Explore topic-wise InterviewSolutions in Current Affairs.

This section includes 7 InterviewSolutions, each offering curated multiple-choice questions to sharpen your Current Affairs knowledge and support exam preparation. Choose a topic below to get started.

1.

Make a survey on the weight of 10 people aged between on 40-60 years. Find mean median and mode from the data collected from the people.

Answer»

Step-by-step explanation:

MAKE ME a BRAINLEAST PLEASE and my NAME is RAGHAV ok BRO

2.

QUESTION BANK MCQ1. A 10 share at 10% premium has a market price of 3a. 20b. oc. 9d. 11​

Answer»

ANSWER:

CORRECT answer is 20

Step-by-step explanation:

10/100×10

20

3.

परीघ 11 cm असेल तर त्यावर्तुळाचा व्यास किती​

Answer»

ANSWER:

PLEAS e

fol LOW me

HELLO

hi

how are you

roselady7

4.

Find the perimeter of a racing track of length 6km breadth 5 km?​

Answer»

Answer:

Perimeter of a rectangle = 2 × ( length + breadth) ... area. (b) TRUE – Track is prepared ALONG the BOUNDARY of the ... Example 5: The length ... By splitting the figure into rectangles, FIND its area.

5.

Place value of 7 in '3627992' is ________.​

Answer»

ANSWER:

thousand is the CORRECT answer in INDIAN COUNTING system

6.

The three zeroes in the below shown graph are​

Answer»

Answer:

where's the graph.

Step-by-step explanation:

it maybe anything like if it is in same LINE horizontal or vertically then it is SAID to be a normal graph and either the graph has CURVE DIRECTION it is know as ogive graph.

7.

Verify the relationship between zeroes and coefficients for this polynomial:√3x^2-8x+4√3

Answer»

Step-by-step EXPLANATION:

Observation is the oldest METHOD of the CONTROLS in which the ACTUAL performance is observed.

8.

3x³+4xy²+y³-2y²x+3x(x²+y²) simplify​

Answer»

ANSWER:

Here is the solution.

If it HELPS MARK as BRAINLIEST

9.

The probability that it will rain the next day is 0.45. What is the probability that it will not rain the next day?

Answer»

EVEN CHANCE,even chance

10.

2log25/24 + 3log9/5 - 4log3/4​

Answer»

Given : 2log25/24 + 3log9/5 - 4log3/4​

To FIND : Simplify

Solution:

2 LOG ( 25 / 24)  + 3 log  (9/5)  - 4 log ( 3/4)

log ( a/ b) = log a - log b

= 2 log 25 - 2 log 24  + 3 log 9  - 3 log 5  - 4 log 3 + 4 log 4

= 2 log 5² - 2 log (8 x 3)   + 3 log 3²  - 3 log 5  - 4 log 3 + 4 log 2²

log aⁿ = n log a

=  4 log 5 - 2 log 8 - 2 log 3 + 6 log 3 - 3 log 5  - 4 log 3 + 8 log 2

=   log 5 - 2 log 2³ + 8 log 2

= log 5 - 6 log 2  + 8 log 2

= log 5   + 2 log 2

= log 5 + log 2 + log 2

= log ( 5 * 2)  + log 2

= log 10 + log 2

= 1 + log 2

= 1 + 0.3010

= 1.3010

2 log ( 25 / 24)  + 3 log  (9/5)  - 4 log ( 3/4) = 1 + log 2

Learn More:

if x=log [a-b]; y=log [a+b] ; z= log [a2 - Brainly.in

brainly.in/question/13214148

If x=log(a+b) y=log(a-b) z=log(a²-b²). Then what is the relation ...

brainly.in/question/13203506

11.

Convert 6 into binary number​

Answer»

ANSWER:

Mark me as a BRAINLIST plz. BECOZ UR ONE Brainlist makes me genious rank.

Step-by-step explanation:

ur answer is there in image

12.

√9x √27is equal to:a. 9√3 b. 3√3 c. 2√2 d. 18​

Answer»

Answer:

9√3 is ANSWERS for this QUESTION

13.

Date 1614,2021. MATHS.J. Let A =2-1-24 -54 내-322O225Compute thethe product matrix AB.​

Answer»

ANSWER:

I don't RECOGNIZE your QUESTIONS

14.

AB=BCAD=CDangle ABD = 9+20angle CBD=50angle ADB =b-5angle CDB=35 find the value of a and b​

Answer»

ANSWER:

jsdj3nndjdjjdn2jdeidjj2neejje

15.

Rani had ` 20.50. She bought one ice-cream for `13.75. How much money does she have now?​

Answer»

ANSWER:

RS 6.75 MONEY will be left

mark me as BRAINLIEST.

16.

1. Represent the following rational numberson a number line. optional (1)3 by 4. (2)7by8. (3)-3by2. (4)1by9.​

Answer»

7 by 8

Step-by-step EXPLANATION:

straight a single LINE that represent the NUMBER on a line

17.

1,1,2,8_____a)60 b)64 c)65 d),90​

Answer»

Answer:

64 is the ANSWERS FOR THIS QUESTION

Step-by-step EXPLANATION:

MARK ME BRAINLIEST

18.

26. The area of a triangle is 170.5cm , and base is 22cm. Find the height of the triangle, In explanation ​

Answer»

Given :-

  • Area of triangle is 170.5cm²
  • Base of triangle is 22cm

To find :-

Formula used :-

Area \: of \: triangle \:  =  \dfrac{1}{2} bh

  • B= base of triangle
  • H = height of triangle

Solution:-

Substituting the values in above formula Then we can find height of triangle

Area \: of \: triangle \:  =  \dfrac{1}{2} bh

170.5 \:  =  \dfrac{1}{2} \times  22 \times h

170.5 = 11h

h =  \dfrac{170.5}{11}

h = 15.5cm

So, height of a triangle is 15.5 cm

Know more :-

  • Perimeter of equilateral triangle = 3a
  • Perimeter of Scalene triangle = a+b+c
  • Perimeter of rhombus = 4a
  • Perimeter of square = 4a
  • Perimeter of CIRCLE = 2πr
  • Perimeter of rectangle = 2(l+b)

Here "a" REPRESENTS SIDE

a, b, c also represents side

19.

-2 whole 5/6+13/-16show the steps​

Answer»

Step-by-step EXPLANATION:

ER ndjafgsnjtaduhSmcATLJGABELEYHTAJWJEUJQTMEUKATKSYHFAJATBWYHDHGFOJENDUNDAHSH

WTHSHMSYHSHKDABDHHSGBAFHSGK

Vagbfjarjargshhwwthatv

20.

H. C. F of two numbers is always a factor of there LCM (a) true (b) false​

Answer»

ANSWER:

true

Step-by-step explanation:

suppose 2,4 are TWO numbers

HCF of 2,4 is 2

LCM of 2,4 is 4

here hcf is FACTOR of lcm

21.

Divide 24 ( X2 - yz + xy2z + xyz2 by 6xyz​

Answer»

Step-by-step explanation:

24(X^2yz+xy^2z+XYZ^2)

24(x*xyz+y*xyz+z*xyz)

4(24/6 = 4)division of 24 and 6 REMAINING xyz

4(x^2yz+xy^2z+xyz^2)

4xy(x+y+z)

= 4(x+y+z)

22.

Evaluate (1 ,1÷4)^2 mixed fraction pls​

Answer»

ANSWER:

1 whole 9 by 16.

1,9/16

Step-by-step EXPLANATION:

Hope this answer HELPS you.

Please MARK it as the brainliest.

23.

Express each of the following as a fraction in simplest form:i) 0.0024​

Answer»

ANSWER:

12/5000

=6/2500 is the answer

24.

*If the angles of a quadrilateral are in the ratio 3: 2: 1: 4 then what is the type of quadrilateral*1️⃣ Parallelogram2️⃣ Trapezium3️⃣ Kite4️⃣ none of these​

Answer»

ANSWER:

FRIEND it's a trapezium

Step-by-step explanation:

because all angles of trapezium is DIFFERENT

25.

Express in p/q form​

Answer»

ANSWER:

LET x=0.8181...(1)

100x=81.8181..(2)

(2)-(1)

100x=81.8181

-. X=0.8181

99x=81

X=81/99

X=9/11

26.

1.Find five rational numbers between 1 and 2.​

Answer»

Step-by-step explanation:

1 and 2

Or.

\frac{1}{1}

and

\frac{2}{1}

We can increase them with a MULTIPLE

I am TAKING the multiple as 10

1/1*10=10/10

2/1*10=20/10

The five rational NUMBERS in between will be

\frac{11}{10}

\frac{12}{10}

\frac{13}{10}

\frac{14}{10}

\frac{15}{10}

27.

9/4, 11/35, 2/7, 13/28 Arrange the following fraction in descending order​

Answer»
28.

ISYour answer13. The order of rotational symmetry for isVour answer​

Answer»
29.

D = {x : x ∈ R, x2 = -1} is it an empty set?

Answer»

Answer:

YES

Step-by-step explanation:

Yes, D is an EMPTY set. Since, when any positive number or negative number is SQUARED, the result is always a positive number. In the case of the above QUESTION, there is no such number which when squared results a negative number.

30.

कन्वर्ट इनटू रूपीस 75 पैसे​

Answer»

good LUCK with you and your friends who is a COMPANY you can do with the RIGHT AMOUNT of time 7jhnhcbh

31.

In How many ways we can calculate EMI ? a) 1 b)2 c)3 d)4​

Answer»

ANSWER:

we can CALCULATE EML WHITH 2

32.

2squard root 3 by 6 -squard root 3 by 6

Answer»

Answer:

6 SQUARE ROOT of 3 square root

33.

Give me all 10th books​

Answer»

ANSWER:

which STATES you WANT BOOKS

34.

Solve the linear inequations​

Answer»

Step-by-step EXPLANATION:

HOPE this HELPS you

please MARK this answer brainliest

FOLLOW too plz bro

35.

Use the symbols /,x,+,or - between numbers to make the statement true:25 _ 7_ 8_ 3=42​​

Answer»

Answer:

25-7+8x3=42

HOPE it HELPS

36.

11. After how many years does a sum ofRs. 500 become double at sinple interestof 20% per annum?1) 3 years 2) 4 yearsSe3) 5 years4) 6 yearsAns: 5 years16​

Answer»

Solution

Given :-

  • P = Rs 500
  • r = 20%

Find :-

  • t = ?

Explanation

Here ,

We want sum be DOUBLE that's MEAN , SIMPLE INTEREST will be same of principal .

Then, let

  • C.I. = Rs 500

Now using C.I. Formula.

\boxed{\underline{\tt{\red{\:C.I.=\:\dfrac{p\times r\times t}{100}}}}}

Keep all values

==> 500 = 500 × 20 × t/100

==> 500 × 100/( 20 × 500) = t

==> t = 100/20

==> t = 5

Hence

  • Sum of Rs 500 became double after 5 years .

________________

37.

Which of the following is singleton set?1) Set of multiples of 72) Single digit perfect number3) Set of all two-digit numbers with digits sum is '3'4) Set of all odd prime numbers.​

Answer»

AnSweR:-

\Large{\red{\sf{Option\;2 - Single\;digit\; perfect\; numbers}}}

For Example:- A = {1,2,3,4,5}

Singleton Sets are {{1},{2},{3},{4},{5}}, All are single

Hope It HELPS You ✌️

38.

3. A sum of money is lent at 8% per annumcompound interest. If the interest for thesecond year exceeds that for the first year by*96, find the sum of money.​

Answer»

\large\underline{\sf{Solution-}}

  • Let the sum of MONEY invested be Rs P.

CASE :- 1

  • Principal = Rs P

  • Rate, r = 8 % per annum

So,

Compound interest is given by

\tt{\implies CI=P\bigg(1+\dfrac{r}{100}\bigg)^{n}-P}

\tt{\longmapsto CI=P\bigg(1+\dfrac{8}{100}\bigg)^{1}-P}

\tt{\longmapsto CI=P\bigg(\dfrac{100 + 8}{100}\bigg)-P}

\tt{\longmapsto CI=P\bigg(\dfrac{108}{100}\bigg)-P}

\tt{\longmapsto CI=P(1.08)-P}

\tt{\longmapsto CI=0.08P} -  - (1)

Case :- 2

  • Principal = Rs P

  • Time, n = 3 year

  • Rate, r = 8 % per annum

So,

Compound interest is given by

\tt{\implies CI=P\bigg(1+\dfrac{r}{100}\bigg)^{n}-P}

\tt{\longmapsto CI=P\bigg(1+\dfrac{8}{100}\bigg)^{2}-P}

\tt{\longmapsto CI=P\bigg(\dfrac{100 + 8}{100}\bigg)^{2}-P}

\tt{\longmapsto CI=P\bigg(\dfrac{108}{100}\bigg)^{2}-P}

\tt{\longmapsto CI=P\bigg(1.08\bigg)^{2}-P}

\tt{\longmapsto CI=P\bigg(1.1664\bigg)-P}

\tt\longmapsto CI=0.1664P -  -  - (2)

Now,

According to statement,

Cimpound interest for the second year exceeds that for the first year by Rs 96.

\tt\longmapsto \: (0.1664P \:  -  \: 0.08 P) -  0.08 P= 96

\tt\longmapsto \: 0.0064P \: = 96

\bf\implies \:P = 15000

Hence,

  • Sum invested is Rs 15000

Additional Information :-

Let us assume that certain sum of money Rs P is invested at the rate of r % per annum compounded annually for n years then Amount is given by

\tt{\implies Amount=P\bigg(1+\dfrac{r}{100}\bigg)^{n}}

Let us assume that certain sum of money Rs P is invested at the rate of r % per annum compounded half yearly for n years then Amount is given by

\tt{\implies Amount=P\bigg(1+\dfrac{r}{200}\bigg)^{2n}}

Let us assume that certain sum of money Rs P is invested at the rate of r % per annum compounded QUARTERLY for n years then Amount is given by

\tt{\implies Amount=P\bigg(1+\dfrac{r}{400}\bigg)^{4n}}

39.

Three resistors of 5, 10 and 15 Ωare connected in a parallel and the combination isconnected to a battery of 30V. Find: (i) the current flowing through it (ii) The effective resistance​

Answer»

Step-by-step explanation:

GIVEN= three resistors R1,R2and R3 are connected in parallel

potential difference,V = 30

(ii) we know that,

1/R1+1R2+1/R3 = 1/5+1/10+1/15

= 1/R= 6+3+2/30 = 11/30

R = 30/11

(i)USING ohm's law,

V = IR

30 = I x 30/11

I = 30 x 11/30 = 330/30 = 11 AMPERE

40.

(3-√2)²+(3+√2)²-(4-√5)² simplify​

Answer»

Step-by-step explanation:

9-6root2+2 +9+6root2+2-16+8root5-5

9+2+9+2-16+8root5-5

1+8root5

41.

23. Determine the present value of perpetuity Rs. 10 per month for infinite period at an effectiverate of interest of 14% p.a.?(a).Rs. 657(c) Rs. 857(b)(d)Rs. 757Rs. 957​

Answer»

ANSWER:

HU ikimokp

hn kya

bikedgtgthnimcfrjgjrjrjrjttjrjjtjrjtjt

42.

Show that 3x 4 + 1 is o(x 4/2) and x 4/2 is o(3x 4 + 1)​

Answer»

SOLUTION

TO PROVE

\displaystyle\sf{3 {x}^{4}  + 1 = O \bigg(  \frac{ {x}^{4} }{2} \bigg) \:  \:  \: and \:  \:  \:  \frac{ {x}^{4} }{2}= O \bigg( 3 {x}^{4}  + 1  \bigg)}

CONCEPT TO BE IMPLEMENTED

If f(x) & g(x) are two real valued function then f(x) = O(g(x)) if there exists a non ZERO real NUMBER c such that f(x) ≤ c g(x)

PROOF

\displaystyle\sf{3 {x}^{4}  + 1 }

\displaystyle\sf{ \leqslant 3 {x}^{4}   +  {x}^{4} }

\displaystyle\sf{ = 4 {x}^{4}  }

\displaystyle\sf{ = 8. \frac{ {x}^{4} }{2} }

\displaystyle\sf{ \therefore \:  \: 3 {x}^{4}  + 1 = O \bigg(  \frac{ {x}^{4} }{2} \bigg) \:}

Again

\displaystyle\sf{\frac{ {x}^{4} }{2}}

\displaystyle\sf{ =  \frac{1}{6}.3 {x}^{4} }

\displaystyle\sf{ \leqslant   \frac{1}{6} \bigg( 3 {x}^{4}  + 1  \bigg) }

\displaystyle\sf{ \therefore \:  \: \frac{ {x}^{4} }{2}= O \bigg( 3 {x}^{4}  + 1  \bigg)}

Hence proved

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. write the expression for gradient and DIVERGENCE

brainly.in/question/32412615

2. prove that the CURL of the gradient of

(scalar function) is zero

brainly.in/question/19412715

43.

* 12 apples and 17 bananas cost Rs 87.25, while 16 apples and 25 bananas cost Rs 119.25How much will 2 bananas and an apple cost?and 1 garlic bread, or 2 pizzas and 4 garlic​

Answer»

Basic Concept Used :-

Writing Systems of Linear Equation from Word Problem.

1. Understand the problem.

  • Understand all the words used in STATING the problem.

  • Understand what you are asked to find.

2. Translate the problem to an equation.

  • Assign a variable (or VARIABLES) to represent the unknown.

  • Clearly state what the variable represents.

3. Carry out the plan and solve the problem.

Let's solve the problem now!!!

Appropriate Question :-

12 apples and 17 bananas cost Rs 87.25, while 16 apples and 25 bananas cost Rs 119.25. How much will 2 bananas and an apple cost?

SOLUTION :-

  • Let cost of 1 apple be Rs x

and

According to statement,

  • 12 apples and 17 bananas cost Rs 87.25

\rm :\longmapsto\:12x + 17y = 87.25 -  -  - (1)

According to statement,

  • 16 apples and 25 bananas cost Rs 119.25.

\rm :\longmapsto\:16x + 25y = 119.25 -  -  - (2)

On multiply equation (1) by 4 and equation (2) by 3, we get

\rm :\longmapsto\:48x + 68y = 349 -  -  - (3)

and

\rm :\longmapsto\:48x + 75y = 357.75 -  -  - (4)

On Subtracting equation (3) from equation (4), we get

\rm :\longmapsto\:7y = 8.75

\bf\implies \:y = 1.25

Now,

On substituting the value of y, in equation (1), we get

\rm :\longmapsto\:12x + 17 \times 1.25 = 87.25

\rm :\longmapsto\:12x + 21.25 = 87.25

\rm :\longmapsto\:12x = 87.25 - 21.25

\rm :\longmapsto\:12x = 66

\bf\implies \:x = 5.5

THUS,

  • Cost of 1 apple = Rs 5.5

and

  • Cost of 1 Banana be = Rs 1.25

Hence

Cost of 2 bananas and an apple = 2×1.25 + 5.5 = Rs 8.

44.

20. A skipping rope is held by two children and its slope is shown in the graphgiven below.ty13hyperbola2-13(a) The shape of the rope is in the form of(i) a hyperbola. (ii) an ellipse. (iii) a parabola. (iv) spiral.b.(b) If the given graph is represented by f(x) = ax2 + bx +\c, thenb(i) a > 0 (ii) a = 0 (iii) a = 1 (iv) a < 0(a(c) The sum of the zeroes of the polynomial representing given graph is(i) 3 (ii) -2 (iii) 4 (iv) 4(d) The product of the zeroes of the polynomial representing given graph is(i) 3 (ii) -3 (iii) 1 (iv) -1​

Answer»

ANSWER:

the graphgiven below.ty13hyperbola2-13(a) The shape of the rope is in the FORM of(i) a hyperbola. (ii) an ellipse. (iii) a parabola. (iv) spiral.B.(b) If the given graph is represented by f(x) = ax2 + bx +\C, thenb(i) a > 0 (ii) a = 0 (iii) a = 1 (iv) a < 0(a(c) The sum of the zeroes of the polynomial representing given graph is(i) 3 (ii) -2 (iii) 4 (iv) 4(d) The product of the zeroes of the polynomial

45.

1) Find the difference between simple interst and compoundinterest on Rs. 6250 in 2 years at 4 p.c.p.a.​

Answer»

\bf \: \underline{Given} : ↴

\sf \implies Principal = Rs. \: 6250

\sf \implies Time = 2 \: years

\sf \implies Rate = 4 \: \%

\bf \: \underline{To \: find} : ↴

\sf \implies Compound \: interest - Simple \: interest = \: ?

\bf \: \underline{\underline{Solution}}

\sf \: First \: of \: all, let's \: find \: the \: value \: of \: (S.I) \: Simple \: interest.

\sf \implies \boxed{ \sf \: \pink{S.I = \dfrac{Principal \times Rate \times Time }{100} }}

\sf \implies\sf \:S.I = \dfrac{6250 \times 4 \times 2 }{100}

\sf \implies\sf \:S.I = \dfrac{625 \times 4 \times 2 }{10}

\sf \implies\sf \:S.I = \dfrac{625 \times 4  }{5}

\sf \implies\sf \:S.I = 125 \times 4

\red{\sf \implies\sf  \: S.I = Rs.  \: 500}

\sf \: Now, we \: will \: find \: the \: value \: of \: (C.I) \: Compound \: interest.

\sf \: For \: that, we \: have \: \: to \: find \: the \: amount.

\sf \implies \boxed{ \sf \: \pink{Amount = P \bigg( 1 + \dfrac{R }{100} \bigg)^{n} }}

\sf \implies \sf 6250 \bigg( 1 + \dfrac{4 }{100} \bigg)^{2}

\sf \implies \sf 6250 \bigg( 1 + \dfrac{1}{25} \bigg)^{2}

\sf \implies \sf 6250 \bigg(\dfrac{25 + 1}{25} \bigg)^{2}

\sf \implies \sf 6250 \bigg(\dfrac{26}{25} \bigg)^{2}

\sf \implies \sf 6250  \times \dfrac{676}{625}

\sf \implies \sf 10 \times 676

\red{\sf \implies \sf Amount = Rs.  \: 6760}

\sf \: Now, Compound \: interest = Amount - Principal

\sf \implies \sf Rs. \: 6760 \: - \: Rs. \: 6250

\red{\sf \implies \sf Compound \: interest = Rs. \: 510}

\sf \: Now, difference \: between \: C.I \: and \: S.I \: is :

\sf \implies \sf Compound \: interest \: - \: Simple \: interest

\sf \implies \sf Rs. \: 510 \: - \: Rs. \: 500

\red{\sf \implies \sf Rs. \: 10}

\underline{ \boxed{ \pink{ \sf \: Hence, the \: difference \: between \: C.I \: and \: S.I = Rs. \: 10}}}

46.

1. Given ∆MAC ≅ ∆BIR, if mÐI = 27 and mÐR = 49, what is the measurementof angle M? Draw and label the triangles and show your solutions.

Answer»

Answer:

given

Step-by-step EXPLANATION:

what is the measurement of ANGEL m DRAW and LABEL

47.

In the adjoining figure, A is a common point of contact of two externally touching circles and line is a common tangent to both the circles touches at B and C. Line mis another common tangent at A and its intersects BC at D.TOProve that: (BAC = 90° (ii) Point D is the mid-point of seg BCand​

Answer»

Answer:

A is a COMMON point of CONTACT of two externally touching circles and line is a common tangent to both the circles TOUCHES at B and C. Line mis another common tangent at A and its intersects BC at D.TOProve that:

48.

for a drill 60 boys and 72 girls have to stand in college consisting of same number of boys and same number of girls find the maximum number of such columns

Answer»

ANSWER:

I THINK THE ANSWER IS 26

Step-by-step EXPLANATION:

49.

Rita purchased a ribbon of 8 1/2 m length. If she divided into 5 pieces of equal length, find the length of one piece.

Answer»

Given :

  • Ribbon = 8 1/2
  • Dividing by = 5

To find :

Solution :

  • Now, we have to divide the 8 1/2 meters of ribbon by 5. So, to do that, we have to FIRST convert the following Mixed Fraction to Improper fraction. And let's do it.

  • First, multiplying 8 and 2. Then, adding 1.

\sf 8 \dfrac{1}{2} \rightarrow \dfrac{8 \times 2 + 1}{2 }

     \sf \rightarrow \dfrac{16 + 1}{2 }

      \sf \rightarrow \dfrac{17}{2 }

Therefore, 8 1/2 when converted to an improper fraction is 17/2.

  • Let's divide 17/2 by 5.
  • Here, multiplying 5 and 2 and we get 10. Then, we must divide it by 10. And we get 1.7 meters as the final answer.

\sf \rightarrow \dfrac{17}{2} \div 5

\sf \rightarrow \dfrac{17}{2 \times 5 }

\sf \rightarrow \dfrac{17}{10 }

\sf \rightarrow 1.7

Therefore, 8 1/2 meters of ribbon when DIVIDED by 5 is 1.7 meters.

50.

Integral x by x minus 1 into x minus 2 into x minus 3

Answer»

Answer:

\[ \int \frac{x \, dx}{(x+1)(x+2)(x+3)}. \]

First, we need to get the partial fraction decomposition of the integrand. To that END write

\[ \frac{x}{(x+1)(x+2)(x+3)} = \frac{A}{x+1} + \frac{B}{x+2} + \frac{C}{x+3}. \]

Then we have the equation

\[ A(x+2)(x+3) + B(x+1)(x+3) + C(x+1)(x+2) = x. \]

Plugging in the values x = -1, x = -2, and x = -3 we obtain the following

\begin{align*} 2A &= -1 & \IMPLIES \qquad A &= -\frac{1}{2} \\ -B &= -2 & \implies \qquad B &= 2 \\ 2C &= -3 & \implies \qquad C &= -\frac{3}{2}. \end{align*}

THEREFORE we have

\begin{align*} \int \frac{x \, dx}{(x+1)(x+2)(x+3)} &= \int \frac{A}{x+1} \, dx + \int \frac{B}{x+2} \, dx + \int \frac{C}{x+3} \, dx \\[9pt] &= -\frac{1}{2} \int \frac{1}{x+1} \, dx + 2 \int \frac{1}{x+2} \, dx - \frac{3}{2} \int \frac{1}{x+3} \, dx \\[9pt] &= -\frac{1}{2} \LOG |x+1| + 2 \log |x+2| - \frac{3}{2} \log |x+3| + C \\ &= \frac{1}{2} \left( \log (x+2)^4 - \log |x+1| - \log |x+3|^3 \right) \\ &= \frac{1}{2} \log \left| \frac{(x+2)^4}{(x+1)(x+3)^3} \right| + C. \end{align*}