Quadratic Equations
Class 10th Mathematics NCERT Exemplar Solution
- Which of the following is a quadratic equation?A. x^2 + 2x + 1 = (4 - x)^2 + 3…
- Which of the following is not a quadratic equation?A. 2(x - 1)^2 = 4x^2 - 2x + 1…
- Which of the following equation has 2 as a root?A. x^2 - 4x + 5 = 0 B. x^2 + 3x…
- If 1/2 is a root of the equation x^2 + kx - 5/4 = 0 , then the value of k isA. 2…
- Which of the following equations has the sum of its roots as 3?A. 2x^2 - 3x + 6…
- Value(s) of k for which the quadratic equation 2x^2 - kx + k = 0 has equal roots…
- Which constant value must be added and subtracted to solve the quadratic…
- The quadratic equation 2x^2 - root 5x+1 = 0 hasA. two distinct real roots B. two…
- Which of the following equations has two distinct real roots?A. 2x^2 - 3 root 2x…
- Which of the following equations has no real roots?A. x^2 - 4x+3 root 2 = 0 B.…
- (x^2 + 1)^2 - x^2 = 0 hasA. four real roots B. two real roots C. no real roots…
- x^2 - 3x + 4 = 0 State whether the following quadratic equations have two…
- 2x^2 + x - 1 = 0 State whether the following quadratic equations have two…
- 2x^2 - 6x + 9/2 = 0 State whether the following quadratic equations have two…
- 3x^2 - 4x + 1 = 0 State whether the following quadratic equations have two…
- (x + 4)^2 - 8x = 0 State whether the following quadratic equations have two…
- (x - root 2)^2 - root 2 (x+1) = 0 State whether the following quadratic…
- root 2x^2 - 3/root 2x x + 1/2 = 0 State whether the following quadratic…
- x (1 - x) - 2 = 0 State whether the following quadratic equations have two…
- (x - 1) (x + 2) + 2 = 0 State whether the following quadratic equations have…
- (x + 1) (x - 2) + x = 0 State whether the following quadratic equations have…
- Write whether the following statements are true or false. Justify your answers.…
- A quadratic equation with integral coefficient has integral roots. Justify your…
- Does there exist a quadratic equation whose coefficients are rational but both…
- Does there exist a quadratic equation whose coefficients are all distinct…
- Is 0.2 a root of the equation x^2 - 0.4 = 0? Justify your answer.…
- If b = 0, c 0, is it true the roots of x^2 + bx + c = 0 are numerically equal…
- 2x^2 - 3x - 5 = 0 Find the roots of the quadratic equations by using the…
- 5x^2 + 13x + 8 = 0 Find the roots of the quadratic equations by using the…
- -3x^2 + 5x + 12 = 0 Find the roots of the quadratic equations by using the…
- -x^2 + 7x - 10 = 0 Find the roots of the quadratic equations by using the…
- x^2 + 2 root 2x-6 = 0 Find the roots of the quadratic equations by using the…
- x^2 - 3 root 5x+10 = 0 Find the roots of the quadratic equations by using the…
- 1x^2/2 - root 11x+1 = 0 Find the roots of the quadratic equations by using the…
- 2x^2 + 5/3 x-2 = 0 Find the roots of the following quadratic equations by…
- 2/5 x^2 - x - 3/5 = 0 Find the roots of the following quadratic equations by…
- 3 root 2x^2 - 5x - root 2 = 0 Find the roots of the following quadratic…
- 3x^2 + 5 root 5x-10 = 0 Find the roots of the following quadratic equations by…
- 21x^2 - 2x + 1/21 = 0 Find the roots of the following quadratic equations by…
- 8x^2 + 2x - 3 = 0 Find whether the following equations have real roots. If real…
- -2x^2 + 3x + 2 = 0 Find whether the following equations have real roots. If…
- 5x^2 - 2x - 10 = 0 Find whether the following equations have real roots. If…
- 1/2x-3 + 1/x-5 = 1 , x not equal 3/2 , 5 Find whether the following equations…
- x^2 + 5 root 5x-70 = 0 Find whether the following equations have real roots. If…
- Find a natural number whose square diminished by 84 is equal to thrice of 8 more…
- A natural number, when increased by 12, equals 160 times its reciprocal. Find…
- A train, travelling at a uniform speed for 360 km, would have taken 48 mins less…
- If Zeba were younger by 5 years than what she really is, then the square of her…
- At present Asha’s age (in years) is 2 more than the square of her daughter…
- In the centre of a rectangular lawn of dimensions 50 m x 40 m, a rectangular…
- At t min past 2 pm, the time needed by the minute hand of a clock to show 3pm…
Exercise 4.1
Question 1.Which of the following is a quadratic equation?
A. x2 + 2x + 1 = (4 - x)2 + 3
B. 
C. 
where k = -1
D. x3 - x2 = (x - 1)3
Answer:
The standard form of a quadratic equation is given by,
ax2 + bx + c = 0, a ≠ 0
A. Given, x2 + 2x + 1 = (4 - x)2 + 3
⇒ x2 + 2x + 1 = 16 - 8x + x2 + 3
∴ 10x – 18 = 0
which is not a quadratic equation.
B. Given, -2x2 = (5 - x) (2x – 
)
⇒ -2x2 = 10x - 2x2 – 2 + 
∴ 52x – 10 = 0
which is not a quadratic equation.
C. Given, (k + 1) x2 + 
= 7, where k = -1
⇒ (-1 + 1) x2 + = 7
∴ 3x – 14 = 0
which is not a quadratic equation.
D. Given, x3 - x2 = (x - 1)3
⇒ x3 - x2 = x3 - 3x2 + 3x - 1
∴ 2x2 - 3x + 1 = 0
which represents a quadratic equation.
Question 2.
Which of the following is not a quadratic equation?
A. 2(x - 1)2 = 4x2 - 2x + 1
B. 2x - x2 = x2 + 5
C. 
D. (x2 + 2x)2 = x4 + 3 + 4x2
Answer:
A quadratic equation is represented by the form,
ax2 + bx + c = 0, a ≠ 0
A. Given, 2(x - 1)2 = 4x2 - 2x + 1
⇒ 2(x2 - 2x + 1) = 4x2 - 2x + 1
∴ 2x2 + 2x – 1 = 0
which is a quadratic equation.
B. Given, 2x - x2 = x2 + 5
∴ 2x2 - 2x + 5 = 0
which is a quadratic equation.
C. Given, 
= 3x2 – 5x
⇒ 
= 3x2 – 5x
∴
which is a quadratic equation.
D. Given, (x2 + 2x)2 = x4 + 3 + 4x2
⇒ x4 + 4x3 + 4x2 = x4 + 3 + 4x2
∴ 4x3 – 3 = 0
which is a cubic equation and not a quadratic equation.
Question 3.
Which of the following equation has 2 as a root?
A. x2 - 4x + 5 = 0
B. x2 + 3x – 12 = 0
C. 2x2 - 7x + 6 = 0
D. 3x2 - 6x – 2 = 0
Answer:
If 2 is a root then substituting the value 2 in place of x should satisfy the equation.
A. Given, x2 - 4x + 5 = 0
⇒ (2)2 - 4(2) + 5 = 1 ≠ 0
So, x = 2 is not a root of x2 - 4x + 5 = 0
B. Given, x2 + 3x – 12 = 0
⇒ (2)2 + 3(2) – 12 = -2 ≠ 0
So, x = 2 is not a root of x2 + 3x – 12 = 0
C. Given, 2x2 - 7x + 6 = 0
⇒ 2(2)2 - 7(2) + 6 = 0
Here, x = 2 is a root of 2x2 - 7x + 6 = 0
D. Given, 3x2 - 6x – 2 = 0
⇒ 3(2)2 - 6(2) – 2 = -2 ≠ 0
So, x = 2 is not a root of 3x2 - 6x – 2 = 0
Question 4.
If 
is a root of the equation 
, then the value of k is
A. 2
B. -2
C. 
D. 
Answer:
If 
is a root of the equation
x2 + kx - 
= 0 then, substituting the value of in place of x should give us the value of k.
Given, x2 + kx - = 0 where, 
⇒ 
⇒ 
∴ k = 2
Question 5.
Which of the following equations has the sum of its roots as 3?
A. 2x2 - 3x + 6 = 0
B. -x2 + 3x – 3 = 0
C. 
D. 3x2 - 3x + 3 = 0
Answer:
The sum of the roots of a quadratic equation ax2 + bx + c = 0, a ≠ 0 is given by,
A. Given, 2x2 - 3x + 6 = 0
∴ Sum of the roots = 
B. Given, -x2 + 3x – 3 = 0
∴ Sum of the roots = 
C. Given, 
⇒ 
∴ Sum of the roots =
D. Given, 3x2 - 3x + 3 = 0
∴ Sum of the roots = 
Question 6.
Value(s) of k for which the quadratic equation 2x2 – kx + k = 0 has equal roots is/are
A. 0
B. 4
C. 8
D. 0,8
Answer:
If a quadratic equation
ax2 + bx + c = 0, a ≠ 0 has two equal roots, then its discriminant value will be equal to zero i.e.,
D = b2 - 4ac = 0
Given, 2x2 – kx + k = 0
For equal roots,
D = b2 - 4ac = 0
⇒ (-k)2 - 4(2)(k) = 0
⇒ k2 - 8k = 0
⇒ k (k - 8) = 0
∴ k = 0,8
Question 7.
Which constant value must be added and subtracted to solve the quadratic equation 
by the method of completing the squares?
A. 
B. 
C.
D. 
Answer:
Given, 
⇒ 
Let 
,
⇒ 
⇒ 
Thus, 
must be added to solve the quadratic equation.
Question 8.
The quadratic equation 
has
A. two distinct real roots
B. two equal real roots
C. no real roots
D. more than two real roots
Answer:
The discriminant value of a quadratic equation ax2 + bx + c = 0, a ≠ 0 is given by,
D = b2 - 4ac = 0
i. If D = b2 - 4ac > 0, then its roots are distinct and real.
ii. If D = b2 - 4ac = 0, then its roots are real and equal.
iii. If D = b2 - 4ac < 0, then its roots are imaginary.
Given, 
∴ D = b2 - 4ac = 0
= -3 < 0
Hence, the roots of the quadratic equation are imaginary.
Question 9.
Which of the following equations has two distinct real roots?
A. 
B. x2 + x – 5 = 0
C. 
D. 5x2 - 3x + 1 = 0
Answer:
Same as the previous one. Let’s check the discriminant value for distinct real roots.
A. Given, 
∴ D = b2 - 4ac = 0
= 18 - 18 = 0
Hence, the roots are real and equal.
B. Given, x2 + x – 5 = 0
∴ D = b2 - 4ac = 0
= (1)2 - 4(1) (-5)
= 1 + 20 = 21 > 0
Hence, the roots are real and distinct.
C. Given, 
∴ D = b2 - 4ac = 0
= 32 - 4(1) (2√2)
Hence, the roots are imaginary.
D. Given, 5x2 - 3x + 1 = 0
∴ D = b2 - 4ac = 0
= (-3)2 - 4(5)(1)
= 9 – 20 < 0
Hence, the roots are imaginary.
Question 10.
Which of the following equations has no real roots?
A. 
B. 
C. 
D. 
Answer:
For the quadratic equation to have no real roots, the discriminant value should be negative i.e.,
D = b2 - 4ac = 0
A. Given, 
∴ D = b2 - 4ac = 0
Hence, the roots are imaginary.
B. Given, 
∴ D = b2 - 4ac = 0
Hence, the roots are real and distinct.
C. Given, 
∴ D = b2 - 4ac = 0
Hence, the roots are real and distinct.
D. Given, 
∴ D = b2 - 4ac = 0
Hence, the roots are real and equal.
Question 11.
(x2 + 1)2 - x2 = 0 has
A. four real roots
B. two real roots
C. no real roots
D. one real root.
Answer:
Let’s simplify the equation,
(x2 + 1)2 - x2 = 0
⇒ x4 + 2x2 + 1 - x2 = 0
⇒ x4 + x2 + 1 = 0
Let x2 = y,
⇒ y2 + y + 1 = 0
D = b2 - 4ac = 0
= 12 - 4(1)(1)
∴ D = -3 < 0
Hence, the equation (x2 + 1)2 - x2 = 0 has no real roots.
Exercise 4.2
Question 1.State whether the following quadratic equations have two distinct real roots. Justify your answer.
x2 - 3x + 4 = 0
Answer:
The equation x2 - 3x + 4 = 0 has no real roots.
∴ D = b2 - 4ac
= (-3)2 - 4(1)(4)
= 9 – 16 < 0
Hence, the roots are imaginary.
Question 2.
State whether the following quadratic equations have two distinct real roots. Justify your answer.
2x2 + x – 1 = 0
Answer:
The equation 2x2 + x – 1 = 0 has two real and distinct roots.
∴ D = b2 - 4ac
= 12 - 4(2) (-1)
= 1 + 8 > 0
Hence, the roots are real and distinct.
Question 3.
State whether the following quadratic equations have two distinct real roots. Justify your answer.
Answer:
The equation 
has real and equal roots.
∴ D = b2 - 4ac
= 36 – 36 = 0
Hence, the roots are real and equal..
Question 4.
State whether the following quadratic equations have two distinct real roots. Justify your answer.
3x2 – 4x + 1 = 0
Answer:
The equation 3x2 – 4x + 1 = 0 has two real and distinct roots.
∴ D = b2 - 4ac
= (-4)2 - 4(3)(1)
= 16 – 12 > 0
Hence, the roots are real and distinct.
Question 5.
State whether the following quadratic equations have two distinct real roots. Justify your answer.
(x + 4)2 - 8x = 0
Answer:
The equation (x + 4)2 - 8x = 0 has no real roots.
Simplifying the above equation,
⇒ x2 + 8x + 16 - 8x = 0
∴ x2 + 16 = 0
∴ D = b2 - 4ac
= (0) - 4(1) (16) < 0
Hence, the roots are imaginary.
Question 6.
State whether the following quadratic equations have two distinct real roots. Justify your answer.
Answer:
The equation 
has two distinct and real roots.
Simplifying the above equation,
⇒ 
⇒ 
∴
∴ D = b2 - 4ac
Hence, the roots are real and distinct.
Question 7.
State whether the following quadratic equations have two distinct real roots. Justify your answer.
Answer:
The equation 
has two real and distinct roots.
∴ D = b2 - 4ac
Hence, the roots are real and distinct.
Question 8.
State whether the following quadratic equations have two distinct real roots. Justify your answer.
x (1 - x) – 2 = 0
Answer:
The equation x (1 - x) – 2 = 0 has no real roots.
Simplifying the above equation,
∴ x2 – x + 2 = 0
∴ D = b2 - 4ac
= (-1)2 - 4(1)(2)
= 1 - 8 < 0
Hence, the roots are imaginary.
Question 9.
State whether the following quadratic equations have two distinct real roots. Justify your answer.
(x - 1) (x + 2) + 2 = 0
Answer:
The equation (x - 1) (x + 2) + 2 = 0 has two real and distinct roots.
Simplifying the above equation,
⇒ x2 – x + 2x – 2 + 2 = 0
∴ x2 + x = 0
∴ D = b2 - 4ac
= 12 - 4(1)(0)
= 1 - 0 > 0
Hence, the roots are real and distinct.
Question 10.
State whether the following quadratic equations have two distinct real roots. Justify your answer.
(x + 1) (x - 2) + x = 0
Answer:
The equation (x + 1) (x - 2) + x = 0 has two real and distinct roots.
Simplifying the above equation,
⇒ x2 + x – 2x – 2 + x = 0
∴ x2 – 2 = 0
∴ D = b2 - 4ac
= (0)2 - 4(1) (-2)
= 0 + 8 > 0
Hence, the roots are real and distinct.
Question 11.
Write whether the following statements are true or false. Justify your answers.
(i) Every quadratic equation has exactly one root.
(ii) Every quadratic equation has at least one real root.
(iii) Every quadratic equation has at least two roots.
(iv) Every quadratic equation has at most two roots.
(v) If the coefficient of x2 and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.
(vi) If the coefficient of x2 and the constant term have the same sign and if the coefficient of x term is zero, then the quadratic equation has no real roots.
Answer:
(i) Every quadratic equation has exactly one root: False
A quadratic equation has exactly two roots.
(ii) Every quadratic equation has at least one real root: False
A quadratic equation may have real or imaginary roots.
(iii) Every quadratic equation has at least two roots: False
A quadratic equation has exactly two roots, no more no less.
(iv) Every quadratic equation has at most two roots: True
A quadratic equation can’t have more than two roots.
(v) If the coefficient of x2 and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots: True
The standard form of a quadratic equation is given by,
ax2 + bx + c = 0, a ≠ 0
So, if coefficient of x2(a) and constant term(c) have opposite signs then, the roots will always be real i.e.,
ac < 0
⇒ b2 - 4ac > 0
(vi) If the coefficient of x2 and the constant term have the same sign and if the coefficient of x term is zero, then the quadratic equation has no real roots: True
The standard form of a quadratic equation is given by,
ax2 + bx + c = 0, a ≠ 0
So, if coefficient of x2(a) and constant term(c) have the same sign and coefficient of x(b) is zero, then the roots will be imaginary.
ac > 0
⇒ b2 - 4ac < 0
Question 12.
A quadratic equation with integral coefficient has integral roots. Justify your answer.
Answer:
No, a quadratic equation with integral coefficients may or may not have integral roots.
Example: Consider the following equation,
8x2 - 2x – 1 = 0
The roots of the given equation are 
which are not integers.
Question 13.
Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational? Justify your answer.
Answer:
Yes, a quadratic equation with rational coefficients may have irrational roots.
Example: Consider the following equation,
x2 + 3x + 1 = 0
The roots of the given equation are 
which is irrational.
Question 14.
Does there exist a quadratic equation whose coefficients are all distinct irrationals but both the roots are rational? Justify.
Answer:
Yes, a quadratic equation whose coefficients are irrational can have rational roots.
Example: Consider the following equation,
The roots of the given equation are 4 and 1 which are rational.
Question 15.
Is 0.2 a root of the equation x2 - 0.4 = 0? Justify your answer.
Answer:
No, 0.2 is not a root of the equation x2 - 0.4 = 0.
If we substitute the value 0.2 in place of x in the equation, x2 - 0.4 = 0
⇒ (0.2)2 - 0.4 ≠ 0
Question 16.
If b = 0, c < 0, is it true the roots of x2 + bx + c = 0 are numerically equal and opposite in sign? Justify your answer.
Answer:
Yes, the roots will be equal and opposite in sign.
Given, x2 + bx + c = 0, b = 0, c < 0
⇒ x2 – c = 0
∴
Hence, the roots are equal and opposite in sign.
Exercise 4.3
Question 1.Find the roots of the quadratic equations by using the quadratic formula in each of the following:
2x2 - 3x – 5 = 0
Answer:
The quadratic formula for finding the roots of quadratic equation
ax2 + bx + c = 0, a ≠ 0 is given by,
Given, 2x2 - 3x – 5 = 0
Question 2.
Find the roots of the quadratic equations by using the quadratic formula in each of the following:
5x2 + 13x + 8 = 0
Answer:
Given, 5x2 + 13x + 8 = 0
By using quadratic formula,
Question 3.
Find the roots of the quadratic equations by using the quadratic formula in each of the following:
-3x2 + 5x + 12 = 0
Answer:
Given, -3x2 + 5x + 12 = 0
By using the quadratic formula,
Question 4.
Find the roots of the quadratic equations by using the quadratic formula in each of the following:
-x2 + 7x – 10 = 0
Answer:
Given, -x2 + 7x – 10 = 0
By using the quadratic formula,
Question 5.
Find the roots of the quadratic equations by using the quadratic formula in each of the following:
Answer:
Given, 
By using the quadratic formula,
Question 6.
Find the roots of the quadratic equations by using the quadratic formula in each of the following:
Answer:
Given, 
By using quadratic formula,
Question 7.
Find the roots of the quadratic equations by using the quadratic formula in each of the following:
Answer:
Given, 
By using the quadratic equation,
Question 8.
Find the roots of the following quadratic equations by factorisation method.
Answer:
Given, 
⇒ 6x2 + 5x – 6 = 0
By splitting the middle term,
⇒ 6x2 + 9x - 4x – 6 = 0
Taking common in the expression,
⇒ 3x (2x + 3) – 2 (2x + 3) = 0
⇒ (2x + 3) (3x - 2) = 0
∴ 2x + 3 = 0 and ∴ 3x – 2 = 0
∴
and ∴
Question 9.
Find the roots of the following quadratic equations by factorisation method.
Answer:
Given, 
⇒ 2x2 - 5x – 3 = 0
By splitting the middle term,
⇒ 2x2 - 6x + x – 3 = 0
Taking common in the expression,
⇒ 2x (x - 3) + 1 (x - 3) = 0
⇒ (2x + 1) (x - 3) = 0
∴ 2x + 1 = 0 and ∴ x – 3 = 0
∴
and ∴
Question 10.
Find the roots of the following quadratic equations by factorisation method.
Answer:
Given, 
By splitting the middle term,
⇒ 
⇒ 
⇒ 
∴
and ∴
∴
and ∴
Question 11.
Find the roots of the following quadratic equations by factorisation method.
Answer:
Given, 
By splitting the middle term,
⇒ 
⇒ 
⇒ 
∴
and ∴
∴
and ∴
Question 12.
Find the roots of the following quadratic equations by factorisation method.
Answer:
Given, 
⇒ 441x2 - 42x + 1 = 0
By splitting the middle term,
⇒ 441x2 - 21x - 21x + 1 = 0
Taking common in the expression,
⇒ 21x (21x - 1) – 1 (21x - 1) = 0
⇒ (21x - 1) (21 x - 1) = 0
∴ 21x – 1 = 0 and ∴ 21x – 1 = 0
∴
and ∴
Exercise 4.4
Question 1.Find whether the following equations have real roots. If real roots exist, find them.
8x2 + 2x – 3 = 0
Answer:
To check whether the quadratic equation has real roots or not, we need to check the discriminant value i.e.,
D = b2 - 4ac
Given, 8x2 + 2x – 3 = 0
∴ D = 22 - 4(8) (-3)
⇒D = 4 + 96 > 0
Hence, the roots are real and distinct.
To find the roots, use the formula,
Question 2.
Find whether the following equations have real roots. If real roots exist, find them.
-2x2 + 3x + 2 = 0
Answer:
To check whether the quadratic equation has real roots or not, we need to check the discriminant value i.e.,
D = b2 - 4ac
Given, -2x2 + 3x + 2 = 0
∴ D = 32 - 4(-2) (2)
⇒D = 9 + 16 > 0
Hence, the roots are real and distinct.
To find the roots, use the formula,
Question 3.
Find whether the following equations have real roots. If real roots exist, find them.
5x2 - 2x – 10 = 0
Answer:
To check whether the quadratic equation has real roots or not, we need to check the discriminant value i.e.,
D = b2 - 4ac
Given, 5x2 - 2x – 10 = 0
∴ D = (-2)2 - 4(5) (-10)
⇒ D = 4 + 200 > 0
Hence, the roots are real and distinct.
To find the roots, use the formula,
Question 4.
Find whether the following equations have real roots. If real roots exist, find them.
Answer:
To check whether the quadratic equation has real roots or not, we need to check the discriminant value i.e.,
D = b2 - 4ac
Given, 
⇒ 
(x - 5) + (2x - 3) = (2x - 3)(x - 5)
⇒ 3x – 8 = 2x2 - 3x - 10x + 15
⇒ 2x2 - 16x + 23 = 0
⇒ D = (-16)2 - 4(2) (23)
⇒ D = 256 – 184 > 0
Hence, the roots are real.
Question 5.
Find whether the following equations have real roots. If real roots exist, find them.
Answer:
To check whether the quadratic equation has real roots or not, we need to check the discriminant value i.e.,
D = b2 - 4ac
Given, 
∴
⇒
Hence, the roots are real and distinct.
To find the roots, use the formula,
Question 6.
Find a natural number whose square diminished by 84 is equal to thrice of 8 more than the given number.
Answer:
Let the natural number be x,
Square of a number and diminished by 84
= (x2 + 84)
Thrice of 8 more than the given number
= 3(x + 8)
Equating both the equations,
⇒ x2 – 84 = 3(x + 8)
⇒ x2 – 84 = 3x + 24
∴ x2 - 3x – 108 = 0
Using the quadratic formula,
∴ The natural number 
. (* A natural number cannot be negative)
Question 7.
A natural number, when increased by 12, equals 160 times its reciprocal. Find the number.
Answer:
Let the number be x.
According to question,
∴
⇒ x2 + 12x – 160 = 0
Using the quadratic formula,
∴ The natural number is 8. (* A natural number cannot be negative)
Question 8.
A train, travelling at a uniform speed for 360 km, would have taken 48 mins less to travel the same distance, if its peed were 5 km/hr more. Find the original speed of the train.
Answer:
Let the original speed of the train be x km/hr
Then, the increased speed of the train = (x + 5) km/hr
According to question,
[*
]
⇒ 
⇒ [5(360x + 1800) – (360x)] = 4x (x + 5)
⇒ 4x2 + 20x – 9000 = 0
⇒ x2 + 5x – 2250 = 0
By using the quadratic formula,
∴ The original speed of the train is 45 km/hr. (* Speed cannot be negative)
Question 9.
If Zeba were younger by 5 years than what she really is, then the square of her age (in years) would have been 11 more than five times her actual age, what is her age now?
Answer:
Let Zeba’s present age be x.
According to question,
⇒ (x - 5)2 = 5x + 11
⇒ x2 - 10x + 25 = 5x + 11
∴ x2 - 15x + 14 = 0
By using the quadratic formula,
∴ Zeba’s age is 14. (* x – 5 = -4 and age cannot be negative.)
Question 10.
At present Asha’s age (in years) is 2 more than the square of her daughter Nisha’s age. When Nisha grows to her mother’s present age, Asha’s age would be one year less than 10 times the present age of Nisha. Find the present ages of both Asha and Nisha.
Answer:
Let Nisha’s age be x.
Then, Asha’s age = x2 + 2
When Nisha grows to Asha’s present age, Asha’s age be one year less than 10 times that of Nisha’s present age i.e.,
(x2 + 2) + [(x2 + 2) – x] = 10x - 1
⇒ 2x2 - 11x + 5 = 0
By using quadratic formula,
∴ For 
,
Nisha’s age is 5 years.
Asha’s age will be 27 years.
∴ For ,
Nisha’s age is 1/2 years.
Asha’s age will be 
years. (not possible)
Question 11.
In the centre of a rectangular lawn of dimensions 50 m x 40 m, a rectangular pond has to be constructed, so that the area of the grass surrounding the pond would be 1184 m2 [see figure]. Find the length and breadth of the pond.
Answer:
Let the distance between the pond and the rectangular lawn from all sides be x m.
Let the length of the rectangular pond be l m
∴ l = (50 - 2x) m
Let the breadth of the rectangular pond be b m
∴ b = (40 - 2x) m
Given,
Area of the grass surrounding the pond = 1184 m2
Area of the rectangular lawn,
= (l * b) = (50m * 40m) = 2000 m2
Area of the rectangular pond,
= (l * b) = (50 - 2x) (40 - 2x) m2
According to question,
⇒ 2000 - (50 - 2x) (40 - 2x) = 1184
⇒ 2000 - (2000 - 80x - 100x + 4x2) = 1184
⇒ 4x2 - 180x + 1184 = 0
∴ x2 - 45x + 296 = 0
Using the quadratic formula,
For x = 37, Length and breadth of the pond will be −24m and −34m respectively. Hence, length and breadth cannot be negative. (Not possible)
∴ Length of the pond = (50 - 2x) = 34m
∴ Breadth of the pond = (40 - 2x) = 24m
Question 12.
At t min past 2 pm, the time needed by the minute hand of a clock to show 3pm was found to be 3 min less than 
min. Find t.
Answer:
Time (in minutes) taken by the minute hand to travel from 2 pm to 3 pm = 60 minutes
According to question,
⇒ 
⇒ 4(60 - t) = t2 - 12
⇒ 240 - 4t = t2 - 12
∴ t2 + 4t – 252 = 0
Using the quadratic formula,
∴ The required value of t is 14 mins. (* Time cannot be negative)