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#### Question 1:

Which of the following are quadratic equation in x ?
(i) ${x}^{2}-x+3=0$
(ii) $2{x}^{2}+\frac{5}{2}x-\sqrt{3}=0$
(iii) $\sqrt{2}{x}^{2}+7x+5\sqrt{2}=0$
(iv) $\frac{1}{3}{x}^{2}+\frac{1}{5}x-2=0$
(v) ${x}^{2}-3x-\sqrt{x}+4=0$
(vi) $x-\frac{6}{x}=3$
(vii) $x+\frac{2}{x}={x}^{2}$
(viii) ${x}^{2}-\frac{1}{{x}^{2}}=5$
(ix) ${\left(x+2\right)}^{3}={x}^{3}-8$
(x) $\left(2x+3\right)\left(3x+2\right)=6\left(x-1\right)\left(x-2\right)$
(xi) ${\left(x+\frac{1}{x}\right)}^{2}=2\left(x+\frac{1}{x}\right)+3$

(ix) ${\left(x+2\right)}^{3}={x}^{3}-8$
$⇒{x}^{3}+6{x}^{2}+12x+8={x}^{3}-8\phantom{\rule{0ex}{0ex}}⇒6{x}^{2}+12x+16=0$
This is of the form ax2 + bx + c = 0.
Hence, the given equation is a quadratic equation.
(x) $\left(2x+3\right)\left(3x+2\right)=6\left(x-1\right)\left(x-2\right)$
$⇒6{x}^{2}+4x+9x+6=6\left({x}^{2}-3x+2\right)\phantom{\rule{0ex}{0ex}}⇒6{x}^{2}+13x+6=6{x}^{2}-18x+12\phantom{\rule{0ex}{0ex}}⇒31x-6=0$
This is not of the form ax2 + bx + c = 0.
Hence, the given equation is not a quadratic equation.
(xi) ${\left(x+\frac{1}{x}\right)}^{2}=2\left(x+\frac{1}{x}\right)+3$
$⇒{\left(\frac{{x}^{2}+1}{x}\right)}^{2}=2\left(\frac{{x}^{2}+1}{x}\right)+3\phantom{\rule{0ex}{0ex}}⇒{\left({x}^{2}+1\right)}^{2}=2x\left({x}^{2}+1\right)+3{x}^{2}\phantom{\rule{0ex}{0ex}}⇒{x}^{4}+2{x}^{2}+1=2{x}^{3}+2x+3{x}^{2}\phantom{\rule{0ex}{0ex}}⇒{x}^{4}-2{x}^{3}-{x}^{2}-2x+1=0$
This is not of the form ax2 + bx + c = 0.
Hence, the given equation is not a quadratic equation.

#### Question 2:

Which of the following are the roots of $3{x}^{2}+2x-1=0$ ?
(i) −1
(ii) $\frac{1}{3}$
(iii) $-\frac{1}{2}$

#### Question 3:

Find the value of k for which x = 1 is root of the equation ${x}^{2}+kx+3=0$.

#### Question 4:

Find the values of a and b for which are the roots of the equation $a{x}^{2}+bx-6=0.$

#### Question 5:

$\left(3x-5\right)\left(2x+3\right)=0$

#### Question 6:

$5{x}^{2}+4x=0$

#### Question 7:

$3{x}^{2}-243=0$

#### Question 8:

${x}^{2}+12x+35=0$

#### Question 9:

${x}^{2}=18x-77$

#### Question 10:

$9{x}^{2}+6x+1=0$

#### Question 11:

$4{x}^{2}-12x+9=0$

#### Question 12:

$6{x}^{2}+11x+3=0$

#### Question 13:

$6{x}^{2}+x-12=0$

#### Question 14:

$3{x}^{2}-2x-1=0$

#### Question 15:

$6{x}^{2}-x-2=0$

#### Question 16:

$48{x}^{2}-13x-1=0$

#### Question 17:

$3{x}^{2}+11x+10=0$

#### Question 18:

$4{x}^{2}-9x=100$

#### Question 19:

$9{x}^{2}-22x+8=0$

#### Question 20:

$15{x}^{2}-28=x$

#### Question 21:

$4-11x=3{x}^{2}\phantom{\rule{0ex}{0ex}}$

#### Question 22:

$4\sqrt{3}{x}^{2}+5x-2\sqrt{3}=0$

#### Question 23:

$\sqrt{3}{x}^{2}+11x+6\sqrt{3}=0$

#### Question 24:

$4\sqrt{3}{x}^{2}+5x-2\sqrt{3}=0$

#### Question 25:

$3\sqrt{7}{x}^{2}+4x-\sqrt{7}=0$

#### Question 26:

$\sqrt{7}{y}^{2}-6y-13\sqrt{7}=0$

#### Question 27:

$4\sqrt{6}{x}^{2}-13x-2\sqrt{6}=0$

#### Question 29:

$10x-\frac{1}{3}=3$

#### Question 30:

$\frac{2}{{x}^{2}}-\frac{5}{x}+2=0$

#### Question 31:

$ab{x}^{2}+\left({b}^{2}-ac\right)x-bc=0$

#### Question 32:

${a}^{2}{b}^{2}{x}^{2}+{b}^{2}x-{a}^{2}x-1=0$

#### Question 33:

$12ab{x}^{2}-\left(9{a}^{2}-8{b}^{2}\right)x-6ab=0$

#### Question 34:

$4{x}^{2}-2\left({a}^{2}+{b}^{2}\right)x+{a}^{2}{b}^{2}=0$

#### Question 36:

$\frac{1}{\left(x-3\right)}-\frac{1}{\left(x+5\right)}=\frac{1}{6},\left(x\ne 3,-5\right)$

#### Question 48:

${3}^{\left(x+2\right)}+{3}^{-x}=10$

#### Question 49:

${4}^{\left(\mathrm{x}+1\right)}+{4}^{\left(1-x\right)}=10$

#### Question 50:

${2}^{2x}-3.{2}^{\left(x+2\right)}+32=0$

#### Question 1:

$2{x}^{2}-7x+6=0$

#### Question 2:

$3{x}^{2}-2x+8=0$

#### Question 3:

$2{x}^{2}-5\sqrt{2}x+4=0$

#### Question 4:

$\sqrt{3}{x}^{2}+2\sqrt{2}x-2\sqrt{3}=0$

#### Question 5:

$1-x=2{x}^{2}$

#### Question 6:

${x}^{2}=4x-c$

#### Question 7:

$6{x}^{2}+7x-10=0$

#### Question 8:

$2{x}^{2}-9x+7=0$

#### Question 9:

$2{x}^{2}+x-6=0$

#### Question 10:

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

${x}^{2}-4x-1=0$

#### Question 11:

${x}^{2}-6x+4=0$

#### Question 12:

${x}^{2}-7x-5=0$

#### Question 13:

$5{x}^{2}-19x+17=0$

#### Question 14:

$3{x}^{2}-32x+12=0$

#### Question 15:

$25{x}^{2}+30x+7=0$

#### Question 16:

$15{x}^{2}-28=x$

#### Question 17:

$4-11x=3{x}^{2}$

#### Question 18:

$16{x}^{2}=24x+1$

#### Question 19:

$3{x}^{2}+2\sqrt{5}x-5=0$

#### Question 20:

$2{x}^{2}-2\sqrt{6}x+3=0$

#### Question 21:

$\sqrt{3}{x}^{2}+10x-8\sqrt{3}=0$

#### Question 22:

$9{x}^{2}-4=0$

#### Question 24:

${p}^{2}{x}^{2}+\left({p}^{2}-{q}^{2}\right)x-{q}^{2}=0$

#### Question 25:

${x}^{2}-2ax+\left({a}^{2}-{b}^{2}\right)=0$

#### Question 26:

$ab{x}^{2}+\left({b}^{2}-ac\right)x-bc=0$

#### Question 27:

$12ab{x}^{2}-\left(9{a}^{2}-8{b}^{2}\right)x-6ab=0$, where a$\ne$0 and b$\ne$0

#### Question 1:

${x}^{2}+8x+16=0$

#### Question 2:

${x}^{2}-6x+6=0$

#### Question 3:

$9{x}^{2}-6x+4=0$

#### Question 4:

$9{x}_{{}^{2}}-6x+4=0$

#### Question 5:

$3{x}^{2}-2\sqrt{6}x+2=0$

#### Question 6:

$12{x}^{2}-4\sqrt{15}x+5=0$

#### Question 7:

Show that the roots of the equation ${x}^{2}+px-{q}^{2}=0$ are real for all real value of p and q.

#### Question 8:

Show that the equation ${x}^{2}+ax-1=0$ has real and distinct roots for all real values of a.

#### Question 9:

Show that the equation ${x}^{2}-x+2=0$ has no real roots.

#### Question 10:

Find the values of k for which the equation $k{x}^{2}+2x+1=0$ has real and distinct roots.

#### Question 11:

Determine the values of p for which the quadratic equation $2{x}^{2}+px+8=0$ has real roots.

#### Question 12:

Find the value of α for which the equation $\left(\alpha -12\right){x}^{2}+2\left(\alpha -12\right)x+2=0$ has equal roots.

#### Question 13:

If the equation $\left(1+{m}^{2}\right){x}^{2}+2mcx+\left({c}^{2}-{a}^{2}\right)=0$ has equal roots, prove that ${c}^{2}={a}^{2}\left(1+{m}^{2}\right).$

#### Question 14:

If the roots of the equation $\left({c}^{2}-ab\right){x}^{2}-2\left({a}^{2}-bc\right)x+\left({b}^{2}-ac\right)=0$ are real and equal, show that either

#### Question 15:

Find the values k for which of roots of $9{x}^{2}+8kx+16=0$ are real and equal

#### Question 16:

Find the values of k for which the roots of the equation $\left(k+4\right){x}^{2}+\left(k+1\right)x+1=0$ are real and equal?

#### Question 17:

For what values of k are the roots of the quadratic equation $3{x}^{2}+2kx+27=0$ real and equal?

#### Question 1:

The sum of two numbers is 8. Determine the numbers, if the sum of their reciprocals is $\frac{8}{15}.$

#### Question 2:

The difference of two numbers is 4. If the difference of their reciprocals is $\frac{4}{21}$, find the number.

#### Question 3:

The sum of two numbers is 18 and the sum of their reciprocals is $\frac{1}{4}$. Find the numbers.

#### Question 4:

The difference of two numbers is 5 and the difference of their reciprocals is $\left(\frac{1}{10}\right)$. Find the numbers.

#### Question 5:

The sum of a number and its reciprocal is $3\frac{41}{80}$. Find the number.

#### Question 6:

Divide 57 into two parts whose product is 782.

#### Question 7:

Find two consecutive positive multiples of 3 whose product is 270.

#### Question 8:

Find two consecutive positive even integers, the sum of whose squares is 340.

#### Question 9:

The sum of a number and its squares is $\frac{63}{4}$. Find the number.

#### Question 10:

The sum of a number and its positive square root is $\frac{6}{25}$. Find the number.

#### Question 11:

Two natural numbers differ by 3 and their product is 504. Find the numbers.

#### Question 12:

Find two consecutive positive integers, the sum of whose squares is 365.

#### Question 13:

The difference of the squares of two natural numbers is 45. The squares of the smaller number is four times the largest number. Find the numbers.

#### Question 14:

Find two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference.

#### Question 15:

Divide 16 into two parts such that twice the squares of the larger part exceeds the square of the smaller part by 164.

#### Question 16:

The denominator of a fraction is 3 more than its numerator. The sum of the fraction and its reciprocal is $2\frac{9}{10}$. Find the fraction.

#### Question 17:

A two-digit number is 4 times the sum of its digits and twice the product of its digit. Find the number.

#### Question 18:

A two-digit number is such that the product of its digits is 14. If 45 is added to the number, the digits interchange their places. Find the number.

​​

#### Question 19:

Out of a number of saras birds, one-fourth of the number are moving about in lots, $\frac{1}{9}\mathrm{th}$ coupled with $\frac{1}{4}\mathrm{th}$ as well as 7 times the square root of the number move on a hill, 56 birds remain in vakula trees. What is the total number of birds?

#### Question 20:

A teacher on attempting to arrange the students for mass drill in the form of a solid square found that 24 students were left. When he increased the size of the square by one student, he found that he was short of 25 students. Find the number of students.

#### Question 21:

300 apples are distributed equally among a certain number of students. Had there been 10 more students, each would have received one apple less. Find the number of student.

#### Question 22:

A man busy a number of pens for Rs 80. If he had bought 4 more pens for the same amount, each pen would have cost him Rs 1 less. How many pens did he buy?

#### Question 23:

In a class test, the sum of Kamal's marks in Mathematics and English is 40. Had he got 3 marks more in Mathematics and 4 marks less in English, the product of the marks would have been 360. Find his marks in two subjects separately.

#### Question 24:

A takes 10 days than the time taken by B to finish a piece of work. If both A and B together can finish the work in 12 days, find the time taken by B to finish the work.

#### Question 25:

A passenger train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km/hr from its usual speed. Find its usual speed.

#### Question 26:

A train travels a distance of 360 km at a uniform speed. If the speed of he train is increased by 5 km an hour, the journey would have taken 1 hour less. Find the original speed of the train.

#### Question 27:

A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hr more, it would have taken 30 minutes less for the journey. Find the original speed of the train.

#### Question 28:

The distance between Mumbai and Pune is 192 km. Travelling by the Deccan Queen, it takes 48 minutes less than another train. Calculate the speed of the Deccan Queen if the speed of the two trains differ by 20 km/hr.

#### Question 29:

A motorboat whose speed is 9 km/hr in still water, goes 15 km downstream and comes back in a total time of 3 hours 45 minutes. Find the speed of the stream.

#### Question 30:

A motor boat whose speed in still water is 18 km/hr, takes 1 hour more to go 24 km upstream than o return to the same spot. Find the speed of the stream.

#### Question 31:

The speed of a boat in still water is 8 km/hr. It can go 15 km upstream and 22 km downstream in 5 hours. Find the speed of the stream.

#### Question 32:

A sailor can row a boat 8 km downstream and return back to the starting point in 1 hour 40 minutes. If the speed of the stream is 2 kmph, find the speed of the boat in still water.

#### Question 33:

Two pipes running together can fill a cistern in $3\frac{1}{13}$ minutes. If one pipe takes 3 minutes more than the other to fill it, find the time in which each pipe would fill the cistern.

#### Question 34:

One year ago, a man was 8 times old as his son. Now his age is equal to the square of his son's age. Find their present ages.

#### Question 35:

The sum of the ages of a man and his son is 45 years. Five years ago, the product of their ages was four times the man's age at that time. Find their present ages.

#### Question 36:

The product of Tanvy's age (in years) 5 years ago and her age 8 years later is 30. Find her present age.

#### Question 37:

The sum of the ages of a boy and his brother is 25 years, and the product of their ages in years is 126. Find their ages.

#### Question 38:

A rectangular field is 16 m long and 10 m wide. There is a path of uniform width all around it, having an area of 120 m2. Find the width of the path.

#### Question 39:

The length of a rectangle is twice its breadth and its area is 288 cm2. Find the dimensions of the rectangle.

#### Question 40:

The length of a rectangular field is three times its breadth. If the area of the field by 147 sq metres, find the length of the field.

#### Question 41:

The length of a hall is 3 metres more than its breadth. If the area of the hall is 238 square metres, calculate its length and breadth.

#### Question 42:

The perimeter of a rectangular plot is 62 m and its area is 228 sq metres. Find the dimensions of the plot.

#### Question 43:

The length of a rectangle is thrice as long as the side of a square. The side of the square is 4 cm more than width of a the rectangle. Their areas being equal, find their dimensions.

#### Question 44:

A farmer prepares a rectangular vegetable garden of area 180 sq metres. With 39 metres of barbed wire, he can fence the three sides of the garden, leaving one of the longer sides unfenced. Find the dimensions of the garden.

#### Question 45:

The area of a right-triangle is  600 cm2. If the base of the triangle exceeds the altitude by 10 cm, find the dimensions of the triangle.

Let the altitude of the triangle be x cm.
Therefore, the base of the triangle will be (+ 10) cm.

#### Question 46:

The area of a right-angled triangle is 96 sq metres. If the base is three times the altitude, find the base.

Let the altitude of the triangle be x m.
Therefore, the base will be 3x m.

Area of a triangle =

The value of cannot be negative.
Therefore, the altitude and base of the triangle are 8 m and (3 $×$ 8 = 24 m), respectively.

#### Question 47:

The area of a right-angled triangle is 165 sq metres. Determine its base and altitude if the latter exceeds the former by 7 metres.

Let the base be $x$ m.
Therefore, the altitude will be m.

Area of a triangle =

The value of $x$ cannot be negative.
Therefore, the base is 15 m and the altitude is {(15 + 7) = 22 m}.

#### Question 48:

The hypotenuse of a right-angled triangle is 20 metres. If the difference between the length of the other sides be 4 metres, find the other sides.

Let one side of the right-angled triangle be $x$ m and the other side be  m.
On applying Pythagoras theorem, we have:

The value of x cannot be negative.
Therefore, the base is 12 m and the other side is {(12 + 4) = 16 m}.

#### Question 49:

The length of the hypotenuse of a right-angled triangle exceeds the length of the base by 2 cm and exceeds twice the length of the altitude by 1 cm. Find the length of each side of the triangle.

Let the base and altitude of the right-angled triangle be $x$ and y cm, respectively.
Therefore, the hypotenuse will be  cm.

Thus, the base, altitude and hypotenuse of the triangle are 15 cm, 8 cm and 17 cm, respectively.

#### Question 50:

The hypotenuse of a right-angled triangle is 1 metre less than twice the shortest side. If the third is 1 metre more than the shortest side, find the sides of the triangle.

Let the shortest side be $x$ m.
Therefore, according to the question:
Hypotenuse =
Third side =  m
On applying Pythagoras theorem, we get:

The length of the side cannot be 0; therefore, the shortest side is 3 m.
Therefore,
Hypotenuse = 5 m
Third side = (3 + 1) = 4 m

#### Question 51:

The sum of the areas of two squares is 640 m2. If the difference in their perimeters be 64 m, find the sides of the two squares.

Let the length of the side of the first and the second square be $x$ and y, respectively.

Putting the value of $x$ in (i), we get:

#### Question 52:

The difference of squares of two numbers is 88. If the larger number is 5 less than twice the smaller number, then find the two numbers.

#### Question 53:

Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46, find the integers.

#### Question 1:

Which of the following is a quadratic equation?
(a) ${x}^{2}-3\sqrt{x}+2=0$
(b) $x+\frac{1}{x}={x}^{2}$
(c) ${x}^{2}+\frac{1}{{x}^{2}}=5$
(d)

(d)

#### Question 2:

Which of the following is a quadratic equation?
(a) $\left({x}^{2}+1\right)=\left(2-x{\right)}^{2}+3$
(b) ${x}^{3}-{x}^{2}=\left(x-1{\right)}^{3}$
(c) $2{x}^{2}+3=\left(5+x\right)\left(2x-3\right)$
(d) None of these

(b)

#### Question 3:

Which of the following is not a quadratic equation?
(a) $3x-{x}^{2}={x}^{2}+5$
(b) $\left(x+2{\right)}^{2}=2\left({x}^{2}-5\right)$
(c) $\left(\sqrt{2}x+3{\right)}^{2}=2{x}^{2}+6$
(d) $\left(x-1{\right)}^{2}=3{x}^{2}+x-2$

(c)

#### Question 4:

If x = 3 is a solution of the equation $3{x}^{2}+\left(k-1\right)x+9=0,$ then k = ?
(a) 11
(b) −11
(c) 13
(d) −13

(b) −11

#### Question 5:

The sum of the roots of the equation ${x}^{2}-6x+2=0$ is
(a) 2
(b) −2
(c) 6
(d) −6

(c) 6

#### Question 6:

If the product of the roots of the equation ${x}^{2}-3x+k=10$ is −2, then the value of k is
(a) −2
(b) −8
(c)  8
(d) 12

(c)  8

#### Question 7:

If one root of the equation $2{x}^{2}+ax+6=0$ is 2, then a = ?
(a) 7
(b) −7
(c) $\frac{7}{2}$
(d) $\frac{-7}{2}$

(b) −7

#### Question 8:

The ratio of the sum and product of the roots of the equation $7{x}^{2}-12x+18=0$
(a) 7 : 12
(b) 7 : 18
(c) 2 : 3
(d) 3 : 2

(c) 2 : 3

#### Question 9:

The roots of the equation $4\sqrt{3}{x}^{2}+5x-2\sqrt{3}=0$ are
(a) $\frac{2\sqrt{3}}{3},\frac{-\sqrt{3}}{4}$
(b) $\frac{-2\sqrt{3}}{3},\frac{\sqrt{3}}{4}$
(c) $\frac{\sqrt{3}}{3},\frac{-\sqrt{3}}{4}$
(d) $\frac{-\sqrt{3}}{3},\frac{\sqrt{3}}{4}$

(b)

#### Question 10:

The root of the equation $\sqrt{2}{x}^{2}+7x+5\sqrt{2}=0$ are
(a) $\sqrt{2},\frac{5\sqrt{2}}{2}$
(b) $-\sqrt{2},\frac{5\sqrt{2}}{2}$
(c) $\sqrt{2},\frac{-5\sqrt{2}}{2}$
(d) $-\sqrt{2},\frac{-5\sqrt{2}}{2}$

(d)

#### Question 11:

The roots of the equation ${3}^{x+2}+{3}^{-x}$ = 10 are
(a) 2, 0
(b) −2, 0
(c) 3, −1
(d) −3, 1

(b) −2, 0

#### Question 12:

The roots of the equation $3{x}^{2}-2\sqrt{6}x+2=0$ are
(a) $\sqrt{\frac{3}{2}},\sqrt{\frac{3}{2}}$
(b) $\sqrt{\frac{2}{3}},\sqrt{\frac{2}{3}}$
(c) $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$
(d) $\frac{2}{\sqrt{3}},\frac{2}{\sqrt{3}}$

(b)

#### Question 13:

The roots of the equation $\frac{x}{x+1}+\frac{x+1}{x}=\frac{34}{15}$ are
(a) $\frac{5}{2},\frac{3}{2}$
(b) $\frac{5}{2},\frac{-3}{2}$
(c) $\frac{-5}{2},\frac{3}{2}$
(d) $\frac{-5}{2},\frac{-3}{2}$

(c)

#### Question 14:

The roots of the equation $\sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}=2\frac{1}{6}$ are
(a) $\frac{9}{13},\frac{7}{13}$
(b) $\frac{9}{13},\frac{4}{13}$
(c) $\frac{7}{13},\frac{4}{13}$
(d) none of these

(b)

#### Question 15:

The roots of $\frac{x+4}{x-4}+\frac{x-4}{x+4}=\frac{10}{3}$ are
(a) ±4
(b) ±6
(c) ±8
(d) $2±\sqrt{3}$

(c) ±8

#### Question 16:

The root of a quadratic equation are 5 and −2. Then, the equation is
(a) ${x}^{2}-3x+10=0$
(b) ${x}^{2}-3x-10=0$
(c) ${x}^{2}+3x-10=0$
(d) ${x}^{2}+3x+10=0$

(b)

#### Question 17:

If the sum of the roots of a quadratic equation is 6 and their product is 6, the equation is
(a) ${x}^{2}-6x+6=0$
(b) ${x}^{2}+6x-6=0$
(c) ${x}^{2}-6x-6=0$
(d) ${x}^{2}+6x+6=0$

(a)

#### Question 18:

If one root of the equation  then the other root is
(a) $\frac{-1}{3}$
(b) $\frac{1}{3}$
(c) −3
(d) 3

(d) 3

#### Question 19:

The quadratic equation whose one root is $\left(3+2\sqrt{3}\right)$ is
(a) ${x}^{2}+6x-3=0$
(b) ${x}^{2}-6x-3=0$
(c) ${x}^{2}+6x+3=0$
(d) ${x}^{2}-6x+3=0$

(b)

#### Question 20:

If the sum of the roots of the equation $k{x}^{2}+2x+3x=0$ is equal to their product, then the value of k is
(a) $\frac{1}{3}$
(b) $\frac{-1}{3}$
(c) $\frac{2}{3}$
(d) $\frac{-2}{3}$

(d) $\frac{-2}{3}$

#### Question 21:

If one root of $5{x}^{2}+13x+k=0$ be the reciprocal of the other root, then the value of k is
(a) 0
(b) 1
(c) 2
(d) 5

(d) 5

#### Question 22:

The roots of the equation $a{x}^{2}+bx+c=0$ will be reciprocal of each other if
(a) a = b
(b) b= c
(c) c = a
(d) none of these

(c) c  =  a

#### Question 23:

If the roots of the equation $a{x}^{2}+bx+c=0$ are equal, then then c = ?
(a) $\frac{-b}{2a}$
(b) $\frac{b}{2a}$
(c) $\frac{-{b}^{2}}{4a}$
(d) $\frac{{b}^{2}}{4a}$

(d) $\frac{{b}^{2}}{4a}$

#### Question 24:

If the equation  has equal roots, then k = ?
(a) 2 or 0
(b) −2 or 0
(c) 2 or −2
(d) 0 only

(c) 2 or −2

#### Question 25:

If the equation ${x}^{2}+2\left(k+2\right)x+9k=0$ has equal rots, then k = ?
(a) 1 or 4
(b) −1 or 4
(c) 1 or −4
(d) −1 or −4

(a) 1 or 4

#### Question 26:

If the equation $4{x}^{2}-3kx+1=0$ has equal roots, then k = ?
(a) $±\frac{2}{3}$
(b) $±\frac{1}{3}$
(c) $±\frac{3}{4}$
(d) $±\frac{4}{3}$

(d) $±\frac{4}{3}$

#### Question 27:

If the equation ${x}^{2}-2x\left(1+3k\right)+7\left(3+2k\right)=0$ has equal roots, then k = ?
(a)
(b)
(c)
(d)

(c)

#### Question 28:

If the equation $\left({a}^{2}+{b}^{2}\right){x}^{2}-2b\left(a+c\right)x+\left({b}^{2}+{c}^{2}\right)=0$ has both roots equal, then
(a) b = ac
(b) $b=\frac{1}{2}\left(a+c\right)$
(c) b2 = ac
(d) $b=\frac{2ac}{\left(a+c\right)}$

(c) b2 = ac

#### Question 29:

The roots of  are real and unequal, if $\left({b}^{2}-4ac\right)$
(a) > 0
(b) = 0
(c) < 0
(d) none of these

(a) >  0

#### Question 30:

In the equation $a{x}^{2}+bx+c=0,$ it is given that $D=\left({b}^{2}-4ac\right)>0.$ Then, the roots of the equation are
(a) real and equal
(b) real and unequal
(c) imaginary
(d) none of these

(b) real and unequal

#### Question 31:

If the equation ${x}^{2}+5kx+16=0$ has no real roots, then
(a) $k>\frac{8}{5}$
(b) $k<\frac{-8}{5}$
(c) $\frac{-8}{5}\frac{8}{5}$
(d) none of these

(c)

#### Question 32:

If the equation ${x}^{2}-kx+1=0$ has no real roots, then
(a) k < −2
(b) k > 2
(c) −2 < k < 2
(d) none of these

(c) −2  <   <  2

#### Question 33:

The roots of the equation $2{x}^{2}-6x+7=0$ are
(a) real, unequal and rational
(b) real, unequal and irrational
(c) real and equal
(d) imaginary

(d) imaginary

#### Question 34:

The roots of the equation $2{x}^{2}-6x+3=0$ are
(a) real, unequal and rational
(b) real, unequal and irrational
(c) real and equal
(d) imaginary

(b) real, unequal and irrational

#### Question 35:

For the equation $a{x}^{2}+bx+c=0$ which of the following statements is incorrect?
(a) If (b2 − 4ac) < 0, the roots are imaginary.
(b) If (b2 − 4ac) = 0, the roots are real and equal.
(c) If (b2 − 4ac) > 0 and (b2 − 4ac) is a perfect square, then the roots are rational and unequal.
(d) (b2 − 4ac) < 0, the roots are irrational.

(d) (b2 − 4ac)  <  0, the roots are irrational.

#### Question 36:

If the roots of $5{x}^{2}-kx+1=0$ are real and distinct, then
(a) $-2\sqrt{5}
(b) $k>2\sqrt{5}$ only
(c) $k<-2\sqrt{5}$ only
(d) either $k>2\sqrt{5}$ or $k<-2\sqrt{5}$

(d) either or

#### Question 37:

The root of the equation $3{x}^{2}+7x+8=0$ are
(a) both real and equal
(b) both real and unequal
(c) both imaginary
(d) none of these

(c) both imaginary

#### Question 38:

The sum of a number and its reciprocal is $2\frac{1}{20}.$ The number is
(a) $\frac{5}{4}$
(b) $\frac{4}{3}$
(c) $\frac{3}{4}$
(d) $\frac{1}{6}$

(a) $\frac{5}{4}$

#### Question 39:

The two parts into which 57 should be divided so that their product is 782, are
(a) 43 and 14
(b) 33 and 24
(c) 34 and 23
(d) 44 and 13

(c) 34 and 23

#### Question 40:

The perimeter of a rectangle is 82 m and its area is 400 m2. The breadth of the rectangle is
(a) 25 m
(b) 20 m
(c) 16 m
(d) 9 m

(c) 16 m

#### Question 41:

Which constant should be added and subtracted to solve the quadratic equation $4{x}^{2}-\sqrt{3}x-5=0$ by the method of completing the square?
(a) $\frac{9}{10}$
(b) $\frac{3}{16}$
(c) $\frac{3}{4}$
(d) $\frac{\sqrt{3}}{4}$

(b) $\frac{3}{16}$

#### Question 42:

The roots of the equation $2x-\frac{3}{x}=1$ are
(a) $\frac{1}{2},-1$
(b) $\frac{3}{2},1$
(c) $\frac{3}{2},-1$
(d) $\frac{-1}{2},\frac{3}{2}$

(c)

#### Question 43:

For what real values of k, the equation $9{x}^{2}+8kx+16=0$ has real and equal roots?
(a) k = 2 or −2
(b) k = 3 or −3
(c) $k=\frac{4}{3}\mathrm{or}\frac{-4}{3}$
(d) None of these

(b) k = 3 or −3

#### Question 44:

For what values of k, the equation $k{x}^{2}-6x-2=0$ has real roots?
(a) $k\le \frac{-9}{2}$
(b) $k\ge \frac{-9}{2}$
(c) $k\le -2$
(d) None of these

(b)

#### Question 45:

If α and β are the roots of the equation
(a) $\frac{-3}{8}$
(b) $\frac{2}{3}$
(c) −4
(d) 4

(c) −4

#### Question 46:

Which of the following is not true?
(a) Every quadratic equation can have at the most two real roots.
(b) Some quadratic equations do not have any real root.
(c) Some quadratic equations may have one real root.
(d) Every quadratic equations has two real roots.

#### Question 47:

Consider the following statements:
I. If the roots of the equation $a{x}^{2}+bx+c=0$ are negative reciprocal of each other, than a + c = 0.
II. A quadratic equation can have at all most two roots.
III. If α, β are the roots of ${x}^{2}-22x+105=0,$ then α + β = 22 and α − β = 8.

Of these statements:
(a) I and II are true and III is false.
(b) I and III are true and II is false.
(c) II and III are true I is false.
(d) I, II and III are all true.

(d) I, II and III are all true.

#### Question 48:

Assertion (A)
The equation x2 + x + 1 = 0 has both real roots.

Reason (R)
The equation ax2 + bx + c = 0, (a ≠ 0) has both real roots, if (b2 − 4ac) > 0.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

(d) Assertion (A) is false and Reason (R) is true.

#### Question 49:

Assertion (A)
The equation $2{x}^{2}-4x+3=0$ has no real roots.

Reason (R)
The equation  has no real roots, if $\left({b}^{2}-4ac\right)<0.$

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

#### Question 50:

Assertion (A)
If −5 is a root of  has equal roots, then $k=\frac{7}{8}.$

Reason (R)
The equation  has equal roots, if $\left({b}^{2}-4ac\right)=0.$

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

#### Question 51:

Assertion (A)
The roots of

Reason (R)
If

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

Thus, Assertion (A) is true and Reason (R) is a correct explanation of Assertion (A).

#### Question 52:

Assertion (A)
If $p{x}^{2}-2x+2=0$ has real roots, then $p\le \frac{1}{2}.$

Reason (R)
The equation $\left({a}^{2}{+}^{b}\right){x}^{2}+2\left(ac+bd\right)x+\left({c}^{2}+{d}^{2}\right)=0$ has no real root, if adbc.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

#### Question 53:

Match the following columns:

 Column I Column II (a) $2{x}^{2}-7x+5=0$ (p) Real and equal (b) $3{x}^{2}-2\sqrt{6}x+2=0$ (q) Real, distinct and irrational (c) $9{x}^{2}-10x+15=0$ (r) Real distinct and rational (d) $\sqrt{3}{x}^{2}+2\sqrt{2}x-2\sqrt{3}=0$ (s) Imaginary

#### Question 1:

If ${x}^{2}-4x+p=0$ has real roots, then
(a) p ≥ 4
(b) p ≤ 4
(c) p ≥ 5
(d) p ≤ 5

(b) p ≤ 4

#### Question 2:

If x = 1 is a common root of
(a) 3
(b) −3
(c) 4
(d) 6

(a) 3

#### Question 3:

If ${x}^{2}+2\left(k+2\right)x+9k=0$ has a repeated root, then k = ?
(a) 1 or 4
(b) −1 or 4
(c) 1 or −4
(d) −1 or −4

(a) 1 or 4

#### Question 4:

If ${x}^{2}+5px+16=0$ has no real root, then
(a) $p>\frac{8}{5}$
(b) $p<\frac{-8}{5}$
(c) $\frac{-8}{5}
(d) none of these

(c)

#### Question 5:

For what values of k, the roots the equation ${x}^{2}-kx+1=0$ are imaginary?

#### Question 6:

For what values of k, the roots of the equation ${x}^{2}+4k+k=0$ are real?

For real roots, the value of k must be lesser than or equal to 4.

#### Question 7:

For what values of k, the roots of the equation $2{x}^{2}+kx+k=0$ are real?

#### Question 8:

Solve: $2{x}^{2}-6x+3=0.$

#### Question 9:

Show that the equation $2{x}^{2}-5x-4=0$ has real and unequal roots.

#### Question 10:

Show that the equation ${x}^{2}-x+2=0$ has no real roots.

#### Question 11:

Solve: $10x-\frac{1}{x}=3.$

#### Question 12:

Solve: $\sqrt{3}{x}^{2}+11x+6\sqrt{3}=0$

#### Question 13:

Solve: $3{x}^{2}+2\sqrt{5}x-5=0.$

#### Question 14:

Find two consecutive positive integers, the sum of whose squares is 25.

#### Question 15:

Solve: ${x}^{2}-4ax+4{a}^{2}-{b}^{2}=0.$

#### Question 16:

Using quadratic formula, solve $ab{x}^{2}+\left({b}^{2}-ac\right)x-bc=0.$

#### Question 17:

The area of a right-angle triangle is 165 m2. Determine its base and altitude if the latter exceeds the former by 7 metres.

#### Question 18:

Solve: $\frac{x+3}{x+2}=\frac{3x-7}{2x-3}.$

Solve: