Chapters
Chapter 2: Polynomials
Chapter 3: Pair of Linear Equations in Two Variables
Chapter 4: Quadratic Equations
Chapter 5: Arithmetic Progression
Chapter 6: Co-Ordinate Geometry
Chapter 7: Triangles
Chapter 8: Circles
Chapter 9: Constructions
Chapter 10: Trigonometric Ratios
Chapter 11: Trigonometric Identities
Chapter 12: Trigonometry
Chapter 13: Areas Related to Circles
Chapter 14: Surface Areas and Volumes
Chapter 15: Statistics
Chapter 16: Probability
Chapter 4: Quadratic Equations
Exercise 4.1 [Pages 4 - 5]
RD Sharma solutions for Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.1 [Pages 4 - 5]
Check whether the following is quadratic equation or not.
x^{2} + 6x − 4 = 0
Check whether the following is quadratic equation or not.
`sqrt(3x^2)-2x+1/2=0`
Check whether the following is quadratic equation or not.
`x^2+1/x^2=5`
Check whether the following is quadratic equation or not.
`x-3/x=x^2`
Check whether the following is quadratic equation or not.
`2x^2-sqrt(3x)+9=0`
Check whether the following is quadratic equation or not.
`x^2 - 2x - sqrtx - 5 = 0`
Check whether the following is quadratic equation or not.
3x^{2} - 5x + 9 = x^{2} - 7x + 3
Check whether the following is quadratic equation or not.
`x+1/x=1`
Check whether the following is quadratic equation or not.
x^{2} - 3x = 0
Check whether the following is quadratic equation or not.
`(x+1/x)^2=3(1+1/x)+4`
Check whether the following is quadratic equation or not.
(2𝑥 + 1)(3𝑥 + 2) = 6(𝑥 − 1)(𝑥 − 2)
Check whether the following is quadratic equation or not.
`x+1/x=x^2`, x ≠ 0
Check whether the following is quadratic equation or not.
16x^{2} − 3 = (2x + 5) (5x − 3)
Check whether the following is quadratic equation or not.
(x + 2)^{3} = x^{3} − 4
Check whether the following is quadratic equation or not.
x(x + 1) + 8 = (x + 2) (x - 2)
In the following, determine whether the given values are solutions of the given equation or not:
x^{2} - 3x + 2 = 0, x = 2, x = -1
In the following, determine whether the given values are solutions of the given equation or not:
x^{2} + x + 1 = 0, x = 0, x = 1
In the following, determine whether the given values are solutions of the given equation or not:
`x^2 - 3sqrt3x+6=0`, `x=sqrt3`, `x=-2sqrt3`
In the following, determine whether the given values are solutions of the given equation or not:
`x+1/x=13/6`, `x=5/6`, `x=4/3`
In the following, determine whether the given values are solutions of the given equation or not:
2x^{2} - x + 9 = x^{2} + 4x + 3, x = 2, x =3
In the following, determine whether the given values are solutions of the given equation or not:
`x^2-sqrt2x-4=0`, `x=-sqrt2`, `x=-2sqrt2`
In the following, determine whether the given values are solutions of the given equation or not:
a^{2}x^{2} - 3abx + 2b^{2} = 0, `x=a/b`, `x=b/a`
In the following, find the value of k for which the given value is a solution of the given equation:
7x^{2} + kx - 3 = 0, `x=2/3`
In the following, find the value of k for which the given value is a solution of the given equation:
x^{2} - x(a + b) + k = 0, x = a
In the following, find the value of k for which the given value is a solution of the given equation:
`kx^2+sqrt2x-4=0`, `x=sqrt2`
In the following, find the value of k for which the given value is a solution of the given equation:
x^{2} + 3ax + k = 0, x = -a
Determine if, 3 is a root of the equation given below:
`sqrt(x^2-4x+3)+sqrt(x^2-9)=sqrt(4x^2-14x+16)`
If x = 2/3 and x = −3 are the roots of the equation ax^{2} + 7x + b = 0, find the values of aand b.
Exercise 4.2 [Page 8]
RD Sharma solutions for Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.2 [Page 8]
The product of two consecutive positive integers is 306. Form the quadratic equation to find the integers, if x denotes the smaller integer.
John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 128. Form the quadratic equation to find how many marbles they had to start with, if John had x marbles.
A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of articles produced in a day. On a particular day, the total cost of production was Rs. 750. If x denotes the number of toys produced that day, form the quadratic equation fo find x.
The height of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, form the quadratic equation to find the base of the triangle.
An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore. If the average speed of the express train is 1 1 km/hr more than that of the passenger train, form the quadratic equation to find the average speed of express train.
A train travels 360 km at a uniform speed. If the speed had been 5 km/hr more, it would have taken 1 hour less for the same journey. Form the quadratic eqiation to find the speed of the train.
Exercise 4.3, Exercise 3.4 [Pages 19 - 21]
RD Sharma solutions for Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.3, Exercise 3.4 [Pages 19 - 21]
Solve the following quadratic equations by factorization:
(x − 4) (x + 2) = 0
Solve the following quadratic equations by factorization:
(2x + 3)(3x − 7) = 0
Solve the following quadratic equations by factorization:
3x^{2} − 14x − 5 = 0
Solve the following quadratic equations by factorization:
9x^{2} − 3x − 2 = 0
Solve the following quadratic equations by factorization:
`1/(x-1)-1/(x+5)=6/7` , x ≠ 1, -5
Solve the following quadratic equations by factorization:
6x^{2} + 11x + 3 = 0
Solve the following quadratic equations by factorization:
5x^{2} - 3x - 2 = 0
Solve the following quadratic equations by factorization:
48x^{2} − 13x − 1 = 0
Solve the following quadratic equations by factorization:
3x^{2} = -11x - 10
Solve the following quadratic equations by factorization:
25x(x + 1) = -4
Solve the following quadratic equations by factorization:
\[16x - \frac{10}{x} = 27\]
Solve the following quadratic equations by factorization:
`1/x-1/(x-2)=3` , x ≠ 0, 2
Solve the following quadratic equations by factorization:
`1/(x+4)-1/(x-7)=11/30` , x ≠ 4, 7
Solve the following quadratic equations by factorization:\[\frac{1}{x - 3} + \frac{2}{x - 2} = \frac{8}{x}; x \neq 0, 2, 3\]
Solve the following quadratic equations by factorization:
a^{2}x^{2} - 3abx + 2b^{2} = 0
Solve the following quadratic equations by factorization:
\[9 x^2 - 6 b^2 x - \left( a^4 - b^4 \right) = 0\]
Solve the following quadratic equations by factorization:
4x^{2} + 4bx - (a^{2} - b^{2}) = 0
Solve the following quadratic equations by factorization:
ax^{2} + (4a^{2} - 3b)x - 12ab = 0
Solve the following quadratic equations by factorization: \[2 x^2 + ax - a^2 = 0\]
Solve the following quadratic equations by factorization: \[\frac{16}{x} - 1 = \frac{15}{x + 1}; x \neq 0, - 1\]
Solve the following quadratic equations by factorization:
`(x+3)/(x+2)=(3x-7)/(2x-3)`
Solve the following quadratic equations by factorization:
`(2x)/(x-4)+(2x-5)/(x-3)=25/3`
Solve the following quadratic equations by factorization:
`(x+3)/(x-2)-(1-x)/x=17/4`
Solve the following quadratic equations by factorization:
`(x-3)/(x+3)-(x+3)/(x-3)=48/7` , x ≠ 3, x ≠ -3
Solve the following quadratic equations by factorization:
`1/(x-2)+2/(x-1)=6/x` , x ≠ 0
Solve the following quadratic equations by factorization:
`(x+1)/(x-1)-(x-1)/(x+1)=5/6` , x ≠ 1, x ≠ -1
Solve the following quadratic equations by factorization:
`(x-1)/(2x+1)+(2x+1)/(x-1)=5/2` , x ≠ -1/2, 1
Solve the following quadratic equations by factorization:
\[\frac{4}{x} - 3 = \frac{5}{2x + 3}, x \neq 0, - \frac{3}{2}\]
Solve the following quadratic equations by factorization: \[\frac{x - 4}{x - 5} + \frac{x - 6}{x - 7} = \frac{10}{3}; x \neq 5, 7\]
Solve the following quadratic equations by factorization:
\[\frac{x - 2}{x - 3} + \frac{x - 4}{x - 5} = \frac{10}{3}; x \neq 3, 5\]
Solve the following quadratic equations by factorization: \[\frac{5 + x}{5 - x} - \frac{5 - x}{5 + x} = 3\frac{3}{4}; x \neq 5, - 5\]
Solve the following quadratic equations by factorization: \[\frac{3}{x + 1} - \frac{1}{2} = \frac{2}{3x - 1}, x \neq - 1, \frac{1}{3}\]
Solve the following quadratic equations by factorization: \[\frac{3}{x + 1} + \frac{4}{x - 1} = \frac{29}{4x - 1}; x \neq 1, - 1, \frac{1}{4}\]
Solve the following quadratic equations by factorization: \[\frac{2}{x + 1} + \frac{3}{2(x - 2)} = \frac{23}{5x}; x \neq 0, - 1, 2\]
Solve the following quadratic equations by factorization: \[\sqrt{3} x^2 - 2\sqrt{2}x - 2\sqrt{3} = 0\]
Solve the following quadratic equations by factorization:
`4sqrt3x^2+5x-2sqrt3=0`
Solve the following quadratic equations by factorization:
`sqrt2x^2-3x-2sqrt2=0`
Solve the following quadratic equations by factorization:
`3x^2-2sqrt6x+2=0`
Find the roots of the quadratic equation \[\sqrt{2} x^2 + 7x + 5\sqrt{2} = 0\].
Solve the following quadratic equations by factorization:
`m/nx^2+n/m=1-2x`
Solve the following quadratic equations by factorization:
`(x-a)/(x-b)+(x-b)/(x-a)=a/b+b/a`
Solve the following quadratic equations by factorization:
`1/((x-1)(x-2))+1/((x-2)(x-3))+1/((x-3)(x-4))=1/6`
Solve the following quadratic equation by factorization: \[\frac{a}{x - b} + \frac{b}{x - a} = 2\]
Solve the following quadratic equations by factorization: \[\frac{x + 1}{x - 1} + \frac{x - 2}{x + 2} = 4 - \frac{2x + 3}{x - 2}; x \neq 1, - 2, 2\]
Solve the following quadratic equations by factorization:
`a/(x-a)+b/(x-b)=(2c)/(x-c)`
Solve the following quadratic equations by factorization:
x^{2} + 2ab = (2a + b)x
Solve the following quadratic equations by factorization:
(a + b)^{2}x^{2} - 4abx - (a - b)^{2} = 0
Solve the following quadratic equations by factorization:
a(x^{2} + 1) - x(a^{2} + 1) = 0
Solve the following quadratic equations by factorization:
x^{2} - x - a(a + 1) = 0
Solve the following quadratic equations by factorization:
`x^2+(a+1/a)x+1=0`
Solve the following quadratic equations by factorization:
abx^{2} + (b^{2} - ac)x - bc = 0
Solve the following quadratic equations by factorization:
a^{2}b^{2}x^{2} + b^{2}x - a^{2}x - 1 = 0
Solve the following quadratic equations by factorization:
`(x-1)/(x-2)+(x-3)/(x-4)=3 1/3`, x ≠ 2, 4
Solve the following quadratic equations by factorization: \[\frac{1}{2a + b + 2x} = \frac{1}{2a} + \frac{1}{b} + \frac{1}{2x}\]
Solve the following quadratic equations by factorization:
\[3\left( \frac{3x - 1}{2x + 3} \right) - 2\left( \frac{2x + 3}{3x - 1} \right) = 5; x \neq \frac{1}{3}, - \frac{3}{2}\]
Solve the following quadratic equations by factorization:
\[3\left( \frac{7x + 1}{5x - 3} \right) - 4\left( \frac{5x - 3}{7x + 1} \right) = 11; x \neq \frac{3}{5}, - \frac{1}{7}\]
Solve the following quadratic equations by factorization:
`(x-5)(x-6)=25/(24)^2`
Solve the following quadratic equations by factorization:
`7x + 3/x=35 3/5`
Exercise 4.4 [Page 26]
RD Sharma solutions for Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.4 [Page 26]
Find the roots of the following quadratic equations (if they exist) by the method of completing the square.
`x^2-4sqrt2x+6=0`
Find the roots of the following quadratic equations, if they exist, by the method of completing the square 2x^{2} – 7x + 3 = 0
Find the roots of the following quadratic equations (if they exist) by the method of completing the square.
3x^{2} + 11x + 10 = 0
Find the roots of the following quadratic equations, if they exist, by the method of completing the square 2x^{2} + x – 4 = 0
Find the roots of the following quadratic equations, if they exist, by the method of completing the square 2x^{2} + x + 4 = 0
Find the roots of the following quadratic equations, if they exist, by the method of completing the square `4x^2 + 4sqrt3x + 3 = 0`
Find the roots of the following quadratic equations (if they exist) by the method of completing the square.
`sqrt2x^2-3x-2sqrt2=0`
Find the roots of the following quadratic equations (if they exist) by the method of completing the square.
`sqrt3x^2+10x+7sqrt3=0`
Find the roots of the following quadratic equations (if they exist) by the method of completing the square.
`x^2-(sqrt2+1)x+sqrt2=0`
Find the roots of the following quadratic equations (if they exist) by the method of completing the square.
x^{2} - 4ax + 4a^{2} - b^{2} = 0
Exercise 4.5 [Page 32]
RD Sharma solutions for Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.5 [Page 32]
Write the discriminant of the following quadratic equations:
2x^{2} - 5x + 3 = 0
Write the discriminant of the following quadratic equations:
x^{2} + 2x + 4 = 0
Write the discriminant of the following quadratic equations:
(x − 1) (2x − 1) = 0
Write the discriminant of the following quadratic equations:
x^{2} - 2x + k = 0, k ∈ R
Write the discriminant of the following quadratic equations:
`sqrt3x^2+2sqrt2x-2sqrt3=0`
Write the discriminant of the following quadratic equations:
x^{2} - x + 1 = 0
In the following, determine whether the given quadratic equation have real roots and if so, find the roots:
16x^{2} = 24x + 1
In the following, determine whether the given quadratic equation have real roots and if so, find the roots:
x^{2} + x + 2 = 0
In the following, determine whether the given quadratic equation have real roots and if so, find the roots:
`sqrt3x^2+10x-8sqrt3=0`
In the following, determine whether the given quadratic equation have real roots and if so, find the roots:
3x^{2} - 2x + 2 = 0
In the following, determine whether the given quadratic equation have real roots and if so, find the roots:
`2x^2-2sqrt6x+3=0`
In the following, determine whether the given quadratic equation have real roots and if so, find the roots:
3a^{2}x^{2} + 8abx + 4b^{2} = 0, a ≠ 0
In the following, determine whether the given quadratic equation have real roots and if so, find the roots:
`3x^2+2sqrt5x-5=0`
In the following, determine whether the given quadratic equation have real roots and if so, find the roots:
x^{2} - 2x + 1 = 0
In the following, determine whether the given quadratic equation have real roots and if so, find the roots:
`2x^2+5sqrt3x+6=0`
In the following, determine whether the given quadratic equation have real roots and if so, find the roots:
`sqrt2x^2+7x+5sqrt2=0`
In the following, determine whether the given quadratic equation have real roots and if so, find the roots:
`2x^2-2sqrt2x+1=0`
In the following, determine whether the given quadratic equation have real roots and if so, find the roots:
3x^{2} - 5x + 2 = 0
Solve for x
`(x-1)/(x-2)+(x-3)/(x-4)=3 1/3`; x ≠ 2, 4
Solve for x
`1/x - 1/(x-2)=3`, x ≠ 0, 2
Solve for x
`x+1/x=3`, x ≠ 0
Solve for x: \[\frac{16}{x} - 1 = \frac{15}{x + 1}, x \neq 0, - 1\]
Solve for x: \[\frac{1}{x - 3} - \frac{1}{x + 5} = \frac{1}{6}, x \neq 3, - 5\]
Exercise 4.6 [Pages 41 - 43]
RD Sharma solutions for Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.6 [Pages 41 - 43]
Find the nature of the roots of the following quadratic equation. If the real roots exist, find them
2x^{2} - 3x + 5 = 0
Find the nature of the roots of the following quadratic equations. If the real roots exist, find them
2x^{2} - 6x + 3 = 0
Determine the nature of the roots of the following quadratic equation:
`3/5x^2-2/3x+1=0`
Find the nature of the roots of the following quadratic equations. If the real roots exist, find them;
`3x^2 - 4sqrt3x + 4 = 0`
Determine the nature of the roots of the following quadratic equation:
`3x^2-2sqrt6x+2=0`
Find the values of k for which the roots are real and equal in each of the following equation:
kx^{2} + 4x + 1 = 0
Find the values of k for which the roots are real and equal in each of the following equation:
`kx^2-2sqrt5x+4=0`
Find the values of k for which the roots are real and equal in each of the following equation:
3x^{2} - 5x + 2k = 0
Find the values of k for which the roots are real and equal in each of the following equation:
4x^{2} + kx + 9 = 0
Find the values of k for which the roots are real and equal in each of the following equation:
2kx^{2} - 40x + 25 = 0
Find the values of k for which the roots are real and equal in each of the following equation:
9x^{2} - 24x + k = 0
Find the values of k for which the roots are real and equal in each of the following equation:
4x^{2} - 3kx + 1 = 0
Find the values of k for which the roots are real and equal in each of the following equation:
x^{2} - 2(5 + 2k)x + 3(7 + 10k) = 0
Find the values of k for which the roots are real and equal in each of the following equation:
(3k+1)x^{2} + 2(k + 1)x + k = 0
Find the values of k for which the roots are real and equal in each of the following equation:
kx^{2} + kx + 1 = -4x^{2} - x
Find the values of k for which the roots are real and equal in each of the following equation:
(k + 1)x^{2} + 2(k + 3)x + (k + 8) = 0
Find the values of k for which the roots are real and equal in each of the following equation:
x^{2} - 2kx + 7k - 12 = 0
Find the values of k for which the roots are real and equal in each of the following equation:
(k + 1)x^{2} - 2(3k + 1)x + 8k + 1 = 0
Find the values of k for which the roots are real and equal in each of the following equation:
(2k + 1)x^{2} + 2(k + 3)x + (k + 5) = 0
Find the values of k for which the roots are real and equal in each of the following equation:
4x^{2} - 2(k + 1)x + (k + 4) = 0
Find the values of k for which the roots are real and equal in each of the following equation:
\[4 x^2 - 2\left( k + 1 \right)x + \left( k + 1 \right) = 0\]
In the following determine the set of values of k for which the given quadratic equation has real roots:
2x^{2} + 3x + k = 0
In the following determine the set of values of k for which the given quadratic equation has real roots: \[2 x^2 + x + k = 0\]
In the following determine the set of values of k for which the given quadratic equation has real roots:
2x^{2} - 5x - k = 0
In the following determine the set of values of k for which the given quadratic equation has real roots:
kx^{2} + 6x + 1 = 0
In the following determine the set of values of k for which the given quadratic equation has real roots:
3x^{2} + 2x + k = 0
Find the value of k for which the following equations have real and equal roots:
\[x^2 - 2\left( k + 1 \right)x + k^2 = 0\]
Find the values of k for which the roots are real and equal in each of the following equation:
k^{2}x^{2} - 2(2k - 1)x + 4 = 0
Find the value of k for which the following equations have real and equal roots:
\[\left( k + 1 \right) x^2 - 2\left( k - 1 \right)x + 1 = 0\]
Find the value of k for which the following equations have real and equal roots:
\[x^2 + k\left( 2x + k - 1 \right) + 2 = 0\]
Find the values of k for which the roots are real and equal in each of the following equation:
2x^{2} + kx + 3 = 0
Find the values of k for which the roots are real and equal in each of the following equation:
kx(x - 2) + 6 = 0
Find the values of k for which the roots are real and equal in each of the following equation:
x^{2} - 4kx + k = 0
Find the values of k for which the roots are real and equal in each of the following equation:
\[kx\left( x - 2\sqrt{5} \right) + 10 = 0\]
Find the values of k for which the roots are real and equal in each of the following equation:\[px(x - 3) + 9 = 0\]
Find the values of k for which the roots are real and equal in each of the following equation:
\[4 x^2 + px + 3 = 0\]
Find the values of k for which the given quadratic equation has real and distinct roots:
kx^{2} + 2x + 1 = 0
Find the values of k for which the given quadratic equation has real and distinct roots:
kx^{2} + 6x + 1 = 0
Find the values of k for which the given quadratic equation has real and distinct roots:
x^{2} - kx + 9 = 0
For what value of k, (4 - k)x^{2} + (2k + 4)x + (8k + 1) = 0, is a perfect square.
Find the least positive value of k for which the equation x^{2} + kx + 4 = 0 has real roots.
Find the values of k for which the quadratic equation
\[\left( 3k + 1 \right) x^2 + 2\left( k + 1 \right)x + 1 = 0\] has equal roots. Also, find the roots.
Find the values of p for which the quadratic equation
If −5 is a root of the quadratic equation\[2 x^2 + px - 15 = 0\] and the quadratic equation \[p( x^2 + x) + k = 0\] has equal roots, find the value of k.
If 2 is a root of the quadratic equation \[3 x^2 + px - 8 = 0\] and the quadratic equation \[4 x^2 - 2px + k = 0\] has equal roots, find the value of k.
If 1 is a root of the quadratic equation \[3 x^2 + ax - 2 = 0\] and the quadratic equation \[a( x^2 + 6x) - b = 0\] has equal roots, find the value of b.
Find the value of p for which the quadratic equation
\[\left( p + 1 \right) x^2 - 6(p + 1)x + 3(p + 9) = 0, p \neq - 1\] has equal roots. Hence, find the roots of the equation.
Disclaimer: There is a misprinting in the given question. In the question 'q' is printed instead of 9.
Determine the nature of the roots of the following quadratic equation:
(x - 2a)(x - 2b) = 4ab
Determine the nature of the roots of the following quadratic equation:
9a^{2}b^{2}x^{2} - 24abcdx + 16c^{2}d^{2} = 0
Determine the nature of the roots of the following quadratic equation:
2(a^{2} + b^{2})x^{2} + 2(a + b)x + 1 = 0
Determine the nature of the roots of the following quadratic equation:
(b + c)x^{2} - (a + b + c)x + a = 0
In the following determine the set of values of k for which the given quadratic equation has real roots:
x^{2} - kx + 9 = 0
In the following determine the set of values of k for which the given quadratic equation has real roots:
2x^{2} + kx + 2 = 0
In the following determine the set of values of k for which the given quadratic equation has real roots:
4x^{2} - 3kx + 1 = 0
In the following determine the set of values of k for which the given quadratic equation has real roots:
2x^{2} + kx - 4 = 0
If the roots of the equation (b - c)x^{2} + (c - a)x + (a - b) = 0 are equal, then prove that 2b = a + c.
If the roots of the equation (a^{2} + b^{2})x^{2} − 2 (ac + bd)x + (c^{2} + d^{2}) = 0 are equal, prove that `a/b=c/d`.
If the roots of the equations ax^{2} + 2bx + c = 0 and `bx^2-2sqrt(ac)x+b = 0` are simultaneously real, then prove that b^{2} = ac.
If p, q are real and p ≠ q, then show that the roots of the equation (p − q) x^{2} + 5(p + q) x− 2(p − q) = 0 are real and unequal.
If the roots of the equation (c^{2} – ab) x^{2} – 2 (a^{2} – bc) x + b^{2} – ac = 0 in x are equal, then show that either a = 0 or a^{3} + b^{3} + c^{3} = 3abc
Show that the equation 2(a^{2} + b^{2})x^{2} + 2(a + b)x + 1 = 0 has no real roots, when a ≠ b.
Prove that both the roots of the equation (x - a)(x - b) +(x - b)(x - c)+ (x - c)(x - a) = 0 are real but they are equal only when a = b = c.
If a, b, c are real numbers such that ac ≠ 0, then show that at least one of the equations ax^{2} + bx + c = 0 and -ax^{2} + bx + c = 0 has real roots.
If the equation \[\left( 1 + m^2 \right) x^2 + 2 mcx + \left( c^2 - a^2 \right) = 0\] has equal roots, prove that c^{2} = a^{2}(1 + m^{2}).
Exercise 4.7, Exercise 4.6 [Pages 51 - 53]
RD Sharma solutions for Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.7, Exercise 4.6 [Pages 51 - 53]
Find the consecutive numbers whose squares have the sum 85.
Divide 29 into two parts so that the sum of the squares of the parts is 425.
Two squares have sides x cm and (x + 4)cm. The sum of this areas is 656 cm2. Find the sides of the squares.
The sum of two numbers is 48 and their product is 432. Find the numbers?
If an integer is added to its square, the sum is 90. Find the integer with the help of quadratic equation.
Find the whole numbers which when decreased by 20 is equal to 69 times the reciprocal of the members.
Find the two consecutive natural numbers whose product is 20.
The sum of the squares of the two consecutive odd positive integers as 394. Find them.
The sum of two numbers is 8 and 15 times the sum of their reciprocals is also 8. Find the numbers.
The sum of a numbers and its positive square root is 6/25. Find the numbers.
The sum of a number and its square is 63/4. Find the numbers.
There are three consecutive integers such that the square of the first increased by the product of the first increased by the product of the others the two gives 154. What are the integers?
The product of two successive integral multiples of 5 is 300. Determine the multiples.
The sum of the squares of two numbers as 233 and one of the numbers as 3 less than twice the other number find the numbers.
Find the consecutive even integers whose squares have the sum 340.
The difference of two numbers is 4. If the difference of their reciprocals is 4/21. Find the numbers.
Let us find two natural numbers which differ by 3 and whose squares have the sum 117.
The sum of the squares of three consecutive natural numbers as 149. Find the numbers
Sum of two numbers is 16. The sum of their reciprocals is 1/3. Find the numbers.
Determine two consecutive multiples of 3, whose product is 270.
The sum of a number and its reciprocal is 17/4. Find the number.
A two-digit number is such that the products of its digits is 8. When 18 is subtracted from the number, the digits interchange their places. Find the number?
A two digits number is such that the product of the digits is 12. When 36 is added to the number, the digits inter change their places determine the number.
A two digit number is such that the product of the digits is 16. When 54 is subtracted from the number the digits are interchanged. Find the number
Two numbers differ by 3 and their product is 504. Find the number
Two number differ by 4 and their product is 192. Find the numbers?
A two digit number is 4 times the sum of its digits and twice the product of its digits. Find the number.
The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find two numbers.
The sum of two numbers is 18. The sum of their reciprocals is 1/4. Find the numbers.
The sum of two number a and b is 15, and the sum of their reciprocals `1/a` and `1/b` is 3/10. Find the numbers a and b.
The sum of two numbers is 9. The sum of their reciprocals is 1/2. Find the numbers.
Three consecutive positive integers are such that the sum of the square of the first and the product of other two is 46, find the integers.
The difference of squares of two number is 88. If the larger number is 5 less than twice the smaller number, then find the two numbers.
The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find two numbers.
Find two consecutive odd positive integers, sum of whose squares is 970.
The difference of two natural numbers is 3 and the difference of their reciprocals is \[\frac{3}{28}\].Find the numbers.
The sum of the squares of two consecutive odd numbers is 394. Find the numbers.
The sum of the squares of two consecutive multiples of 7 is 637. Find the multiples.
The sum of the squares of two consecutive even numbers is 340. Find the numbers.
The numerator of a fraction is 3 less than the denominator. If 2 is added to both the numerator and the denominator, then the sum of the new fraction and the original fraction is \[\frac{29}{20}\].Find the original fraction.
Exercise 4.8 [Pages 58 - 59]
RD Sharma solutions for Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.8 [Pages 58 - 59]
The speed of a boat in still water is 8 km/hr. It can go 15 km upstream and 22 km downstream in 5 hours. Find the speed of the stream.
A passenger train takes 3 hours less for a journey of 360 km, if its speed is increased by 10 km/hr from its usual speed. What is the usual speed?
A fast train takes one hour less than a slow train for a journey of 200 km. If the speed of the slow train is 10 km/hr less than that of the fast train, find the speed of the two trains.
A passenger train takes one hour less for a journey of 150 km if its speed is increased by 5 km/hr from its usual speed. Find the usual speed of the train.
The time taken by a person to cover 150 km was 2.5 hrs more than the time taken in the return journey. If he returned at a speed of 10 km/hr more than the speed of going, what was the speed per hour in each direction?
A plane left 40 minutes late due to bad weather and in order to reach its destination, 1600 km away in time, it had to increase its speed by 400 km/hr from its usual speed. Find the usual speed of the plane.
An aeroplane take 1 hour less for a journey of 1200 km if its speed is increased by 100 km/hr from its usual speed. Find its usual speed.
A passenger train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km/hr from its usual speed. Find the usual speed of the train.
A train travels at a certain average speed for a distance 63 km and then travels a distance of 72 km at an average speed of 6 km/hr more than the original speed, If it takes 3 hours to complete total journey, what is its original average speed?
A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hour more, it would have taken 30 minutes less for a journey. Find the original speed of the train.
A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.
An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11km/h more than that of the passenger train, find the average speed of the two trains.
An aeroplane left 50 minutes later than its scheduled time, and in order to reach the destination, 1250 km away, in time, it had to increase its speed by 250 km/hr from its usual speed. Find its usual speed.
While boarding an aeroplane, a passenger got hurt. The pilot showing promptness and concern, made arrangements to hospitalise the injured and so the plane started late by 30 minutes to reach the destination, 1500 km away in time, the pilot increased the speed by 100 km/hr. Find the original speed/hour of the plane.
A motor boat whose speed in still water is 18 km/hr takes 1 hour more to go 24 km up stream that to return down stream to the same spot. Find the speed of the stream.
A car moves a distance of 2592 km with uniform speed. The number of hours taken for the journey is one-half the number representing the speed, in km/hour. Find the time taken to cover the distance.
Exercise 4.9 [Pages 61 - 62]
RD Sharma solutions for Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.9 [Pages 61 - 62]
Ashu is x years old while his mother Mrs Veena is x^{2} years old. Five years hence Mrs Veena will be three times old as Ashu. Find their present ages.
The sum of ages of a man and his son is 45 years. Five years ago, the product of their ages was four times the man's age at the time. Find their present ages.
The product of Shikha's age five years ago and her age 8 years later is 30, her age at both times being given in years. Find her present age.
The product of Ramu's age (in years) five years ago and his age (in years) nice years later is 15. Determine Ramu's present age.
Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.
A girls is twice as old as her sister. Four years hence, the product of their ages (in years) will be 160. Find their present ages.
The sum of the reciprocals of Rehman's ages, (in years) 3 years ago and 5 years from now is 1/3. Find his present age.
Exercise 4.1 [Page 64]
RD Sharma solutions for Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.1 [Page 64]
The hypotenuse of a right triangle is 25 cm. The difference between the lengths of the other two sides of the triangle is 5 cm. Find the lengths of these sides.
The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.
The hypotenuse of a right triangle is `3sqrt10`. If the smaller leg is tripled and the longer leg doubled, new hypotenuse wll be `9sqrt5`. How long are the legs of the triangle?
A pole has to be erected at a point on the boundary of a circular park of diameter 13 meters in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 meters. Is it the possible to do so? If yes, at what distances from the two gates should the pole be erected?
Exercise 4.7 [Pages 70 - 71]
RD Sharma solutions for Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.7 [Pages 70 - 71]
The perimeter of a rectangular field is 82 m and its area is 400 m^{2}. Find the breadth of the rectangle.
The length of a hall is 5 m more than its breadth. If the area of the floor of the hall is 84 m^{2}, what are the length and breadth of the hall?
Two squares have sides x cm and (x + 4)cm. The sum of this areas is 656 cm2. Find the sides of the squares.
The area of a right angled triangle is 165 m^{2}. Determine its base and altitude if the latter exceeds the former by 7 m.
Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m^{2}? If so, find its length and breadth.
Is it possible to design a rectangular park of perimeter 80 and area 400 m^{2}? If so find its length and breadth.
Sum of the areas of two squares is 640 m^{2}. If the difference of their perimeters is 64 m. Find the sides of the two squares.
Sum of the area of two squares is 400 cm^{2}. If the difference of their perimeters is 16 cm, find the sides of two squares.
The area of a rectangular plot is 528 m^{2}. The length of the plot (in metres) is one metre more then twice its breadth. Find the length and the breadth of the plot.
[Pages 73 - 74]
RD Sharma solutions for Class 10 Maths Chapter 4 Quadratic Equations [Pages 73 - 74]
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.
If two pipes function simultaneously, a reservoir will be filled in 12 hours. One pipe fills the reservoir 10 hours faster than the other. How many hours will the second pipe take to fill the reservoir?
Two water taps together can fill a tank in `9 3/8`. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
Two pipes running together can fill a tank in `11 1/9` minutes. If one pipe takes 5 minutes more than the other to fill the tank separately, find the time in which each pipe would fill the tank separately.
To fill a swimming pool two pipes are used. If the pipe of larger diameter used for 4 hours and the pipe of smaller diameter for 9 hours, only half of the pool can be filled. Find, how long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool?
[Pages 80 - 81]
RD Sharma solutions for Class 10 Maths Chapter 4 Quadratic Equations [Pages 80 - 81]
A piece of cloth costs Rs. 35. If the piece were 4 m longer and each meter costs Rs. one less, the cost would remain unchanged. How long is the piece?
Some students planned a picnic. The budget for food was Rs. 480. But eight of these failed to go and thus the cost of food for each member increased by Rs. 10. How many students attended the picnic?
A dealer sells an article for Rs. 24 and gains as much percent as the cost price of the article. Find the cost price of the article.
Out of a group of swans, 7/2 times the square root of the total number are playing on the share of a pond. The two remaining ones are swinging in water. Find the total number of swans.
If the list price of a toy is reduced by Rs. 2, a person can buy 2 toys more for Rs. 360. Find the original price of the toy.
Rs. 9000 were divided equally among a certain number of persons. Had there been 20 more persons, each would have got Rs. 160 less. Find the original number of persons.
Some students planned a picnic. The budget for food was Rs. 500. But, 5 of them failed to go and thus the cost of food for each member increased by Rs. 5. How many students attended the picnic?
A pole has to be erected at a point on the boundary of a circular park of diameter 13 meters in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 meters. Is it the possible to do so? If yes, at what distances from the two gates should the pole be erected?
In a class test, the sum of the marks obtained by P in Mathematics and science is 28. Had he got 3 marks more in mathematics and 4 marks less in Science. The product of his marks would have been 180. Find his marks in two subjects.
In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects
A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.
[Page 82]
RD Sharma solutions for Class 10 Maths Chapter 4 Quadratic Equations [Page 82]
Write the value of k for which the quadratic equation x^{2} − kx + 4 = 0 has equal roots.
What is the nature of roots of the quadratic equation 4x^{2} − 12x − 9 = 0?
If \[1 + \sqrt{2}\] is a root of a quadratic equation will rational coefficients, write its other root.
Write the number of real roots of the equation x^{2} + 3 |x| + 2 = 0.
Write the sum of real roots of the equation x^{2} + |x| − 6 = 0.
Write the set of value of 'a' for which the equation x^{2} + ax − 1 = 0 has real roots.
Is there any real value of 'a' for which the equation x^{2} + 2x + (a^{2} + 1) = 0 has real roots?
Write the value of λ for which x^{2} + 4x + λ is a perfect square.
Write the condition to be satisfied for which equations ax^{2} + 2bx + c = 0 and \[b x^2 - 2\sqrt{ac}x + b = 0\] have equal roots.
Write the set of value of k for which the quadratic equations has 2x^{2} + kx − 8 = 0 has real roots.
Write a quadratic polynomial, sum of whose zeros is \[2\sqrt{3}\] and their product is 2.
Show that x = −3 is a solution of x^{2} + 6x + 9 = 0.
Show that x = −2 is a solution of 3x^{2} + 13x + 14 = 0.
Find the discriminant of the quadratic equation \[3\sqrt{3} x^2 + 10x + \sqrt{3} = 0\].
If \[x = - \frac{1}{2}\],is a solution of the quadratic equation \[3 x^2 + 2kx - 3 = 0\] ,find the value of k.
[Pages 83 - 85]
RD Sharma solutions for Class 10 Maths Chapter 4 Quadratic Equations [Pages 83 - 85]
If the equation x^{2} + 4x + k = 0 has real and distinct roots, then
k < 4
k > 4
k ≥ 4
k ≤ 4
If the equation x^{2} − ax + 1 = 0 has two distinct roots, then
|a| = 2
|a| < 2
|a| > 2
None of these
If the equation 9x^{2} + 6kx + 4 = 0 has equal roots, then the roots are both equal to
\[\pm \frac{2}{3}\]
\[\pm \frac{3}{2}\]
0
±3
If ax^{2} + bx + c = 0 has equal roots, then c =
\[\frac{- b}{2a}\]
\[\frac{b}{2a}\]
\[\frac{- b^2}{4a}\]
\[\frac{b^2}{4a}\]
If the equation ax^{2} + 2x + a = 0 has two distinct roots, if
a = ±1
a = 0
a = 0, 1
a = −1, 0
The positive value of k for which the equation x^{2}^{ }+ kx + 64 = 0 and x^{2} − 8x + k = 0 will both have real roots, is
4
8
12
16
The value of \[\sqrt{6 + \sqrt{6 + \sqrt{6 +}}} . . . .\] is
4
3
-2
3.5
If 2 is a root of the equation x^{2} + bx + 12 = 0 and the equation x^{2} + bx + q = 0 has equal roots, then q =
8
-8
16
-16
If the equations \[\left( a^2 + b^2 \right) x^2 - 2\left( ac + bd \right)x + c^2 + d^2 = 0\] has equal roots, then
ab = cd
ad = bc
\[ad = \sqrt{bc}\]
\[ab = \sqrt{cd}\]
If the roots of the equations \[\left( a^2 + b^2 \right) x^2 - 2b\left( a + c \right)x + \left( b^2 + c^2 \right) = 0\] are equal, then
2b = a + c
b^{2} = ac
\[b = \frac{2ac}{a + c}\]
b = ac
If the equation x^{2} − bx + 1 = 0 does not possess real roots, then
−3 < b < 3
−2 < b < 2
b > 2
b < −2
If x = 1 is a common roots of the equations ax^{2} + ax + 3 = 0 and x^{2} + x + b = 0, then ab =
3
3.5
6
-3
If p and q are the roots of the equation x^{2} − px + q = 0, then
p = 1, q = −2
p = 1, q = −2
p = −2, q = 0
p = −2, q = 1
If a and b can take values 1, 2, 3, 4. Then the number of the equations of the form ax^{2} +bx + 1 = 0 having real roots is
10
7
6
12
The number of quadratic equations having real roots and which do not change by squaring their roots is
4
3
2
1
If \[\left( a^2 + b^2 \right) x^2 + 2\left( ab + bd \right)x + c^2 + d^2 = 0\] has no real roots, then
ab = bc
ab = cd
ac = bd
ad ≠ bc
If the sum of the roots of the equation x^{2} − x = λ(2x − 1) is zero, then λ =
−2
2
\[- \frac{1}{2}\]
\[\frac{1}{2}\]
If x = 1 is a common root of ax^{2} + ax + 2 = 0 and x^{2} + x + b = 0, then, ab =
1
2
3
4
The value of c for which the equation ax^{2} + 2bx + c = 0 has equal roots is
\[\frac{b^2}{a}\]
\[\frac{b^2}{4a}\]
\[\frac{a^2}{b}\]
\[\frac{a^2}{4b}\]
If \[x^2 + k\left( 4x + k - 1 \right) + 2 = 0\] has equal roots, then k =
\[- \frac{2}{3}, 1\]
\[\frac{2}{3}, - 1\]
\[\frac{3}{2}, \frac{1}{3}\]
\[- \frac{3}{2}, - \frac{1}{3}\]
If the sum and product of the roots of the equation kx^{2} + 6x + 4k = 0 are real, then k =
\[- \frac{3}{2}\]
\[\frac{3}{2}\]
\[\frac{2}{3}\]
\[- \frac{2}{3}\]
If sin α and cos α are the roots of the equations ax^{2}^{ }+ bx + c = 0, then b^{2} =
a^{2} − 2ac
a^{2} + 2ac
a^{2} − ac
a^{2} + ac
If 2 is a root of the equation x^{2} + ax + 12 = 0 and the quadratic equation x^{2} + ax + q = 0 has equal roots, then q =
12
8
20
16
If the sum of the roots of the equation \[x^2 - \left( k + 6 \right)x + 2\left( 2k - 1 \right) = 0\] is equal to half of their product, then k =
6
7
1
5
If a and b are roots of the equation x^{2} + ax + b = 0, then a + b =
1
2
-2
-1
A quadratic equation whose one root is 2 and the sum of whose roots is zero, is
x^{2} + 4 = 0
x^{2} − 4 = 0
4x^{2} − 1 = 0
x^{2} − 2 = 0
If one of the equation ax^{2} + bx + c = 0 is three times times the other, then b^{2} : ac =
3 : 1
3 : 16
16 : 3
16 : 1
If one root the equation 2x^{2} + kx + 4 = 0 is 2, then the other root is
6
-6
-1
1
If one of the equation x^{2} + ax + 3 = 0 is 1, then its other root is
3
-3
2
1
If one root of the equation 4x^{2} − 2x + (λ − 4) = 0 be the reciprocal of the other, then λ =
8
-8
4
-4
If y = 1 is a common root of the equations \[a y^2 + ay + 3 = 0 \text { and } y^2 + y + b = 0\], then ab equals
3
-7/2
6
-3
The values of k for which the quadratic equation \[16 x^2 + 4kx + 9 = 0\] has real and equal roots are
\[6, - \frac{1}{6}\]
36, −36
6, −6
\[\frac{3}{4}, - \frac{3}{4}\]
Chapter 4: Quadratic Equations
RD Sharma solutions for Class 10 Maths chapter 4 - Quadratic Equations
RD Sharma solutions for Class 10 Maths chapter 4 (Quadratic Equations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 10 Maths solutions in a manner that help students grasp basic concepts better and faster.
Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.
Concepts covered in Class 10 Maths chapter 4 Quadratic Equations are Quadratic Equations Examples and Solutions, Quadratic Equations, Roots of a Quadratic Equation, Nature of Roots, Relation Between Roots of the Equation and Coefficient of the Terms in the Equation Equations Reducible to Quadratic Form, Solutions of Quadratic Equations by Factorization, Solutions of Quadratic Equations by Completing the Square, Formula for Solving a Quadratic Equation, Relationship Between Discriminant and Nature of Roots, Situational Problems Based on Quadratic Equations Related to Day to Day Activities to Be Incorporated, Quadratic Equations Examples and Solutions, Quadratic Equations, Solutions of Quadratic Equations by Factorization, Solutions of Quadratic Equations by Completing the Square, Nature of Roots.
Using RD Sharma Class 10 solutions Quadratic Equations exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 10 prefer RD Sharma Textbook Solutions to score more in exam.
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