots for the following quadratic equation

9a²b²x² – 24abcdx + 16c²d² = 0, a ≠ 0, b ≠ 0

Find the value of ‘k’ to identify the roots of the following equation is real and equal

(5k – 6)x² + 2kx + 1 = 0

Find the value of ‘k’ to identify the roots of the following equation is real and equal

kx² + (6k + 2)x + 16 = 0

If the roots of (a – b)x² + (b – c)x + (c – a) = 0 are real and equal, then prove that b, a, c are in arithmetic progression

If a, b are real then show that the roots of the equation (a – b)x² – 6(a + b)x – 9(a – b) = 0 are real and unequal

If the roots of the equation (c² – ab)x² – 2(a² – bc)x + b² – ac = 0 are real and equal prove that either a = 0 (or) a³ + b³ + c³ = 3abc

Write the following expression in terms of α + β and αβ

`alpha/(3beta) + beta/(3alpha)`

Write the following expression in terms of α + β and αβ

`1/(alpha^2beta) + 1/(beta^2alpha)`

Write the following expression in terms of α + β and αβ

(3α – 1) (3β – 1)

Write the following expression in terms of α + β and αβ

`(alpha + 3)/beta + (beta + 3)/alpha`

The roots of the equation 2x² – 7x + 5 = 0 are α and β. Without solving for the roots, find `1/alpha + 1/beta`

The roots of the equation 2x² – 7x + 5 = 0 are α and β. Without solving for the roots, find `alpha/beta + beta/alpha`

The roots of the equation 2x² – 7x + 5 = 0 are α and β. Without solving for the roots, find `(alpha + 2)/(beta + 2) + (beta + 2)/(alpha + 2)`

The roots of the equation x² + 6x – 4 = 0 are α, β. Find the quadratic equation whose roots are α² and β²

The roots of the equation x² + 6x – 4 = 0 are α, β. Find the quadratic equation whose roots are `2/alpha` and `2/beta`

The roots of the equation x² + 6x – 4 = 0 are α, β. Find the quadratic equation whose roots are α²β and β²α

If α, β are the roots of 7x² + ax + 2 = 0 and if β – α = `(-13)/7` Find the values of a

If one root of the equation 2y² – ay + 64 = 0 is twice the other then find the values of a

If one root of the equation 3x² + kx + 81 = 0 (having real roots) is the square of the other then find k

Graph the following quadratic equation and state their nature of solution

x² – 9x + 20 = 0

Graph the following quadratic equation and state their nature of solution

x² – 4x + 4 = 0

Graph the following quadratic equation and state their nature of solution

x² + x + 7 = 0

Graph the following quadratic equation and state their nature of solution

x² – 9 = 0

Graph the following quadratic equation and state their nature of solution

x² – 6x + 9 = 0

Graph the following quadratic equation and state their nature of solution

(2x – 3) (x + 2) = 0

Draw the graph of y = x² – 4 and hence solve x² – x – 12 = 0

Draw the graph of y = x² + x and hence solve x² + 1 = 0

Draw the graph of y = x² + 3x + 2 and use it to solve x² + 2x + 1 = 0

Draw the graph of y = x² + 3x – 4 and hence use it to solve x² + 3x – 4 = 0

Draw the graph of y = x² – 5x – 6 and hence solve x² – 5x – 14 = 0

Draw the graph of y = 2x² – 3x – 5 and hence solve 2x² – 4x – 6 = 0

Draw the graph of y = (x – 1) (x + 3) and hence solve x² – x – 6 = 0

In the matrix A = `[(8, 9, 4, 3),(- 1, sqrt(7), sqrt(3)/2, 5),(1, 4, 3, 0),(6, 8, -11, 1)]`, write The number of elements

In the matrix A = `[(8, 9, 4, 3),(- 1, sqrt(7), sqrt(3)/2, 5),(1, 4, 3, 0),(6, 8, -11, 1)]`, write The order of the matrix

In the matrix A = `[(8, 9, 4, 3),(- 1, sqrt(7), sqrt(3)/2, 5),(1, 4, 3, 0),(6, 8, -11, 1)]`, Write the elements a₂₂, a₂₃, a₂₄, a₃₄, a₄₃, a₄₄

If a matrix has 18 elements, what are the possible orders it can have? What if it has 6 elements?

Construct a 3 × 3 matrix whose elements are given by a_ij = |i – 2j|

Construct a 3 × 3 matrix whose elements are given by a_ij = `("i" + "j")^3/3`

If A = `[(5, 4, 3),(1, -7, 9),(3, 8, 2)]` then find the transpose of A

If A = `[(sqrt(7), - 3),(- sqrt(5), 2),(sqrt(3), -5)]` then find the transpose of – A

If A = `[(5, 2, 2),(-sqrt(17), 0.7, 5/2),(8, 3, 1)]` then verify (A^T)^T = A

Find the values of x, y and z from the following equation

`[(12, 3),(x, 3/2)] = [(y, z),(3, 5)]`

Find the values of x, y and z from the following equation

`[(x + y, 2),(5 + z, xy)] = [(6, 2),(5, 8)]`

Find the values of x, y and z from the following equation

`[(x + y + z),(x + z),(y + z)] = [(9),(5),(7)]`

If A = `[(1, 9),(3, 4),(8, -3)]`, B = `[(5, 7),(3, 3),(1, 0)]` then verify that A + B = B + A

If A = `[(1, 9),(3, 4),(8, -3)]`, B = `[(5, 7),(3, 3),(1, 0)]` then verify that A + (– A) = (– A) + A = 0

If A = `[(4, 3, 1),(2, 3, -8),(1, 0, -4)]`, B = `[(2, 3, 4),(1, 9, 2),(-7, 1, -1)]` and C = `[(8, 3, 4),(1, -2, 3),(2, 4, -1)]` then verify that A + (B + C) = (A + B) + C

Find X and Y if X + Y = `[(7, 0),(3, 5)]` and X – Y = `[(3, 0),(0, 4)]`

If A = `[(0, 4, 9),(8, 3, 7)]`, B = `[(7, 3, 8),(1, 4, 9)]` find the value of B – 5A

If A = `[(0, 4, 9),(8, 3, 7)]`, B = `[(7, 3, 8),(1, 4, 9)]` find the value of 3A – 9B

Find the values of x, y, z if `[(x - 3, 3x - z),(x + y + 7, x + y + z)] = [(1, 0),(1, 6)]`

Find the values of x, y, z if `[(x), (y  – z), (z + 3)] + [(y), (4), (3)] = [(4), (8), (16)]`

Find x and y if `x[(4),(-3)] + y[(-2),(3)] = [(4),(6)]`

Find the non-zero values of x satisfying the matrix equation

`x[(2x, 2),(3, x)] + 2[(8, 5x),(4, 4x)] = 2[(x^2 + 8, 24),(10, 6x)]`

Solve for x, y : `[(x^2),(y^2)] + 2[(-2x),(-y)] = [(5),(8)]`

Find the order of the product matrix AB if

If A is of order p × q and B is of order q × r what is the order of AB and BA?

A has ‘a’ rows and ‘a + 3’ columns. B has ‘b’ rows and ‘17 − b’ columns, and if both products AB and BA exist, find a, b?

If A = `[(2, 5),(4, 3)]`, B = `[(1, -3),(2, 5)]` find AB, BA and verify AB = BA?

Given that A = `[(1, 3),(5, -1)]`, B = `[(1, -1, 2),(3, 5, 2)]`, C = `[(1, 3, 2),(-4, 1, 3)]` verify that A(B + C) = AB + AC

Show that the matrices A = `[(1, 2),(3, 1)]`, B = `[(1, -2),(-3, 1)]` satisfy commutative property AB = BA

Let A = `[(1, 2),(1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, 2)]` Show that A(BC) = (AB)C

Let A = `[(1, 2),(1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, 2)]` Show that (A – B)C = AC – BC

Let A = `[(1, 2),(1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, 2)]` Show that (A – B)^T = A^T – B^T

If A = `[(costheta, theta),(0, costheta)]`, B = `[(sintheta, 0),(0, sintheta)]` then show that A² + B² = I

If A = `[(costheta, sintheta),(-sintheta, costheta)]` prove that AA^T = I

Verify that A² = I when A = `[(5, -4),(6, -5)]`

If A = `[("a", "b"),("c", "d")]` and I = `[(1, 0),(0, 1)]` show that A² – (a + d)A = (bc – ad)I₂

If A = `[(5, 2, 9),(1, 2, 8)]`, B = `[(1, 7),(1, 2),(5, -1)]` verify that (AB)^T = B^T A^T

If A = `[(3, 1),(-1, 2)]` show that A² – 5A + 7I₂ = 0

A system of three linear equation in three variables is inconsistent if their planes

The solution of the system x + y – 3z = – 6, – 7y + 7z = 7, 3z = 9 is

If (x – 6) is the HCF of x² – 2x – 24 and x² – kx – 6 then the value of k is 

`(3y - 3)/y ÷ (7y - 7)/(3y^2)` is

`y^2 + 1/y^2` is not equal to

`x/(x^2 - 25) - 8/(x^2 + 6x + 5)` gives

The square root of `(256x^8y^4z^10)/(25x^6y^6z^6)` is equal to

Which of the following should be added to make x⁴ + 64 a perfect square

The solution of (2x – 1)² = 9 is equal to

The values of a and b if 4x⁴ – 24x³ + 76x² + ax + b is a perfect square are 

If the roots of the equation q²x² + p²x + r² = 0 are the squares of the roots of the equation qx² + px + r = 0, then q, p, r are in __________

Graph of a linear equation is a _______

The number of points of intersection of the quadratic polynomial x² + 4x + 4 with the X axis is

For the given matrix A = `[(1, 3, 5, 7),(2, 4, 6, 8),(9, 11, 13, 15)]` the order of the matrix A^T is 

If A is a 2 × 3 matrix and B is a 3 × 4 matrix, how many columns does AB have

If the number of columns and rows are not equal in a matrix then it is said to be a

Transpose of a column matrix is

Find the matrix X if 2X + `[(1, 3),(5, 7)] = [(5, 7),(9, 5)]`

Which of the following can be calculated from the given matrices A = `[(1, 2),(3, 4),(5, 6)]`, B = `[(1, 2, 3),(4, 5, 6),(7, 8, 9)]`,
(i) A²
(ii) B²
(iii) AB
(iv) BA

If A = `[(1, 2, 3),(3, 2, 1)]`, B = `[(1, 0),(2, -1),(0, 2)]` and C = `[(0, 1),(-2, 5)]` Which of the following statements are correct?

(i) AB + C = `[(5, 5),(5, 5)]`

(ii) BC = `[(0, 1),(2, -3),(-4, 10)]`

(iii) BA + C = `[(2, 5),(3, 0)]`

(iv) (AB)C = `[(-8, 20),(-8, 13)]`

Solve `1/3` (x + y – 5) = y – z = 2x – 11 = 9 – (x + 2z)

One hundred and fifty students are admitted to a school. They are distributed over three sections A, B and C. If 6 students are shifted from section A to section C, the sections will have equal number of students. If 4 times of students of section C exceeds the number of students of section A by the number of students in section B, find the number of students in the three sections

In a three-digit number, when the tens and the hundreds digit are interchanged the new number is 54 more than three times the original number. If 198 is added to the number, the digits are reversed. The tens digit exceeds the hundreds digit by twice as that of the tens digit exceeds the unit digit. Find the original number

Find the least common multiple of xy(k² + 1) + k(x² + y²) and xy(k² – 1) + k(x² – y²)

Find the GCD of the following by division algorithm

2x⁴ + 13x³ + 27x² + 23x + 7, x³ + 3x² + 3x + 1, x² + 2x + 1

Reduce the given Rational expression to its lowest form

`(x^(3"a") - 8)/(x^(2"a") + 2x^"a" + 4)`

Reduce the given Rational expression to its lowest form

`(10x^3 - 25x^2 + 4x - 10)/(-4 - 10x^2)`

Simplify `(1/("p") + 1/("q" + "r"))/(1/"p" - 1/("q" + "r")) xx [1 + ("q"^2 + "r"^2 - "p"^2)/(2"qr")]`

Arul, Madan and Ram working together can clean a store in 6 hours. Working alone, Madan takes twice as long to clean the store as Arul does. Ram needs three times as long as Arul does. How long would it take each if they are working alone?

Find the square root of 289x⁴ – 612x³ + 970x² – 684x + 361

Solve `sqrt(y + 1) + sqrt(2y - 5)` = 3

A boat takes 1.6 hours longer to go 36 kms up a river than down the river. If the speed of the water current is 4 km per hr, what is the speed of the boat in still water?

Is it possible to design a rectangular park of perimeter 320 m and area 4800 m²? If so find its length and breadth.

At t minutes past 2 pm, the time needed to 3 pm is 3 minutes less than `("t"^2)/4`. Find t.

The number of seats in a row is equal to the total number of rows in a hall. The total number of seats in the hall will increase by 375 if the number of rows is doubled and the number of seats in each row is reduced by 5. Find the number of rows in the hall at the beginning.

If α and β are the roots of the polynomial f(x) = x² – 2x + 3, find the polynomial whose roots are α + 2, β + 2

If α and β are the roots of the polynomial f(x) = x² – 2x + 3, find the polynomial whose roots are `(alpha - 1)/(alpha + 1), (beta - 1)/(beta + 1)`

If – 4 is a root of the equation x² + px – 4 = 0 and if the equation x² + px + q = 0 has equal roots, find the values of p and q.

Two farmers Thilagan and Kausigan cultivates three varieties of grains namely rice, wheat and ragi. If the sale (in ₹) of three varieties of grains by both the farmers in the month of April is given by the matrix.

          `{:("April sale in"  ₹)/("rice"   "wheat"   "ragi"):}`

A = `[(500, 1000, 1500),(2500, 1500, 500)]"Thilagan"/"Kausigan"`

and the May month sale (in ₹) is exactly twice as that of the April month sale for each variety.

What is the average sales for the months of April and May

Two farmers Thilagan and Kausigan cultivates three varieties of grains namely rice, wheat and ragi. If the sale (in ₹) of three varieties of grains by both the farmers in the month of April is given by the matrix.

          `{:"April sale in"  ₹)/("rice"   "wheat"   "ragi":}`

A = `[(500, 1000, 1500),(2500, 1500, 500)]"Thilagan"/"Kausigan"`

and the May month sale (in ₹) is exactly twice as that of the April month sale for each variety.

If the sales continue to increase in the same way in the successive months, what will be sales in the month of August?

If `cos theta [(cos theta, sin theta),(-sin theta, cos theta)] + sin theta[(x, -cos theta),(cos theta, x)]` = I₂, find x.

Given A = `[("p", 0),(0, 2)]`, B = `[(0, -"q"),(1, 0)]`, C = `[(2, -2),(2, 2)]` and if BA = C², find p and q.

A = `[(3, 0),(4, 5)]`, B = `[(6, 3),(8, 5)]`, C = `[(3, 6),(1, 1)]` find the matrix D, such that CD – AB = 0