Free Quadratic Equations 01 Practice Test - 10th Grade
Question 1
With respect to the roots of x2–2x–3=0, we can say that
both of them are natural numbers
both of them are integers
one of the root is zero
roots are not real
SOLUTION
Solution : B
Step 1:- For x2–2x–3=0, value of discriminant
D=(−2)2–4(−3)(1)=4+12=16.Step 2:- Since, D is a perfect square, roots are rational and unequal.
Step 3:- Solving the equation,
x=−(−2)+√(16)2 or −(−2)−√(16)2
x=3,−1
Thus, both of them are integers.
Question 2
If the equation 9x2+6kx+4=0 has equal roots, then find the value of k =
0
2 , 0
-2, 0
± 2
SOLUTION
Solution : D
Given, the roots of quadratic equation 9x2+6kx+4=0 are real and equal.
Discriminant, Δ=b2−4acΔ=36k2–4(4)(9)=36k2–4(36)The roots of quadratic equation are real and equal then Δ=036k2–4(36)=0k2−4=0k2=4k=√4∴k=± 2
Question 3
The equation is a quadratic equation : x+51=2x+10x−6
True
False
SOLUTION
Solution : A
x+51=2x+10x−6
⇒(x + 5)(x - 6) = 2x + 10
⇒x2−6x+5x−30=2x+10
⇒x2−3x−40=0 is of the form of ax2+bx+cHence, is a quadratic equation.
Question 4
Taylor purchased a rectangular plot of area 634 m2. The length of the plot is 2 m more than thrice its breadth. The length and breadth respectively is _____ (approximate values).
34.6 m & 11.20 m
88 m & 24 m
32 m & 16 m
44.6 m & 14.20 m
SOLUTION
Solution : D
Let, Length = x
Breadth = y
Given,
Area of Rectangle = 634 m2
Length : x
Thrice the breadth : 3y
2 more that thrice : 3y+2
So, x=3y+2
Area of the rectangle=length×breadth
634=xy634=(2+3y)y634=2y+3y2∴3y2+2y−634=0
This equation resembles the general form of quadratic equation ax2+bx+c=0.
Lets find the values of y satisfying the equation.(Roots of the equation)
y=−b ±√b2− 4ac2a=−2 ±√22 − 4×3(−634)2 × 3y=−2 ±√4 + 76086=−2 ±√76126y=−2 ± 87.2466y=−2+87.2466 or −2−87.2466y=14.20 or −14.87
Length is always positive.
∴y=14.20 m
x=2+3y∴x=2+3(14.20)=44.6 m
Length =44.6 m
Breadth =14.2 m
Question 5
Find the two consecutive positive integers, such that the sum of their squares is 365.
17 , 18
13 , 14
9 , 10
12 , 13
SOLUTION
Solution : B
Let the numbers be x & x+1
Square of consecutive number: x2 & (x+1)2
Sum of square: x2+(x+1)2
∴x2+(x+1)2=365
Lets solve
x2+x2+2x+1=3652x2+2x−364=0x2+x−182=0
(Dividing by 2)
Factorise,
x2+14x−13x−182=0x(x+14)−13(x+14)=0(x+14)(x−13)=0∴x=13 & −14
Since numbers are positive integers, x=13
So, required consecutive numbers are 13 & 14.
Question 6
Solve following equation for x:
x2−3x−10=0
-2 and -6
10 and 5
5 and -2
7 and 6
SOLUTION
Solution : C
By using the method of factorisation,
x2−3x−10=0
⇒ x2−5x+2x−10=0
⇒x(x−5)+2(x−5)=0⇒ (x−5)(x+2)=0
⇒x=5 and x=−2
Question 7
Find the roots of the equation 5x2–6x–2=0 by the method of completing the square.
5±√193
3±√195
5
3
SOLUTION
Solution : B
Multiply the equation by 5.
we get 25x2–30x–10=0
(5x)2–[2×(5x)×3]+32–32–10=0
(5x–3)2–9–10=0(5x–3)2–19=0(5x–3)2=195x−3=±√19x=3±√195
Roots of the equation are 3+√195 & 3−√195
Question 8
Find the numbers such that the square of number is 15 lesser than twice the number.
8
9
-3
5
SOLUTION
Solution : C and D
Let the number be x.
Square of number: x2
Lesser by 15: x2−15
of two times the number: x2−15=2xWe get, x2−15=2x
Lets solve, x2−2x−15=0x2−5x+3x−15=0x(x−5)+3(x−5)=0(x−5)(x+3)=0∴x=5 & x=−3
The number satisfying the given statement are 5 & −3.
Question 9
If x=23 is a solution of the quadratic equation 7x2+mx−3=0, find the value of m.
m=16
m=−13
m=−16
m=13
SOLUTION
Solution : C
Given: x=23 is a solution of the quadratic equation 7x2+mx−3=0.
Substituting x=23 in the given quadratic equation we get
⇒ 7(23)2+m(23)−3=0
⇒ 7(49)+m(23)−3=0
⇒ 289+2m3−3=0
⇒ 28+6m−27=0
⇒ m=−16
Question 10
For what value of m does the equation, x2+2x+m=0 have two distinct real roots?
SOLUTION
Solution : B
Step 1 : For, x2+2x+m=0,
⇒ a=1, b=2, c=m
We know that
D=b2–4ac
⇒D=(2)2–4m
=4−4m
Step 2 : The roots of a quadratic equation are real and distinct only when D>0Step 3 : 4−4m>0
4>4m
m<1