1.### Which term of the progression 4,9,14,…. is 109?

Answer:
22

Explanation:

tn= a+(n-1)d

Subtract the first term, 4, from 109: 109 - 4 = 105

Divide the result by the common difference, which is 5: 105 / 5 = 21

Add 1 to account for the progression starting at term 1: 21 + 1 = 22

So, the 22nd term of the sequence is 109.

Subtract the first term, 4, from 109: 109 - 4 = 105

Divide the result by the common difference, which is 5: 105 / 5 = 21

Add 1 to account for the progression starting at term 1: 21 + 1 = 22

So, the 22nd term of the sequence is 109.

2.### If 1+2+….+k=55, then the value of k is

Answer:
10

Explanation:

Start with the Sum Formula: We begin with the formula for the sum of an arithmetic series, which is given as:

Sn=n/2(a1+an)Sn=2n(a1+an)

Sn represents the sum of the series.

n is the number of terms in the series.

a1 is the first term of the series.

an is the nth term of the series.

Plug in the Known Values: In this problem:

Sn is 55 because we want the sum to be equal to 55.

a1 is 1 because the first term in the series is 1.

So, we have:

55=k/2(1+an)55=2k(1+an)

Simplify: Since a1=1, we simplify further:

55=k/2(1+k)

Multiply by 2 to Eliminate Fraction: To get rid of the fraction, we multiply both sides by 2:

110=k(1+k)

Quadratic Equation: We now have a quadratic equation in the form k2+k−110=0

Factor or Use Quadratic Formula: To solve this equation, you can either factor it or use the quadratic formula. Factoring it, we get:

(k+11)(k−10)=0

Solve for k: Set each factor equal to zero and solve for k:

k+11=0 → −11

k−10=0 → k=10

Choose the Positive Solution: Since we're looking for a positive value of k that represents the sum of the series, the value of k is 10.

So, the value of k in the equation 1+2+…+k=55 is 10, which means that the sum of the first 10 positive integers is equal to 55.

Sn=n/2(a1+an)Sn=2n(a1+an)

Sn represents the sum of the series.

n is the number of terms in the series.

a1 is the first term of the series.

an is the nth term of the series.

Plug in the Known Values: In this problem:

Sn is 55 because we want the sum to be equal to 55.

a1 is 1 because the first term in the series is 1.

So, we have:

55=k/2(1+an)55=2k(1+an)

Simplify: Since a1=1, we simplify further:

55=k/2(1+k)

Multiply by 2 to Eliminate Fraction: To get rid of the fraction, we multiply both sides by 2:

110=k(1+k)

Quadratic Equation: We now have a quadratic equation in the form k2+k−110=0

Factor or Use Quadratic Formula: To solve this equation, you can either factor it or use the quadratic formula. Factoring it, we get:

(k+11)(k−10)=0

Solve for k: Set each factor equal to zero and solve for k:

k+11=0 → −11

k−10=0 → k=10

Choose the Positive Solution: Since we're looking for a positive value of k that represents the sum of the series, the value of k is 10.

So, the value of k in the equation 1+2+…+k=55 is 10, which means that the sum of the first 10 positive integers is equal to 55.

3.### The 10th term of the sequence, whose nth term is (3n-2), is

Answer:
28

Explanation:

To find the 10th term of the sequence where the nth term is described by the formula (3n - 2), you can substitute 10 for n in the formula:

Multiply 3 by 10: 3 * 10 = 30

Subtract 2 from the result: 30 - 2 = 28

Therefore, the 10th term of the sequence is 28.

Multiply 3 by 10: 3 * 10 = 30

Subtract 2 from the result: 30 - 2 = 28

Therefore, the 10th term of the sequence is 28.

4.### If cot A = 12/5, then the value of (sin A + cos A) x cosec A is :

Answer:
17/5

Explanation:

To solve for the value of (sin A + cos A) x cosec A given that cot A = 12/5, we need to first find the values of sin A, cos A, and cosec A.

Step 1: Calculate sin A and cos A From the cotangent identity, cot A = cos A / sin A, and given cot A = 12/5, we can express cos A and sin A using a right triangle where:

The adjacent side (to angle A) = 12 (corresponding to cos A)

The opposite side (to angle A) = 5 (corresponding to sin A)

To find the hypotenuse (h), we use the Pythagorean theorem: Hypotenuse squared = Adjacent squared + Opposite squared h squared = 12 squared + 5 squared h squared = 144 + 25 h squared = 169 h = 13

Thus, the values of sin A and cos A are: sin A = Opposite / Hypotenuse = 5 / 13 cos A = Adjacent / Hypotenuse = 12 / 13

Step 2: Calculate cosec A cosec A = 1 / sin A = 13 / 5

Step 3: Calculate (sin A + cos A) x cosec A sin A + cos A = 5/13 + 12/13 = 17/13 (sin A + cos A) x cosec A = 17/13 x 13/5 = 17 x 13 / 13 x 5 = 17 / 5

Therefore, the value of (sin A + cos A) x cosec A is 17/5.

Step 1: Calculate sin A and cos A From the cotangent identity, cot A = cos A / sin A, and given cot A = 12/5, we can express cos A and sin A using a right triangle where:

The adjacent side (to angle A) = 12 (corresponding to cos A)

The opposite side (to angle A) = 5 (corresponding to sin A)

To find the hypotenuse (h), we use the Pythagorean theorem: Hypotenuse squared = Adjacent squared + Opposite squared h squared = 12 squared + 5 squared h squared = 144 + 25 h squared = 169 h = 13

Thus, the values of sin A and cos A are: sin A = Opposite / Hypotenuse = 5 / 13 cos A = Adjacent / Hypotenuse = 12 / 13

Step 2: Calculate cosec A cosec A = 1 / sin A = 13 / 5

Step 3: Calculate (sin A + cos A) x cosec A sin A + cos A = 5/13 + 12/13 = 17/13 (sin A + cos A) x cosec A = 17/13 x 13/5 = 17 x 13 / 13 x 5 = 17 / 5

Therefore, the value of (sin A + cos A) x cosec A is 17/5.

5.### Given that sin A = 1/2 and cos B= 1/ then the value of (A+B) is :

Answer:
75 Degree

Explanation:

sin A = sin 30 = 1/2 => A = 30 degree

cos B = cos 45 = 1/ /2 => B = 45 degree

So, A+B => 30 +45 = 75 degree

cos B = cos 45 = 1/ /2 => B = 45 degree

So, A+B => 30 +45 = 75 degree

6.### When a point is observed, the angle formed by the line of sight with the horizontal level where the point being viewed is above the horizontal plane is known as:

Answer:
Angle of elevation

Explanation:

The angle formed by the line of sight with the horizontal level when the point being viewed is above the horizontal plane is known as the angle of elevation. This geometric concept is commonly used in various fields such as surveying, navigation, astronomy, and architecture to describe the angle at which an observer must look upwards to see an object that is higher than the level at which the observer is located.

7.### The HCF of two numbers, a and b, is 30, while their LCM is 45. What is the value of (a x b)?

Answer:
1350

Explanation:

HCF (a,b) = 30

LCM (a,b) = 45

We know that, HCF (a,b) x LCM (a,b) = a x b

Thus, a x b => 30 x 45 = 1350

LCM (a,b) = 45

We know that, HCF (a,b) x LCM (a,b) = a x b

Thus, a x b => 30 x 45 = 1350

8.### Solve for x : 2x + y = 8, y= - 6

Answer:
7

Explanation:

The given equation is

2x + y = 8,

Plug in y = - 6

2x-6 = 8

Add 6 on both sides of the equation 2x = 8 + 6 => 14

To get x by itself, divide 2 on both sides of the equation

x = 14/2 = 7

2x + y = 8,

Plug in y = - 6

2x-6 = 8

Add 6 on both sides of the equation 2x = 8 + 6 => 14

To get x by itself, divide 2 on both sides of the equation

x = 14/2 = 7

9.### In a lottery, there are 10 prizes and 20 blanks. What is the probability of getting a prize?

Answer:
1/3

Explanation:

In the context of a lottery ticket, where the total possible outcomes are calculated by adding the outcomes of not getting a prize (20) to the outcomes of getting a prize (10), the sum equals 30 possible outcomes. To find the probability of winning a prize, we divide the number of favorable outcomes (getting a prize) by the total number of outcomes. This calculation, 10 divided by 30, simplifies to 1/3. Therefore, the probability of securing a prize when purchasing a single ticket in this scenario is 1/3.

10.### The probability of an event can't be ________________.

Answer:
Negative

Explanation:

The probability of an event represents the likelihood of that event occurring and is expressed as a number between 0 and 1, inclusive. A probability of 0 means the event will not occur, while a probability of 1 indicates certainty that the event will occur. Negative probabilities do not exist in this context as they fall outside the defined range and have no meaningful interpretation within the theory of probability.

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