If one root of the equation 2x2+ax+6=0 is 3, then the value of a is ___________.

A.

-8

B.

7

C.

3

D.

5

#### SOLUTION

Solution : A

If one root of the equation 2x2+ax+6=0 is 3, then substituting x=3 will satisfy the equation.

2(3)2+3a+6=0

18+3a+6=0

24+3a=0

a=8

The roots of 2x26x+8=0 are ___________.

A.

real, unequal and rational

B.

real, unequal and irrational

C.

real and equal

D.

imaginary

#### SOLUTION

Solution : D

Step 1: For 2x26x+8=0, value of discriminant
D=(6)24(2)(8)=3664=28

Step 2: Since D<0, roots are imaginary.

If the equation x2+2(k+2)x+9k=0 has equal roots, then values of k are __________.

A.

1 or 4

B.

–1 or 5

C.

1 or –5

D.

–1 or –4

#### SOLUTION

Solution : A

Step 1:- For, x2+2(k+2)x+9k=0, value of discriminant D=[2(k+2)]24(9k)=4(k2+45k)

Step 2:- The roots of quadratic equation are real and equal only when D=0
k2+45k=0
k25k+4=0
k2k4k+4=0
k(k1)4(k1)=0
(k1)(k4)=0

Step 3:-   k=4 or 1

Using the method of completion of squares find one of the roots of the equation 2x27x+3=0. Also, find the equation obtained after completion of the square.

A.

6, (x74)22516=0

B.

3, (x74)22516=0

C.

3, (x72)22516=0

D.

13, (x72)22516=0

#### SOLUTION

Solution : B

2x27x+3=0
Dividing by the coefficient of x2, we get
x272x+32=0; a=1, b=72, c=32

Adding and subtracting the square of b2=74, (half of coefficient of x)

we get,
[x22(74)x+(74)2](74)2+32=0

The equation after completing the square is :
(x74)22516=0

Taking square root, (x74)=(±54)
Taking positive sign 54, x=3
Taking negative sign 54, x=12

If the equation x2+2(k+2)x+9=0 has equal roots, then find the values of k.

A.

1, 4

B.

–1, 5

C.

1, –5

D.

–1, –4

#### SOLUTION

Solution : C

Step 1:-  x2+2(k+2)x+9=0  a=1,b=2(k+2),c=9

D=4(k+2)24(9)=4(k2+4k5)       [D=b24ac ]

Step 2:- The roots of quadratic equation are real and equal only when D=0

4(k2+4k5)=0, k2+4k5=0

Step 3:-    k2+4k5=0

k2+5kk5=0

k(k+5)1(k+5)=0

(k1)(k+5)=0

k=1 or k=5

A can do a piece of work in x days and B can do the same work in x+16 days. If both working together can do it in 15 days, find the value of x.

A.

22

B.

20

C.

24

D.

40

#### SOLUTION

Solution : C

Given: A can do a piece of work in x days and B in x+16 days.
Work done by A in one day = 1x
Work done by B in one day = 1x+16
Work done by A and B together in one day = 115
1x1x+16115
2x+16x(x+16)=115
x2+16x=15(2x+16)
x214x240=0
x224x+10x240=0
x(x24)+10(x24)=0
(x24)(x+10)=0
x=24 or x=10
x=24 as x cannot be negative.

Solve for x if  4(2x+3)2(2x+3)14=0.

A.

x=12,198

B.

x=12,198

C.

x=12,198

D.

x=12,198

#### SOLUTION

Solution : A

Given:  4(2x+3)2(2x+3)14=0
Substitute (2x+3)=y, Hence the given equation reduces to
4y2y14=0
4y28y+7y14=0
4y(y2)+7(y2)=0
(4y+7)(y2)=0
y=74 or y=2
When  y=74
(2x+3)=74
2x=194
x=198
When  y=2
(2x+3)=2
2x=1
x=12

If one root of the quadratic equation 2x2+ax6=0 is 2, find the value of a.

A.

-1

B.

3

C.

5

D.

-5

#### SOLUTION

Solution : A

Since, x = 2 is a root of the given equation 2x2+ax6=0

2(2)2+a×26=0

8+2a6=0

2a=2

a=1

The product of 2 consecutive natural numbers is 72. Find the numbers.

A.

8,9

B.

-8, -9

C.

36, 2

D.

18, 4

#### SOLUTION

Solution : A

Let the smaller number be x. Then the larger number is x+1.

Their product is x(x+1).

x(x+1)=72

x2+x72=0

x2+9x8x72=0

x(x+9)8(x+9)=0

(x8)(x+9)=0

x=8 or x=9

Since the given numbers are natural numbers, we get x=8

Hence the two consecutive natural numbers are 8 and 9.

If the speed of a train is reduced by 40 km/hr, it takes 20 minutes more to cover 1200 km.  Find the speed of the train.

A. 400 km/hr
B. 360 km/hr
C. 420 km/hr
D. 380 km/hr

#### SOLUTION

Solution : A

Let the speed of the train be x km/hr.

Time=DistanceSpeed

Time taken to cover 1200 km =  1200x

Reduced speed  = (x40) km/hr

Time taken to cover 1200 km in reduced speed =  1200x40

Relation between the time taken to cover 1200 km with speed x and with speed (x - 40) km/hr is as given below

1200x401200x = 13     (20 minutes=13 hours)
48000x(x40) = 13
144000=x240x
x240x144000=0
x2400x+360x144000=0
x(x400)+360(x400)=0
(x+360)(x400)=0
x=360 or x=400
Since speed cannot be negative, we get x = 400 km/hr.
The speed of the train is 400 km/hr.