## Binary number 10101 is equivalent to its decimal equivalent ___?

A. 10
B. 20
C. 21
D. 111

To convert the binary number 10101 to its decimal equivalent, you can use the positional notation system. Each digit in a binary number represents a power of 2, starting from the rightmost digit as 2^0, then increasing by one power for each subsequent digit to the left.

So, 10101 in binary is calculated as:

1 * 2^4 + 0 * 2^3 + 1 * 2^2 + 0 * 2^1 + 1 * 2^0 = 16 + 0 + 4 + 0 + 1 = 21

C. 21

## If 20% of an electricity bill is deducted, then Rs. 100 is till to be paid. How much was the original bill?

A. Rs. 115
B. Rs. 110
C. Rs. 125
D. None of these

To solve this problem, let’s denote the original electricity bill amount as .

If 20% of the bill is deducted, the remaining amount to be paid is 100. This means 80% of the bill remains.

We can set up the equation as follows:
0.80×x=100
Now, let’s solve for :
x=100​/0.80=125

So, the original electricity bill was Rs. 125.

Therefore, the correct answer is C. Rs. 125.

## A shopkeeper earns 15% profit on a shirt even after allowing 31% discount on the list price. If list price is Rs125, then cost price of shirt is?

A. 87
B. 80
C. 75
D. None of these

Let’s break down the problem step by step:

1. The shopkeeper earns a 15% profit on the shirt.
2. The shopkeeper offers a 31% discount on the list price.

First, let’s find out the selling price (SP) after applying the discount: Discount = 31% of the list price = 0.31 * 125 = Rs 38.75 So, the selling price after the discount = List price – Discount = Rs 125 – Rs 38.75 = Rs 86.25

Now, we know that the shopkeeper earns a 15% profit on the selling price. Let’s represent the cost price (CP) as x.

The selling price (SP) after profit = Cost price (CP) + Profit So, SP = CP + 15% of CP = CP + 0.15 * CP = 1.15 * CP

Given that SP = Rs 86.25, we can write the equation as: 1.15 * CP = 86.25

Now, we can solve for the cost price (CP): CP = 86.25 / 1.15 = Rs 75

So, the cost price of the shirt is Rs 75.

Therefore, the correct answer is C. 75.

## How many numbers up to 450 are divisible by 4, 6 and 8 together?

A. 19
B. 18
C. 17
D. None of these

To find the numbers up to 450 that are divisible by 4, 6, and 8 together, we need to find the numbers that are divisible by the least common multiple (LCM) of 4, 6, and 8.

The LCM of 4, 6, and 8 is the smallest number that is divisible by all three numbers, which can be found by calculating the prime factorization of each number:

• 4 = 2^2
• 6 = 2 * 3
• 8 = 2^3

To find the LCM, we take the highest power of each prime factor that appears in any of the numbers: 2^3 * 3 = 24.

So, any number that is divisible by 24 is also divisible by 4, 6, and 8.

Now, to find how many numbers up to 450 are divisible by 24, we divide 450 by 24:

450 ÷ 24 = 18 with a remainder of 18.

So, there are 18 whole multiples of 24 up to 450.

Therefore, the correct answer is B. 18.

## If the roots of the equation are -7/4 , 4/7 then the equation will be?

A. 28x^2+43x-28=0
B. 28x^2-43x-28=0
C. 28x^2+49x-16=0
D. 28x^2-33x-28=0

Correct Answer: C. 28x^2 + 49x -16 = 0

### Solution

If the roots of the equation are -7/4 and 4/7, then the equation can be written as:

(x + 7/4)(x – 4/7) = 0

Multiplying out the terms gives:

x^2 + (7/4)x – (4/7)x – 7/4 * 4/7 = 0
x^2 + (49x – 16)/28 = 0

Multiplying both sides by 28 to get rid of the denominator, we get:

28x^2 + 49x – 16 = 0

Therefore, the equation with roots -7/4 and 4/7 is 28x^2 + 49x - 16 = 0.

## If S=the sum of the roots of the equation and P= Product of the roots of the equation, then their equation can be written as:

A. x^2+Sx+P=0
B. x^2+Sx-P=0
C. x^2-Sx-P=0
D. x^2-Sx+P=0

### Solution

If we have a quadratic equation of the form ax^2 + bx + c = 0, the sum of its roots is given by:

S = -b/a

and the product of its roots is given by:

P = c/a

To see why these formulas are true, we can use Vieta’s formulas, which state that for a quadratic equation of the form ax^2 + bx + c = 0, the sum of the roots is equal to the negation of the coefficient of the linear term divided by the coefficient of the quadratic term, and the product of the roots is equal to the constant term divided by the coefficient of the quadratic term.

So, we can write the quadratic equation as:

a(x – r1)(x – r2) = 0

where r1 and r2 are the roots of the equation. Expanding the product and comparing the coefficients with the equation ax^2 + bx + c = 0, we get:
a(x^2 – (r1 + r2)x + r1r2) = 0
Equating the coefficients of corresponding powers of x, we get:
r1 + r2 = -b/a
r1r2 = c/a

Therefore, we have:

S = -b/a
P = c/a
Thus, if S is the sum of the roots of the equation and P is the product of the roots of the equation, we can write the equation as:
x^2 – Sx + P = 0

## If the discriminant of ax2+bx+c=0 is not a perfect square then its roots are:

A. Equal
B. Rational
C. Irrational
D. Real

### Solution

If the discriminant of the quadratic equation ax^2 + bx + c = 0 is not a perfect square, then its roots are irrational and real.

The discriminant of the quadratic equation is given by the expression b^2 - 4ac. If this value is not a perfect square, then the roots of the equation will be of the form:

(-b ± √(b^2 – 4ac)) / 2a

Since the discriminant is not a perfect square, the square root of b^2 - 4ac will be irrational, and the roots will be of the form (-b ± irrational number) / 2a, which are both real and irrational. Therefore, the answer is option C: Irrational.

## The sum of the roots of the equation 122-7x+4=0 is:

A. 12/7
B. -7/122
C. 1/3
D. 7/122

### Solution

122x^2 – 7x + 4 = 0

We can find the sum of the roots of this equation using the formula:

sum of roots = -b/a

where a and b are the coefficients of x^2 and x, respectively.

Comparing the given equation with the standard form ax^2 + bx + c = 0, we can see that a = 122 and b = -7.

So, the sum of the roots of the equation 122x^2 – 7x + 4 = 0 is:

sum of roots = -b/a = -(-7) / 122 = 7/122

Therefore, the sum of the roots of the equation 122x^2 – 7x + 4 = 0 is 7/122.

## If the discriminant of the quadratic equation is +ve then the roots are:

A. Real & Equal
B. Rational & equal
C. Real & unequal
D. Complex

Correct Answer: C. Real & unequal

### Solution

If the discriminant of a quadratic equation ax^2 + bx + c = 0 is positive (i.e., b^2 – 4ac > 0), then the roots of the quadratic equation are real and unequal.

This is because the discriminant is used to determine the nature of the roots of a quadratic equation. Specifically, if the discriminant is positive, then the roots of the quadratic equation are real and unequal. This is because in this case, the quadratic formula (-b ± sqrt(b^2 – 4ac)) / 2a will yield two distinct real solutions for x.

In contrast, if the discriminant is zero (i.e., b^2 – 4ac = 0), then the roots of the quadratic equation are real and equal. This is because the quadratic formula will yield a single real solution for x.

If the discriminant is negative (i.e., b^2 – 4ac < 0), then the roots of the quadratic equation are complex conjugates. This means that the roots will have the form of a+bi and a-bi, where a and b are real numbers and i is the imaginary unit.

## Sum of the roots of equation ax^2+bx+c=0 is:

A. -b/a
B. -c/a
C. b/a
D. c/a

### Solution

The sum of the roots of a quadratic equation of the form ax^2 + bx + c = 0 is given by -b/a.

To see why this is the case, let the roots of the quadratic be x1 and x2. Then we can write the quadratic equation as:

ax^2 + bx + c = a(x – x1)(x – x2) = 0

Expanding the right-hand side gives:

a(x^2 – (x1 + x2)x + x1x2) = 0

Comparing the coefficients of x^2, x, and the constant term on both sides, we get:

x^2 coefficient: a = a x coefficient: -a(x1 + x2) = b constant term: a(x1)(x2) = c

Solving for x1 + x2, we have:

x1 + x2 = -b/a

Therefore, the sum of the roots of the quadratic equation ax^2 + bx + c = 0 is -b/a.