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Let \( S=\left\{ 1,2,3,…,100 \right\} \). The number of nonempty subsets \( A \) of \( S \) such that the product of elements in \( A \) is even __________.
The set \( A=\left\{ x:x\in R,\ x^2=25\text{ and } 2x=8\right\} \) is equal to _________.
If \( \left A \right=3, \left B \right=4 \), and \( \left C \right=x \), and \( \left A\times B\times C \right=60 \), then \( x \) is equal to _________.
The total number of subsets of a finite set \( A \) has \( 124 \) more elements than the total number of subsets of another finite set \( B \). What is the number of elements in set \( A \)?
\( \left A \right=3,\ \left B \right=4 \), then the number of relations from \( A\text{ to }B \) is ___________.
If \( R \) is an equivalence relation on a set \( A \), then \( R^{1} \) is __________.
If \( \left A \right=9,\ \text{ and } \left B \right=13 \), then the number of oneone function from \( A\text{ to }B \) is _________.
Find the domain of the function \( f(x)=\dfrac{7x}{x^29} \).
If \( f(x)=\dfrac{x+1}{x1} \) , then _________.
The statement \( \sim \left( p\wedge \sim q \right) \) is equivalent to __________.
Find the truth value of \( P\to \left( p\wedge q \right) \) is __________.
Express in propositional logic “If Sita will come to party, if and only if Gita will come and Anu will not come.”
For any natural number \( ‘n’,\ 13^n8^n \) is divisible by __________.
\( 10^{2n1}1 \) is divisible by, where \( n\in N \) _________.
\( n(n+2)(n+4) \) is divisible by __________.
How many words can be formed by using all letters of the word ‘CHAPTER’
A box contain \( 5 \) red, \( 3 \) white and \( 4 \) blue balls.
Three balls are drawn at random. Find out the number of ways of selecting the balls of different colours?
Find out the number of ways in which \( 5 \) rings of different types can be worn in \( 2 \) fingers?
In how many ways \( 9 \) people can be seated in a circular order?
How many triangles can be formed by joining the vertices of an ocatgon?
If the expansion of \( \left( 1+x \right)^{27} \), the coefficient of \( x^r \) and \( x^{r+1} \) be equal, then \( r \) is equal to _________.
Find the \( 13^{th} \) term in the expansion of \( \left( 2x\dfrac{1}{x^2} \right)^{14} \)
If \( T_{r+1} \) in expansion of \( \left( 2x^2+\dfrac{5}{x^2} \right)^{10} \) is a constant, then \( (r+1)^2 \) is __________.
If \( \dfrac{x2}{x+3} \) > \( \dfrac{1}{2} \), then \( x \) lies in the interval __________.
What is the solution set for \( \dfrac{\left x5 \right}{x5} \) > \( 0 \)
The mark secured by Ram in two exams were \( 65 \) and \( 72 \).
Find the minimum marks he should score in the third exam to have an average of atleast \( 68 \)?
Given that \( x \) is an integer, find the value/s of \( x \) which satisfy both the inequations \( 3x+4 \) > \( 7 \) and \( x+5 \) < \( 8 \).
Find the foot of the perpendicular drawn from the point \( P(2, 3,4) \) on \( xz \)plane.
If the origin is the centroid of \( \Delta ABC \) having vertices \( A\ (2,a,3), B\ (b,8,5) \) and \( C\ (4,7,c) \), then find \( a, b, \) and \( c \).
The distance between two points \( \left( 4, a, 10 \right) \) and \( \left( 3,6,8 \right) \) is \( 3 \), find \( a \)?
The equation of straight line is passing through the point \( (2,3) \) and parallel to the line \( y=4x+3 \) is __________.
Find the slope of the line \( 5x+3y7=0 \) is __________.
\( x \)intercept of the line \( 5x7y+35=0 \) is __________.
The equation of the line passing through the points \( (2,3) \) and \( (1,5) \) is __________.
Find the incentre of the triangle \( ABC \) having vertices \( A(0,0), B(4,0), C(0,3) \) is __________.
Find the equation of a hyperbola with foci on the \( x \)axis is __________.
if \( (a,b) \) is the midpoint of a chord passing through the vertex of the parabola \( y^2=4x \), then _________.
A hyperbola, having the transverse axis of length \( 2\sin\theta \), is confocal with the ellipse \( 3x^2+4y^2=12 \). Then its equation is __________.
The parametic form of the ellipse \( 4(x+1)^2+(y1)^2=4 \) is _________.
The axis of a parabola is along the line \( y=x \) and the distances of its vertex and focus from origin are \( \sqrt{2} \) and \( 2\sqrt{2} \) respectively. If vertex and focus both lie in the first quadrant, then the equation of the parabola is _________.
For any \( \theta\in \left( \dfrac{\pi}{4}, \dfrac{\pi}{2} \right) \), the expression \( 3\left( \sin\theta\cos\theta \right)^4+6\left( \sin\theta+\cos\theta \right)^2+4\sin^6\theta \) equals __________.
If \( 0\leq x \) < \( \dfrac{\pi}{2} \), then the number of values of \( x \) for which \( \sin x\sin 2x+\sin3x=0 \) is __________.
If \( f_k(x)=\dfrac{1}{k}\left( \sin^kx+\cos^kx \right) \), for \( k=1,2,3,…, \), then \( \forall x\in R \), the value of \( f_4(x)f_6(x) \) is equal to __________.
All the pairs \( (x,y) \) satisfy the inequality \( 2^\sqrt{\sin^2x2\sin x+5}\ \cdot \dfrac{1}{y^{\sin2y}}\le 1 \) also satisfy the equation __________.
The number of solutions of the equation \( 1+\sin^4x=\cos^23x,\ x \in \left[ \dfrac{5\pi}{2}, \dfrac{5\pi}{2} \right] \) is __________.
\( \lim_{x \to 7} \dfrac{x^249}{x7} \) is equal to __________.
\( \lim_{x \to \infty } \left( \dfrac{x+5}{x} \right)^x \) is equal; to __________.
Solve: \( \lim_{x \to 0} \dfrac{\sin4x}{\sin5x} \)
Solve: \( \lim_{x \to 3} \dfrac{\ln\left( 4x5 \right)}{x3} \)
Evaluate: \( \lim_{x \to 0} \dfrac{\left( x+3 \right)^{\frac{1}{2}}3^{\frac{1}{2}}}{x} \)
The value of \( \sum_{n=1}^{13}\left( i^{n+1}+i^n \right) \) equals __________.
Let \( \alpha \) and \( \beta \) be two roots of the equation \( x^2+2x+2=0 \), then \( \alpha^{15}+\beta^{15} \) equals __________.
The number of possible integral values of \( \alpha \) for which the roots of the quadrative equation \( 6x^211x+\alpha=0 \) are rational numbers is __________.
The value of \( \lambda \) such that sum of the squares of the roots of the quadratic equation, \( x^2+\left( 3\lambda \right)x+2=\lambda \), has the least value is __________.
If \(w \) be a cube root of unity and \( (1+w^4)=(1+w^2)^n \), then the least value of \( n \) is __________.
If \( a_1,a_2,a_3,… \) are in A.P. such that \( a_1,a_7,a_{16}=40 \), then the sum of \( 1{\text{st}}~15 \) terms of this A.P. is __________.
Let \( a,b \) and \( c \) are in geometric progression with common ratio \( r \), where \( a\neq 0 \) and \( 0 \) < \( r\le \dfrac{1}{2} \).
If \( 3a,\ 7b \) and \( 15c \), are the \( 1^{st} 3\) terms of an arithmetic progression, then the \( 4^{th} \) term of this arithmetic progression is ___________.
Let the sum of first \( n \) terms of a nonconstant arithmetic progression, \( a_1,a_2,a_3,… \) be \( 50n+\dfrac{n\left( n7 \right)}{2}A \), where \( A \) is a constant.
If \( d \) is the common difference of this arithmetic progression, then the ordered pair \( \left( d, a_{50} \right) \) equals __________.
If the sum and product of the first three terms in an arithemtic progression are \( 33 \) and \( 1155 \), respectively, then a value of its \( 11^{th} \) term is __________.
Let \( x,y \) be positive real numbers and \( m,n \) positive integers. The maximum value of the expression \( \dfrac{x^my^n}{\left( 1+x^{2m} \right)\left( 1+y^{2n} \right)} \) is __________.
Three randomly chosen nonnegative integers \( x,y \) and \( z \) are found to satisfy the equation \( x+y+z=10 \), Then the probability that \( 2 \) is even, is ___________.
Four persons independently solve a certain problem correctly with probabilities \(\dfrac{1}{2},\dfrac{3}{4},\dfrac{1}{4},\dfrac{1}{8} \). Then the probability that the problem is solved correctly by at least one of them is __________.
An experiment has \( 10 \) likely outcomes. Let \( A \) and \( B \) be twoempty event sof the experiment. If \( A \) consists of \( 4 \) outcomes, the number of outcomes that \( B \) must have so that \( A \) and \( B \) are independent, __________.
Let \( E^c \) denote the complement of an event \( E \). Let \( E,F,G \) be pairwise independent events with \( P(G) \) > \( 0 \) and \( P\left( E\cap F\cap G \right)=0 \), then \( P\left( \dfrac{E^c\cap F^c}{G} \right) \) equals __________.
A fair die is rolled. The probability that the first time \( 1 \) occurs at the even number of trials is __________.
In an experiment with \( 15 \) observations on \( x \), the following results were available.
\( \Sigma x^2=2830, \Sigma x=170 \). One observation that was \( 20 \) was found to be wrong and was replaced by the correct value \( 30 \). Then the corrected variance is __________.
In a frequency distribution, the mean and median are \( 21 \) and \( 22 \) respectively, then its mode is approximately __________.
The avergae marks of boys in class is \( 52 \) and that of girls is \( 42 \).
The average marks of boys and girls combined is \( 50 \). The percentage of boys in the class is __________.
The mean of the data set comprising of \( 16 \) observations is \( 16 \). If one of the observation valued \( 16 \) is deleted and three new observations valued \( 3,4 \) and \( 5 \) are added to the data, then the mean of the resultant data, is __________.
The variance of first \( 50 \) even natural numbers is __________.