Class 10 maths subjective questions with solutions for the term 2 exams based on standard and basic maths.

## Class 10 – Maths Subjective Chapter – 4 Quadratic Equations

### Quiz -1 (2 Mark Questions)

### Both Standard and Basic maths

Q- (1) Find two consecutive odd positive integers, sum of whose squares is 290.

Solution: Let the first odd number be = x and the 2^{nd} be = x+2

Given that sum of the square of numbers is 290

So x² + (x + 2)² = 290

Solving the numbers now x² + x² + 4x + 4 = 290

2x² + 4x + 4 = 290

2x² + 4x + 4 – 290 = 0

2x² + 4x – 286 = 0

2(x² + 2x – 143) = 0

Solving by factorization method

x² + 13x – 11x – 143 = 0

x(x + 13) -11(x + 13) = 0

(x – 11) (x + 13) = 0

x = 11, -13

So the numbers are 11 and 13.

Q- (2) A motor boat whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

Solution: let the speed of stream be = x km/h

The speed of motor boat be = 18 km/h

So the downstream speed of boat be = x + 18 km/h

The upstream speed of boat be = x – 18 km/h

Distance = 24 km

Time (t1) = 24/(x + 18) ( For downstream)

Time (t2) = 24/(x – 18) ( For upstream)

In the upstream journey, 1 hour more is taken –

t2 – t1 = 1

Q- (3) If one root of 5x² + 13x + k = 0 is the reciprocal of the other root, then find the value of k.

Solution: Let a be the first root and 1/a be the second root

Product of roots = c/a

a x 1/a = k/5

1 = k/5

k = 5

Q- (4) Find the roots of the following quadratic equations –

(a) √3x² – 8x + 4√3 = 0

Solution: By quadratic formula

Q- (5) Find the value of k, for which the quadratic equation 4x² + 4√3 x + k = 0 has equal roots.

Solution: In this quadratic equation, there are equal roots –

b² – 4ac = 0

(4√3)² – 4 x 4 x k = 0

48 – 16k = 0

48 = 16k

k = 3

Q- (6) The product of two consecutive positive integers is 210. Find the integers.

Solution: Let the first integer be = x and 2^{nd} integer be = (x+1)

Product of two numbers x(x + 1) = 210

x² + x – 210 = 0

By factorization method x² + 15x – 14x – 210 = 0

x(x + 15) -14(x + 15) = 0

(x – 14)(x + 15) = 0

x = 14, -15

First number is 14 and 2^{nd} number is 15.

Q- (7) If the roots of the quadratic equation (c² – ab)x² – 2(a² – bc)x + b² – ac = 0 in x are equal, then show that either a = 0 or a³ + b³ + c³ = 3abc.

Solution:

Q- (8) Find the value of p, for which one root of the quadratic equation px² – 14x + 8 = 0 is 6 times the other.

Q- (9) Solve the following quadratic equation for x : 4x² – 4a²x + (a⁴ – b⁴) = 0

Q- (10) Solve for x : 16/x – 1 = 15/(x+1)

Class 10 maths Subjective questions

Q- (11) Find the roots of the equation – ax² + a = a²x + x

Q- (12) Find the value of k for which the given equation has equal roots.

(k – 12)x² + 2(k – 12)x + 2 = 0

Q- (13) Show that (x² + 1)² – x² = 0 has no real roots.

Q- (14) The speed of a boat in still water is 15km/hr. It can go 30 km upstream and return downstream to the original point in 4 h and 30 min. Find the speed of stream.

Q- (15) If the discriminant of the equation 6x² – bx + 2 = 0 is 1, then find the value of b.

Q- (16) Two water taps together can fill a tank in 1 7/8 h. (one whole 7/8 h). The tap with longer diameter takes 2 h less than the tap with smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately.

Q- (17) Find the solution of the equation –

Q- (18) If y = -2 and y = 3 are roots of the equation 3y² – 2ay + 2b = 0,then find the value of a and b.

Q- (19) Without solving the following quadratic equation, find the value of p, for which the roots are equal.

px² – 4x – 3 = 0

Q- (20) Solve the following equation for y, 9(y² – 2) – 9y – 52 = 0.

Q- (21) Solve for x :

Q- (22) The roots a and b of the quadratic equation x² – 5x + 3(k – 1) = 0 are such that a – b = 1, Find the value of k.

Q- (23) For what value of ‘a’ quadratic equation 30ax² – 6x + 1 = 0 has no real roots?

**Case Study** –

Q- (1) p(x) is a quadratic polynomial i.e. q(x) = ax² + bx + c, a ≠ 0, then q(x) = 0 is called a quadratic equation and answer the following questions.

(a) Which condition is required to become the equation ax² + bx + c = 0, a quadratic equation is 6x² + (5x – 1)(x) = (x² + 1)(11) of the form of ax² + bx + c = 0 ?

(b) Find the degree of the following equation and state which are quadratic equation among x(x + 3) + 7 = 5x – 11 and (x + 2)(x – 3) = 6 + (x + 4)².