Question 1 :
Check whether the following is a quadratic equation: $(x – 3)(2x +1) = x(x + 5)$
Question 2 :
Find the roots of the quadratic equation (by using the quadratic formula): $x^2+2\sqrt{2}x-6=0$
Question 3 :
Check whether the following is quadratic equation : $x^2 - 2x = (-2)(3-x)$
Question 4 :
Justify why the following quadratic equation has two distinct real roots: $2x^2+x-1=0$
Question 5 :
A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.
Question 6 :
State True or False: A real number α is said to be a root of the quadratic equation a$x^2$ + bx + c = 0, if a$α^2$ + bα + c = 0.
Question 8 :
Find the discriminant of the quadratic equation $2x^2 – 4x + 3 = 0$.
Question 9 :
Find the nature of the roots of the following quadratic equation: $2x^2 – 3x + 5 = 0$.
Question 10 :
Represent the following situation in the form of quadratic equations: Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.
Question 12 :
Is the following situation possible? If so, determine their present ages.The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.
Question 13 :
Find the roots of the quadratic equations, if they exist, by applying quadratic formula: $2x^2 – 7x + 3 = 0$
Question 14 :
Find two numbers whose sum is 27 and product is 182.
Question 15 :
Find the roots of the quadratic equations, if they exist, by applying quadratic formula: $2x^2 + x – 4 = 0$
Question 16 :
A train, travelling at a uniform speed for 360 km, would have taken 48 minutes less to travel the same distance if its speed were 5 km/h more. Find the original speed of the train.
Question 19 :
Represent the following situation in the form of quadratic equations: A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.
Question 20 :
Find the roots of the quadratic equation (by using the quadratic formula): $5x^2+13x+8=0$
Question 21 :
A train travels at a certain average speed for a distance of 63 km and then travels a distance of 72 km at an average speed of 6 km/h more than its original speed. If it takes 3 hours to complete the total journey, original average speed of the train is?
Question 22 :
State True or False whether the following quadratic equation has two distinct real roots: $\sqrt{2}x^2-\frac{3}{\sqrt{2}}x+\frac{1}{\sqrt{2}}=0$
Question 23 :
Check whether the following is a quadratic equation: $(x – 2)(x + 1) = (x – 1)(x + 3)$
Question 24 :
Find the roots of the equation $2x^2 – 5x + 3 = 0$, by factorisation.
Question 25 :
State True or False whether the following quadratic equation has two distinct real roots: $\left(x+4\right)^2-8x=0$
Question 27 :
State True or False: The expression $b^2$ + $4ac$ is called the discriminant of the quadratic equation.
Question 28 :
State True or False whether the following quadratic equation has two distinct real roots: $x\left(1-x\right)-2=0$
Question 29 :
Find the roots of the following quadratic equation (by the factorisation method): $3\sqrt{2}x^2-5x-\sqrt{2}=0$
Question 30 :
Sum of the areas of two squares is $468 m^2$. If the difference of their perimeters is 24 m, find the sides of the two squares.
Question 32 :
The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.
Question 33 :
John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. Write an equation to find out how many marbles they had to start with.
Question 34 :
Find the roots of the following quadratic equation (by the factorisation method): $2x^2+\frac{5}{3}x-2=0$
Question 35 :
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.
Question 36 :
An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11km/h more than that of the passenger train, find the average speed of the express train.
Question 37 :
Justify why the following quadratic equation has two distinct real roots: $3x^2-4x+1=0$
Question 39 :
Justify why the following quadratic equation has no two distinct real roots: $x^2-3x+4=0$
Question 40 :
Check whether the following is quadratic equation : (2x- 1)(x -3)=(x +5)(x -1)
Question 41 :
Which constant must be added and subtracted to solve the quadratic equation $9x^2+\frac{3}{4}x-\sqrt{2}=0$ by the method of completing the square?
Question 42 :
State true or false:
$b^2 – 4ac$ is called the discriminant of the quadratic equation $ax^2 + bx + c = 0$.
Question 43 :
Find the roots of the quadratic equation (by using the quadratic formula): $2x^2-3x-5=0$
Question 44 :
A rectangular park is to be designed whose breadth is 3 m less than its length. Its area is to be 4 $m^2$ more than the area of a park that has already been made in the shape of an isosceles triangle with its base as the breadth of the rectangular park and of altitude 12 m. Find its length and breadth.
Question 45 :
The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is $\frac{1}{3}$. Find his present age.
Question 47 :
John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. Find out how many marbles they had to start with.
Question 48 :
Check whether the following is a quadratic equation: $x^2 – 2x = (–2) (3 – x)$
Question 49 :
Which constant should be added and subtracted to solve the quadratic equation $4x^2-\sqrt{3}x-5=0$ by the method of completing the square?
