Verification of Truth of the Statement
Verification of Truth of the Statement
‘Verification of truth of the statement’ type questions are asked to check the perfect observation about statement, the object and its characteristics. Candidate should always check all the possibilities for the statement which is always true. The alternatives other than the correct answer also seem to bear a strong relationship with the thing mentioned. So, absolute truth is to be followed.
Example:
A camera always has______
(a) reels (b) flash
(c) photograph (d) lens
(e) stand
Ans. (d)
Explanation: Clearly, though all the alternatives may form part of the camera, the lens is the most vital part, without which camera cannot work. So the correct answer is D.
A book always has______
(a) pages (b) contents
(c) illustrations (d) chapters
(e) pictures
Ans. (a)
Explanation: A book may contain all the alternatives but without page there can be no book. So the correct answer is A.
A man always has:
(a) teeth (b) hand
(c) hair (d) brain
(e) None of these
Ans. (d)
Explanation: Clearly, all the alternatives are the part of the human body but a human cannot survive without brain. So the correct answer is D.
A window always has:
(a) curtain (b) panes
(c) grill (d) All of these
(e) None of these
Ans. (e)
Explanation: There can be windows without curtain, panes and grill. So the correct answer is (e).
Snap Test
Which of the following a ‘drama’ must have
(a) stage (b) director
(c) story (d) actors
(e) None of these
Ans. (c)
Explanation: A drama may have no stage, director and actors. But there can’t be a drama without story.
A mirror always
(a) retracts (b) refracts
(c) reflects (d) distorts
(e) None of these
Ans. (c)
Explanation: A mirror always reflects.
Which one of the following is always associated with justice?
(a) Magnanimity (b) Hypocrisy
(c) Diminutiveness (d) Legitimate
(e) None of these
Ans. (d)
Explanation: Justice is always associated with legitimate.
A river always has
(a) tributaries (b) delta
(c) banks (d) boats
(e) None of these
Ans. (c)
Explanation: A river may be without tributaries, delta and boast. But there can’t be a river without banks.
A tree always has:
(a) fruits (b) roots
(c) branches (d) leaves
(e) None of these
Ans. (b)
Explanation: A tree can exist without fruits, branches, and more...
Logical Sequence
Logical Sequence
In this type of question, a group of words/things/facts is given. You have to arrange these words in a meaningful order such as the sequence of occurrence of events, sequence of form of a part to whole, sequence of increasing/decreasing size, weight, value, intensity etc. and then choose the correct option accordingly.
Example:
Arrange the following in a meaningful sequence.
Sowing 2. Ploughing
Grain 4. Irrigation
Harvesting
(a) Sowing (b) Ploughing
(c) Grain (d) Irrigation
(a) 5, 1, 2, 3, 4 (b) 4, 2, 3, 1, 5
(c) 4, 3, 2, 5, 1 (d) 2, 1, 4, 5, 3
(e) None of these
Ans. (d)
Explanation: To produce the crop, first of all field is ploughed then seed is scattered in the field. After some time filed is irrigated. And when the crop is matured, it is harvested and grain is obtained from the crop.
If the words below organized according to size (smaller to big) which would come in last?
(a) Planet (b) Star
(c) Comet (d) Galaxy cluster
(e) Galaxy
Ans. (d)
Explanation: If the arrange there celestial bodies according to their increasing size/ the correct sequence will be. Planet, Star, Galaxy cluster. So option (d) is correct answer.
Arrange the following in a meaningful order, from particular to general.
Human body 2. Tissue
Organ 4. Cell
Organ system
(a) 4, 2, 3, 5, 1 (b) 5, 2, 3, 4, 1
(c) 4, 3, 2, 1, 5 (d) 5, 4, 3, 2, 1
(e) None of these
Ans. (a)
Explanation: Clearly, tissue is made of cells, organ is made of tissues, organ system is made of organs and human body is made of many organ systems. So correct answer is A.
If the designations of police in India below are organized according to their rank (higher to lower), which would come first?
(a) IGP (b) DGP
(c) DIG (d) SSP
(e) IPS
Ans. (b)
Explanation: DGP is the highest commissioned rank in Indian police force. So correct option is (b).
Snap Test
Arrange the following in meaningful sequence.
Design 2. Analysis
Planning 4. Development & Implementation
Maintenance 6. Testing
(a) 3, 2, 1.4,6,5 (b) 3, 1, 2, 4, 5, 6
(c) 1, 3, 5, 2, 6, 4 (d) 6, 5, 3, 1, 2, 4
(e) None of these
Answer: (a)
Explanation: The process of software development goes through a series of in step wise fashion that almost every developing company follows known as the ‘software development more...
Decision Making
Decision Making
In this type of questions certain categories of information are given followed by certain criteria. The candidates are given a few cause that needs to be analysed. They are required to take the right decision, from the given options, after comparing the information in each c.se with the given information and criteria.
Example:
Directions: Read the following information to answer the given questions,
Following are the criteria for selecting candidates for Research Fellowship.
The candidate must-
(i) be a post-graduate with minimum of 60% marks.
(ii) not be more than 32 years as on 15.04.2015
(iii) have at least 3 years research experience.
(iv) have diploma in Statistics.
(v) have secured at least 70% marks in the entrance test.
(vi) have finalised the topic for research.
However, in case a candidate who fulfil all other criteria except -
(a) (iii) above but has M. Phil degree should be referred to Chairman.
(b) (iv) above should be referred to Dean.
(c) (i) above but has at least 55% marks in post-graduation and 75% in the entrance test should be given fellowship.
Matrices
Matrices
In this type of questions a set of numbers or letters are given in a figure which is divided into different cells. These numbers or letters form a pattern by applying some logic - either row- wise or column-wise. One cell of the matrix is left empty. The candidate is required to analyse the matrix to find out the logic applied in the pattern and select one from the given options which will complete it.
Example:
Replace the question mark with the correct number.
Dot Situation
Dot Situation
The problems on Dot Situation involve a combination of three or more geometrical shapes usually triangle, square, rectangle or/and circle having one or more dots placed at any point inside the combination. This combination is followed by a set of four alternative figures each composed of a combination of same type of figures. Now, for each dot we have to observe the region in which it is enclosed, i.e. to which of the geometrical figures this region is common. Then, one has to look for such a region in the four alternatives. Once this is found, the procedure for other dots if any is repeated. The alternative figure which contains all such regions is the answer
Example:
1. Select the figure from the following options which satisfies the same conditions of placement of the dot as in fig. (X)
(X) (a) (b) (c) (d)
Ans. (c)
Explanation: In fig. (X), the dot is placed in the region common to all the three geometrical shapes- the square, the triangle and the circle. Among the given options, only in figure (c), we have a region common to the square, the triangle and the circle. So option (c) is correct.
2. Select the figure from the following options which satisfies the same condition of placement of the dots as in fig. (X)
(X) (a) (b) (c) more...
Natural Numbers: Counting numbers \[1,\,\,2,\,\,3,\,\,4,\text{ }5,\text{ }6,....etc.\] are called Natural Numbers.
Whole Numbers: Counting numbers with 0 are whole numbers, i.e. \[0,1,2,3,4,....\] etc. are Whole numbers.
Integers: All natural numbers, the negatives of all natural numbers and zero are collectively known as integers, i.e. \[...-4,\text{ }-3,\text{ }-2,\text{ }-1,\text{ }0,\text{ }1,\text{ }2,\text{ }3,\text{ }4,...\] etc. are integers.
Rational Numbers: The numbers that can be expressed in the form \[\frac{p}{q}\], where p and q are integers are called rational numbers. Each rational number can be expressed either in a terminating or in a non-terminating repeating decimal form.
Irrational Numbers: The numbers which when expressed in decimal form are expressible as non-terminating and non-repeating decimals are known as irrational numbers.
Euclid’s Division Lemma or Euclid’s Division Algorithm
For any two given positive integer a and b there exist unique integer q and r satisfying
\[a\text{ }=\text{ }bq\text{ }+\text{ }r\], where \[0\,\,\le \,\,r\,\,<\text{ }b\]
Here a is known as dividend, b as divisor, q as quotient and r as remainder.
Fundamental Theorem of Arithmetic: Every composite number can be expressed (factorised) as a product of primes and this factorisation is unique apart from order in which prime factors occurs.
To find the H.C.F. and L.C.M. of numbers using the method of Fundamental Theorem of Arithmetic (Prime Factorisation Method):
Express each one of the given numbers as a product of prime factors. Then,
H.C.F. = Product of the smallest powers of each common prime factor in the numbers
L.C.M. = Product of the greatest powers of each prime factor, involved in the numbers
To test whether a given rational number is a terminating or a repeating decimal:
A rational number, in the simplest form \[\frac{p}{q}\], where p and q are integers and \[q\,\,\ne \,\,0\] is:
A terminating decimal if prime factorisation of q is of the form \[\left( {{2}^{m}}\text{ }\times \text{ }{{5}^{n}} \right)\], where m and n are non-negative integers.
(ii) A non-terminating repeating decimal if prime factorisation of q is not of the form \[\left( {{2}^{m}}\text{ }\times \text{ }{{5}^{n}} \right)\], where m and n are non-negative integers.
Snap Test
Using Prime Factorisation method, find the H.C.F. of 9775 and 11730.
Without actual division find whether the rational number \[\frac{\mathbf{45}}{\mathbf{37500}}\] is a terminating or a non-terminating repeating decimal.
Polynomials: If x is a variable, n be a positive integer and \[{{a}_{0}},\text{ }{{a}_{1}},\text{ }{{a}_{2}}\ldots .,\text{ }{{a}_{n}}\] are real number, then an expression of the form \[p\left( x \right)\text{ }=\text{ }{{a}_{0}}+\text{ }{{a}_{1}}x\text{ }+\text{ }{{a}_{2}}{{x}^{2}}+\text{ }{{a}_{n}}{{x}^{n}}\] is called polynomial, in the variable x. In a polynomial, \[p\left( x \right)\text{ }=\text{ }{{a}_{0}}+\text{ }{{a}_{1}}x+\text{ }{{a}_{2}}{{x}^{2}}+\text{ }\ldots \ldots .\text{ }+\text{ }{{a}_{n}}{{x}^{n}},\text{ }{{a}_{0}},\text{ }{{a}_{1}}x,\text{ }{{a}_{2}}{{x}^{2}},\ldots .,\text{ }{{a}_{n}}{{x}^{2}}\] are known as the terms of the polynomial and \[{{a}_{0}},\,\,{{a}_{1}},\text{ }{{a}_{2}}\ldots \ldots ..\], an are known as their coefficients
Degree of a polynomial: Let p(x) be a polynomial in x. Then, the highest power of x in p(x) is called the degree of the polynomial p(x). Thus, the degree of the polynomial, \[p\left( x \right)\text{ }=\text{ }{{a}_{0}}+\text{ }{{a}_{1}}x\text{ }+\text{ }{{a}_{2}}{{x}^{2}}+\text{ }\ldots \ldots .\text{ }+\text{ }{{a}_{n}}{{x}^{n}}\], where an \[\ne \] 0 is n.
Constant polynomial: A polynomial of degree zero is called a constant polynomial e.g. \[p\left( x \right)\text{ }=\text{ }-\text{ }5\] is a constant polynomial.
Zero polynomial: The constant polynomial \[p\left( x \right)\text{ }=\text{ }0\] is called the zero polynomial. The degree of the zero polynomial is not defined since \[p\left( x \right)\,\,=\,\,0\,\,=\,\,0.x\,\,=\,\,0.{{x}^{2}}\,\,=\,\,0.{{x}^{3}}\,\,=\]… etc.
Linear polynomial: A polynomial of degree 1 is called a linear polynomial A linear polynomial is of the form \[p\left( x \right)~~=\text{ }ax\text{ }+\text{ }b\], where \[a~\,\,\ne \,\,0\] e.g. \[5x\text{ }+\text{ }1\], \[-\frac{5}{2}x,\,\,2\sqrt{3}x-\sqrt{2}\,\,etc.\] are linear polynomials.
Quadratic polynomial: A polynomial of degree 2 is called a quadratic polynomial. A quadratic polynomial is of the form \[p\left( x \right)\text{ }=\text{ }a{{x}^{2}}+\text{ }bx\text{ }+\text{ }c\], where\[a~\,\,\ne \,\,0\].
e.g.\[{{x}^{2}}-5,\,\,\,5\sqrt{2}{{x}^{2}}-\frac{1}{\sqrt{3}}x,\,\,7{{x}^{2}}\,\,+\,\,\sqrt{5}\] etc. are quadratic polynomials.
Cubic polynomial: A polynomial of degree 3 is called a cubic polynomial A cubic polynomial is of the form \[p\left( x \right)~~=\text{ }a{{x}^{2}}+\text{ }b{{x}^{2}}+\text{ }cx\text{ }+\text{ }d\], where\[a~\,\,\ne \,\,0\].
e.g. \[{{x}^{3}}-20,\,\,\sqrt{5}{{x}^{3}}\,\,-\,\,\frac{1}{9}x,\,\,\frac{7}{2}{{x}^{3}}-\frac{1}{2}{{x}^{2}}-4\]etc. are cubic polynomials.
Biquadratic polynomial: A polynomial of degree 4 is called a biquadratic polynomial. A biquadratic polynomial is of the form\[p\left( x \right)\text{ }=\text{ }a{{x}^{4}}+\text{ }b{{x}^{3}}+\text{ }c{{x}^{2}}dx\text{ }+\text{ }e\], where \[a~\,\,\ne \,\,0\].
e.g. \[{{x}^{4}}\text{ }23,\,\sqrt{3}{{x}^{4}}-\frac{1}{9}x,\,\,\frac{1}{2}{{x}^{4}}+\frac{3}{4}x-\frac{1}{8}\] etc. are biquadratic polynomials.
Zeros of a polynomial: A real number k is said to be a zero of the polynomial p(x), if\[p\left( k \right)\text{ }=\text{ }0\].
Relationship between the Zeros and Coefficients of a Linear Polynomial: The zero of a linear polynomial
\[p\left( x \right)\text{ }=\text{ }ax\text{ }+\text{ }b\] is given by\[\alpha =\frac{-b}{a}=\frac{-(constant\,\,term)}{(coefficient\,\,of\,\,x)}\].
A linear polynomial can have at the most one zero.
Relationship between the Zeros and Coefficients of a Quadratic Polynomial:
(i) If \[\alpha \] and \[\beta \] are the zeros of a quadratic polynomial \[p\left( x \right)\text{ }=\text{ }a{{x}^{2}}+\text{ }bx\text{ }+\text{ }c,\text{ }a\,\,\ne \,~0\] then
\[\alpha +\beta \,\,=\,\,\frac{-b}{a}\,\,=\,\,\frac{-(coefficient\,\,of\,\,x)}{(coefficien\,\,of\,\,{{x}^{2}})};\]
(ii) A quadratic polynomial whose zeroes are \[\alpha \] and \[\beta \] is given by:
\[p\left( x \right)\text{ }=\text{ }{{x}^{2}}\,-\,\,(\alpha +\beta )\,\,+\text{ }(\alpha \beta )\]
Pair of Linear Equations in Two Variables
Pair of Linear Equations in Two Variables
Linear equation in two variables: An equation which can be put in the form \[ax\text{ }+\text{ }by\text{ }+\text{ }c\text{ }=\text{ }0\], where a, b, c are real numbers \[(a~\,\,\ne \,\,0,\text{ }b\,\,\ne \,\,~0)\] is called a linear equation in two variables x and y.
Simultaneous linear equations in two variables: A pair of linear equations in two variables is said to form a system of simultaneous linear equation.
Solution of a given system of two simultaneous equations: A pair of value of the variable x and y satisfying each of the equations in a given system of two simultaneous equations in x and y is called a solution of the system.
Consistent system: A system of simultaneous linear equations. Is said to be consistent if it has at least one solution.
Inconsistent system: A system of simultaneous linear equations is said to be inconsistent if it has no solution.
If a pair of linear equations is given by \[{{a}_{1}}x\text{ }+\text{ }{{b}_{1}}y\text{ }+\text{ }{{c}_{1}}=\text{ }0\] and \[{{a}_{2}}x\text{ }+\text{ }{{b}_{2}}y\text{ }+\text{ }{{c}_{2}}=\text{ }0\] then the following situations can arise:
(i) if \[\frac{{{a}_{1}}}{{{a}_{2}}}\ne \frac{{{b}_{1}}}{{{b}_{2}}}\], the pair of linear equations is consistent.
(ii) if \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}\ne \frac{{{c}_{1}}}{{{c}_{2}}}\] the pair of linear equations is inconsistent.
(iii) if \[\frac{{{a}_{1}}}{{{a}_{2}}}\,\,=\,\,\frac{{{b}_{1}}}{{{b}_{2}}}\,\,=\,\,\frac{{{c}_{1}}}{{{c}_{2}}}\] the pair of linear equations is dependent and consistent.
Snap Test
Find the value of x and y from the following equations.
\[\mathbf{x+}\frac{\mathbf{6}}{\mathbf{y}}\mathbf{=6;}\,\,\,\,\mathbf{3x-}\frac{\mathbf{8}}{\mathbf{y}}\mathbf{=5}\]
(a) \[x\text{ }=\text{ }3,\text{ }y\text{ }=\text{ }2\]
(b) \[x\text{ }=\text{ }2,\text{ }y\text{ }=\text{ }5\]
(c) \[x\text{ }=\text{ }7,\text{ }y\text{ }=\text{ }3\]
(d) \[x\,\,\text{=}\,\,4,\text{ }y\,\,=\,\,6\]
(e) None of these
Ans. (a)
Explanation: Given equations are
\[x+\frac{6}{y}=6\] ..... (i) and \[3x-\frac{8}{y}=5\] ..... (ii)
Putting \[\frac{1}{y}\,\,=\,\,z\] in (i) and (ii), we get:
\[x\text{ }+\text{ }6z\text{ }=\text{ }6\] ..... (iii)
\[3x\text{ }-\text{ }8z\text{ }=\text{ }5\] ….. (iv)
Multiplying (iii) by 3 and subtracting (iv) from it we get:
\[26z\,\,=\,\,13~~~\Rightarrow \,\,~~~z=\frac{1}{2}\]
\[\therefore \,\,\,\,z=\frac{1}{y}\,\,\,\,\,\,\,\Rightarrow \,\,\,\,\,\,y\,\,=\,\,\frac{1}{z}\,\,=\,\,2\]
Substituting \[y\text{ }=\text{ }2\] in (i) we get: \[x\text{ }=\text{ }3\]
Find the value of k for which the system of equations:
\[\mathbf{kx}\text{ }-\text{ }\mathbf{4y}\text{ }=\text{ }\mathbf{3};\text{ }\mathbf{6x}\text{ }-\text{ }\mathbf{12y}\text{ }=\text{ }\mathbf{9}\] has an infinite number of solutions.
(a) \[k\text{ }=\text{ }5\]
(b) \[k\text{ }=\text{ }6\]
(c) \[k\text{ }=\text{ }2\]
(d) \[k\text{ }=\text{ }4\]
(e) None of these
Ans. (c)
Explanation: From the given equation: \[{{a}_{1}}=\text{ }k,\text{ }{{b}_{1}}=\text{ }-\text{ }4,\text{ }{{c}_{1}}=\text{ }-\text{ }3\] and \[{{a}_{2}}=\text{ }6,\text{ }{{b}_{2}}=\text{ }-\text{ }12,\text{ }{{c}_{2}}=\text{ }-9\]
We know that for the infinite number more...
Quadratic Equation: An equation of the form \[a{{x}^{2}}+\text{ }bx\text{ }+\text{ }c\text{ }=\text{ }0\], where a, b, c real numbers and \[(a~\,\,\ne \,\,0)\], is called a quadratic equation in variable x or if p(x) is a quadratic polynomial then \[p\left( x \right)\text{ }=\text{ }0\] is called a quadratic equation.
Roots of a Quadratic Equation: A real number is called a root of the quadratic equation \[a{{x}^{2}}+\text{ }bx\text{ }+\text{ }c\text{ }=\text{ }0\text{ }(a\,\,\ne \,~0)\text{ }\] if \[a{{\alpha }^{2}}+\text{ }b\alpha \text{ }+\text{ }c\,\,=\text{ 0}\]
Note: If \[\alpha \] is a root of the quadratic equation \[\mathbf{a}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{+ bx + c = 0}\], then \[\alpha \] is called a zero of the polynomial \[\mathbf{a}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{+ bx +c}\mathbf{.}\]
Solving a Quadratic Equation: Solving a quadratic equation means finding its roots.
Discriminant: If \[a{{x}^{2}}+\text{ }bx\text{ }+\text{ }c\text{ }=\text{ }0,\text{ }(a\,\,\ne \,~0)\] is a quadratic equation then the expression \[{{b}^{2}}-\text{ }4ac\] is called the discriminant. It is denoted by D.
Sequence: Certain numbers arranged in a definite order, according to a definite rule, are said to form a sequence.
e.g. (i) A rule defined as \[{{T}_{n}}\text{ }=\text{ }5n\text{ }+\text{ }1\] gives
\[{{T}_{1}}=\text{ }6,\text{ }{{T}_{2}}=\text{ }11,\text{ }{{T}_{3}}=\text{ }16,\text{ }{{T}_{4}}=\text{ }21\], .............
Thus, the numbers \[6,\text{ }11,\text{ }16,\text{ }21..........\] from a sequence.
Progressions: Sequences which follow a definite pattern are called progressions.
Arithmetic Progression (A.P.): An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number d to the proceeding term, except the first term. The fixed number d is called the common difference.
The general form of an AP is; \[a,\text{ }a\text{ }+\text{ }d,\text{ }a\text{ }+\text{ }2d,\text{ }a\text{ }+\text{ }3d...\]
Arithmetic Series: By adding the terms of an A.P. we get the corresponding arithmetic series.
g. On adding the terms of an A.P. \[5,\text{ }8,\text{ }11,\text{ }14,\text{ }17.......\]
we get the arithmetic series \[5\text{ }+\text{ }8\text{ }+\text{ }11\text{ }+\text{ }14\text{ }+\text{ }17..........\]
Arithmetic Mean: If a, b and c are in A.P., then \[b=\frac{a+c}{2}\] and b is called the arithmetic mean of a and c.
In an AP with first term a and common difference d, the nth term (or the general term) is given by: \[{{T}_{n}}=a+\left( n-1 \right)d\]
If I is the last term of the finite AP, say the nth term, then the sum of all terms of the
APIs given by: \[S=\frac{n}{2}(a+l)\,\,\,\,\,or\,\,S=\frac{n}{2}[2a+(n-1)d]\]
Snap Test
How many terms are there in the sequence \[\mathbf{3},\text{ }\mathbf{6},\text{ }\mathbf{9},\text{ }\mathbf{12},\text{ }.......,\text{ }\mathbf{111}\]?
(a) 37 (b) 35
(c) 40 (d) 30
(e) None of these
Ans. (a)
Explanation: In the given sequence, \[a\text{ }=\text{ }3\text{ }and\text{ }d\text{ }=\text{ }6\text{ }-\text{ }3\text{ }=\text{ }3.\]
Let there be n terms in the given sequence, then \[{{t}_{n}}=\text{ }111\].
\[\therefore \] \[a\text{ }+\text{ }\left( n\text{ }-\text{ }1 \right)\text{ }d\text{ }=\text{ }111\]
\[\Rightarrow \,\,\,\,3+\left( n-1 \right)~\,\times \,\,3=111\,\,\,\Rightarrow \,\,~3\text{ }\left( n\text{ }-\text{ }1 \right)=108\,\,\,\Rightarrow \,\,~n\text{ }-\text{ }1\text{ }=\text{ }36\,\,\,\Rightarrow \,\,~n\text{ }=\text{ }37.\]
The 7th term of an A.P. is 20 and its 13th term is 32. Find the A.P.
(a) \[5,\text{ }10,\text{ }15,\text{ }20,\text{ }25..........~\]
(b) \[8,\text{ }10,\text{ }12,\text{ }14,\text{ }16........\]
(c) \[8,\text{ }10,\text{ }13,\text{ }17,\text{ }22.........~\]
(d) \[5,\text{ }10,\text{ }10,\text{ }14,\text{ }12\ldots \ldots .\]
(e) None of these
Ans. (b)
Explanation: Let the first term of the A.P. be a and its common difference be d
We know that; \[{{t}_{n}}=\text{ }a\text{ }+\text{ }\left( n\text{ }-\text{ }1 \right)\text{ }d\]
Now, \[{{t}_{7}}=\text{ }20~~~~~\Rightarrow ~~~~a\text{ }+\text{ }6d\text{ }=\text{ }20\] …... (i)
And \[{{t}_{13}}=\text{ }32~~~~~\Rightarrow ~~~~~a\text{ }+\text{ }12d\text{ }=\text{ more...
(X)
(a) 
(b) 
(c) 
(d) 
Ans. (c)
Explanation: In fig. (X), the dot is placed in the region common to all the three geometrical shapes- the square, the triangle and the circle. Among the given options, only in figure (c), we have a region common to the square, the triangle and the circle. So option (c) is correct.
2. Select the figure from the following options which satisfies the same condition of placement of the dots as in fig. (X)
(X)
(a) 
(b) 
(c) more... 
