## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Chapter Test

Solve the following equations (1 to 4) by factorisation:

Question 1.

(i) x² + 6x – 16 = 0

(ii) 3x² + 11x + 10 = 0

Solution:

Question 2.

(i) 2x² + ax – a² = 0

(ii) √3x² + 10x + 7√3 = 0

Solution:

Question 3.

(i) x(x + 1) + (x + 2)(x + 3) = 42

(ii) \(\frac { 6 }{ x } -\frac { 2 }{ x-1 } =\frac { 1 }{ x-2 } \)

Solution:

Question 4.

(i) \(\sqrt { x+15 } =x+3 \)

(ii) \(\sqrt { { 3x }^{ 2 }-2x-1 } =2x-2\)

Solution:

Solve the following equations (5 to 8) by using formula :

Question 5.

(i) 2x² – 3x – 1 = 0

(ii) \(x\left( 3x+\frac { 1 }{ 2 } \right) =6\)

Solution:

Question 6.

(i) \(\frac { 2x+5 }{ 3x+4 } =\frac { x+1 }{ x+3 } \)

(ii) \(\frac { 2 }{ x+2 } -\frac { 1 }{ x+1 } =\frac { 4 }{ x+4 } -\frac { 3 }{ x+3 } \)

Solution:

Question 7.

(i) \(\frac { 3x-4 }{ 7 } +\frac { 7 }{ 3x-4 } =\frac { 5 }{ 2 } ,x\neq \frac { 4 }{ 3 } \)

(ii) \(\frac { 4 }{ x } -3=\frac { 5 }{ 2x+3 } ,x\neq 0,-\frac { 3 }{ 2 } \)

Solution:

Question 8.

(i) x² + (4 – 3a)x – 12a = 0

(ii) 10ax² – 6x + 15ax – 9 = 0, a≠0

Solution:

Question 9.

Solve for x using the quadratic formula. Write your answer correct to two significant figures: (x – 1)² – 3x + 4 = 0. (2014)

Solution:

Question 10.

Discuss the nature of the roots of the following equations:

(i) 3x² – 7x + 8 = 0

(ii) x² – \(\\ \frac { 1 }{ 2 } x\) – 4 = 0

(iii) 5x² – 6√5x + 9 = 0

(iv) √3x² – 2x – √3 = 0

Solution:

Question 11.

Find the values of k so that the quadratic equation (4 – k) x² + 2 (k + 2) x + (8k + 1) = 0 has equal roots.

Solution:

Question 12.

Find the values of m so that the quadratic equation 3x² – 5x – 2m = 0 has two distinct real roots.

Solution:

Question 13.

Find the value(s) of k for which each of the following quadratic equation has equal roots:

(i) 3kx² = 4(kx – 1)

(ii) (k + 4) x² + (k + 1)x + 1 = 0

Also, find the roots for that value (s) of k in each case.

Solution:

Question 14.

Find two natural numbers which differ by 3 and whose squares have the sum 117.

Solution:

Question 15.

Divide 16 into two parts such that the twice the square of the larger part exceeds the square of the smaller part by 164.

Solution:

Question 16.

Two natural numbers are in the ratio 3 : 4. Find the numbers if the difference between their squares is 175.

Solution:

Question 17.

Two squares have sides A cm and (x + 4) cm. The sum of their areas is 656 sq. cm. Express this as an algebraic equation and solve it to find the sides of the squares.

Solution:

Question 18.

The length of a rectangular garden is 12 m more than its breadth. The numerical value of its area is equal to 4 times the numerical value of its perimeter. Find the dimensions of the garden.

Solution:

Question 19.

A farmer wishes to grow a 100 m² rectangular vegetable garden. Since he has with him only 30 m barbed wire, the fences three sides of the rectangular garden letting the compound wall of his house act as the fourth side fence. Find the dimensions of his garden.

Solution:

Question 20.

The hypotenuse of a right-angled triangle is 1 m less than twice the shortest side. If the third side is 1 m more than the shortest side, find the sides of the triangle.

Solution:

Question 21.

A wire 112 cm long is bent to form a right-angled triangle. If the hypotenuse is 50 cm long, find the area of the triangle.

Solution:

Question 22.

Car A travels x km for every litre of petrol, while car B travels (x + 5) km for every litre of petrol.

(i) Write down the number of litres of petrol used by car A and car B in covering a distance of 400 km.

(ii) If car A uses 4 litres of petrol more than car B in covering 400 km. write down an equation, in A and solve it to determine the number of litres of petrol used by car B for the journey.

Solution:

Question 23.

The speed of a boat in still water is 11 km/ hr. It can go 12 km up-stream and return downstream to the original point in 2 hours 45 minutes. Find the speed of the stream

Solution:

Question 24.

By selling an article for Rs. 21, a trader loses as much per cent as the cost price of the article. Find the cost price.

Solution:

Question 25.

A man spent Rs. 2800 on buying a number of plants priced at Rs x each. Because of the number involved, the supplier reduced the price of each plant by Rupee 1. The man finally paid Rs. 2730 and received 10 more plants. Find x.

Solution:

Question 26.

Forty years hence, Mr Pratap’s age will be the square of what it was 32 years ago. Find the present age.

Solution: