Quadratic Equation

Q.

Question 4
Classify the following numbers as rational or irrational with justification.
$\left(i\right)\sqrt{196}$
(ii)$3\sqrt{18}$
(iii)$\sqrt{\frac{9}{27}}$
(iv)$\frac{\sqrt{28}}{343}$
(v)$-\sqrt{0.4}$
(vi)$\frac{\sqrt{12}}{\sqrt{75}}$
(vii) 0.5918
(viii) $(1+\sqrt{5})-(4+\sqrt{5})$
(ix) 10.124124 ....
(x) 1.010010001....
To classify, use the definition a rational number is in the form of $\frac{p}{q}$, where p and q are integers and $q\ne 0$ and otherwise it is an irrational number.

Q.

What is the meaning of real and distinct roots, and real and equal roots?

Q.

Find the sum of the cubes of the first $n$ natural numbers.

1. $[\frac{n(n+1)}{2}{]}^2$

2. $n^2(n+1{)}^2$

3. $\left(n\right)(n+1)(2n+1)$

4. None of these

Q. The speed of an express train is x km/h and the speed of an ordinary train is 12 km/h less than that of the express train. If the ordinary train takes one hour longer than the express train to cover a distance of 240 km, find the speed of the express train. [4 MARKS]

Q. Question 8
A rational number between $\sqrt{2}$ and $\sqrt{3}$ is

A) $\frac{\sqrt{2}+\sqrt{3}}{2}$
B) $\frac{\sqrt{2}.\sqrt{3}}{2}$
C) 1.5
D) 1.8

Q. Let $(x_0,y_0)$ be the solution of the following equations $(2x{)}^{ln2}=(3y{)}^{ln3}$ , $3^{lnx}=2^{lny}$ then $x_0$ is

1. $\frac{1}{6}$
2. 6
3. $\frac{1}{3}$
4. $\frac{1}{2}$

Q.

Condition for a$x^2$ + bx + c = 0 to be a quadratic equation is

1. a≠ 0; a, b, c are real.

2. a≥ 0; a, b, c are real

3. a< 0; a, b, c are unreal

4. a≤ 0; a, b, c are real

Q. Why a quadratic equation is in form- xsquare -(alpha +beta)x +alphabeta

Q.

Write the general form of a quadratic polynomial

1. ax² + bx + c or bx + ax²+ c or c+ bx + ax²

2. ax² + bx + c where a, b and c are real numbers

3. ax² + bx + c = 0

4. ax² + bx + c where a, b and c are real numbers and a is not equal to zero

Q.

Check whether the following are quadratic equations ${\mathrm{x}}^2-2\mathrm{x}=(-2)(3-\mathrm{x})$

Q. if the quadratic expression px^2+qx+r has the same sign as that of p , then find the relation among p, q and r.

Q.

If α and β are zeroes of the polynomial x² ​​​​​​ - a(x + 1) - b such that (α + 1) ( β + 1) = 0, find the value of b.

Q.

ax^2 +bx +c=k(x-alpha)(x-beta)

What does this equation mean why k has been taken as a constant and for more details alpha and beta are the zeroes of the polynomial p(x)=ax^2 +bx+c

Q. The discriminant of the quadratic equation $3x^2-4x-2=0$ is

1. 20
2. 40
3. 48

Q. Find the sum and product of the zeros of 2(x^{​​​​​​2}​​​​​-x)+1

Q.

The difference of squares of two numbers is $180$. The square of the smaller number is $8$ times the larger number. Find the two numbers.

Q.

_________ is called the discriminant of quadratic equation ax^2 + bx + c = 0.

1. a^2 - 4ac
2. b^2 - ac
3. b^2 + 4bc
4. b^2 - 4ac

Q. Write the standard form of a quadratic polynomial with real coefficients.

Q.

The sum of the squares of two natural numbers is $52$. If the first number is $8$ less than twice the second number, find the numbers.

Q. A motor boat whose speed is 18 km/hr in still water takes 1 hr more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
OR
A train travels at a certain average speed for a distance of 63 km and then travels at a distance of 72 km at an average speed of 6 km/ hr more than its original speed. If it takes 3 hours to complete total journey, what is the original average speed ?

Q. Write any two quadratic equations.

Q. Check whether the following are quadratic equation:
$(x-3)(2x+1)=x(x+5)$

Q.

If - √5 and √5 are the roots of the quadratic polynomial. Find the quadratic polynomial.

1. x-5

2. (x-5)(x+5)

3. x² - 25

4. x² - 5

Q.

H.C.F of $x^2+5x+6$ and $x^3+27$ is:

1. $x+2$

2. $x-2$

3. $x-3$

4. $x+3$

Q. Check whether the following are quadratic equation:
$x^2+2x+1=(4–x{)}^2+3$

Q. Which of the following are quadratic equations?

(i) x² + 6x − 4 = 0
(ii) $\sqrt{3}{x}^2-2x+\frac{1}{2}=0$
(iii) $x^2+\frac{1}{x^2}=5$
(iv) $x-\frac{3}{x}=x^2$
(v) $2x^2-\sqrt{3x}+9=0$
(vi) $x^2-2x-\sqrt{x}-5=0$
(vii) 3x² − 5x + 9 = x² − 7x + 3
(viii) $x+\frac{1}{x}=1$
(ix) x² − 3x = 0
(x) ${\left(x+\frac{1}{x}\right)}^2=3\left(x+\frac{1}{x}\right)+4$
(xi) (2x + 1) (3x + 2) = 6(x − 1) (x − 2)
(xii) $x+\frac{1}{x}=x^2,x\ne 0$
(xiii) 16x² − 3 = (2x + 5) (5x − 3)
(xiv) (x + 2)³ = x³ − 4
(xv) x(x + 1) + 8 = (x + 2) (x − 2)

Q. The quadratic polynomial p(x) has the following properties: p(x) can be positive or zero for all real numbers, p(1)=0 and p(2)=2. Then find the quadratic polynomial

Q.

Sir, I am not able to understand difference between word problems of quadratic equation and linear equation with two varuables. Please let me understand this.

Q.

Factorise the given quadratic equation

$x^2+5x+6=0.$

1. $(x+2)(x+3)$

2. $(x-2)(x-3)$

3. $(x-1)(x-3)$

4. $(x+1)(x+3)$

Q.

If $x=1$and $y=6$ is a solution of $8x-ky+k^2=0$, find the values of $k$.