Vardaan Learning Institute

Which of the following is a quadratic equation? [1M]

1. $x^3 - 4x + 1 = 0$
2. $2x^2 + 3 = 0$
3. $x + \dfrac{1}{x} = 2$
4. $x(x+1) + 1 = (x-2)(x-5)$

The roots of the equation $x^2 - 3x - 10 = 0$ are: [1M]

1. $5$ and $-2$
2. $-5$ and $2$
3. $5$ and $2$
4. $-5$ and $-2$

The discriminant of $2x^2 - 4x + 3 = 0$ is: [1M]

1. $-8$
2. $8$
3. $28$
4. $-28$

If the discriminant of a quadratic equation is zero, the roots are: [1M]

1. Real and distinct
2. Equal and real
3. Imaginary
4. Irrational

The value of $k$ for which $kx^2 + 2x + 1 = 0$ has equal roots is: [1M]

1. $1$
2. $-1$
3. $2$
4. $-2$

The roots of $2x^2 - 8x + 8 = 0$ are: [1M]

1. $4$ and $-4$
2. $2$ and $2$
3. $-2$ and $-2$
4. $4$ and $4$

Which of the following equations has $x = 1$ as a root? [1M]

1. $x^2 + x - 2 = 0$
2. $x^2 - x + 1 = 0$
3. $2x^2 - 3x + 1 = 0$
4. Both (a) and (c)

If one root of $3x^2 - 10x + k = 0$ is $\dfrac{1}{3}$, then $k$ equals: [1M]

1. $1$
2. $-1$
3. $3$
4. $-3$

The equation $(x-1)^2 + x^2 = 0$ has: [1M]

1. Two distinct real roots
2. Two equal real roots
3. No real roots
4. Infinitely many roots

The nature of roots of $9x^2 - 6x + 1 = 0$ is: [1M]

1. Real and distinct
2. Equal
3. No real roots
4. Irrational

For $p(x) = 3x^2 - 5x + 2$, the value of $p\!\left(\dfrac{2}{3}\right)$ is: [1M]

1. $0$
2. $1$
3. $-1$
4. $2$

The sum of roots of $5x^2 - 3x - 2 = 0$ is: [1M]

1. $\dfrac{3}{5}$
2. $-\dfrac{3}{5}$
3. $\dfrac{2}{5}$
4. $-\dfrac{2}{5}$

Check whether the following are quadratic equations: [2M]

1. $(x+1)^2 = 2(x-3)$
2. $x^2 - 2x = (-2)(3-x)$
3. $(x-2)(x+1) = (x-1)(x+3)$
4. $(x+2)^3 = 2x(x^2-1)$

Represent the following situations in the form of quadratic equations: [2M]

1. The area of a rectangular plot is 528 m². The length of the plot is one more than twice its breadth.
2. The product of two consecutive positive integers is 306. Find the integers.

Is $x = -3$ a solution of $x^2 + 6x + 9 = 0$? Verify. [1M]

Find the roots of $2x^2 + x - 6 = 0$ by factorisation. [2M]

Find the roots of $\sqrt{2}\,x^2 + 7x + 5\sqrt{2} = 0$ by factorisation. [2M]

Find the roots of $100x^2 - 20x + 1 = 0$ by factorisation. [2M]

Find the roots of $x^2 - \dfrac{11}{4}x + \dfrac{15}{8} = 0$ by factorisation. [2M]

Find the roots of $4x^2 - 4a^2x + (a^4 - b^4) = 0$ by factorisation. [3M]

Solve: $4\sqrt{3}\,x^2 + 5x - 2\sqrt{3} = 0$. [2M]

Solve: $\dfrac{1}{x+1} + \dfrac{2}{x+2} = \dfrac{4}{x+4}$, $x \neq -1, -2, -4$. [3M]

Solve: $x - \dfrac{1}{x} = 3$, $x \neq 0$. [2M]

Solve: $\dfrac{x+1}{x-1} - \dfrac{x-1}{x+1} = \dfrac{5}{6}$, $x \neq 1, -1$. [3M]

Solve: $(x^2 + 3x)^2 - 2(x^2 + 3x) - 8 = 0$. [3M]

Find the roots of $2x^2 - 7x + 3 = 0$ using the quadratic formula. [2M]

Find the roots of $2x^2 + x - 4 = 0$ using the quadratic formula. [2M]

Find the roots of $4x^2 + 4\sqrt{3}\,x + 3 = 0$ using the quadratic formula. [2M]

Find the roots of $x^2 - 3\sqrt{5}\,x + 10 = 0$ using the quadratic formula. [2M]

Find the roots of $3x^2 - 2\sqrt{6}\,x + 2 = 0$ using the quadratic formula. [3M]

Find the roots (if they exist) of $x^2 + 4x + 5 = 0$ by the quadratic formula. [2M]

Solve: $\dfrac{2x}{x-3} + \dfrac{1}{2x+3} + \dfrac{3x+9}{(x-3)(2x+3)} = 0$, $x\neq 3, -\dfrac{3}{2}$. [3M]

Find the discriminant and state the nature of roots of: [2M]

1. $2x^2 - 3x + 5 = 0$
2. $3x^2 - 4\sqrt{3}\,x + 4 = 0$
3. $2x^2 - 6x + 3 = 0$

Find the value of $k$ for which the quadratic equation $kx^2 + kx + 1 = 0$ has equal roots. [2M]

Find the value of $k$ for which the equation $2x^2 + kx + 3 = 0$ has two equal roots. [2M]

Find the value(s) of $k$ for which the quadratic equation $(k-1)x^2 + 2(k-1)x + 1 = 0$ has equal roots. [3M]

Find the value of $k$ for which $kx(x-2) + 6 = 0$ has equal roots. [2M]

Find the value of $p$ for which the equation $3x^2 - px + 3 = 0$ has real roots. [2M]

If $-4$ is a root of $x^2 + px - 4 = 0$ and the equation $x^2 + px + q = 0$ has equal roots, find the value of $p$ and $q$. [3M]

Find the values of $k$ for which the equation $(k+4)x^2 + (k+1)x + 1 = 0$ has equal roots. [3M]

Is it possible to design a rectangular mango grove whose length is twice its breadth and the area is 800 m²? If so, find its length and breadth. [3M]

The product of two consecutive positive integers is 306. Find the integers. [3M]

The product of two consecutive odd positive integers is 483. Find the integers. [3M]

Find two numbers whose sum is 27 and product is 182. [3M]

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. [3M]

Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. Find Rohan's present age. [3M]

The sum of the reciprocals of Rehman's ages (in years) 3 years ago and 5 years from now is $\dfrac{1}{3}$. Find his present age. [3M]

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. [4M]

Two water taps together can fill a tank in $9\dfrac{3}{8}$ hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank. [4M]

A motorboat, whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream. [4M]

An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore. If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains. [4M]

In a flight of 600 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/h and the time of flight increased by 30 minutes. Find the duration of the flight. [4M]

The area of a rectangular plot is 528 m². The length (in metres) of the plot is one more than twice its breadth. Find the length and breadth of the plot. [3M]

The sum of areas of two squares is 468 m². If the difference of their perimeters is 24 m, find the sides of the two squares. [4M]

A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was ₹90, find the number of articles produced. [3M]

The hypotenuse of a right triangle is 6 m more than twice the shortest side. If the third side is 2 m less than the hypotenuse, find the sides of the triangle. [4M]

A right triangle's sides (other than hypotenuse) are 5 cm and 12 cm. A square is inscribed in the triangle with one side on the hypotenuse. Find the side of the square. [4M]

Assertion (A): $x^2 - 4 = 0$ is a quadratic equation with roots $2$ and $-2$.
Reason (R): A quadratic equation of the form $x^2 - a^2 = 0$ has roots $a$ and $-a$. [1M]

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is NOT the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.

Assertion (A): The equation $x^2 + 2x + 5 = 0$ has no real roots.
Reason (R): A quadratic equation has no real roots when its discriminant is negative. [1M]

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is NOT the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.

Assertion (A): The roots of $3x^2 - 4\sqrt{3}\,x + 4 = 0$ are equal.
Reason (R): If discriminant $D = 0$, then the roots are real and equal. [1M]

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is NOT the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.

Assertion (A): For $k = 4$, the equation $kx^2 + kx + 1 = 0$ has equal roots.
Reason (R): Equal roots condition requires $b^2 - 4ac = 0$. [1M]

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is NOT the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.

Case Study: The Cricket Ground
The length of a rectangular cricket ground is 30 m more than its width. The area of the ground is 3400 m². [4M]

1. Represent the situation as a quadratic equation in terms of width $w$.
2. Solve the equation to find the width of the ground.
3. Find the length of the ground.
4. Find the perimeter of the ground.

Case Study: The Swimming Pool
A swimming pool is being built by a contractor. The total cost is ₹$57,600$. He decides to put tiles only on the bottom of the pool, whose area is $x$ sq. m. The cost per sq. m. is $\dfrac{57600}{x}$ rupees. If the length of the pool is $2x - 60$ m and breadth is $x - 20$ m: [4M]

1. Write the quadratic equation using area = length × breadth.
2. Find the length and breadth of the pool.
3. Find the cost per sq. m. if total cost is ₹57,600.
4. Is it possible to reduce cost to ₹40,000? Justify using discriminant.

Case Study: The Scholarship Problem
A class teacher has 7 packets of chocolates, each containing the same number of chocolates. She kept 7 chocolates aside and distributed the remaining equally among 32 students. Each student got as many chocolates as the number of packets. [4M]

1. Let the number of chocolates in each packet be $x$. Write the quadratic equation.
2. Solve the equation to find $x$.
3. Find the total number of chocolates.
4. How many chocolates did each student receive?

Case Study: The Bridge Design
An engineer designs a parabolic arch bridge. The height $h$ (in metres) of the arch above the road at a horizontal distance $x$ (in metres) from the centre is given by $h = -x^2 + 25$. [4M]

1. At what distances from the centre does the arch meet the road level ($h = 0$)? Form and solve the quadratic equation.
2. What is the maximum height of the arch?
3. Find the discriminant of the equation in part (i) and state the nature of roots.
4. Is there any point where the height is $-5$ m? Justify using the discriminant.