Neev Classes

Assignments

Quadratic Equations (Factorisation Method)

Solve the following quadratic equations:

1. 𝒙𝟐 + 11𝑥 + 30 = 0
2. 𝑥 2 + 7𝑥 − 18 = 0
3. 𝑦 2 − 4𝑦 + 3 = 0
4. 𝑥 2 − 11𝑥 − 80 = 0
5. 𝑧 2 − 32𝑧 − 105 = 0
6. 6 − 𝑥 − 𝑥 2 = 0
7. 𝑚2 + 17𝑚𝑛 − 84𝑛2 = 0
8. 6𝑥 2 + 17𝑥 + 12 = 0
9. 14𝑥 2 + 9𝑥 + 1 = 0
10. 2𝑥 2 + 11𝑥 − 21 = 0
11. 18𝑥 2 + 3𝑥 − 10 = 0
12. 6𝑥 2 + 11𝑥 − 10 = 0
13. 24𝑥 2 − 41𝑥 + 12 = 0
14. 6𝑥 2 + 11𝑥 − 10 = 0
15. 5𝑥 2 − 16𝑥 − 21 = 0
16. 15𝑥 2 − 𝑥 − 28 = 0
17. 5𝑥 2 + 33𝑥𝑦 − 14𝑦 2 = 0
18. 𝑥 2 + 9𝑥 + 18 = 0
19. 𝑥 2 − 4𝑥 − 21 = 0
20. 2𝑥 2 − 7𝑥 − 39 = 0
21. 6𝑥 2 + 40 = 31𝑥
22. 8𝑥 2 − 22𝑥 − 21 = 0
23. 4 3𝑥 2 + 5𝑥 − 2 3 = 0

Solve the following by Factorisation method:

1. 2𝑥 2 + 7𝑥 + 5 2 = 0
4 5 −3
2. 𝑥
− 3 = 2𝑥+3 , 𝑥 ≠ 0 , 2
𝑥+3 3𝑥−7
3. 𝑥+2
= 2𝑥−3
1 1 1 1
4. 𝑎+𝑏+𝑥
= 𝑎 +𝑏 +𝑥 , 𝑥 ≠ 0 , −(𝑎 + 𝑏)
(𝑥+1) (2−𝑥) 3
5. 5 +5 =5 +1
6. 22𝑥 − 3.2(𝑥+2) + 32 = 0
7. 3(𝑥+2) + 3−𝑥 = 10
2 5
8. 𝑥 2 − 𝑥 + 2 = 0
9. 4 3𝑥 2 + 5𝑥 − 2 3 = 0
10. 7𝑥 2 − 6𝑥 − 13 7 = 0
11. 𝑥 2 − 1 + 2 𝑥 + 2 = 0
𝑥 2 𝑥
12. −5 + 6 = 0, (𝑥 ≠ −1)
𝑥+1 𝑥+1
𝑥−1 𝑥+3
13. 2 𝑥+3 − 7 2𝑥−1 = 5 , (𝑥 ≠ −3, 1)
𝑎 𝑏 1 1
14. 𝑎𝑥 −1 + 𝑏𝑥 −1 = 𝑎 + 𝑏, 𝑥 ≠ 𝑎 , 𝑏
2𝑥 2𝑥−5 25
15. 𝑥−4 + 𝑥−3 = 3 , (𝑥 ≠ 4 , 3)

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1 2 6
16. + = , (𝑥 ≠ 2, 1)
𝑥−2 𝑥−1 𝑥
1 1 4
17. + = , (𝑥 ≠ 2, 4)
𝑥−2 𝑥−4 3
2𝑥 1 3𝑥+9
18. 𝑥−3 + 2𝑥+3
+ 𝑥−3 (2𝑥+3)
=0
2 2 2 2 2
19. 4𝑥 − 2 𝑎 + 𝑏 𝑥 + 𝑎 𝑏 = 0
20. 4𝑥 2 − 4𝑎2 𝑥 + 𝑎4 − 𝑏 4 = 0
21. 𝑥 2 + 𝑥 − 𝑎 + 1 𝑎 + 2 = 0
22. 𝑎2 𝑏 2 𝑥 2 + 𝑏 2 𝑥 − 𝑎2 𝑥 − 1 = 0
34
23. 𝑥 − 3 𝑥 − 4 = 2
(33)
24. (𝑎 + 𝑏)2 𝑥 2 − 4𝑎𝑏𝑥 − (𝑎 − 𝑏)2 = 0
𝑥−𝑎 𝑥−𝑏 𝑎 𝑏
25. + = +
𝑥−𝑏 𝑥−𝑎 𝑏 𝑎

Method of completing the square

Solve the following quadratic equation (if they exist) by the method of completing the
square:

1. 8𝑥 2 − 22𝑥 − 21 = 0
2. 4 3𝑥 2 + 5𝑥 − 2 3 = 0
3. 9𝑥 2 − 15𝑥 + 6 = 0
4. 4𝑥 2 + 3𝑥 + 5 = 0
5. 4𝑥 2 + 4𝑏𝑥 − 𝑎2 − 𝑏 2 = 0
6. 𝑥2 − 3 + 1 𝑥 + 3 = 0
7. 𝑥 2 − 2+1 𝑥+ 2=0
8. 2𝑥 2 − 3𝑥 − 2 2 = 0
9. 2𝑥 2 + 𝑥 + 4 = 0
10. 3𝑥 2 + 11𝑥 + 10 = 0
11. 5𝑥 2 − 19𝑥 + 17 = 0
12. 2𝑥 2 − 9𝑥 + 7 = 0
13. 𝑥 2 − 4 2𝑥 + 6 = 0

Method of Quadratic Formula:

Show that each of the following equations has real roots, and solve each by using the
quadratic formula:

1. 9𝑥 2 + 7𝑥 − 2 = 0
2. 2𝑥 2 + 5 3𝑥 + 6 = 0
3. 𝑎2 𝑏 2 𝑥 2 − 4𝑏 4 − 3𝑎4 𝑥 − 12𝑎2 𝑏 2 = 0
4. 4𝑥 2 − 2 𝑎2 + 𝑏 2 𝑥 + 𝑎2 𝑏 2 = 0
5. 4𝑥 2 − 4𝑎2 𝑥 + 𝑎4 − 𝑏 4 = 0
6. 4 3𝑥 2 + 5𝑥 − 2 3 = 0
7. 7𝑥 2 − 6𝑥 − 13 7 = 0
8. 𝑥 2 − 1 + 2 𝑥 + 2 = 0
9. 𝑥 2 − 2𝑥 + 1 = 0
10. 3𝑎2 𝑥 2 + 8𝑎𝑏𝑥 + 4𝑏 2 = 0, 𝑎 ≠ 0
11. 3𝑥 2 − 2𝑥 + 2 = 0

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12. 𝑥 2 + 𝑥 + 2 = 0
13. 25𝑥 2 + 20𝑥 + 7 = 0
14. 𝑥 2 + 5𝑥 + 5 = 0
15. 𝑎𝑏𝑥 2 + 𝑏 2 − 𝑎𝑐 𝑥 − 𝑏𝑐 = 0
16. 12𝑎𝑏𝑥 2 − 9𝑎2 − 8𝑏 2 𝑥 − 6𝑎𝑏 = 0
17. 2𝑥 2 − 7𝑥 − 15 = 0
18. 10𝑥 2 − 9𝑥 − 7 = 0
19. 𝑧 2 − 32𝑧 − 105 = 0
20. 6 − 𝑥 − 𝑥 2 = 0

Nature of Roots:

1. Find the value of k for which the quadratic equation 2𝑥 2 + 𝑘𝑥 + 3 = 0 has two real equal
roots.
2. Find the value of k for which the quadratic equation 4𝑥 2 − 3𝑘𝑥 + 1 = 0 has two real equal
roots.
3. If - 5 is a root of the equation 2𝑥 2 + 𝑝𝑥 − 15 = 0 and the equation 𝑝 𝑥 2 + 𝑥 + 𝑘 = 0 has
equal roots, find the value of k.
4. Find the value of k for which the quadratic equation 𝑘 2 𝑥 2 − 2 𝑘 − 1 𝑥 + 4 = 0 has two
real equal roots.
5. Prove that both the roots of the equation 𝑥 − 𝑎 𝑥 − 𝑏 + 𝑥 − 𝑏 𝑥 − 𝑐 + 𝑥 − 𝑐 𝑥 −
𝑎=0 are real but they are equal only when 𝑎=𝑏=𝑐.
6. Find the value of k for which the quadratic equation 𝑘𝑥 2 − 6𝑥 − 2 = 0 has two real roots.
7. Find the value of k for which the quadratic equation 2𝑥 2 + 𝑘𝑥 + 2 = 0 has two real roots.
8. Show that the equation 2 𝑎2 + 𝑏 2 𝑥 2 + 2 𝑎 + 𝑏 𝑥 + 1 = 0 has no real roots, when 𝑎 ≠ 𝑏.
9. Find the value of p for which the quadratic equation 2𝑥 2 + 𝑝𝑥 + 8 = 0 has two real and
distinct roots.
10. If the roots of the equation 𝑐 2 − 𝑎𝑏 𝑥 2 − 2 𝑎2 − 𝑏𝑐 𝑥 + 𝑏 2 − 𝑎𝑐 = 0 are real and equal,
show that either 𝑎 = 0 𝑜𝑟 𝑎3 + 𝑏 3 + 𝑐 3 = 3𝑎𝑏𝑐
11. Find the value of k for which the quadratic equation 𝑘 + 4 𝑥 2 + 𝑘 + 1 𝑥 + 1 = 0 has
two real equal roots
12. If the roots of the equation 𝑥 2 + 2𝑐𝑥 + 𝑎𝑏 = 0 are real unequal, prove that the equation
𝑥 2 − 2 𝑎 + 𝑏 + 𝑎2 + 𝑏 2 + 2𝑐 2 = 0 has no real roots
13. Find the value of k for which the quadratic equation 𝑘 + 4 𝑥 2 + 𝑘 + 1 𝑥 + 1 = 0 has
equal roots.
14. Find the value of k for which the quadratic equation 𝑘 2 𝑥 2 − 2 2𝑘 − 1 𝑥 + 4 = 0 has real
and equal roots.
15. Find the value of k for which the quadratic equation 4 − 𝑘 𝑥 2 + 2𝑘 + 4 𝑥 + 8𝑘 + 1 =
0 has real and equal roots

Word Problems Category Wise

Number Based Questions (Direct Questions)

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1
1. The difference of two numbers is 5 and the difference of their reciprocals is . Find the
10
numbers.
2. The difference of the squares of two numbers is 45. The squares of the smaller number are
4times the larger number. Find the numbers.
3. The denominator of a fraction is 3 more than its numerator. The sum of the fraction and
9
its reciprocal is 2 10 . Find the fraction.
4. Two numbers differ by 3 and their product is 504. Find the numbers.
1
5. The sum of two numbers is 16 and the sum of their reciprocals is . Find the numbers.
3
3
6. The sum of two numbers is 25 and the sum of their reciprocals is 10
. Find the numbers.
41
7. The sum of a number and its reciprocal is 3 80
. Find the numbers.
8. Find two natural numbers, the sum of whose squares is 25 times their sum and also equal
to 50 times their difference.

Two Digit Problems:

1. A two digit number is such that the product of its digits is 12. When 36 is added to the
number, the digits are reversed. Find the number.
2. A two digit number is four times the sum and twice the product of its digits. Find the
number.
3. A two digit number is such that the product of its digits is 18. When 63 is subtracted from
the number. The digits interchange their places. Find the number.
4. A two digit number is such that the product of its digits is 8. When 18 is subtracted from
the number, the digits are reversed. Find the number.
5. A two digit number is 5 times the sum of its digits and is also equal to 5 more than twice
the product of its digits. Find the number.

Age Related Questions

1. The sum of ages of a father and his son is 45 years. Five years ago, the product of their
ages in years was 124. Find their present ages.
2. The product of Rohit’s age five years ago with his age 9 years later is 15 in years. Find his
present age.
3. The sum of the ages of a man and his son is 45 years. Five years ago, the product of their
ages in years was four times the man’s age at that time. Find their present ages.
4. The sum of the ages of a boy and his brother is 12 years and the sum of the square of their
ages is 74 in years. Find their ages
5. The difference of the ages of a boy and his brother is 3 and the product of their ages in
years is 504. Find their ages .

Speed, Distance and time related Questions

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

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2. 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.
3. A plane left 30 minutes later than the schedule time and in order to reach its destination
1500 km away in time it has to increase its speed by 250km/hr from its usual speed. Find
its usual speed.
4. A train travels 360 km at a uniform speed. If the speed had been 5km/h more, it would
have taken 1 hour less for the same journey. Find the speed of the train.
1
5. The time taken by a man to cover 300 km on a scooter was 1 hours more than the time
2
taken by him during the return journey. If the speed in returning be 10 km/hr more than
the speed in going, find its speed in each direction.
6. 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.
7. 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 km/hr, find the speed of the boat in still water.
8. 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
speeds of the two trains differ by 20km/hr.

Geometrical Figures Related Questions:

1. 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.
2. A pole has to be erected at a point on the boundary of a circular park of diameter 13
metres in such a way that the differences of its distances from two diametrically opposite
fixed gates A and B on the boundary is 7 metres. Is it possible to do so? If yes, at what
distances from the two gates should the pole be erected?
3. The hypotenuse of a right triangle is 3 5 cm. If the smaller side is tripled and the longer
sides doubled, new hypotenuse will be 15 cm. How long are the sides of the triangle?
4. The hypotenuse of a right triangle is 25 cm. The difference between the lengths of the
other two sides of the triangle is 5 cm. Find the lengths of these sides.
5. The perimeter of a right triangle is 60 cm. Its hypotenuse is 25 cm. Find the area of the
triangle.
6. The length of the rectangle exceeds its breadth by 8 cm and the area of the rectangle is 240
𝑐𝑚2 . Find the dimensions of the rectangle?
7. A rectangular field is 25 m long and 16m broad. There is a path of equal width all around
inside it. If the area of the path is 148 𝑚2 , find the width of the path.
8. A farmer prepares a rectangular vegetable garden of area 180 𝑚2 . With 39 m 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.
9. The area of right triangle is 600 𝑐𝑚2 . If the base of the triangle exceeds the altitude by 10
cm, find the dimensions of the triangle.
10. The length of the hypotenuse of a right 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.

Time and work related Questions:

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3
1. Two water taps together can fill a tank in 9 hours. The tap of larger diameter takes 10
8
hours less than the smaller one to fill the tank separately. Find the time in which each tap
can separately fill the tank.
1
2. Two pipes running together can fill a cistern in 3 hours. If one pipe takes 3 minutes
13
more than the other to fill the cistern. Find the time in which each can separately fill the
cistern.
3. If two pipes function simultaneously, a reservoir will be filled in 12 hours. One pipe fills
the reservoir 10 hours faster than the other. How many hours will the second pipe take to
fill the reservoir?

Reasoning Based Questions:

1. In a class test, the sum of Ranjitha’s marks in mathematics and English is 40. Had she got
3 marks more in mathematics and 4 marks less in English, the product of the marks
would have been 360. Find her marks in two subjects separately.
2. A teacher 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 1 student,
he found that he was short of 25 students. Find the number of students.
3. John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the
product of the number of marble they now have is 124. We would like to find out how
many marbles they had to start with.
4. 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 students.
5. One-fourth of a herd of camels was seen in the forest. Twice the square root of the herd
had gone to mountains and the remaining 15 camels were seen on the bank of a river.
Find the total number of camels.
6. In a class test, the sum of the marks obtained by P in mathematics and science is 28. Had
he got 3 more marks in mathematics and 4 marks less in science, the product of marks
obtained in the two subjects would have been 180. Find the marks obtained by him in the
two subjects separately.
7. A peacock is sitting on the top of a pillar, which is 9m high. From a point 27 m away from
the bottom of the pillar, a snake is coming to its hole at base of pillar. Seeing the snake the
peacock pounches on it. If their speeds are equal at what distance from the whole is the
snake caught?
8. If the list price of a toy is reduced by Rs. 2, a person can buy 2 toys more for Rs. 360. Find
the original price of the toy.

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Answer Key

Solve the following quadratic equations:

1. 𝑥 = −5, −6
2. 2, −9
3. 3,1
4. 16, −5
5. 35, −3
6. −3,2
7. 4𝑛, −21n
8. −4/3,−3/2
9. −1/2,−1/7
3
10. , −7
2
2 −5
11. ,
3 6
2 −5
12. ,
3 2
3 4
13. ,
8 3
2
14. 3 , −5/2
15. −1,21/5
7
16. , −4/3
5
17. −7𝑦, −2𝑦/5
18. 𝑥 = −3,−6
19. 𝑥 = 7, −3
13
20. 𝑥 = 2
, −3
8 5
21. 𝑥 = ,
3 2
7
22. , −3/4
2
3 −2
23. 𝑥 = 4
, 3

Solve the following by Factorisation method:

1. − 2, −5/ 2
2. 1, −2
3. −1,5
4. 𝑥 = −𝑏, −𝑎
5. 𝑥=2
6. 𝑥 = 2,3
7. 𝑥 = 0, −2
8. 𝑥 = 2,1/2
−2
9. 𝑥 = 3
, 3/4
10. 𝑥 = − 7, 13/ 7
11. 𝑥 = 2 or 1
12. 𝑥 = −2, −3/2

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13. 𝑥 = −1, −23/5

2
14. 𝑥 = 𝑎 + 𝑏/𝑎𝑏 or 𝑥 =
𝑎+𝑏
15. 𝑥 = 6,40/13
16. 𝑥 = 3,4/3
5
17. 𝑥 = 2 , 𝑥 = 5
18. 𝑥 = −3/2, 𝑥 = −1
𝑎2 𝑏2
19. 𝑥 = ,
2 2
𝑎 +𝑏 2
2 𝑎 2 −𝑏 2
20. 𝑥 = or
2 2
21. 𝑥 = 𝑎 + 1, −(𝑎 + 2)
1 −1
22. 𝑥 = 𝑏 2 , 𝑎 2
133 98
23. 𝑥 = ,
33 33
−(𝑎−𝑏)2
24. 𝑥 = 1,
(𝑎+𝑏)2
25. 𝑥 = 0, (𝑎 + 𝑏)

Method of completing the square

7
1. , −3/4
2
3
2. 4 3
, −2/ 3
3. 1,2/3
4. −3 ± 71𝑖 /8
5. (−𝑏 ± 𝑎)/2
6. 3, 1
7. 2, 1
8. 2 2, −1/ 2
9. −1 ± 31𝑖 /4
10. −2,−5/3
11. 19 ± 21 /10
12. 7/2,1
13. 3 2, 2

Method of Quadratic Formula:

1. −1,2/3
2. −2 3, − 3/2
4𝑏 2 −3𝑎 2
3. 𝑎2
, 𝑏2
2 2
4. 𝑎 /2, 𝑏 /2
5. (𝑎2 ± 𝑏 2 )/2
3
6. 4
, −2/ 3
7. − 7, 13/ 7
8. 1, 1
9. 1 ± 2

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−2𝑏
10. , −2𝑏/𝑎
3𝑎
11. (1 ± 5𝑖)/3
12. 1, −2
13. (−2 ± 3𝑖)/5
14. (5 ± 5)/2
𝑐
15. , −b/a
𝑏
3𝑎
16. , −2𝑏/3𝑎
4𝑏
17. 5, −3/2
7
18. , −1/2
5
19. −3,35
20. 2, −3

Nature Of Roots
1. 𝑘 ± 2 6
4
2. 𝑘 = ±
3
3. 𝑘 = 7/4
4. 𝑘 = −1,1/3
6. 𝑘 = −9/2
7. 𝑘 = ±4
9. 𝑝 ∈ (−∞, 8) ∪ (8, ∞)
11. 𝑘 = −3,5
13. 𝑘 = −3,5
14. 𝑘 = 1/4
15. 𝑘 = 0,3

Word Problems Category Wise

Number Based Questions (Direct Questions)

1. 5,10
2. 9,6 and 9, −6
2 −5
3. 5
or −2
4. 21,24 or −21, −24
5. 4,12
75±5 15
6.
6
5 16
7. ,
16 5
8. 30,10

Two Digit Problems

1. 26
2. 36
3. 92
4. 42
5. 45

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Age Related Questions:

1. son = 9 years
father = 36 years
2. 6 years
3. son = 10 years
father = 35 years
4. 5 , 7 years
5. boy = 24 years
friend = 21 years

Speed, distance and time related questions:

1. 6 𝑘𝑚/ℎ𝑟
2. 15 𝑘𝑚/ℎ𝑟
3. 750 𝑘𝑚/ℎ𝑟
4. 45 𝑘𝑚/ℎ𝑟
5. 40 𝑘𝑚/ℎ𝑟
50 𝑘𝑚/ℎ𝑟
6. 3 𝑘𝑚/ℎ𝑟
7. 10 𝑘𝑚/ℎ𝑟
8. 80 𝑘𝑚/ℎ𝑟

Geometrical Figures Related Questions

1. 24 m, 8m
2. 12 m
3. 3 cm, 6 cm
4. 15 cm, 20 cm
5. 150 𝑐𝑚2
6. 20, 12 cm
7. 17, 15, 8 cm
8. (15,12) or (12, 7.5)
9. Base = 40 cm, Altitude = 30 cm
10. 17,15,8 cm

Time & Work related Questions

1. smaller tap = 25 hrs, larger tap = 15 hrs
2. 5 minutes, 8 minutes
3. 30 hours

Reasoning Based Questions:

2. 600
3. 9,36

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4. 50 students
5. 36
6. 12 in mathematics, science =16 or maths – 9, science – 19
7. 12 meter
8. Rs. 20

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