M l aggarwal understanding icse mathematics class 10 Solutions for Chapter 4 quadratic equations in one variable

Exercises in Chapter 4 quadratic equations in one variable Grade 10

Quadratic Equations in One Variable Exercise 5.1

Quadratic Equations in One Variable Exercise 5.2

Quadratic Equations in One Variable Exercise 5.3

Quadratic Equations in One Variable Exercise 5.4

Quadratic Equations in One Variable Exercise 5.5

Questions in Quadratic Equations in One Variable Exercise 5.1

Check whether the following are quadratic equations:

(i) $\sqrt{3} x^{2}-2 x+3 / 5=0$

(ii) $(2 x+1)(3 x-2)=6(x+1)(x-2)$

(iii) $(x-3)^{3}+5=x^{3}+7 x^{2}-1$

(iv) $x-3 / x=2, x \neq 0$

(v) $x+2 / x=x^{2}, x \neq 0$

(vi) $x^{2}+1 / x^{2}=3, x \neq 0$

In each of the following, determine whether the given numbers are roots of the

given equations or not;

(i) $x^{2}-x+1=0 ; 1,-1$

(ii) $x^{2}-5 x+6=0 ; 2,-3$

(iii) $3 x^{2}-13 x-10=0 ; 5,-2 / 3$

(iv) $6 x^{2}-x-2=0 ;-1 / 2,2 / 3$

In each of the following, determine whether the given numbers are solutions of

the given equation or not:

(i) $x^{2}-3 \sqrt{3} x+6=0 ; x=\sqrt{3},-2 \sqrt{3}$

(ii) $x^{2}-\sqrt{2} x-4=0 ; x=-\sqrt{2}, 2 \sqrt{2}$

(i) If –1/2 is a solution of the equation $3 x^{2}+2 k x-3=0$ , find the value of k.

(ii) If 2/3 is a solution of the equation $7 x^{2}+k x-3=0$ , find the value of k.

(i) If √2 is a root of the equation $k x^{2}+\sqrt{2} x-4=0$ , find the value of k.

(ii) If a is a root of the equation $x^{2}-(a+b) x+k=0$ , find the value of k.

If 2/3 and -3 are the roots of the equation $p x^{2}+7 x+q=0$ , find the values of p and

Questions in Quadratic Equations in One Variable Exercise 5.2

Solve the questions by factorization:

(i) $4 x^{2}=3 x$

(ii) $\left(x^{2}-5 x\right) / 2=0$

Solve the following equations by factorization :

(i) (x – 3) (2x + 5) = 0

(ii) x (2x + 1) = 6

Solve the following equations by factorization :

(i) $x^{2}-3 x-10=0$

(ii) x(2x + 5) = 3

Solve the following equations using factorization:

(i) $3 x^{2}=x+4$

(ii) x(6x – 1) = 35

Solve the following equations using factorization:

(i) $6 p^{2}+11 p-10=0$

(ii) $2 / 3 x^{2}-1 / 3 x=1$

Solve the following equations using factorization:

(i) $(x-4)^{2}+5^{2}=13^{2}$

(ii) $3(x-2)^{2}=147$

Solve the following equations using factorization:

(i) $1 / 7(3 x-5)^{2}=28$

(ii) $3\left(y^{2}-6\right)=y(y+7)-3$

Solve the following equation by factorization :

$x^{2}-4 x-12=0, \text { when } x \in N$

Solve the following equation by factorization :

$2 x^{2}-8 x-24=0 \text { when } x \in \mathbf{I}$

Solve the following equation by factorization :

$5 x^{2}-8 x-4=0 \text { when } x \in Q$

Solve the following equations by factorization :

$2 x^{2}-9 x+10=0$ , when

(i) x ∈ N

(ii) x ∈ Q

Solve the following equations by factorization :

(i) $a^{2} x^{2}+2 a x+1=0, a \neq 0$

(ii) $x^{2}-(p+q) x+p q=0$

Solve the following equation by factorization :

$\mathbf{a}^{2} \mathbf{x}^{2}+\left(\mathbf{a}^{2}+\mathbf{b}^{2}\right) \mathbf{x}+\mathbf{b}^{2}=\mathbf{0}, \mathbf{a} \neq \mathbf{0}$

Questions in Quadratic Equations in One Variable Exercise 5.3

Solve the following equations by using the formula :

(i) $2 x^{2}-7 x+6=0$

(ii) $2 x^{2}-6 x+3=0$

Find the discriminate of the following equations and hence find the nature of roots:

(i) $3 x^{2}-5 x-2=0$

(ii) $2 x^{2}-3 x+5=0$

(iii) $7 x^{2}+8 x+2=0$

(iv) $3 x^{2}+2 x-1=0$

(v) $16 x^{2}-40 x+25=0$

(vi) $2 x^{2}+15 x+30=0$

(i) $x^{2}+7 x-7=0$

(ii) (2x + 3) (3x - 2) + 2 = 0

Solve the following equations using the formula :

(i) $256 x^{2}-32 x+1=0$

(ii) $25 x^{2}+30 x+7=0$

Solve the following equations by using the formula :

(i) $2 x^{2}+\sqrt{5 }x-5=0$

(ii) $\sqrt{3} x^{2}+10 x-8 \sqrt{3}=0$

Solve the following equations by using the formula :

(i) $\frac{x-2}{x+2}+\frac{x+2}{x-2}=4$

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

Solve the following equations by using the formula :

(i) $\mathbf{a}\left(\mathbf{x}^{2}+\mathbf{1}\right)=\left(\mathbf{a}^{2}+\mathbf{1}\right) \mathbf{x}, \mathbf{a} \neq \mathbf{0}$

(ii) $4 x^{2}-4 a x+\left(a^{2}-b^{2}\right)=0$

Solve the following equations by using the formula :

(i) x−1/x = 3, x ≠ 0

(ii) 1/ x + 1/(x−2) = 3, x ≠ 0, 2

Solve the following equation by using the formula :

$\frac{1}{x-2}+\frac{1}{x-3}+\frac{1}{x-4}=0$

Solve for x:

$2\left(\frac{2 x-1}{x+3}\right)-3\left(\frac{x+3}{2 x-1}\right)=5, x \neq-3, \frac{1}{2}$

Solve the following equation by using quadratic equations for x.

(i) $x^{2}-5 x-10=0$

(ii) 5x(x + 2) = 3

Solve the following equations by using the quadratic formula and give your answer

correct to 2 decimal places:

(i) $4 x^{2}-5 x-3=0$

(ii) 2x – 1/x = 1

Solve the following equation: x−18/x = 6. Give your answer correct to two x significant figures. (2011)

Solve the equation $5 x^{2}-3 x-4=0$ and give your answer correct to 3 significant figures:

Questions in Quadratic Equations in One Variable Exercise 5.4

Discuss the nature of the roots of the following quadratic equations:

(i) $x^{2}-4 x-1=0$

(ii) $3 x^{2}-2 x+1 / 3=0$

(iii) $3 x^{2}-4 \sqrt{3} x+4=0$

(iv) $x^{2}-1 / 2 x+4=0$

(v) $-2 x^{2}+x+1=0$

(vi) $2 \sqrt{3} x^{2}-5 x+\sqrt{3}=0$

Find the nature of the roots of the following quadratic equations:

(i) $x^{2}-1 / 2 x-1 / 2=0$

(ii) $x^{2}-2 \sqrt{3} x-1=0$ , If real roots exist, find them.

Without solving the following quadratic equation, find the value of ‘p’ for which

the given equations have real and equal roots:

(i) $\mathbf{p} \mathbf{x}^{2}-\mathbf{4} \mathbf{x}+3=0$

(ii) $x^{2}+(p-3) x+p=0$

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

equal roots:

(i) $k x^{2}-4 x-5=0$

(ii) $(k-4) x^{2}+2(k-4) x+4=0$

Find the value(s) of m for which each of the following quadratic equation has real

and equal roots:

(i) $(3 m+1) x^{2}+2(m+1) x+m=0$

(ii) $x^{2}+2(m-1) x+(m+5)=0$

Find the values of k for which each of the following quadratic equation has equal roots:

(i) $9 x^{2}+k x+1=0$

(ii) $x^{2}-2 k x+7 k-12=0$

Also, find the roots for those values of k in each case.

Find the value(s) of p for which the quadratic equation (2p + 1)x² – (7p + 2)x + (7p – 3) = 0 has equal roots. Also, find these roots.

If – 5 is a root of the quadratic equation 2x² + px – 15 = 0 and the quadratic equation p(x² + x) + k = 0 has equal roots, find the value of k.

Find the value(s) of p for which the equation 2x² + 3x + p = 0 has real roots.

Find the least positive value of k for which the equation x² + kx + 4 = 0 has real roots.

Find the values of p for which the equation 3x² – px + 5 = 0 has real roots.

Questions in Quadratic Equations in One Variable Exercise 5.5

(i) Find two consecutive natural numbers such that the sum of their squares is 61.

(ii) Find two consecutive integers such that the sum of their squares is 61.

(i) If the product of two positive consecutive even integers is 288, find the integers.

(ii) If the product of two consecutive even integers is 224, find the integers.

(iii) Find two consecutive even natural numbers such that the sum of their squares is 340.

(iv) Find two consecutive odd integers such that the sum of their squares is 394.

The sum of two numbers is 9 and the sum of their squares is 41. Taking one number as x, form ail equation in x and solve it to find the numbers.

Five times a certain whole number is equal to three less than twice the square of the number. Find the number.

The Sum of two natural numbers is 8 and the difference of their reciprocals is 2/15. Find the numbers.

The difference between the squares of two numbers is 45. The square of the smaller number is 4 times the larger number. Determine the numbers.

There are three consecutive positive integers such that the sum of the square of the first and the product of the other two is 154. What are the integers?

(i) Find three successive even natural numbers, the sum of whose squares is 308.

(ii) Find three consecutive odd integers, the sum of whose squares is 83.

In a certain positive fraction, the denominator is greater than the numerator by 3. If 1 is subtracted from both the numerator and denominator, the fraction is decreased by 1/14. Find the fraction.

The sum of the numerator and denominator of a certain positive fraction is 8. If 2 is added to both the numerator and denominator, the fraction is increased by 4/35. Find the fraction.

A two-digit number contains the bigger at ten’s place. The product of the digits is 27 and the difference between two digits is 6. Find the number.

A two-digit positive number is such that the product of its digits is 6. If 9 is added to the number, the digits interchange their places. Find the number. (2014)

A rectangle of area 105 cm² has its length equal to x cm. Write down its breadth in terms of x. Given that the perimeter is 44 cm, write down an equation in x and solve it to determine the dimensions of the rectangle.

A rectangular garden 10 m by 16 m is to be surrounded by a concrete walk of uniform width. Given that the area of the walk is 120 square meters, assuming the width of the walk to be x, form an equation in x and solve it to find the value of x. (1992)

(i) Harish made a rectangular garden, with its length 5 meters more than its width. The next year, he increased the length by 3 meters and decreased the width by 2 meters. If the area of the second garden was 119 sqm, was the second garden larger or smaller?

(ii) The length of a rectangle exceeds its breadth by 5 m. If the breadth was doubled and the length reduced by 9 m, the area of the rectangle would have increased by 140 m². Find its dimensions.