100525 + Interview Questions in GENERAL QA IN GENERAL AWARENESS

1.	Directions: The following question is accompanied by two statements (I), (II). You have to determine which statements(s) is/are sufficient/necessary to answer the questions.What is the time P and Q to complete the work?Statement I: P and R take 10 hours to complete the work.Statement II: Q and R take 12 hours to complete the work.1.Statement I alone is sufficient to answer the question but statement II alone is not sufficient. 2.Statement II alone is sufficient to answer the question but statement I alone is not sufficient. 3.Both statements I and statement II together are needed to answer the question.4.Neither statement I and nor statement II is sufficient to answer the question. 5. Either of the statement I and statement II together is needed to answer the question.
Answer» Correct Answer - Option 4 : Neither statement I and nor statement II is sufficient to answer the question. Formula used: Total work = Efficiency × Time period Calculation: Statement I: P and R can complete the work in 10 hr From statement I alone we cannot answer the question Statement II: Q and R can complete the work in 12 hr From statement II alone we cannot answer the question Combining both the statements I and II P and R can complete the work in 10 hr Q and R can complete the work in 12 hr Taking the L.C.M of 10 and 12 ⇒ L.C.M = 60 ⇒ Efficiency of P and R = (60/10) = 6 ⇒ Efficiency of Q and R = (60/12) = 5 As few values are missing we cannot find the solution after combining both the statements It would be possible to solve if the time taken by R alone to complete the work were given ∴ Neither statement I and nor statement II is sufficient to answer the question.

1.

Directions: The following question is accompanied by two statements (I), (II). You have to determine which statements(s) is/are sufficient/necessary to answer the questions.What is the time P and Q to complete the work?Statement I: P and R take 10 hours to complete the work.Statement II: Q and R take 12 hours to complete the work.1.Statement I alone is sufficient to answer the question but statement II alone is not sufficient. 2.Statement II alone is sufficient to answer the question but statement I alone is not sufficient. 3.Both statements I and statement II together are needed to answer the question.4.Neither statement I and nor statement II is sufficient to answer the question. 5. Either of the statement I and statement II together is needed to answer the question.

Answer» Correct Answer - Option 4 :

Neither statement I and nor statement II is sufficient to answer the question.

Formula used:

Total work = Efficiency × Time period

Calculation:

Statement I:

P and R can complete the work in 10 hr

From statement I alone we cannot answer the question

Statement II:

Q and R can complete the work in 12 hr

From statement II alone we cannot answer the question

Combining both the statements I and II

P and R can complete the work in 10 hr

Q and R can complete the work in 12 hr

Taking the L.C.M of 10 and 12

⇒ L.C.M = 60

⇒ Efficiency of P and R = (60/10) = 6

⇒ Efficiency of Q and R = (60/12) = 5

As few values are missing we cannot find the solution after combining both the statements

It would be possible to solve if the time taken by R alone to complete the work were given

∴ Neither statement I and nor statement II is sufficient to answer the question.

Discussion

2.	Directions: The following question is accompanied by two statements (I) and (II). You have to determine which statements(s) is/are sufficient/necessary to answer the questions.Find the total hours to make Rs. 2200?I) A man earn Rs. 1500 for his first 30 hours in a week.II) If After 30 hours he gets seven-fifths times his regular hourly rate for any additional hours then the total hours to make Rs. 2200.1. The statement I alone is sufficient to answer the question, but the statement II alone is not sufficient. 2. The statement II alone is sufficient to answer the question, but the statement I alone is not sufficient. 3. Both the statements I and II together are needed to answer the question. 4. Either statement I alone or statement II alone is sufficient to answer the question. 5. Neither statement I nor statement II is sufficient to answer the question.
Answer» Correct Answer - Option 3 : Both the statements I and II together are needed to answer the question. Concept used: Using the elementary mathematics. Statement (I): A man earn Rs. 1500 for his first 30 hours in a week. In 1 hour he earn = 1500/30 = Rs. 50 ∴ The statement (I) alone is not sufficient to answer the question. Statement (II): If After 30 hours he gets seven-fifths times his regular hourly rate for any additional hours then the total hours to make Rs. 2200. Total money he earned = Rs. 2200 Earn money for extra hour = (7/5) × 50 = Rs. 70 ∴ The statement (I) alone is not sufficient to answer the question. ∴ Both the statements I and II together are needed to answer the question.

2.

Directions: The following question is accompanied by two statements (I) and (II). You have to determine which statements(s) is/are sufficient/necessary to answer the questions.Find the total hours to make Rs. 2200?I) A man earn Rs. 1500 for his first 30 hours in a week.II) If After 30 hours he gets seven-fifths times his regular hourly rate for any additional hours then the total hours to make Rs. 2200.1. The statement I alone is sufficient to answer the question, but the statement II alone is not sufficient. 2. The statement II alone is sufficient to answer the question, but the statement I alone is not sufficient. 3. Both the statements I and II together are needed to answer the question. 4. Either statement I alone or statement II alone is sufficient to answer the question. 5. Neither statement I nor statement II is sufficient to answer the question.

Answer» Correct Answer - Option 3 : Both the statements I and II together are needed to answer the question.

Concept used:

Using the elementary mathematics.

Statement (I):

A man earn Rs. 1500 for his first 30 hours in a week.

In 1 hour he earn = 1500/30 = Rs. 50

∴ The statement (I) alone is not sufficient to answer the question.

Statement (II):

If After 30 hours he gets seven-fifths times his regular hourly rate for any additional hours then the total hours to make Rs. 2200.

Total money he earned = Rs. 2200

Earn money for extra hour = (7/5) × 50 = Rs. 70

∴ The statement (I) alone is not sufficient to answer the question.

∴ Both the statements I and II together are needed to answer the question.

Discussion

3.	Directions: The following question is accompanied by two statements (I) and (II). You have to determine which statements(s) is/are sufficient/necessary to answer the questions.Find the discount percentage.I) The marked price of an article is Rs. 3800. But due to sale a certain percent of discount is declared.II) Amit availed the opportunity and bought the article at Rs. 3200 and thereby made a profit of (100/7)%.1. The statement I alone is sufficient to answer the question, but the statement II alone is not sufficient.2. The statement II alone is sufficient to answer the question, but the statement I alone is not sufficient. 3. Both the statements I and II together are needed to answer the question. 4. Either statement I alone or statement II alone is sufficient to answer the question. 5. Neither statement I nor statement II is sufficient to answer the question.
Answer» Correct Answer - Option 3 : Both the statements I and II together are needed to answer the question. Formula used: Discount % = (Discount/M.P) × 100 Selling price = Marked price - Discount Statement (I): The marked price of an article is Rs. 3800. Discount percent is given. ∴ Statement (I) is not sufficient to answer the question. Statement (II): Amit availed bought the article at Rs. 3200. Profit percent = (100/7)% ∴ Statement (II) is not sufficient to answer the question. Statement (I & II): MP & CP - Given P% - Given MP/CP = (100 + P%)/(100 - D%) ∴ Both the statements I and II together are needed to answer the question.

3.

Directions: The following question is accompanied by two statements (I) and (II). You have to determine which statements(s) is/are sufficient/necessary to answer the questions.Find the discount percentage.I) The marked price of an article is Rs. 3800. But due to sale a certain percent of discount is declared.II) Amit availed the opportunity and bought the article at Rs. 3200 and thereby made a profit of (100/7)%.1. The statement I alone is sufficient to answer the question, but the statement II alone is not sufficient.2. The statement II alone is sufficient to answer the question, but the statement I alone is not sufficient. 3. Both the statements I and II together are needed to answer the question. 4. Either statement I alone or statement II alone is sufficient to answer the question. 5. Neither statement I nor statement II is sufficient to answer the question.

Answer» Correct Answer - Option 3 : Both the statements I and II together are needed to answer the question.

Formula used:

Discount % = (Discount/M.P) × 100

Selling price = Marked price - Discount

Statement (I):

The marked price of an article is Rs. 3800.

Discount percent is given.

∴ Statement (I) is not sufficient to answer the question.

Statement (II):

Amit availed bought the article at Rs. 3200.

Profit percent = (100/7)%

∴ Statement (II) is not sufficient to answer the question.

Statement (I & II):

MP & CP - Given

P% - Given

MP/CP = (100 + P%)/(100 - D%)

∴ Both the statements I and II together are needed to answer the question.

Discussion

4.	Directions: The following question is accompanied by two statements (I) and (II). You have to determine which statements(s) is/are sufficient/necessary to answer the questions.Find the height of cylinder.I) A rectangular piece of paper of breadth 22 cm is rolled along breadth to form a cylinder. If the curved surface area of a cylinder is 264 cm2. II) The volume of cylinder is 462 cm3.1. The statement I alone is sufficient to answer the question, but the statement II alone is not sufficient. 2. The statement II alone is sufficient to answer the question, but the statement I alone is not sufficient. 3. Both the statements I and II together are needed to answer the question.4. Either statement I alone or statement II alone is sufficient to answer the question. 5. Neither statement I nor statement II is sufficient to answer the question.
Answer» Correct Answer - Option 1 : The statement I alone is sufficient to answer the question, but the statement II alone is not sufficient. Formula used: The curved surface area of a cylinder = 2πrh Volume of cylinder = πr2h Statement (I): The curved surface area of a cylinder = 264 cm2. A rectangular piece of paper of breadth 22 cm is rolled along breadth to form a cylinder. The rectangular side which is rolled to form a cylinder will become its Height. So, Height of Cylinder = 22 cm ∴ The statement I alone is sufficient to answer the question Statement (II): Volume of cylinder = πr2h The volume of cylinder = 462 cm3. ⇒ πr2h = 462 cm3 ∴ Statement II alone is not sufficient. ∴ The statement I alone is sufficient to answer the question, but the statement II alone is not sufficient.

4.

Directions: The following question is accompanied by two statements (I) and (II). You have to determine which statements(s) is/are sufficient/necessary to answer the questions.Find the height of cylinder.I) A rectangular piece of paper of breadth 22 cm is rolled along breadth to form a cylinder. If the curved surface area of a cylinder is 264 cm2. II) The volume of cylinder is 462 cm3.1. The statement I alone is sufficient to answer the question, but the statement II alone is not sufficient. 2. The statement II alone is sufficient to answer the question, but the statement I alone is not sufficient. 3. Both the statements I and II together are needed to answer the question.4. Either statement I alone or statement II alone is sufficient to answer the question. 5. Neither statement I nor statement II is sufficient to answer the question.

Answer» Correct Answer - Option 1 : The statement I alone is sufficient to answer the question, but the statement II alone is not sufficient.

Formula used:

The curved surface area of a cylinder = 2πrh

Volume of cylinder = πr2h

Statement (I):

The curved surface area of a cylinder = 264 cm2.

A rectangular piece of paper of breadth 22 cm is rolled along breadth to form a cylinder.

The rectangular side which is rolled to form a cylinder will become its Height.

So, Height of Cylinder = 22 cm

∴ The statement I alone is sufficient to answer the question

Statement (II):

Volume of cylinder = πr2h

The volume of cylinder = 462 cm3.

⇒ πr2h = 462 cm3

∴ Statement II alone is not sufficient.

∴ The statement I alone is sufficient to answer the question, but the statement II alone is not sufficient.

Discussion

5.	If \({A \over 3} = {B \over 4} = {C \over 5}\), then A : B : C is:1. 3 : 5 : 62. 3 : 4 : 63. 3 : 4 : 54. 2 : 3 : 5
Answer» Correct Answer - Option 3 : 3 : 4 : 5 Given: A/3 = B/4 = C/5 Calculation: Let A/3 = B/4 = C/5 = K [Where K is a constant] ⇒ A = 3K, B = 4K, C = 5K ⇒ A : B : C = 3K : 4K : 5K ⇒ A : B : C = 3 : 4 : 5 ∴ A : B : C is 3 : 4 : 5

5.

If \({A \over 3} = {B \over 4} = {C \over 5}\), then A : B : C is:1. 3 : 5 : 62. 3 : 4 : 63. 3 : 4 : 54. 2 : 3 : 5

Answer» Correct Answer - Option 3 : 3 : 4 : 5

Given:

A/3 = B/4 = C/5

Calculation:

Let A/3 = B/4 = C/5 = K [Where K is a constant]

⇒ A = 3K, B = 4K, C = 5K

⇒ A : B : C = 3K : 4K : 5K

⇒ A : B : C = 3 : 4 : 5

∴ A : B : C is 3 : 4 : 5

Discussion

6.	Let `Z` be the set of all integers and `R` be the relation on `Z` defined as `R={(a, b); a, b in Z,` and `(a-b)` is divisible by `5}`. Prove that `R` is an equivalence relation.
Answer» Here, `R = {(a,b):a,b in R and (a-b)` is divisible by `5}` For all `a in R`, `=> (a-a) =0` and `0` is divisible by `5`. `:. R` is refexive. Since in `R` for every `(a,b) in R` `=> (a-b)` is divisible by `5`. `=> (-(b-a))` is divisible by `5`. `=> (b-a)` is also divisble by `5`. `:. (b,a) in R`. `:. R` is symmetric. Since `(a,b) in R and (b,c) in R` `=> (a-b)` is divisible by `5` & `(b-c)` is divisible by `5`. `=> (a-b+(b-c))` is divisible by `5`. `=> (a-c)` is divisible by `5`. `:. (a,c) in R`. `:. R` is transitive. As `R` is reflexive, symmetric and transitive, `R` is an equivalence relation.

Discussion

7.	Find the Cartesianequation of the plane passing through the points `A (0,0,0)`and `b(3,-1,2)`and parallel to the line `(x-4)/1=(y+3)/(-4)=(z+1)/7`
Answer» Vector joining point `A` and `B` wil be, `veca = 3hati-hatj+2hatk` Given line is, `(x-4)/1 = (y+3)/(-4) = (z+1)/7` So, `vecb = hati-4hatj+7hatk` is the another vector in the plane. So, a normal to the plane will be, `vec a xx vec b`. Now, `vec a xx vec b = \|[hati,hatj,hatk],[3,-1,2],[1,-4,7]\|` `=hati(-7+8)-hatj(21-2)+hatk(-12+1)` `=hati-19hatj-11hatk` So, equation of plane will be, `vecr (vec a xx vecb) = 0` Let `vecr = xhati+yhatj+zhatk` So, required equation will be, `(xhati+yhatj+zhatk)(hati-19hatj-11hatk) = 0` `=>x-19y-11z = 0`, which is the equation of the plane.

Discussion

8.	If 3x/2y = 48/36, find the value of x : y.1. 7 : 82. 7 : 93. 8 : 94. 9 : 7
Answer» Correct Answer - Option 3 : 8 : 9 Given: 3x/2y = 48/36 Calculation: ⇒ 3x/2y = 48/36 ⇒ x/y = (48× 2)/(3× 36) ⇒ x/y = 16/18 ⇒ x/y = 8/9 ∴ The value of x ∶ y is 8 ∶ 9.

8.

If 3x/2y = 48/36, find the value of x : y.1. 7 : 82. 7 : 93. 8 : 94. 9 : 7

Answer» Correct Answer - Option 3 : 8 : 9

Given:

3x/2y = 48/36

Calculation:

⇒ 3x/2y = 48/36

⇒ x/y = (48× 2)/(3× 36)

⇒ x/y = 16/18

⇒ x/y = 8/9

∴ The value of x ∶ y is 8 ∶ 9.

Discussion

9.	On a multiple choice examination with three possibleanswers (out of which only one is correct) for each of the five questions,what is the probability that a candidate would get four or more correctanswers just by guessing?
Answer» Let probability of getting a correct answer is `P(A) `. Here, `P(A) = 1/3` `:. P(barA) = 1-1/3 = 2/3` Here, `P(barA)` is the probability of getting an incorrect answer. Now, there are two cases when a candidate can answer four or more answers correctly. Case 1: When he answers 4 questions correcly and one incorrect. In this case, probability will be `= C(5,4)(1/3)^4(2/3) = 51/811/3 = 10/243` Case 2: When he answers all 5 questions correcly . In this case, probability will be `= C(5,5)(1/3)^5 = 1*1/243 = 1/243` `:.` Required probaility ` = 10/243+1/243 = 11/243.`

Discussion

10.	An article is sold at certain price. By selling it at 2/3 of its price one losses 10%,find the gain at original price?
Answer» Let the original S.P be Rs x. then now S.P = Rs. 2x/3,loss=10% Now C.P = Rs 20x/2727/20x100)%=35%

10.

An article is sold at certain price. By selling it at 2/3 of its price one losses 10%,find the gain at original price?

Answer»

Let the original S.P be Rs x. then now S.P = Rs. 2x/3,loss=10%

Now C.P = Rs 20x/27*27/20x*100)%=35%

Discussion

11.	Ram sold a watch for Rs.376 at a loss of 6%. Find the cost price of the watch.
Answer» Cost price = selling price (100/100-Loss%) = Rs.376 × (100/100-6) = Rs.376 ×100/94 =Rs.400

11.

Ram sold a watch for Rs.376 at a loss of 6%. Find the cost price of the watch.

Answer»

Cost price = selling price (100/100-Loss%)

= Rs.376 × (100/100-6)

= Rs.376 ×100/94

=Rs.400

Discussion

12.	Monika purchased a pressure cooker at 9/10th of its selling price and sold it at 8% more than its S.P .find her gain percent.
Answer» Let the S.P be Rs. X .then C.P = Rs.9x/10,Receipt=108% of Rs.x=Rs 27x/25 Gain=Rs (27x/259x/10)=Rs(108x-90x/100)=Rs18x/100 Gain%=(18x/10010/9x*100)%=20%

12.

Monika purchased a pressure cooker at 9/10th of its selling price and sold it at 8% more than its S.P .find her gain percent.

Answer»

Let the S.P be Rs. X .then C.P = Rs.9x/10,Receipt=108% of Rs.x=Rs 27x/25

Gain=Rs (27x/25*9x/10)=Rs(108x-90x/100)=Rs18x/100

Gain%=(18x/100*10/9x*100)%=20%

Discussion

13.	Describe the rules for the athletes of track events?
Answer» The following tare the rules for the athletes of track events: They should wear a clear outfit which does not hinder or obstract their movement. They can run either wearing spikes or base footed. An athlete is disqualified if he obstructs a fellow athlete or hinders his progress. Every athlete wears his number clearly at the back and the front. The athlete must stick to their allotted lanes throughout the race. If an athlete delibrately leaves his lane, he is disqualified. The jugdes can accommodate a player appropriately if both track and field events start simultaneously. The athletes must not use banned drugs and intoxicants. If any athlete is found using such a substance, he is disqualified. The start of 800 metres race is announced in native language “On Your Mark” and the whistle is blown to start the race. The athlete shouldn’t touch the ground or the starting-line when “On Your Mark” has been announced. An athlete, who makes two foul starts, is disqualified from the race. The winner of a race is decided at the ‘finishing line’. The athlete, who touches the line first by any part of his body, is declared the winner. If in a hurdles race the referee thinks that a particular athlete delibrately throws a hurdle down by his hand or foot he can duly disqualify that athlete. If there are a large number of athletes in a ‘throw’ event, the referee sets a qualifying mark and gives six chances to each athlete.

13.

Describe the rules for the athletes of track events?

Answer»

The following tare the rules for the athletes of track events:

They should wear a clear outfit which does not hinder or obstract their movement.
They can run either wearing spikes or base footed.
An athlete is disqualified if he obstructs a fellow athlete or hinders his progress.
Every athlete wears his number clearly at the back and the front.
The athlete must stick to their allotted lanes throughout the race.
If an athlete delibrately leaves his lane, he is disqualified.
The jugdes can accommodate a player appropriately if both track and field events start simultaneously.
The athletes must not use banned drugs and intoxicants. If any athlete is found using such a substance, he is disqualified.
The start of 800 metres race is announced in native language “On Your Mark” and the whistle is blown to start the race.
The athlete shouldn’t touch the ground or the starting-line when “On Your Mark” has been announced.
An athlete, who makes two foul starts, is disqualified from the race.
The winner of a race is decided at the ‘finishing line’. The athlete, who touches the line first by any part of his body, is declared the winner.
If in a hurdles race the referee thinks that a particular athlete delibrately throws a hurdle down by his hand or foot he can duly disqualify that athlete.
If there are a large number of athletes in a ‘throw’ event, the referee sets a qualifying mark and gives six chances to each athlete.

Discussion

14.	Outline that mean and variance of the Poisson distribution are same.
Answer» The mean and the variance of the Poisson distribution are the same, which is equal to the average number of successes that occur in the given interval of time.

Discussion

15.	How many ‘foul starts’ lead to the disqualification of the athlete for the race?
Answer» Two foul starts in races and 3 such starts in tholon and decatholon make an athlete ineligible to take part in that event.

Discussion

16.	3) Find the sum of the numbers which are not divisible by 5. between 5 and 50 .
Answer» 6,7,8,9,11,12,13,14,16,17,18,19,21,22,23,24,26,27,28,29,31,32,33,34,36,37,38,39,41,42,43,44,46,47,48,49 these are the numbers which are not divisible by 5and 50

Discussion

17.	find \( A B \)
Answer» where is the question and figure

Discussion

18.	Write dobun the minors of \( -2 \) and 4 in\[\left\|\begin{array}{rrr}2 & 1 & 1 \\1 & -2 & -3 \\3 & 2 & 4\end{array}\right\|\]
Answer» Minor of element -2 or a₂₂ in matrix is M₂₂ = \(\begin{vmatrix}2&1\\3&4\end{vmatrix}\) = 8 - 3 = 5 Minor of element 4 or a₃₃ in given matrix is M₃₃ = \(\begin{vmatrix}2&1\\1&-2\end{vmatrix}\) = -4 - 1 = -5.

18.

Write dobun the minors of \( -2 \) and 4 in\[\left|\begin{array}{rrr}2 & 1 & 1 \\1 & -2 & -3 \\3 & 2 & 4\end{array}\right|\]

Answer»

Minor of element -2 or a₂₂ in matrix is

M₂₂ = \(\begin{vmatrix}2&1\\3&4\end{vmatrix}\) = 8 - 3 = 5

Minor of element 4 or a₃₃ in given matrix is

M₃₃ = \(\begin{vmatrix}2&1\\1&-2\end{vmatrix}\) = -4 - 1 = -5.

Discussion

19.	A cubical die faces marked 1, 2, 3,.... 6 tossed such that the probability of throwing the number t is proportional to t2. The probability that the number 5 has appeared given that when the die is rolled the number turned up is not even is
Answer» Given that probability of throwing a number t is proportional to t². i.e., p(t) \(\propto\) t² ⇒ p(t) = k.t², where k is proportionally constant. i.e., P(1) = k, P(2) = 4k, P(3) = 9k P(u) = 16k, P(5) = 25k & P(6) = 36 k Let event A be has appeared and event B be number turned up is not even number P(B) = P(1 or 3 or 5) = P(1) + P(3) + P(5) (\(\because\) All outcomes are independent) = k + 9k + 25k = 35k P(A \(\cap\) B) = P(5) = 25k \(\therefore\) P\((\frac{A}B)=\frac{P(A\cap B)}{P(B)}\) = \(\frac{25k}{35k}\) = \(\frac{25}{35}\) = \(\frac57\) Hence, probability that number 5 has appeared when given that the die is rolled the number turned up is not even is P(A/B) = 5/7

19.

A cubical die faces marked 1, 2, 3,.... 6 tossed such that the probability of throwing the number t is proportional to t2. The probability that the number 5 has appeared given that when the die is rolled the number turned up is not even is

Answer»

Given that probability of throwing a number t is proportional to t².

i.e., p(t) \(\propto\) t²

⇒ p(t) = k.t², where k is proportionally constant.

i.e., P(1) = k, P(2) = 4k, P(3) = 9k

P(u) = 16k, P(5) = 25k & P(6) = 36 k

Let event A be has appeared and event B be number turned up is not even number

P(B) = P(1 or 3 or 5) = P(1) + P(3) + P(5)

(\(\because\) All outcomes are independent)

= k + 9k + 25k

= 35k

P(A \(\cap\) B) = P(5) = 25k

\(\therefore\) P\((\frac{A}B)=\frac{P(A\cap B)}{P(B)}\) = \(\frac{25k}{35k}\) = \(\frac{25}{35}\) = \(\frac57\)

Hence, probability that number 5 has appeared when given that the die is rolled the number turned up is not even is P(A/B) = 5/7

Discussion

20.	\( \int \frac{3 x-2}{\sqrt{3+2 x-4 x^{2}}} d x \)
Answer» \(\int\frac{3x-2}{\sqrt{3+2x-4x^2}}dx\) = \(\frac38\int\frac{(8x-16/3)dx}{\sqrt{3+2x-4x^2}}\) \(=-\frac38\int\frac{(-8x+16/3)dx}{\sqrt{3+2x-4x^2}}\) \(=-\frac38\frac{(-8x+2+10/3)dx}{\sqrt{3+2x-4x^2}}\) \(=-\frac38\int\frac{-8x+2}{\sqrt{3+2x-4x^2}}dx\)\(-\frac38\times\frac{10}3\int\cfrac{1}{\sqrt{\frac{13}4-(2x-\frac12)^2}}dx\) \(=-\frac38\times2\sqrt{3+2x-4x^2}-\frac52sin^{-1}(\frac{2x-1/2}{\sqrt{13}/2})+c\) \(=-\frac34\sqrt{3+2x-4x^2}-\frac52sin^{-1}(\frac{4x-1}{\sqrt{13}})+c\)

20.

\( \int \frac{3 x-2}{\sqrt{3+2 x-4 x^{2}}} d x \)

Answer»

\(\int\frac{3x-2}{\sqrt{3+2x-4x^2}}dx\)

= \(\frac38\int\frac{(8x-16/3)dx}{\sqrt{3+2x-4x^2}}\)

\(=-\frac38\int\frac{(-8x+16/3)dx}{\sqrt{3+2x-4x^2}}\)

\(=-\frac38\frac{(-8x+2+10/3)dx}{\sqrt{3+2x-4x^2}}\)

\(=-\frac38\int\frac{-8x+2}{\sqrt{3+2x-4x^2}}dx\)\(-\frac38\times\frac{10}3\int\cfrac{1}{\sqrt{\frac{13}4-(2x-\frac12)^2}}dx\)

\(=-\frac38\times2\sqrt{3+2x-4x^2}-\frac52sin^{-1}(\frac{2x-1/2}{\sqrt{13}/2})+c\)

\(=-\frac34\sqrt{3+2x-4x^2}-\frac52sin^{-1}(\frac{4x-1}{\sqrt{13}})+c\)

Discussion

21.	Insert one rational number between 2/3 and 4/5
Answer» The rational numbers between 2/3 and 4/5 are 61/90, 62/90, 63/90, 64/90, 65/90.

Discussion

22.	There is a hemispherical bowl. A cone is to be made such that, if it is filled with water twice and the water is poured in the bowl, it will be filled just completely. State how will you decide the radius and perpendicular height of the cone.
Answer» Volume of hemisphere = 2/3 πR³ Volume of cone = 1/3 πr² × h By the given condition ; 2 × volume of cone = volume of hemisphere ∴ 2 × 1 / 3 πr² h = 2/3 πR³ ∴r² h = R³ ∴ if r = h = R .......then both sides will be equal. ∴ if radius of base of the cone is R and its height is R, which is equal to radius of the bowl, then a cone satisfying the given condition can be made.

22.

There is a hemispherical bowl. A cone is to be made such that, if it is filled with water twice and the water is poured in the bowl, it will be filled just completely. State how will you decide the radius and perpendicular height of the cone.

Answer»

Volume of hemisphere = 2/3 πR³

Volume of cone = 1/3 πr² × h

By the given condition ;

2 × volume of cone = volume of hemisphere

∴ 2 × 1 / 3 πr² h = 2/3 πR³

∴r² h = R³

∴ if r = h = R .......then both sides will be equal.

∴ if radius of base of the cone is R and its height is R, which is equal to radius of the bowl, then a cone satisfying the given condition can be made.

Discussion

23.	Solve x+y=7 and x-y=1
Answer» Adding 2 eqns 2x=8 x=4 y=x-1=4-3=1

Discussion

24.	Solve x^2+y^2=1 and x=1
Answer» 1^2+y^2=1 y^2=0 y=0 x=1

Discussion

25.	What is equation of tangent of y^2=4ax in parametric form
Answer» yt=x+at^2 where t is the parameter such that points on parabola are give by (at^2,2at)

Discussion

26.	What is the maximum no. of normals that can be drawn to a parabola from a point.?
Answer» Maximum 3 normals can be drawn

Discussion

27.	If 7 and 2 are two roots of the equation \|(x,3,7),(2,x,2),(7,6,x)\| = 0, then the third root is -(a) -9 (b) 14 (c) 1/2(d) None of these
Answer» Answer is (a) -9

Discussion

28.	lf a,b,c are in A.P. then -\|(x + 1,x + 2,x + a),(x + 2,x + 3,x + b),(x + 3,x + 4,x + c)\|(a) 3(b) -3(c) 0(d) 1
Answer» Answer is (c) 0

Discussion

29.	Find dy/dx if y = cos√sin x.
Answer» Given that y = cos√sin x Differentiating both sides with respect to x dy/dx = (d(cos√sin x)/d(√sin x)) x (d(√sin x)/d(sin x)) x (d(sin x)/dx) = (- sin√sin x) x (1/(2√sin x)) x (cos x) = (((sin x√sin x) x (cos x))/(2√sin x))

29.

Find dy/dx if y = cos√sin x.

Answer»

Given that y = cos√sin x

Differentiating both sides with respect to x

dy/dx = (d(cos√sin x)/d(√sin x)) x (d(√sin x)/d(sin x)) x (d(sin x)/dx)

= (- sin√sin x) x (1/(2√sin x)) x (cos x)

= (((sin x√sin x) x (cos x))/(2√sin x))

Discussion

30.	tan-1 (1) + cos-1 (-1/2) + sin-1 (-1/2) = (a) π/4(b) 3π/4(c) -π/4(d) π/2
Answer» Answer is (b) 3π/4

Discussion

31.	2cot-1 3 + cot-1 7 =(a) π/2(b) π/4(c) π(d) π/6
Answer» Answer is (b) π/4

Discussion

32.	cos-1 (2x - 1) =(a) 2 cos-1 x(b) cos-1 √x(c) 2 cos-1 √x(d) None of these
Answer» Answer is (c) 2 cos^-1 √x

Discussion

33.	The distance of the plane 2x - 3y + 6z + 7 = 0 from the point (2,-3,-1) is(a) 4 (b) 3 (c) 2 (d) 1/5
Answer» Answer is (c) 2

Discussion

34.	cos-1 (cos(8π/5)) =(a) 8π/5(b) 12π/5(c) 2π/5(d) 4π/5
Answer» Answer is (c) 2π/5 Cos^-1(cos8π/5)= 2π-8π/5=2π/5

34.

cos-1 (cos(8π/5)) =(a) 8π/5(b) 12π/5(c) 2π/5(d) 4π/5

Answer»

Answer is (c) 2π/5

Cos^-1(cos8π/5)= 2π-8π/5=2π/5

Discussion

35.	If P(A) = 3/8, P(B) = 1/2, P(A ∩ B) = 1/4, then P(A ∪ B) = (a) 2/3(b) 1/3(c) 1/2(d) 5/8
Answer» Answer is (d) 5/8

Discussion

36.	What type of a relation is "Less than" in the set of real numbers?(a) only symmetric (b) only transitive (c) only reflexive (d) equivalence relation
Answer» Answer is (b) only transitive

Discussion

37.	If A and B are two independent events, then
Answer» Answer is (a) P(A ∪ B) = 1 - P(A') P(B')

Discussion

38.	Evaluate- ∫cot2 x dx
Answer» Let I = ∫cot² x dx = ∫(cosec² x - 1) dx = ∫cosec² x dx - ∫dx = - cot x - x + C

38.

Evaluate- ∫cot2 x dx

Answer»

Let I = ∫cot² x dx

= ∫(cosec² x - 1) dx

= ∫cosec² x dx - ∫dx = - cot x - x + C

Discussion

39.	If A = {1,2,3}, B = {6,7,8} and f : A → B is a function such that f(x) = x + 5, then what type of a function is f ?(a) into(b) one-one onto(c) many-one onto(d) Constant function
Answer» Answer is (b) one-one onto

Discussion

40.	\|(3,4,5),(0,2,3),(0,0,7)\| = (a) 40(b) 50(c) 42(d) 15
Answer» Answer is (c) 42

Discussion

41.	Let A={l,2,3} which of the following function f : A → A does not have an inverse function?(a) {(1, 1),(2, 2),(3, 3)} (b) {(1, 2),(2, 1),(3, 1)} (c) {(1, 3),(3, 2),(2, 1)} (d) {(1, 2),(2, 3),(3, 1)}
Answer» Answer is (b) {(1, 2),(2, 1),(3, 1)}

Discussion

42.	The inverse of A = [(2,3),(5,k)] will not be obtained if k has the value(a) 2(b) 3/2(c) 5/2(d) 15/2
Answer» Answer is (d) 15/2

Discussion

43.	The value of cot-1 (tan(π/7)) is .....
Answer» The value of cot^-1 (tan(π/7)) is 5π/14

Discussion

44.	Let A = {5,6}; how many binary operations can be defined on this set?(a) 8 (b) 10 (c) 16 (d) 20
Answer» Answer is (c) 16

Discussion

45.	Verify Lagrange's Mean Value Theorem in interval [0,4), where function f(x) = (x - 1) (x - 2) (x - 3).
Answer» Given function f(x) = (x - 1) (x - 2) (x - 3) Hence, f(a) = (-1) (-2) (-3) = 6 ∴ f(0) = 6 ∵ Given function is polynomial Hence function is. (i) Continuous in closed interval (0,4) (ii) Differentiable in open interval (0, 4)

45.

Verify Lagrange's Mean Value Theorem in interval [0,4), where function f(x) = (x - 1) (x - 2) (x - 3).

Answer»

Given function f(x) = (x - 1) (x - 2) (x - 3)

Hence, f(a) = (-1) (-2) (-3) = 6

∴ f(0) = 6

∵ Given function is polynomial Hence function is.

(i) Continuous in closed interval (0,4)

(ii) Differentiable in open interval (0, 4)

Discussion

46.	cos-1 ((1 - x2)/(1 + x2)) = (a) 2cos-1 x(b) 2sin-1 x(c) 2tan-1 x(d) cot-1 2x
Answer» Answer is (c) 2sin^-1 x

Discussion

47.	For any unit of Matrlx I.(a) I2 = I(b) \|I\| = 0(c) \|I\| = 2(d) \|I\| = 5
Answer» Answer is (a) I² = I

Discussion

48.	The value of sin(sin-1 (1/2) + cos-1 (1/2)) is ......
Answer» The value of sin(sin^-1 (1/2) + cos^-1 (1/2)) is 1

Discussion

49.	Solve : x dy + y dx = 0
Answer» ∵ x dy + y dx = 0 x dy = - y dx dy/y = - dx/x Integrating both sides ∫dy/y = - ∫dx/x log y = - log x + log c log x + log y = log c log x . y = log c x . y = c

49.

Solve : x dy + y dx = 0

Answer»

∵ x dy + y dx = 0

x dy = - y dx

dy/y = - dx/x

Integrating both sides

∫dy/y = - ∫dx/x

log y = - log x + log c

log x + log y = log c

log x . y = log c

x . y = c

Discussion

50.	If, x > a, ∫dx/(x2 - a2) =
Answer» Answer is (a) (1/2a) log((x - a)/(x + a))

Discussion

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