f: R x R → R such that f(x + iy) = Then, f is
Since, f (x + iy) = f (x - iy)
∴f is many – one
Also, Range =R+and codomain = R.
∴Range ⊏codomain ⇒f is into.
Hence, f is many – one into.
Let f (x) = sinx + ax + b. Then f(x) = 0 has
f'(x) = - cosx + a, if a>1,then f(x) entirely increasing. So f(x) =0 has only one real root, which is positive if f(0) < 0 and negative if f(0) > 0.
Similarly when a <-1. Then f(x) entirely decreasing. So f(x) has only one real root which is negative if f(0) < 0 and positive if f(0) > 0
If sin y = x sin (a + y) and then the value of A is
If x:y=2:3 and y:z=4:3, then x:y:z=
0, 7, 26, ?, 124, 215
Clearly the given series is 13-1,23-1,33-1,43-1,53-1,63-1. So, the missing number is 43-1=63.
Complete the given series.
AYBZC, DWEXF, GUHVI, JSKTL, (?), POQPR
If x and y are two unequal positive numbers, then which of the following is definitely true?
If A = 2, M = 26, Z = 52, then BET will be equal to
The ratio of two quantities is 3:4. if the antecedent of its inverse ratio is 100, then its consequent is
5, 16, 51, 158, .....?
16 = 5 X 3 + 1, 51 = 16 X 3 +3 , 158 = 51 X 3 + 5
:. Next term = 158 x 3 + 7 = 481
The annual gathering of a college was organised on a day. Six different programmes - drama , singing , mimicry ,speech , story -telling and dance are to be performed by six student A , B , C , D , E , and F , not necessarily in the same order. the programme begins with a song not sung by B and ends with a dance. C performs mimicry immediately after the speech. E performs drama just before the dance. D or F are not available for the last performance. The speech is not given by A. An interval of 30 minutes is given immediately after mimicry with three more items remaining to be performed . D performs immediately after the interval .
Q. Which items is performed by F ?
Data inadequate, Since F can be neither song or speech.
Option (d) is correct.
If the angles of a triangle are in the ratio 2 : 7 : 11, then their respective degree measures are
Let in ΔPQR,m∠P:m∠Q=m∠R =2:7:11 let x be the common multiple
∴m∠P=2x°,m∠Q=7x°,m∠R=11x°⇒m∠P+m∠Q+m∠R=180° ⇒2x°+7x°+11x°=180° ⇒20x°=180° ⇒x°=9°
∴ Required measures of angles are 18°,63°,99°
The number of values of x where the function f(x) = 2 (cos 3x + cos √3x attains itsmaximum, is
If the function f : [1,∞) → [1,∞) is defined by f (x) = 2x(x-1), then f-1(x) is
A function y =f(x) has a second order derivative f"=6(x-1). If its graph passes through the point(2,1) and at that point the tangent to the graph is y =3x -5, then the function is
Find the range of values of x that satisfy the inequalities x2-x-6>0 and x2-6x-7<0
Let f: R → R, g: R → R, be two functions, such that f(x) = 2x - 3, g (x) = x3 + 5 the function (fog)-1 (x) is equal to
A function f such that f (a) = f"(a) = ........... f2n (a) = 0 and f has a local maximum value b at x = a, if f(x) is
The solution of (y(1 + x-1) + siny) dx + (x + logx + x cosy) dy = 0 is
This given equation can be written as
The post office is to the east of the school while my house is to the south of the school. The market is to the north of the post office. If the distance of the market from the post office is equal to the distance of my house from the school, then in which direction is the market with respect to my school?
Nine persons A, B, C, D, E, F, G, H, and I are sitting in a row not necessarily in the same order. A has Rs. 2 with him. B, C and D have Rs. 3 each. E, F, G, H and I have Rs. 5 each . There are four persons from exactly two adjacent pairs who have the same amount with them. No two persons having Rs.3 each are sitting together. The persons having Rs. 2 with him is not sitting adjacent to the persons having Rs.3. The following information is known about them.
(i) F is the only one between H and a person having Rs.3.
(ii) D is sitting to the immediate left of F.
(iii) A persons having Rs.3 is sitting to the immediate left of I.
(iv) Persons having Rs.5 is the only one between B and the one having Rs.2.
(v) C and A are sitting at extreme ends and G is to the right of F.
F is at the middle of the row.
If South-East becomes North, North-East becomes West and so on. What will West become?
x2+8x-65>0 ∴x2+13x – 5x – 65>0
Hence, x>-13 and x>5 (Case I)
or x<-13 and x<5 (Case II)
∴x>5 or x<-13
If the function f(x) = 2x3 - 9ax2 + 12a2 x + 1, where a> 0, attains its maximum and minimum at p and q respectively such that p2 = q, then a equals
Seven cities - A, B, C, D, E, F and G are connected as follows. The cities which are connected by one way route are as follows. A to D, C to B, G to C, F to E, E to B, A to F , D to C and A to G. The cities which are connected by two way routes are as follows. C and E; G and F; D and G.
If a person, who has to reach city B, stared his Journey from city F touching the maximum possible number of cities exactly once, then which city he cannot touch?
A man walks 5 km toward south and then turns to the right. After walking 3 km he turns to the left and walks 5 km. Now in which direction is he from the starting place?
The numbers 14, 16, 35, 42 are not in proportion. the fourth term for which they will be in proportion is for which they will be in proportion is :
Which of the following values of x satisfy the inequality
The order and degree of the differential equation
The annual gathering of a college was organised on a day. Six different programmes - drama , singing , mimicry ,speech , story -telling and dance are to be performed by six student A , B , C , D , E , and F , not necessarily in the same order . the programme begins with a song not sung by B and ends with a dance. C performs mimicry immediately after the speech .E performs drama just before the dance. D or F are not available for the last performance. The speech is not given by A. An interval of 30 minutes is given immediately after mimicry with three more items remaining to be performed. D performs immediately after the interval .
Q. who performed the dance ?
If x2+9x-112<0; then
If f: R→R and g: R→R are give by f(x) = |x| and g(x) = [x], then g(f(x)) ≤ f(g(x) is true for -
This is true for xϵR..
If α, β are non real numbers satisfying x3 - 1 = 0 them the value of is equal to
f is continuous at ‘0’ and f' (0-) >0 and f' ( 0 +) < 0 . Thus f has a local maximum at ‘0’.
240, ... 120, 40, 10, 2
Ratios of two consecutive terms are 1, 1/2, 1/3, 1/4, and 1/5 respectively.
8, 15, 28, 53, .....?
8 x 2 - 1 = 15, 15 x 2 -2 = 28, 28 x 2 - 3 = 53, 53 x 2 - 4 = 102
1 If in a certain code, LUTE is written as MUTE and FATE is written as GATE, then how will BLUE be written in that code?
If the equation x2-(p+4)x+2p+5=0 has equal roots, then: p=
∴a=1,b=-(p+4),c=2p+5.: roots are equal
If sin 2x = 1, then equals
Ramu walks 5kms starting from her house towards west then turns right and walks 3km. Thereafter she takes left turn and walks 2km. Further, she turn left and walks 3km. Finally, she turns right and walks 3kms. in what direction she is now from her house?
It's clear from the diagram Ramu is West of his house
198, 194, 185, 169, ?
One morning Udai and Vishal were talking to each other face to face at a crossing. If Vishal's shadow was exactly to the left of Udai, which direction was Udai facing?
If -2 < x < 7 and 3 < y < 5, then which of the following is true?
-2<x<7 and 3<y<5. Hence, -4<2x<14-15<-3y<-9. Since the second inequality is multiplied by -3, the inequality changes.
Adding both the inequalities, we get, -19<(2x-3y)<5. Note that 2x-3y is calculated first because it is present in 2 options.
This makes the process of elimination faster.
The order of the differential equation of all tangent lines to the parabola y=x2 is
Find the range of ab/c if 1<a<3, 0<b<2 and -2<c<-1.
Since both a and b are positive, mulitplying them keeps the sense of the inequality as it is.
∴0<ab<6. As c is negative, dividing by c reverses the sense of the inequality.
The derivative of sin-1 ( 2x/1+x2 ) with respect to tan -1 ( 2x/ 1-x2 )
The order and degree of the differential equation
Here power on the differential coefficient is fractional, therefore change it into positive integer, so
Hence order is 2 and degree is 3.
Y is in the East of X which is in the North of Z. If P is in the South of Z, then in which direction of Y, is P?
The annual gathering of a college was organised on a day. Six different programmes - drama , singing , mimicry ,speech , story -telling and dance are to be performed by six student A , B , C , D , E , and F , not necessarily in the same order. the programme begins with a song not sung by B and ends with a dance. C performs mimicry immediately after the speech .E performs drama just before the dance. D or F are not available for the last performance. The speech is not given by A. An interval of 30 minutes is given immediately after mimicry with three more items remaining to be performed . D performs immediately after the interval.
Q. Who was the first performer ?
Again could be A or B. Hence, data inadequate. Options (d) is correct.
∴x<18 Note that since we are multiplying by -1 on both sides, the inequality sign changes.
If f: R →R, g: R →R and h: R →R are such that f(x) = x2, g(x) = tan x and h(x) = log x, then the value of (ho (gof )) (x) if will be
If sin (x + y) = loge (x + y), then dy/dx is equal to
The annual gathering of a college was organised on a day. Six different programmes - drama , singing , mimicry ,speech , story -telling and dance are to be performed by six student A , B , C , D , E , and F , not necessarily in the same order. the programme begins with a song not sung by B and ends with a dance. C performs mimicry immediately after the speech .E performs drama just before the dance. D or F are not available for the last performance. The speech is not given by A. An interval of 30 minutes is given immediately after mimicry with three more items remaining to be performed. D performs immediately after the interval.
Q. Who was the last performer ?
Again could be A or B. Hence, data inadequate. Options (d) is correct.
How often is the Census of India carried out?
Census of India is carried out once every 10 years.
Which type(s) of presentation would you opt to present the annual expenditures of a company?
If textual was used, one has to go through the entire text for comprehension. In order to present data such as a company's annual expenditures/sales, one can use either diagrammatic presentation as it saves a lot of time and can be easily interpreted.
The below table shows the profit made by a group of shops in mall. Then the median is
Find the modal class of the following distribution.
The grouping and analysis table for the distribution is given below.
From the analysis table, the class 500-600 has the maximum score. Hence, 500 - 600 is the modal class.
Sales of cement manufacturing companies is
Sales of cement manufacturing companies is an example of continuous variable because sales can be represent in the continuous variable form like 102.26, 142.14, etc. and continuous variate is capable to assume all the values in the given range.
Identify the type of classification of the below table.
Since the given distribution is classified on the basis of number of hours (of labour), this is a quantitative classification.
For which of the following sets of data is the mode and the median the same?
Arrange the data in the ascending order.
A) 1, 1, 2, 3, 5, 7, 7, 7, 8, 10
B) 1, 2, 2, 2, 3, 3, 4, 5, 6, 9
C) 0, 1, 1, 2, 2, 2, 3, 7, 7, 9
D) 2, 3, 5, 5, 6, 7, 7, 7, 8, 8
It can be seen that for option C, the median, as well as the mode, is 2.
What is the root cause of all economic problems?
Limited resources and unlimited human wants lead to scarcity. This is the root of all economic problems.
Identify the type of classification of the below table.
In case of earthquake the best method is to collect data is
Personal interview is better option because coming in contact with the individual is more reliable.
Numerical data is referred to as
Numerical data refers to as variate because the numerical characteristic is termed as variate. It is also a quantitative characteristic.
For a distribution,
Q. If the mean of the distribution is 6, what is the value of x?
The measures used to mitigate or solve economic problems are known as ___ .
The measures used to mitigate or solve economic problems are known as policies.
The heights of six students in a class are 169 cm, 172 cm, 178 cm, 170 cm, 166 cm and 174 cm. Find the standard deviation of heights using the assumed mean method.
Let 170 be the assumed mean (A)
The co-efficient of M.D. from median for the following set 150, 151, 152, 156, 154, 160, 158 is
Which of the following is not an example of production?
A travel agent booking tickets is involved in distribution. Everything else are examples of production.
Calculate the mean of the following data.
Steven Gerrard, arguably one of the most consistent and skilled players in the English premier league was given the following monthly ratings by a football website for the past 40 months.
Q. Calculate how his rating is deviating from his average performance in terms of mean deviation about mean.
The class mark of the C.I.'s 10-19, 20-29, 30-39 are
Class-mark =LCL+UCL/2=10+19/2=14.5 Same for other
class interval so the answer is 14.5,24.5,34.5.
The mean of the following distribution is
(A) Primary data(B) Secondary data(C)
Report on an epidemic from WHO is secondary data because the report received is not originally collected by the WHO.
AM = 10 & GM = 8. Therefore the 2 observations are
Let the two numbers be x & y. A.M=10
∴x+y=20...(i) G.M=8 ∴√xy=8
∴xy=64...(ii) See the option & use trail and correct method.
Option C is the correct answer
Observe the scatter plot and identify the type of correlation.
Candidate appearing at the F.Y.commerce Examination can offer English at either the higher level or at the lower level. They have to offer Elements of Government or Mathematics as an option subject. Out of 1000 candidate from college A, 50% have offered English at the higher level, 80% of the candidate with English at the lower level have offered Mathematics while the different between number of candidate offering Mathematics and offering Elements of Government in the entire college is 600. In college B from, from which 1500 candidates have appeared equal numbers, have offered elements of Government and Mathematics. The number of candidate with the combination English at the higher level and Mathematics is 200 which are 25% of the number offering English at the lower level.
Q. Find the number of students opting for English higher level and elements of governments.
Total No. of candidates = 1000
and 50 % has opted English at higher level
Therefore, No. of students for English
= 1000 - 50 % of 1000
Problem of choice arises when there are _____ .
The problem of choices arises when there are multiple alternatives and limited resources. In this case, a choice between one or the other has to be made.
Which of these comes under the study of production?
Production deals with the choice of produce, given costs and prices when the resources are limited.
Which method of data collection does not allow for the clarification of ambiguous questions?
In the mailing questionnaire method, since there is no verbal communication between the responder and the enumerator, ambiguous questions cannot be clarified.
Statistical calculations in classified data are based on
The items in classified data cannot be exactly measured. Hence, class midpoints are calculated.
Less than ogive is a
For less than ogive, the less than cumulative frequencies are plotted against the upper class boundaries of that class. Here, frequencies are added starting from the upper limit of the 1st class interval of the frequency distribution. The cumulative total tends to increase or remain the same. Hence, less than ogive is a non-decreasing graph.
Find the range of the following data
Range is the difference between the largest and the smallest observation.
The method of presentation least used by Statistician is
The method of presentation least used by Statistician is textual as per the statistical law.
The intersection point of less than ogive and more than ogive gives the ___ of a frequency distribution.
The intersection of less than and more than ogives is the median of the corresponding frequency distribution.
Graphical representation of frequency distribution is done with
Histogram means a graphical presentation of the data in which the continuous class interval. The class frequencies are represented by vertical adjacent rectangles.
(I) The SENSEX is a free-float market-weighted stock market of 30 well established and financially sound companies listed on Bombay stock exchange.
(II) The NIFTY 50 is NSE’s benchmark broad based stock market index for the Indian equity market. It represents the simple average of 50 Indian company stocks in 12 sectors.
SENSEX and NIFTY 50 are the weighted average stock markets of 30 and 50 blue chip stocks respectively.
In case Quartiles the number of points of division are
Which data can be used to plot an arithmetic line graph?
A continuous arithmetic line graph can be plotted only when the data is continuous.
The mark distribution of 40 students in an exam is shown below. Find the standard deviation of the distribution.
Horizontal bar diagram is used for
Horizontal bar diagram is used for presentation of qualitative data and data varying over space as the meaning says.
In which of the following modes of data collection is the possibility of influencing the responder the most?
The possibility of influencing the responder is most in the 'personal interview' mode because there is direct personal interaction involved.
Which of the following expressions is used to determine the mode?