CBSE Class 10 Maths HOTs Quadratic Equations

For the things of this world cannot be made known without a knowledge of mathematics.

Question. The sum of areas of two squares is 468m2 If the difference of their perimeters is 24cm, find the sides of the two squares.
Ans: Let the side of the larger square be x.
Let the side of the smaller square be y.
APQ x²+y² = 468
Cond. II 4x-4y = 24
⇒ x – y = 6
⇒ x = 6 + y
x² + y² = 468
⇒ (6+y)² +y² = 468
on solving we get y = 12
⇒ x = (12+6) = 18 m
∴ sides are 18m & 12m.

Question. A dealer sells a toy for Rs.24 and gains as much percent as the cost price of the toy.
Find the cost price of the toy.
Ans: Let the C.P be x
∴ Gain = x%
⇒ Gain = x. x/100
S.P = C.P +Gain
SP = 24 
x + x2/100 = 24
On solving x=20 or -120 (rej)
∴ C.P of toy = Rs.20

Question. A fox and an eagle lived at the top of a cliff of height 6m, whose base was at a distance of 10m from a point A on the ground. The fox descends the cliff and went straight to the point A. The eagle flew vertically up to a height x metres and then flew in a straight line to a point A, the distance traveled by each being the same.
Find the value of x.
Ans: Distance traveled by the fox = distance traveled by the eagle 

(6+x)² + (10)² = (16 – x)²
on solving we get
x = 2.72m. 

Question. A lotus is 2m above the water in a pond. Due to wind the lotus slides on the side and only the stem completely submerges in the water at a distance of 10m from the original position. Find the depth of water in the pond.
Ans: (x+2)² = x² + 102
x² + 4x + 4 = x² + 100
⇒ 4x + 4 = 100
⇒ x = 24
Depth of the pond = 24m

Question. Solve x = √6 + √6 + √6.........
Ans: x = √6 + √6 + √6 +
⇒ x = 6 + x
⇒ x² = 6 + x
⇒ x² - x – 6 = 0
⇒ (x -3) (x + 2) = 0
⇒ x = 3

Question. The hypotenuse of a right triangle is 20m. If the difference between the length of the other sides is 4m. Find the sides.
Ans: APQ
x² + y² = 202
x² + y² = 400
also x - y = 4
⇒ x = 404 + y
(4 + y)² + y² = 400
⇒2y² + 8y – 384 = 0
⇒ (y + 16) (y – 12) = 0
⇒ y = 12 y = – 16 (N.P)
∴ sides are 12cm & 16cm

Question. The positive value of k for which x² +Kx +64 = 0 & x² - 8x + k = 0 will have real roots .
Ans: x² + Kx + 64 = 0
⇒ b² -4ac ≥ 0
K² ≥ 256 > 0
K > 16 or K ≤ - 16 ……………(1)
x² - 8x + K = 0
64 – 4K ≥ 0
⇒ 4K < 64
K ≥ 16 ……………(2)
From (1) & (2) K = 16

Question. A teacher on attempting to arrange the students for mass drill in the form of a solid square found that 24 students were left over. When he increased the size of the square by one student he found he was short of 25 students. Find the number of students.
Ans: Let the side of the square be x.
No. of students = x² + 24
New side = x + 1
No. of students = (x + 1)² – 25
APQ ⇒ x² + 24 = (x + 1)² – 25
⇒ x² + 24 = x² + 2 x + 1 - 25
⇒ 2x = 48
⇒ x = 24
∴ side of square = 24
No. of students = 576 + 24
= 600

Question. A pole has to be erected at a point on the boundary of a circular park of diameter 13m in such a way that the differences of its distances from two diametrically opposite fixed gates A & B on the boundary in 7m. Is it possible to do so? If answer is yes at what distances from the two gates should the pole be erected. 

Ans: AB = 13 m
BP = x
⇒ AP – BP = 7
⇒ AP = x + 7
APQ
⇒ (13)² = (x + 7)² + x²
⇒ x² +7x – 60 = 0
(x + 12) (x – 5) = 0
⇒ x = - 12 N.P
x = 5
∴ Pole has to be erected at a distance of 5m from gate B & 12m from gate A.

Question. If the roots of the equation (a-b)x² + (b-c) x+ (c - a)= 0 are equal. Prove that 2a=b+c.
Ans: (a-b)x² + (b-c) x+ (c - a) = 0
T.P 2a = b + c
B2 – 4AC = 0
(b-c)² – [4(a-b) (c - a)] = 0
b²-2bc + c² – [4(ac-a² – bc + ab)] = 0
⇒ b²-2bc + c² – 4ac + 4a² + 4bc - 4ab = 0
⇒ b²+ 2bc + c² + 4a² – 4ac – 4ab= 0
⇒ (b + c - 2a)² = 0
⇒ b + c = 2a

Question. X and Y are centers of circles of radius 9cm and 2cm and XY = 17cm. Z is the centre of a circle of radius 4 cm, which touches the above circles externally. Given that ∠XZY=90^o, write an equation in r and solve it for r.
Ans: Let r be the radius of the third circle
XY = 17cm ⇒ XZ = 9 + r YZ = 2
APQ
(r + 9)² + (r + 2)² = (1 + r)² 
⇒ r² + 18r + 81 + r² + 4r + 4 = 289S  

⇒ r² + 11r - 10r = 0
(r + 17) (r – 6) = 0
⇒ r = - 17 (N.P)
r = 6 cm
∴ radius = 6cm.

ADDITIONAL QUESTIONS

More MCQs for NCERT Class 10 Mathematics Quadratic Equations..

1. If α, β, be the roots of the equation ax² + bx + c = 0, then ax² + bx + c =.

(A) (x – α)(x – β) 

(B) a(x – α)(x – β)

(C) a (x – β) (x + α) 

(D) a(x + α)(x + β)

Answer : (C)

2. The condition that the equation ax² + bx + c = 0 has one positive and other negative root is:

(A) a and c will have same sign 

(B) a and c will have opposite signs

(C) b and c will have same sign 

(D) b and c will have opposite signs

Answer : (B)

3. The condition that both roots of the equation ax² + bx + c = 0 are negative is:

(A) a, b, c are of the same sign 

(B) a and b are of opposite signs

(C) b and c are of opposite signs 

(D) the absolute term is zero

Answer : (A)

4. If one root of 5x² + 13x + k = 0 is reciprocal of the other, then the value of k is.

(A) 5 

(B) 7

(C) 4 

(D) None of these

Answer : (A)

5. If the equation x² - bx /ax - c = m -1 / m +1 has roots equal in magnitude but opposite in sign, then m is equal to.

(A) a + b /a - b

(B) a - b /a + b

(C) Both (A) & (B)

(D) None of these

Answer : (B)

6. The set of values of p for which the roots of the equation 3x² + 2x + (p – 1)p = 0 are of opposite sign, is:

(A) (0, 1) 

(B) (-1,1) 

(C) (-2,2) 

(D) None of these

Answer : (A)

7 Find the roots of the equation f(x) = (b – c) x² + (c – a) x + (a – b) = 0

(A) a + b /b - c and 1 

(B) a - b /b + c and 1

(C) a - b /b + c and 1 

(D) None of these

Answer : (A)

8 The number of real roots of the equation (x – 1)² + (x – 2)² + (x – 3)² = 0 is:

(A) 2 

(B) 1 

(C) 0 

(D) 3

Answer : (C)

9 √x /1 - x + √1 - x/x = 13/6 ; find the roots

(A) 13/4,5 

(B) 4/13 , 9/13

(C) 4 /13 ,6

(D) None of these

Answer : (B)

10 Determine k such that the quadratic equation x² + 7(3 +2k) – 2x(1 + 3k) = 0 has equal roots:

(A) 2, 7 

(B) 7, 5 

(C) 2,-10/9

(D) None of these

Answer : (C)

Please refer to link below for CBSE Class 10 Mathematics HOTs Quadratic Equations Set A

Please refer to attached file for CBSE Class 10 Mathematics HOTs Quadratic Equations

Quadratic Equation

 1. Find the sum & product of the roots of the equation x²- √3 =0
 2. Divide 51 into two parts whose product is378.
 3. Determine K, so that the equation x²-4x+k=0 has no real roots.
 4. Find the equation whose roots are 5+ √2 and 5- √2
 5. For the given equation √3 x²-2 √2 x-2 √3 =0
 6. Find the value of a and b such that x=1, and x= -2 are solutions of the quad. Equation x²+ax+b=0
 7. Find the value of K for which the following quad. Equation have equal roots

(a). (k-4) x²+2(k-4) x+4=0 (b). kx (x-2) +6=0 

 8. For what value of ‘P’, the following equations have real roots 

(a) Px²r +4x +1=0 (b) 4x² + 8x-P=0 (c) 2x²+Px +8=0 

9. If ∝ and β are the roots of the equation x2 – 4x –5=0. Find ∝^-1+ β^-1 
10. If one root of a quad. Equation is 2 −√3/2 , then what is the other root? 

2/3 MARKS

1. Solve the following by factorization method

(a) 2x²+ax-a²=0, a∈ R (b) 4x²-4ax+(a²-b²)=0 (a,b∈ R) 

2. Using quadratic formula, solve the following equation for x abx²+ (b2-ac) x-bc=0 (a,b,c ∈ R) 
3. Solve for x, x+ 1/x = 25 1/25. 
4. If one root of the quadratic equation 2x²+kx -6 =0 is 2, find the value of k also find the other root 
5. If -5 is a root of 2x²+ px-15=0 & the quad. Equation f (x²+x)+k = 0 has equal root find the value of k 
6. If the roots of the equation (b-c) x²+(c-a) x+ (a-b)=0 are equal then prove that 2b= a+c 
7. Find two consecutive multiple of three whose product is 270 
8. Two no. differ by 4 and their product is 192, find the numbers.
9. Is the following situation possible ?if so determine their present ages The sum of the ages of two friend are 20 years. 4 years ago, the product of their age in years was 48. 
10.If the list price of a toy is reduced by Rs 2 a person can by 2 toys more for Rs360.find original price of the toy.

Answers 

1 Marks Questions 

(1) sum = 0 , Product = -√13

(2) 42,9

(3) K > 4

(4) X² – 10x + 23 = 0

(5) √6 , −√6/3 

(6) 4 /3 , −4/ 3

(7) (a) P ≤ 4 (b) P ≥ −8 (c) P ≥ 8 , ≤ −8 

(8) −4 /5

(9) 2 + √3/2

2/3 Marks Questions 

(1) (a) x = -a , a /2 (b)a ± b/2 

(2) X = −b/ a , x = c/ b 

(3) 25 , 1/25 

(4) 2 , −3/2 

(5) P=7 , k = 7/4 

(6)

(7) 15,18 ; -15, -18 

(8) 12,16 

(9) Not possible 

(10) Rs.20

MORE QUESTIONS

Quadratic Equation

Key points

1. The equation ax²+ bx + c = 0 , a ≠ 0 is the standard form of a quadratic equation, where a, b, c are real numbers.

2. A real numbers α is said to be a root of the quadratic equation ax2 + bx + c = 0 . If 9α 2 + bα + c = 0 , the zeroes of the quadratic polynomial ax²+ bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same.

3. If we can factorize ax2 + bx + c = 0 , a ≠ 0 into a product of two linear factors, then the roots of the quadratic equation ax2 + bx + c = 0 can be found by equating each factors to zero.

4. A quadratic equation can also be solved by the method of completing the square.

5. A quadratic formula : The roots of a quadratic equation ax²+ bx + c = 0 are given by -b±√b²-4ac / 2a provided that b2 – 4ac ≥ o .

6. A quadratic equation ax2 + bx + c = 0 has :-

(i) Two distinct and real roots if b² – 4ac > 0

(ii) Two equal and real roots, if b² – 4ac = 0

(iii) Two roots are not real, if b² – 4ac < 0

Ans.1. (i) Yes (ii) No (iii) No (iv) Yes

Ans.2. (i) 32 (ii) –53 /9 (iii) 65/4

Ans.3. (i) Real and equal (b) Not real

3. 4 One of the roots of the equation 2x²– 5x + k = 0 is 3. Find the value of k.

Ans.4. K = –3

36. 5 The sum of roots of a quadratic equations is 13 and product of its roots is 36. Write the quadratic equation.

x²–13x + 36 = 0

1. 6 Find the value of k for which the quadratic equation x²– kx + 4 = 0 has real and equal roots.

6. k = 4

1. 7 For what value of P, the given equation has real roots :-

(i) Px² + 4x +1 = 0

(ii) 2x² + px + 3 = 0

7. (i) p ≤ n (ii) p ≥ 2 √6 or p ≤ 2 1√6

1. 8 Represent the following statements in the form of an equation :-

(i) The sum of squares of two consecutive even numbers is 100.

(ii) 10 is divided into two parts such that the sum of their reciprocals is 5/ 12 .

(iii) The sum of reciprocals of Rehman's age 3 years ago and 5 years from now is 1/3 .

8. (i) x²+ x –1.2 = 0

(ii) x² –10x + 20 = 0

(iii) x² – 4x – 21 = 0

9 Write discriminants of the quadratic equation 3x²– 2 2 (x +1) – 2 2 = 0

104

10 State whether roots of quadratic equation 3a2x²+ 8abx + 4b2 = 0 are real or not.

Real roots

11 Some students arranged a picnic, the budget for food was Rs. 240. Because four students of group failed to go. The cost of food for each student increased by Rs. 5 frame quadratic equation for above statement.

x²– 4x –192 = 0

2 marks questions (Question No. 22 to 25 under HOTS) :-

12 If one of the roots of 2x²+ 3x + k = 0 is 1/ 2 find the value of k and other root.

k = –2

ANSWER

19 The product of two consecutive multiples of 5 is 500. Find the numbers.

20, 25

20 The two numbers differ by 3 and their product is 504. Find the numbers.

21, 24

21 If x = 2 is a common root of the equation px²+ px + 3 = 0 and x²+ x + q = 0 . Find q/ p .

 q/p =12

22 Find the value of α such that the quadratic equation (α –12) x²+ 2(α –12) x + 2 = 0 has equal roots.

α =14

23 The difference of squares of two natural number is 45. The square of the small number is four times the larger number. Find the numbers.

9, 16

24 Thirty six years hence the age of a man will be square of what he was thirty six years ago. What is his present age?

Present age 45 years

25 If the roots of the equation (b – c) x²+ (c – a) x + (a – b) = 0 are equal than prove that 2b = a + c .

6 marks questions (Questions no. 35 to 39 under HOTS)

26 The denominator of a fraction exceeds the numberator by 2. If 3 is added to each of them the new fraction exceeds the original by 3/ 20 . Find the original fraction.

fraction 3 / 5

27 The diagonal of a rectangular field is 60 meters more than its shorter side. If the larger sideis 30 meters more than shorter side. Find the area of the field.

10800 sq. meters (L = 120, B = 90)

28 The perimeter of a right triangle is 60 units and hypotenuse is 25 units. Find the other two sides of the triangle.

Sides of triangles are 15, 20, 25

29 By increasing the list price of a book by Rs 10, A person can buy 10 less books for Rs. 1200. Find the original price of the book.

Price of the Book = Rs 30

30 A Shtabdi train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore. The average speed of the Shatabdi train is 11km/h more that of the passenger train. Find the average speed of the two trains.

Speed of Shtabdi train = 44 km/h

Speed of Passanger train = 33 km/h

31 A motor boat whose speed is 15 km/h in still water goes 30 km down stream and comes back in a total time of 4 hours and 30 minutes. Find the speed of the stream.

Speed of stream = 5 km/h

32 An aeroplane left 30 minutes late than its scheduled time and in order to reach its destination 1500 km away in time, it has to increase its speed by 250 km/h from its usual speed. Determine its usual speed of the aeroplane.

Usual speed of the aeroplane 750 k/h

33 A two digit number is five times the sum of its digits and is also equal to 5 more than twice the product of its digits. Find the numbers.

Number = 45

34 If denominator of a fraction is one more than twice the numerator and the sum of the fraction and its reciprocal is 2*16 /21 . Find the fraction.

fraction = 3/7

35 The hypotenuse of a right angle triangle is 3 √10cm . If the smaller side is tripled and longer side is doubled its new hypotenuse will be 9 5cm . How long is each side?

sides of triangles = 3, 9, 3 √10

36 Seven years ago, Varun's age was five times the squares of Swati's age. Three years hence Swati's age will be two fifth of Varun's age. Find their present ages.

Varun's Age = 27 years

Swati's Age = 9 years

37 A person on attempting to arrange the chairs for a function in the form of a square found that 4 chairs were left over. When he increased the size of the square by one chair. He found that he was short of 25 chairs. Find the number of chairs.

200 chairs

38 Two pipes running together can fill a tank in2 *8/ 11 minutes. If one pipe takes 1 minute more than the other to fill the tank. Find the time in which each pipe would fill the tank.

Faster pipe = 5 minutes, slower pipe =
6 mimutes.
39 A takes 10 days less 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.

30 day

More MCQs for NCERT Class 10 Mathematics Quadratic Equations..

1 The real values of a for which the quadratic equation 2x² – (a³ + 8a – 1)x + a2 – 4a = 0 possesses roots of opposite signs are given by:

(A) a > 6 

(B) a > 9 

(C) 0 < a < 4 

(D) a < 0

Answer : (C)

2 The number of real solution of the equation 23x² - 7x + 4= 1 is:

(A) 0 

(B) 4 

(C) 2 

(D) Infinitely many

Answer : (C)

3 The maximum value of – 3x² + 4x – 5 is at x =

(A) 2/3

(B) 1/3

(C) - 33/9

(D) None of these

Answer : (A)

4 Given that (x + 1) is a factor of x² + ax + b and x² + cx – d, then

(A) a + d = b + c 

(B) a = b + c + d 

(C) a + c = b - d 

(D) None of these

Answer : (B)

5 The zeroes of x² – bx + c are each decreased by 2 The resulting polynomial is x² – 2x + 1 Then

(A) b = 6, c = 9 

(B) b = 6, c = 3 

(C) b = 3, c = 6 

(D) b = 9, c = 6

Answer : (A)

6 If the zeroes of x² + Px + t are two consecutive even numbers find the relation between P and t
(A) P² – 4t + 4 = 0 

(B) 4t – P² + 4 = 0 

(C) - 4t² – 4 – P² = 0 

(D) None of these

Answer : (B)

7 Given that 7 – 3i is a zero of x² + px + q, find the value of 3q + 4p

(A) 14 

(B) 58 

(C) 118 

(D) - 14

Answer : (C)

8 One zero of x² – bx + C is the kth power of the other zero, then 1 k Ck+1+Ck+1 is equal to

(A) - b

(B) C 

(C) - C 

(D) b

Answer : (D)

9 α, β, γ, δ are zeroes of x⁴ + 5x3 + 5x² + 5x – 6, then find the value of 1/α + 1/β + 1/γ + 1/δ

(A) 5/6

(B) -6/5

(C) -5/6

(D) None of these

Answer : (A)

10 Given that α is a zero of x⁴ + x² – 1, find the value of (α⁶ + 2α⁴)1000.

(A) 1 

(B) 0 

(C) Either 0 or 1 

(D) None of these

Answer : (A)