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Find the zeroes of the quadratic polynomial 4s2 – 4s + 1 and verify the relationship between the zeroes and the coefficients.
Concept: Relation Between Zeroes (Roots) and Coefficients of a Quadratic Equation
If α and β are the zeroes of the polynomial x2 + x − 2, then find the value of `alpha/beta+beta/alpha`
Concept: Geometrical Meaning of the Zeroes of a Polynomial
Form the pair of linear equation in the following problem, and find its solution (if they exist) by the elimination method:
If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes `1/2` if we only add 1 to the denominator. What is the fraction?
Concept: Algebraic Methods of Solving a Pair of Linear Equations >> Elimination Method
Form the pair of linear equation in the following problem, and find its solution (if they exist) by the elimination method:
A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs 27 for a book kept for seven days, while Susy paid Rs 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.
Concept: Algebraic Methods of Solving a Pair of Linear Equations >> Elimination Method
The sum of a two-digit number and the number formed by reversing the order of digit is 66. If the two digits differ by 2, find the number. How many such numbers are there?
Concept: Algebraic Methods of Solving a Pair of Linear Equations >> Elimination Method
A fraction becomes `1/3` when 2 is subtracted from the numerator and it becomes `1/2` when 1 is subtracted from the denominator. Find the fraction.
Concept: Algebraic Methods of Solving a Pair of Linear Equations >> Elimination Method
Rehana went to a bank to withdraw ₹ 2000. She asked the cashier to give her ₹ 50 and ₹ 100 notes only. Rehana got 25 notes in all. Find how many notes of ₹ 50 and ₹ 100 did she received.
Concept: Algebraic Methods of Solving a Pair of Linear Equations >> Elimination Method
Solve for x : ` 2x^2+6sqrt3x-60=0`
Concept: Nature of Roots of a Quadratic Equation
Find the values of k for which the quadratic equation 9x2 - 3kx + k = 0 has equal roots.
Concept: Nature of Roots of a Quadratic Equation
If -5 is a root of the quadratic equation 2x2 + px – 15 = 0 and the quadratic equation p(x2 + x)k = 0 has equal roots, find the value of k.
Concept: Nature of Roots of a Quadratic Equation
Solve for x: `1/(x+1)+2/(x+2)=4/(x+4), `x ≠ -1, -2, -3
Concept: Nature of Roots of a Quadratic Equation
Find the values of k for which the quadratic equation (k + 4) x2 + (k + 1) x + 1 = 0 has equal roots. Also find these roots.
Concept: Nature of Roots of a Quadratic Equation
Find that value of p for which the quadratic equation (p + 1)x2 – 6(p + 1)x + 3(p + 9) = 0, p ≠ –1 has equal roots. Hence, find the roots of the equation.
Concept: Nature of Roots of a Quadratic Equation
Solve the following quadratic equation for x :
9x2 − 6b2x − (a4 − b4) = 0
Concept: Nature of Roots of a Quadratic Equation
Find the values of k for which the quadratic equation (3k + 1) x2 + 2(k + 1) x + 1 = 0 has equal roots. Also, find the roots.
Concept: Nature of Roots of a Quadratic Equation
Find the value of p for which the quadratic equation (2p + 1)x2 − (7p + 2)x + (7p − 3) = 0 has equal roots. Also find these roots.
Concept: Nature of Roots of a Quadratic Equation
If ad ≠ bc, then prove that the equation (a2 + b2)x2 + 2(ac + bd)x + (c2 + d2) = 0 has no real roots.
Concept: Nature of Roots of a Quadratic Equation
Find the positive value(s) of k for which quadratic equations x2 + kx + 64 = 0 and x2 – 8x + k = 0 both will have real roots ?
Concept: Nature of Roots of a Quadratic Equation
Solve for x :
x2 + 5x − (a2 + a − 6) = 0
Concept: Nature of Roots of a Quadratic Equation
If x = −2 is a root of the equation 3x2 + 7x + p = 1, find the values of p. Now find the value of k so that the roots of the equation x2 + k(4x + k − 1) + p = 0 are equal.
Concept: Nature of Roots of a Quadratic Equation
