Quadratic Formula

Q.

Find a quadratic polynomial whose zeroes are $-4$ and $-5$.

Q. Using quadratic formula, find the value of x.
$p^2{x}^2+(p^2-q^2)x-q^2=0,p\ne 0$

1. $\frac{q^2}{p^2},-1$
2. $-\frac{q^2}{p^2},-1$
3. $0,-1$
4. $\frac{q^2}{p^2},1$

Q. Find the roots of the quadratic equation $4x^2+4\sqrt{3}x+3=0$ by using quadratic formula.

1. $-\frac{3}{2},\frac{3}{2}$
2. $-\frac{3}{2},-\frac{3}{2}$
3. $-\frac{\sqrt{3}}{2}$, $+\frac{\sqrt{3}}{2}$
4. $-\frac{\sqrt{3}}{2}$, $-\frac{\sqrt{3}}{2}$

Q.

Let $\lambda \ne 0$be in R. If $\alpha ,\beta$are the roots of the equation, $x^2-x+2\lambda =0$ and $\alpha ,\gamma$ are the roots of the equation, $3x^2-10x+27\lambda =0$, then $\frac{\beta \gamma }{\lambda }$ is equal to:

1. $27$

2. $9$

3. $18$

4. $36$

Q.

Two water taps together can fill a tank in $9\frac{3}{8}hours$. The tap of larger diameter takes $10hrs$ less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Q. Find the zeros of the polynomial $f\left(x\right)=x^3-5x^2-16x+80$, if its two zeros are equal in magnitude but opposite in sign. [4 MARKS]

Q.

Factorise: $2x^2+7x+3$

Q.

The zeroes of the polynomial
$P\left(x\right)=a{x}^2+bx+c$ are 1 and 2.
Find the value of $a,b\text{and}c$.

1. a = 3, b = -1, c = 1

2. a = -1 , b = -3, c = -2

3. a = 1, b = -3 , c = 1

4. a = 1 , b = -3, c = 2

Q.

Three positive numbers are in the ratio $\frac{1}{2}:\frac{1}{3}:\frac{1}{4}$. Find the numbers if the sum iof their squares is 244.

Q.

Check whether the following are quadratic equations: ${(\mathrm{x}+1)}^2=2(\mathrm{x}-3)$

Q.

Solve each of the following quadratic equations:
$x^2+6x-(a^2+2a-8)=0$

Q. The sum of two numbers is 18. The sum of their reciprocals is 1/4. Find the numbers.

Q.

Solve :

$3x^2-6x+2=0$

Q. In the quadratic equation p(x)=0 with real coefficients has purely imaginary roots. Then the equation p[p(x)]=0 has

1. only purely imaginary roots
2. all real roots
3. two real and two purely imaginary roots
4. neither real nor purely imaginary roots

Q.

Solve the equation $x^2–2x–3=0$ using quadratic formula.

1. both of them are integers

2. one of the root is zero.

3. roots are not real

4. both of them are natural numbers

Q. Find the roots of the quadratic equation $2x^2-7x+3=0$ using quadratic formula. Select all the correct options.

1. $x=\frac{3}{2}$
2. $x=\frac{1}{2}$
3. $x=3$
4. $x=2$

Q. The sum of a number and its reciprocal is 10/ 3. Find the number.

Q.

Find the roots of the quadratic equation $6x^2-13x+6=0$ by using the quadratic formula.

1. $\frac{2}{3},\frac{7}{2}$

2. $\frac{2}{5},\frac{3}{2}$

3. $\frac{2}{3},\frac{3}{2}$

4. $\frac{4}{3},\frac{3}{2}$

Q.

Solve the equation $5x^2-3x-4=0$ and give your answer to correct two decimals.

Q.

Let $S$ be the set of all real roots of the equation, $3^x\left(3^x-1\right)+2=\left|3^x-1\right|+\left|3^x-2\right|$. Then $S$:

1. is a singleton

2. is an empty set

3. Contains at least four elements

4. Contains exactly two elements

Q. The sum of a natural number and its reciprocal is $\frac{65}{8}$. Find the number.

Q.

The number of real solutions of the equation $\sin \left(e^x\right)=5^x+5^{-x}$ is

1. $1$

2. $0$

3. $2$

4. $3$

Q.

Question 1(i)
Evaluate:

$3^{-2}$

Q. Rs 480 is divided equally among 'x' children. If the number of children were 20 more, then each would have got Rs 12 less. Find 'x'.
[4 MARKS]

Q.

Solve each of the following equation:

$(x^2+5x+4)(x^2+5x+6)=120$

Q.

Solve the following quadratic equation by factorization.
$x-\frac{1}{x}=3,x\ne 0$

Q. The sum of a number and its reciprocal is 17/4. Find the number.

Q.

Question 61

Solve the following:

5(x-1)-2(x+8)=0

Q.

If the sum of the roots of the equation $a{x}^2+bx+c=0$ is equal to the sum of the squares of their reciprocals then $b{c}^2,c{a}^2,a{b}^2$are in

1. AP

2. GP

3. HP

4. AGP

Q. Find whether the following equation have real roots. If real roots exist, find them
(i) $8x^2+2x-3=0$
(ii) $-2x^2+3x+2=0$
(iii) $5x^2-2x-10=0$
(iv) $\frac{1}{2x-3}+\frac{1}{x-5}=1,x\ne \frac{3}{2},5$
(v) $x^2+5\sqrt{5}x-70=0$