Math Assignment Class X
Chapter 4 Quadratic Equations

Extra questions of chapter 4 class 10 Quadratic Equations with answer and hints to the difficult questions. Important and useful math assignment for the students of class 10

ASSIGNMENT FOR 10 STANDARD QUADRATIC EQUATIONS

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Question 1. Check whether the given values of x are the solution of equation or not.

a) x² + √2x – 4 = 0 : At x = √2, x = -2√2 : Ans [yes]

b) (2x + 3)(3x - 2) = 0 : At x = 2/3 : Ans [yes]

c) 6x²– x – 1 = 0 : At x = 1/2, x = 3/2

Ans [At x=1/2 yes, at x= 3/2 no]

Question 2. Find k if the value of x is the solution of Q. E.

a) 3x²- 2kx + 5 = 0 : At x = 2

Ans [k = 17/4]

b) x²- (a + b)x + k = 0 : At x = a

Ans [ k = ab ]

c) kx²+ √3x + 6 = 0 : At x = √3

Ans [ k = - 3]

Question 3. If x = - 2 and x = -1/5 are the solution of 5x²+ k x + p = 0 then find k and p.

Ans [K = 11, P = 2]

Question 4. Check whether 3 is the root of this equation.

$\mathbf{\sqrt{x^{2}-4x+3}+\sqrt{x^{2}-9}=\sqrt{4x^{2}-14x+6}$

Ans [yes]

Question 5. Solve the following by factorisation method

a) y²- 5 = 0 [Ans ± √5 ]

b) (2x + 3)² = 81 [Ans -6 & 3]

c) ax²- 2bx = 0 [Ans 0, 2b/a]

d) 4√5x² + 7x - 3√5 = 0 [Ans -3/√5, √5/4]

e) 5x²- 17/2 x + 3/2 = 0 [Ans [3/2, 1/5]

f) 3a²x²+ 8abx + 4b²= 0 [Ans -2b/a, -2b/3a]

g) a²b²x²+ b²x - a²x – 1 = 0 [Ans 1/b², -1/a²]

h) ax² + (4a²- 3b)x - 12ab = 0 [Ans -4a, 3b/a]

i). $\mathbf{\frac{x-1}{x-2}+\frac{x-3}{x-4}=3\frac{1}{3}$ [Ans 5, 5/2]

j). ab - x² = (a - b)x [Ans -a, b]

Question 6. If - 4 is a root of the equation x²+ px – 4 = 0 and the equation x²+ p x + k = 0 has equal roots. Find k.

Ans [k = 9/4 ]

Question 7. If one root of the equation x²– k x - 2 = 0 is 2 then find k and other root.

Ans [k = 3, other root -1].

Question 8. Find the value of m for which the roots of the equation mx(6x + 10) + 25 = 0 are equal.

Ans [m = 6]

Question 9. Discuss the nature of the roots of the following

a) 9x²- 12x + 4 = 0 b) x²– 3 x = 0

c) x²- 1/3x +3/4 = 0 d) x²- 2√2x - 2√3 = 0

e) √3x²- 2√2x - 2√3 = 0 f) √2x²- √5x + 3 = 0

Question 10. Find the quadratic equation whose roots are

a) -8 & -3 [Ans x²+ 11x + 24 = 0 ]

b) 1 + √2 and 1 - √2 [Ans x²- 2x - 1 = 0 ]

c) $\mathbf{\frac{4-\sqrt{5}}{2},\;\frac{4+\sqrt{5}}{2}$ [Ans 4x²- 16x + 11 = 0 ]

Question 11. For what value of p will the following equation have real roots

a) 9x²- 5x + (p + 1) = 0 [Ans p ≤ -11/36]

b) 2x²– p x – 4 = 0 [Ans No Value of P]

Question 12. For what value of k the equations have real and repeated roots(or equal roots)

a) (3k + 1)x²+ 2(k + 1)x + k = 0 [ Ans K = -1/2, 1]

b) x²- 2(k + 1)x + k² = 0 [Ans k = -1/2]

c) 4x²- 2(k + 1) + k + 4 = 0 [Ans k = 5, k = - 3]

d) 9x²+ 8kx + 16 =0 [Ans k= ±3]

e) (k-12)x²+ 2(k - 12)x + 2 = 0 [Ans k = 12]

f) (k + 4)x²+ (k +1)x + 1 = 0 [Ans k = 5, -3]

Question 13. Find the roots of the following by using quadratic formula.

a) 3x²+ 2√5x – 5 = 0 Ans [√5/3, - √5]

b) p²x² + (p²- q²)x - q²= 0 : Ans [q²/p², -1]

c) 12abx²- (9a²- 8b²)x - 6ab = 0 : Ans [3a/4b, -2b/3a]

d) (a + b)²x²+ 8(a²- b²)x + 16(a - b)²= 0 [Ans $\mathbf{\frac{-4(a-b)}{a+b}$ , $\mathbf{\frac{-4(a-b)}{a+b}$

e) 4x²- 4a²x + (a⁴- b⁴) = 0 Ans $\mathbf{\left[\frac{a^{2}+b^{2}}{2},\:\frac{a^{2}-b^{2}}{2}\right]$

f) 9x²- 9(a + b)x + 2a²+ 5ab + 2b²= 0 Ans $\mathbf{\left[\frac{2a+b}{3},\frac{a+2b}{3}\right]$

g) 4x²- 2(a²+ b²)x + a²b² = 0 Ans [a²/2 , b²/2]

h) $\mathbf{\frac{x+2}{6}-\frac{1}{x+2}=\frac{1}{6}$ [Ans 1/-4]

i) 3y² + (6 + 4a)y + 8a = 0 Ans [-2, - 4a/3]

j) (a + b)²x² + 8(a² - b²)x + 16(a - b)² = 0 : Ans $\mathbf{\left[\frac{-4(a-b)}{a+b}\right]$

Question 14. If 𝞪 and 𝞫 are the roots of the quadratic equation ax²+ bx + c = 0, then find the value of:

a) 𝞪²+ 𝞫² [Ans $\mathbf{\left[\frac{b^{2}-2ac}{a^{2}}\right]$

Hint: Using the identity : a² +b² = (a + b)² – 2ab

b) 𝞪³+ 𝞫³ Ans $\mathbf{\left[\frac{-b(b^{2}-3ac)}{a^{3}}\right]$

Hint : Using the identity : a³ + b³ = (a + b)³ - 3ab(a + b)

c) $\mathbf{\frac{\alpha}{\beta}+\frac{\beta}{\alpha}$ Ans $\mathbf{\left[\frac{b^{2}-2ac}{ca}\right]$

Question 15. If and are the roots of 2x²+ 5x – 4 = 0 then find the value of

a) 𝞪²+ 𝞫²- 4 𝞪𝞫 Ans [73/4]

b) 𝞪³+ 𝞫³Ans[-245/8]

Question 16. Solve: a) √x + 2x =1 Ans (1, 1/4)

b) $\mathbf{\sqrt{2x+9}$ + x = 13 : Ans (20, 8)

Hint: Bring all terms without square root to the RHS. Squaring on both side to get a QE and then solve it.

Question 17. Solve the following for the value of x : $\mathbf{\frac{1}{a+b+x}=\frac{1}{a}+\frac{1}{b}+\frac{1}{x}$

Ans : x = - a, - b

Hint: Bring 1/x to the LHS

Now taking LCM and solving the QE for the value of x

Question 18. Solve: (a) $\mathbf{\left(\frac{2x-1}{x+1}\right)-15\left(\frac{x+1}{2x-1}\right)=-2$

Ans (-4, - 4/7)

Solution Hint: Putting $\mathbf{\frac{2x-1}{x+1}=y$ and $\mathbf{\frac{x+1}{2x-1}=\frac{1}{y}$ and find the QE in terms of y

Solve the QE and find the value of y. Find the value of y the replace y into x. Again solve the equation for the value of x.

Question 19. Two pipes running together can fill a tank in $\mathbf{2\frac{8}{11}$ minutes. If one pipe takes 1 minute more than the other to fill the tank. Find the time taken by each pipe.

[Ans 5 & 6 minutes ]

Question 20. Rs.6500 were divided equally among a certain no. of persons. Had there been 15 more persons, each would have got 30 less. Find the no. of persons.

Question 21. A peacock is sitting on the top of the pillar, which is 9m high. From a point 27m away from the bottom of the pillar, a snake is coming to its hole at the base of the pillar. Seeing the snake, the peacock pounces on it. If their speeds are equal, at what distance from the hole is snake caught.

Ans [12m

Question 22. The side of square exceeds the side of another square by 4cm and the sum of their areas is 400cm². Find the sides of two squares.

Ans [16cm & 12cm ]

Question 23. The difference of two numbers is 4. If the difference of their reciprocals is . Find two numbers.

Ans [3, 7]

Question 24. A motor boat whose speed in still water is 5km/h, takes 1 hour more 12 km upstream than to downstream to the same spot. Find the speed of the stream.

Ans [1km/h],

Question 25. If -5 is the root of the quadratic equation 2x² + px – 15 = 0 and p(x² + x) + k = 0 has equal roots then find the value of p and k.

Ans [p = 7 & k = 7/4 ]

Question 26. Divide 12 into 2 parts such that the sum of their squares is 74.

Ans [7, 5]

Question 27. A year ago, the father was 8 times as old as his son. Now his age is the square of his son’s age. Find their present ages.

Ans [7, 49]

Question 28. The speed of the boat in still water is 8km/h. It can go 15km upstream and 22km downstream in 5hrs. Find the speed of stream.

Ans [3km/h]

Question 29. Side of a square exceeds the side of another square by 4cm. And the sum of the areas of the two squares is 400cm². Find the dimensions of the square.

Ans [12, 16]