Chapter 4 Quadrilateral Equations NCERT Exemplar Solutions Exercise 4.1 Class 10 Maths

Chapter 4 Quadrilateral Equations NCERT Exemplar Solutions Exercise 4.1 Class 10 Maths

Chapter Name

NCERT Maths Exemplar Solutions for Chapter 4 Quadrilateral Equations Exercise 4.1

Book Name

NCERT Exemplar for Class 10 Maths

Other Exercises

  • Exercise 4.2
  • Exercise 4.3
  • Exercise 4.4

Related Study

NCERT Solutions for Class 10 Maths

Exercise 4.1 Solutions

Choose the correct answer from the given four options in the following questions:

1. Which of the following is a quadratic equation?
(A) x2 +2x + 1 = (4 – x)2 + 3
(B) -2x2 = (5 – x)(2x – 2/5)
(C) (k + 1)x2 + (3/2)x = 7, where k = -1 
(D) x3 - x2 = (x - 1)3 

Solution

Correct answer is (D) x3 - x2 = (x - 1)3 
Justification : 
We have quadratic equation:
ax2 + bx + c = 0, 

Above equation represents a quadratic equation.  


2. Which of the following is not a quadratic equation ? 
(A) 2(x - 1)2 = 4x2 - 2x + 1
(B) 2x - x2 = x2  + 5 
(C) (√2x + √3)2 + x2 = 3x2 - 5x 
(D) (x2 + 2x)2 = x4 + 3 + 4x3 

Solution

Correct answer is (D) (x2 + 2x)2 = x4 + 3 + 4x3 
Equation to be quadratic, it should be in the form, 
ax2 + bx + c = 0, a ≠ 0 

This equation represents a cubic equation.


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

Solution

Correct answer is (C) 2x2 - 7x + 6 = 0 
As 2 is a root then putting value 2 in place of x, we should get zero. 

Therefore, x = 2 is not a root of 3x2 - 6x - 2 = 0 


4. If 1/2 is a root of the equation x2 + kx - 5/4 = 0 , then the value of k is 
(A) 2 
(B) -2
(C) 1/4 
(D) 1/2 

Solution

Correct answer is (A) 2. 
As, 1/2 is a root of the equation x2 + kx - 5/4 = 0. 
Putting the value of 1/2 in place of x gives us the value of k. 
As, 
x = 1/2 
(1/2)2 + k(1/2) - (5/4) = 0 
⇒ (k/2) = (5/4) - 1/4
So, k = 2 


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) √2x2 - 3/√2x + 1 = 0
(D) 3x2 - 3x + 3 = 0

Solution

Correct answer is (B) -x2 + 3x - 3 = 0. 
The sum of the roots of a quadratic equation ax2 + bx + c = 0, a ≠ 0 is given by,  
Coefficient of x/coefficient of x2 = -(b/a) 
(A) We have, 
2x2 - 3x + 6 = 0 
Sum of the roots = -b/a 
= - (-3/2) 
Sum of the roots = 3/2 
(B) We have, 
-x2 + 3x - 3 = 0 
Sum of the roots = -b/a 
= -(3/-1) = 3 


6. Value(s) of k for which the quadratic equation 2x2 – kx + k = 0 has equal roots is
(a) 0 only
(b) 4
(c) 8 only
(d) 0, 8

Solution

(D) The condition for equal roots of quadratic equation ax2 + bx + c = 0 is 
b2 - 4ac = 0.

We have, 
2x2 - kx + k = 0 
Condition for equal roots, 
b2 - 4ac = 0 
⇒ (-k)2 - 4(2)(k) = 0 
⇒ k2 - 8k = 0 
⇒ k(k - 8) = 0 
⇒ k = 0 
Or 
k - 8 = 0 
⇒ k = 8 
As, the values of k are 0 and 8. 
The answer is (D) .


7. Which constant must be added and subtracted to solve the quadratic equation
9x2 + 3x/4 - √2 = 0 
by the method of completing the square ? 
(a) 1/8 
(b) 1/64 
(c) 1/4 
(d) 9/64 

Solution

(b) 1/64 
The given equation is 
9x2 + 3x/4 - √2 = 0 
So, to make the expression a complete square, we have to subtract 1/64. 
9x2 + 3x/4 + 1/64 - √2 - 1/64 = 0 


8. The quadratic equation has : 
2x2 - √5x + 1 = 0 
(A) two distinct real roots
(B) two equal real roots 
(C) no real roots 
(D) more than two real roots 

Solution

(C) no real roots
We have, 
2x2 - √5x + 1 = 0 
Now, 
D = b2 - 4ac, 
Checking the following conditions: 
(i) for no real roots D < 0 
(ii) for two equal roots, D = 0 
(iii) for two distinct roots D > 0 and any quadratic equation must have only roots. 
The equation is : 
2x2 - √5x + 1 = 0 
So, 
D = b2 - 4ac 
Where, 
a = 2 , 
b = -√5
c = 1 
D = 5 - 8 
⇒ D = - 3 
As D < 0 So, the given equations has no real roots.


9. Which of the following equations has two distinct real roots ? 
(a) 2x2 - 3√2x + 9/4 = 0 
(b) x2 + x - 5 = 0 
(c) x2 + 3x + 2√2 = 0 
(d) 5x2 - 3x + 1 = 0
Solution

Correct answer is (b) x2 + x - 5 = 0 
We have, 
For real distinct roots D > 0 
(A) Equation is 2x2 - 3√2x + 9/4 = 0 
D = b2 - 4ac 
⇒ D = 9 × 2 - 18 
⇒ D = 0 
For, D = 0, the given equation has two real equal roots. 

(B) Equation is x2 + x - 5 = 0 
D = b2 - 4ac 
⇒ D = (1)2 - 4(1)(-5)  (where, a = 1, b = 1, c = -5)
⇒ D = 1 + 20 
⇒ D = 21 
For, D > 0, the given equation has two distinct real roots. 


10. Which of the following equations has no real roots.
(a) x2 - 4x + 3√2 = 0
(b) x2 + 4x - 3√2 = 0
(c) x2 - 4x - 3√2 = 0
(d) 3x2 + 4√3x + 4 = 0
Solution

Correct answer is (a) x2 - 4x + 3√2 = 0 
Given equation is x2 - 4x + 3√2 = 0 
D = b2 - 4ac 


11. (x2 + 1)2 - x2 = 0 has 
(a) four real roots 
(b) two real roots
(c) no real roots
(d) one real root

Solution

(c) no real roots 
We have 

So, the given equation y2 + y + 1 = 0 has no values of y in equation y2 + 1y + 1 = 0 or if y is not real then x2 will not be real so no values of x are real or the given equation has no real roots.

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