Aptitude Preparation
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Logarithms
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Quadratic Equation
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Probability
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H.C.F and L.C.M
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Number Systems
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Averages
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Alligation
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Profit and Loss
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Ratio and proportion
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Interest
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Functions
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Alligation or Mixture
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Discount
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Clocks and Calendar
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Square Root and Cube Root
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Coding-Decoding
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Number-Ranking-Time Sequence
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Percentage
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Volume and Surface Area
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Boats and Streams
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Partnership
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Height and Distance
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Area
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Profit and Loss
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Problems on Ages
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Decimal Fraction
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Average
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Pipes and Cistern
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Permutation and Combination
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Problems on H.C.F and L.C.M
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Compound Interest
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Probability
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Chain Rule
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Stocks and Shares
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Banker’s Discount
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Time and Distance
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Simple Interest
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Problems on Trains
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Time and Work
Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity).
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Function : Meaning
- Let A and B be two non-empty sets, then a rule f which associates each element of A with a unique element of B is called a function or mapping from A to B.
- If f is a mapping from A to B, we write f : A → B (read as f is a mapping from A to B).
- If f associates x ∈ A to y ∈ B then we say that y is the image of the element x under the map (or function) f and we write y = f (x). And, the element x is called the pre-image of y.
Types of Functions
- One-One Function (Injective) : A function f : A → B is said to be a one-one function or an injective if different elements of A have different images in B.
Thus, f : A → B is one-one. ⇔ a≠b ⇒ f(a) ≠ f(b) for all a,b ∈ A or f(a) = f(b)⇔ a=b for all a,b ∈ A
- Onto-Function (Surjective) : A function f : A → B is said to be an onto function or surjective if and only if each element of B is the image of some element of A i.e. for every element y ∈ B there exists some x ∈ A such that y = f (x). Thus f is onto if range of f = co-domain of f.
- One-one and onto Function (Bijective Function) : A function is a bijective if it is one-one as well as onto
Composition of Functions
- Suppose A, B and C be three non-empty sets and let f : A → B, g : B → C be two functions. since f is a function from A to B, therefore for each x ∈ A there exists a unique element
- f(x) ∈ B Again, since g is a function from B to C, therefore corresponding to f(x) ∈ B there exists a unique element g(f(x)) ∈ C Thus, for each fx ∈ A there exists a unique element g(f(x)) ∈ C .
- From above discussion it follows that when f and g are considered together they define a new function from A to C. This function is called the composition of f and g and is denoted by gof.
- i.e. gof(x) = g(f (x)), for all x ∈ A
Inverse of a function
- If f and g be two functions satisfying f (g(x)) = x, for every x in the domain of g, and g(f(x)) = x for every x in the domain of f.
- We say that f is the inverse of g and also, g is the inverse of f. We write f = g−1or g = f−1
- Tips: To find the inverse of f, write down the equation y= f(x) and then solve x as a function of y.
- Thus the resulting equation is x = f−1(y) .
Note :
- (i) If f is one-one function then f has inverse defined on its range
- (ii) If f and g are one-one function then (fog)−1 = g−1of−1
- (iii) If f is bijective then f−1 is bijective and (f−1)−1 = f.
Binary Operation and Properties of Binary Operations
- Binary Operation : A binary operation ∗ on a set A is a function ∗: A × A . We denote ∗ (a, b) by a ∗ b.
- It follows from the definition of a function that a binary operation on a set A associates each ordered pair (a, b) ,
- (a,b) ∈ A × A to a unique element (a, b) in A.
Properties of Binary Operations
- (i) Commutativity: A binary operation ∗ on a set of A is said to be commutative binary operation, if a ∗ b =b ∗ a for all a,b ∈ A.
The binary operations addition (+) and multiplication (×) are commutative binary operation on z.
- (ii) Associativity: A binary operation ∗: A × A → Ais said to be associative if
(a∗b)∗c =a∗(b∗c) , ∀ a,b,c ∈ A The binary operation of addition (+) and multiplication (×) are associative binary operation on z. However, the binary operation of subtraction is not associative binary operation.
- (iii) Identity Element: Let ′∗′ be a binary operation on a set A and if there exists an element e ∈ A, such that a∗e = a = e∗a, for all a ∈ A, then e is called an identity element for the binary operation ′∗′ on set A and it is unique.
- (iv) Inverse of an element: Let ′∗′ be a binary operation on a set A, and let e be the identity element in A for the binary operation ∗ on S.Then, an element a ∈ A is called an invertible element if there exists an element b ∈ A such that a×b = e = b∗a.
- The element b is called the inverse of an element a, and it is unique.
- Note: The inverse of a element is generally denoted by a-1
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Question 1 of 30
1. Question
Find the domain of the definition of the function y=│x│.
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Question 2 of 30
2. Question
Find the domain of the definition of the function y=√x.
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Question 3 of 30
3. Question
Find the domain of the definition of the function y=│√x│.
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Question 4 of 30
4. Question
If f(x) is an even function, then the graph y=f(x) will be symmetrical about
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Question 5 of 30
5. Question
If f(x) is an odd function, then the graph y=f(x) will be symmetrical about
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Question 6 of 30
6. Question
For what value of x, x²+10x+11 will gie the minimum value?
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Question 7 of 30
7. Question
In the above question, what will be the minimum value of the function?
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Question 8 of 30
8. Question
Find the maximum value of the function 1/(x²-3x+2).
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Question 9 of 30
9. Question
Find the minimum value of the function f(x)=log₂ (x²-2x+5).
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Question 10 of 30
10. Question
Find the value of (4#3)@(2!3).
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Question 11 of 30
11. Question
Which of the following has a value of 0.25 for a=0 and b=0.5?
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Question 12 of 30
12. Question
Which of the following expressions has a value of 4 for a=5 and b=3?
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Question 13 of 30
13. Question
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Question 14 of 30
14. Question
If u(t)=40-5, v(t)=t² and f(t)=1/2, then the formula for u(f(v(t))) is
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Question 15 of 30
15. Question
If f(x)=x² and g(x)= logₑ x, f(x)+ g(x) will be
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Question 16 of 30
16. Question
If f(x)=1/g(x), then which of the following is correct
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Question 17 of 30
17. Question
If f(x)=1/g(x), then the minimum value of f(x)+g(x), f(x)>0 and g(x)>0, will be
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Question 18 of 30
18. Question
If x is not an integer, what is the value of ([x}-{x})?
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Question 19 of 30
19. Question
If x is not an integer, then ({x}+[x]) is
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Question 20 of 30
20. Question
What is the value of x if 5
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Question 21 of 30
21. Question
If f(t)=t²+2 and g(t)=(1/t)+2, then for t=2, f[g(t)]-g[f(t)]=?
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Question 22 of 30
22. Question
Given f(t)=kt+1 and g(t)=3t=2. If fog= gof, find k.
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Question 23 of 30
23. Question
If the function R(x)=max (x²-8, 3x,8), then what is the max value of R(x)?
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Question 24 of 30
24. Question
If the function R(x)=min(x²-8, 3x,8), what is the max value of R(x)?
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Question 25 of 30
25. Question
The minimum value of ax²+bx+c is 7/8 at x=5/4. Find the value of the expression at x=5, if the value of the expression at x=1 is 1.
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Question 26 of 30
26. Question
Find the range of the function f(x)=(x+4)(5-x)(x+1).
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Question 27 of 30
27. Question
Find the domain of the definition of the function y=log₁₀[1-log₁₀(x²-5x+16)].
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Question 28 of 30
28. Question
If f(t)=(t-1)/(t+1), then f(f(t)) will be equal to
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Question 29 of 30
29. Question
The function y=1/x shifted 1 unit down and 1 unit right is given by
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Question 30 of 30
30. Question
If f(x)=│x-2│, then which of the following is always true?
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