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If \( f(x)+2f\left( \dfrac{1}{x} \right)=3x,x\neq 0 \), and \( S=\{x\in R\because f(x)=f(x)\} \), then \( S \)
Let \( A \) and \( B \) be two sets containing \( 2 \) elements and \( 4 \) elements respectively. The number of subsets of \( A\times B \) having \( 3 \) or more elements is_________
If \( A \) and \( B \) be any sets and \( A=2,B=3 \), then \( P(A\times B) \) is __________
For any set \( A \) and \( B,(A\cap B)A \) is________
For any set \( P \) and \( Q, P\le Q \), then_________
If \( R=\{(x,y):x,y\in W,x^2+y^2=25\} \) then domain of \( R \) is_______
If \( R_3=\{(x,x):x\in R\} \) is a relation, then find range of \( R_3 \)_______
If \( A=\{1,2,3,4,5\},S=\{(x,y):x\in A,y\in A\} \) then find the ordered which satisfy the condition \( x+y=5 \).
If \( A=\{1,1,0\},B=\{2,3\}, \) Determine \( A\times B \).
Find \( a \) and \( b \) where \( \left( \dfrac{a}{4},a2b \right)=(0,6+b) \).
Solve: \( \sin \dfrac{\pi}{3}+\dfrac{1}{\csc \dfrac{\pi}{6}} \)
If \( \tan\alpha=\dfrac{1\cos\beta}{\sin\beta} \), then \( \tan 2 \alpha \) is equal to _______
If \( \sin x+\cos x=P \), then \(\sin x\cos x= \)_______
Given, \( x > 0 \), the value of the function \( f(x)=3\cos\sqrt{3+x+x^2} \) lie in the interval______
Which of the following condition is false?
Solve: \( 1\cos 60^o \).
For any natural number \( n,~11^n8^n \) is divisible by_____
\( 1+2+2^2+…+2^n=2^{n+1}1 \), for all_______
Solve: \( 1+5+9+…+(4n3) \) is equal to _______
If \( x^n1 \) is divisible by \( xk \), then the least positive value of \( k \) is________
Let \( Z_0 \) be a root of the quadratic equation, \( x^2+x+1=0 \). If \( Z=3+6iZ_0^{81}3iZ_0^{93} \), then \( \arg(Z) \) is equal to _______
Let \( \left(2\dfrac{1}{3}i\right)^3=\dfrac{x+iy}{27} \), where \( x,y\in R \), then \( yx \) equals________
Let \( Z \) be a complex number such that \( Z+Z=3+i \), then \( Z \) is equal to_______
Let \( Z_1 \) and \( Z_2 \) be two complex numbers satisfying \( Z_1=9 \) and \( Z_234i=4 \). The minimum value of \( Z_1Z_2 \) is______
The equation \( Zi=Z1 \), represents_______
Solve the system of inequalities \(2 \) < \(13x \) < \( 7 \).
Find the solution for the pair of inequation \( x > 2 \) and \( x > 2 \).
The solution to \( 3x1+1 \) < \( 3 \) is_______
Find the value of \( x \), when \( x\in R \) and \( 20x\) < \( 90 \).
If \( x+2> 5 \), then ______
It is required to seat \( 4 \) men and \( 3 \) women in a row so that the women occupy the even places. The number of ways such that possible arrangement are _______
Six boys and six] girls sit along a line alternately in \( x \) ways and along a circle (again alternatively in \( y \) ways), then _______
How many \( 3 \)letter words with of without meaning, can be formed out of the letters of the word, LOGARITHMS, if repetition of letters is not allowed
If \( ^nC_{15}=^nC_6 \), then the value of \( ^nC_{21} \) is________
If \( ^{n+1}C_3=2^nC_2 \), then the value \( fn \) is _______
If the fractional part of the number \( \dfrac{2^{403}}{15} \) is \( \dfrac{k}{15} \), then \( k \) is equal to _______
The coefficient of \( t^4 \) In the expansion of \( \left(\dfrac{1t^6}{1t}\right)^3 \) is_______
The negation of compound statement \( P\vee(~P\vee q) \) is_______
If \( P \): He is intteligent, \( q \): He is strong. Then, “It is wrong that he is strong or intelligent” is________
Which of the following is a tautology?
If \( a,b \) and \( c \) are in geometric progression, then the value of \( \dfrac{ab}{bc} \) is equal to _______
Let \( a,b \) and \( c \) be the \( 7^{th},11^{th} \) and \( 13^{th} \) terms respectively of a nonconstant arithmetic progression. If these are also the three consecutive terms of geometric progression, then \( \dfrac{a}{c} \) is equal to______
If \( \sum_{i=1}^{20}\left(\dfrac{^{20}Ci1}{^{20}C_i+^{20}C_{i1}}\right)^3=\dfrac{k}{21} \), then \( k \) is_______
The sum of all two digit positive numbers which divided by \( 7 \) yield \( 2 \) or \( 5 \) as reminder is ________
If \( 19^{th} \) term of a nonzero arithmetic progression is zero, then its \( 49^{th} \) term: \( 29^{th} \) term is_______
For each \( x\in R \), Let \( [x] \) be the greatest integer less than or equal to \( x \). Then \( \lim_{x \to 0^}\dfrac{x([x]+x)\sin[x]}{x} \) equals_______
Let \( [x] \) denote the greatest integer less than or equal to \(x \), then \( \lim_{x\to 0}\dfrac{\tan(\pi\sin ^2x+\{x\sin(x[x])\}^2}{x^2} \)____
\( \lim_{x\to 0}\dfrac{x\cot(4x}{\sin^2x\cot^2(2x)} \) is equal to______
\( \lim_{x\to 0},\dfrac{\sin^2x}{\sqrt{2}\sqrt{1+\cos x}} \) equals_______
If \( \lim_{x\to 1}\dfrac{x^41}{x1}=\lim_{x\to k}\dfrac{x^3k^3}{x^3k^2} \), then \( k \) is _______
Consider the set of all lines \( Px+qy+r=0,3P+2q+4r=0 \). Then, which one statement is true?
Suppose that the points \( (h,k),(1,2) \) and \( (3,4) \) lie on the line \( L_1 \). If a line \( L_2 \) passing through the points \( (h,k) \) and \( (4,3) \) is perpendicular to \( L_1 \), then \( \dfrac{k}{h} \) equals _______
If a straight line passing through the point \( P(3,4) \) is such that its intercepted portion between the coordinate axes is bisected at \( P \), then its equation is_______.
If the straight line \( 2x3y+17=0 \) is perpendicular to the line passing through the points \( (7,17) \) and \( (15,B) \) the \( B \) equals_______
A point \( P \) moves on the line \( 2x4y+4=0 \) If \( Q(1,4) \) and \( R(3,2) \) are fixed positions, then the locus of the centroid of \( \bigtriangleup PQR \) is a_______
Axis of a parabola lies along \( x\text{axis} \). Its vertex and focus are at distance \( 2 \) and \( 4 \) respectively from the origin, on the positive \( x\text{axis} \) then which of the following points does not lie on it?
Equation of a common tangent to the parabola \( y^2=4x \) and the hyperbola \( xy=2 \) is______
In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is \( 10 \) and one of the foci is at \( (0,5\sqrt{3}) \), then the length of its latus rectum is______
If one end of a focal chord of the parabola, \( y^2=16x \) is at \( (1,4) \), then the lengths of this focal chord is______
If \( 5x+9=0 \) is the directrix of the hyperbola \( 16x^29y^2=144 \), then its corresponding focus is_______
In a nonleap year, the probability of having \( 53 \) Tuesday or \( 53 \) Wednesday is______
Three numbers are chosen from \( 1 \) to \( 20 \). Find the probability that they are not consecutive?
While shuffling a pack of \( 52 \) playing cards, \( 2 \) are accidentally dropped. Find the probability that the missing cards to be of different colours.
If seven persons are to be seated in a row. Then, the probability that two particular persons sit next to each other is_____
Find the probability of the sum of \( 5 \) appear in a single toss of a pair of fair dice.
A data consists of \( ‘n’ \) observations: \( x_1,x_2,…,x_n. \) If \( \sum_{i=1}^{n}(x_i+1)^2=9n \) and \( \sum_{i=1}^{n}(x_i1)^2=5n, \) then the standard deviation of this data is_______
If the sum of the deviation of \( 50 \) observations from \( 30 \) is \( 50, \) then the mean of these observations is _______
The mean and variance of seven observations are \( 8 \) and \( 16, \) respectively. If \( 5 \) of the observations are \( 2,4,10,12,14, \) then the product of the remaining two observations is________
The mean and the median of the following ten numbers in increasing order \( 10,22,26,29,34,x,42,67,70,y \) are \( 42 \) and \( 45 \) respectively, then \( \dfrac{y}{x} \) is equal to________
If for some \( x\in R, \) the frequency distribution of the marks obtained by \( 20 \) students in a test is:
then the mean of the marks is_________