**Question**

If both 11

^{2}and 3

^{3}are factors of the number a * 4

^{3}* 6

^{2}* 13

^{11}, then what is the smallest possible value of a?

- 121
- 3267
- 363
- 33

**(3)**. Correct Answer -

**(363)**

**Explanatory Answers**

11

^{2}is a factor of the given number. The number does not have a power or multiple of 11 as its factor. Hence, "a" should include 11

^{2}

3

^{3}is a factor of the given number. 6

^{2}is a part of the number. 6

^{2}has 3

^{2}in it. Therefore, if 3

^{3}has to be a factor of the given number a * 4

^{3}* 6

^{2}* 13

^{11}, then we will need at least another 3.

Therefore, if "a" should be at least 11

^{2}* 3 = 363 if the given number has to have 11

^{2}and 3

^{3}as its factors.

## Question

The equation 2x^{2}+ 2(p + 1)x + p = 0, where p is real, always has roots that are

(1) Equal

(2) Equal in magnitude but opposite in sign

(3) Irrational

(4) Real

(5) Complex Conjugates

Correct Choice -

**(4)**. Correct Answer is

**The Roots are Real**

## Explanatory Answer

The value of the discriminant of a quadratic equation will determine the nature of the roots of a quadratic equation.The discriminant of a quadratic equation ax

^{2}+ bx + c = 0 is given by

**b**.

^{2}- 4ac- If the value of the discriminant is positive, i.e. greater than '0', then the roots of the quadratic equation will be real.
- If the value of the discriminant is '0', then the roots of the quadratic equation will be real and equal.
- If the value of the discriminant is negative, i.e. lesser than '0', then the roots of the quadratic equation will be imaginary. The two roots will be complex conjugates of the form p + iq and p - iq.

In this question, a = 2, b = 2(p + 1) and c = p

Therefore, the disciminant will be (2(p + 1))

^{2}- 4*2*p = 4(p + 1)

^{2}- 8p

= 4[(p + 1)

^{2}- 2p]

= 4[(p

^{2}+ 2p + 1) - 2p]

= 4(p

^{2}+ 1)

For any real value of p, 4(p

^{2}+ 1) will always be positive as p

^{2}cannot be negative for real p.

Hence, the discriminant b

^{2}- 4ac will always be positive.

When the discriminant is greater than '0' or is positive, the roots of a quadratic equation will be real.

Therefore, the answer choice is 4.

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A father and his son are waiting at a bus stop in the evening. There is a lamp post behind them. The lamp post, the father and his son stand on the same straight line. The father observes that the shadows of his head and his son's head are incident at the same point on the ground. If the heights of the lamp post, the father and his son are 6 metres, 1.8 metres and 0.9 metres respectively, and the father is standing 2.1 metres away from the post, then how far (in metres) is the son standing from his father?

- a. 0.9
- b. 0.75
- c. 0.6
- d. 0.45

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For what value of ‘m’ will the quadratic equation x

^{2}– mx + 4 = 0 have real and equal roots?

(A) | 16 |

(B) | 8 |

(C) | 2 |

(D) | -4 |

(E) | Choice (B) and (C) |

**Correct Answer**Choice (D)

**Solution:**

Any quadratic equation of the form ax

^{2}+ bx + c = 0 will have real and equal roots if its discriminant b

^{2}– 4ac = 0.

In the given equation x

^{2}– mx + 4 = 0, a = 1, b = -m and c = 4.

Therefore, b

^{2}– 4ac = m

^{2}– 4(4)(1) = m

^{2}– 16.

As we know, the roots of the given equation are real and equal.

Therefore, m

^{2}– 16 = 0 or m

^{2}= 16 or m = +4 or m = -4.

Hence, answer choice (D) is correct.

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*what percent of 1800 is 400?*

**1800 -> 100%**

400 -> x

x = (400*100) / 1800

x = 22.22 %400 -> x

x = (400*100) / 1800

x = 22.22 %

**---------------------------------------------------------------------------------------------------------------------------------**

**When the positive integer A is divided by 5 and 7, the remainder is 3 and 4, respectively. When the positive integer B is divided by 5 and 7, the remainder is 3 and 4, respectively. Which of the following is a factor of A-B?**(A) 12

(B) 24

(C) 35

(D) 16

(E) 30

*Solution---*A has the form of 5K+3 or 7N+4.

Similarly, B has the form of 5L+3 or 7M+4.

K,L,M and N are integers.

A and B can be -- 18,53,88 and so on..

So,A-B is always divisible by 35.

Ans - C

--------------------------------------------------------------------------------------------------------------Similarly, B has the form of 5L+3 or 7M+4.

K,L,M and N are integers.

A and B can be -- 18,53,88 and so on..

So,A-B is always divisible by 35.

Ans - C

**Find the mean of 3, 6, 11, and 8.**

We add all the numbers, and divide by the number of numbers in the list, which is 4.

(3 + 6 + 11 + 8) ÷ 4 = 7

So the mean of these four numbers is 7.

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**The students in Bjorn's class have the following ages: 4, 29, 4, 3, 4, 11, 16, 14, 17, 3. Find the median of their ages?**

Answer---Placed in order, the ages are 3, 3, 4, 4, 4, 11, 14, 16, 17, 29. The number of ages is 10, so the middle numbers are 4 and 11, which are the 5th and 6th entries on the ordered list. The median is the average of these two numbers:

(4 + 11)/2 = 15/2 = 7.5

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**The students in Bjorn's class have the following ages: 5, 9, 1, 3, 4, 6, 6, 6, 7, 3. Find the mode of their ages.**

Answer---The most common number to appear on the list is 6, which appears three times. No other number appears that many times. The mode of their ages is 6.

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**A retailer sell cameras at profit of 10% if he had paid 10% less for it and sold it for $10 more he would have made a profit of 25%.how much did it cost him and how much did he sell if for?**

Assume a cost price and then work backwards.

Let us say the cost price of the camera is $100. Then his initial selling price is $110.

His new cost would be $90. He will make a profit of 25% as stated if he had sold it for 90 + 25% of 90 = 90 + 22.5 = 112.5

i.e., his new selling price is $2.5 more than the old SP.

However, the question states that the new SP is $10 more than the old SP. Therefore, your assumption of CP ($100) has to be revised by the same factor of 4. So, the CP should be $400.