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If \( A \) and \( B \) any set where \( B \) is nonempty, \( A\cup B=B \) and \( A\cap B=\phi \) then \( \left A \right \) is equal to ___________.
\( \left( BA^\prime \right)\left( AB^\prime \right)^\prime \) is equal to ___________.
If \( \left A \right=5, \left B \right=4, \left A\cup B \right=9 \), then \( \left A\Delta B \right \) is equal to __________.
If \( \left A \right=x^y \) and \( \left B \right=z^y \), then \( \left A\times B \right \) is equal to __________.
If \( f(x)=x^6x^y+2x^2 \), then \( f(x) \) is a/an ___________.
The period of \( f(\theta)=43\sin\theta+4\sin^3\theta \) is __________.
Range of the function that \( f(x)=3\sin\sqrt{\dfrac{\pi^2}{36}x^2} \) is __________.
If \( f:R\to R \) is given by \( f(x)=x^327 \) __________.
\( x^2=16 \) iff \( x=\pm 4 \). This proposition is a __________.
Inverse of the statement \( \sim \left( p\wedge q \right)\to \left( \sim r\vee \sim s \right) \) is __________.
The proposition \( \left( p\wedge q \right)\vee \left( \sim p\vee \sim z \right) \) is __________.
\( (p\wedge q)\vee (\sim q\wedge r)\vee (\sim p\wedge q) \) is equivalence to __________.
\( f(n)=1^3+2^3+3^3+…+n^3 \) is equal to __________.
For all \( n\in N,\ n(n+1)(2n+1) \) is divisible by __________.
\( p(n):n^2\ge 81 \) is true _________.
Find the value of \( m \) where \( ^{11}P_3=\ ^{11}C_3\times m \).
In how many ways can \( 5 \) different coloured balls be distributed among \( 7 \) different boxes when any box can have any number of balls?
How many numbers formed between \( 100 \) and \( 1000 \), by using the digits \( 2,3,4,7,0,9 \) (No repeitition allowed) ___________.
Find out the number of triangles can be drawn out of \( 7 \) given points on a circle?
In how many ways can the letters of the word ‘ATTRACT’ be arranged?
Find the coefficient of \( x \) in the expansion of \( (13x+7x^2)(1x)^{16} \).
Find the coefficient of \( x^{15} \) in the expansion of \( (xx^2)^{10} \).
If \( A \) and \( B \) are coefficient of \( x^n \) in the expansions of \( (1+x)^{2n} \) and \( (1+x)^{2n1} \) respectively, then \( \dfrac{A}{B} \) equals to ___________.
Middle term in the expansion of \( (a^3+ba)^{28} \) is ___________.
Solve: \( 4x+3\ge 2x+17 \)
Solve: \( \dfrac{1}{\left x \right5}\ge \dfrac{1}{3} \).
If \( \left x1 \right \) > \( 6 \), then __________.
If \( x,y \) and \( b \) are real numbers and \( x \)<\( y,\ b \)<\( 0 \), then __________.
Find the equation of the straight line which passes through the point \( (3, 2) \) and cuts off equal intercept from axes.
Find the slope of a line which cuts off intercepts of equal lengths on the axes is __________.
The distance between the lines \( y=mx+5 \) and \( y=mx3 \) is______
Equation of the line passing through \( (2,3) \) and parallel to the line \( y=5x1 \) is ___________.
Let \( P(2,3) \) be a point on the hyperbola \( \dfrac{x^2}{a^2}\dfrac{y^2}{b^2}=1 \). If the normal at the point \( ‘P’ \) intersects the \( x \)axis at \( (4,0) \), then the eccentricity \( (e) \) of the hyperbola is ___________.
The eccentricity of the ellipse \( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1,\ (a \)<\( b) \) then __________.
The length of the latus rectum of the ellipse \( \dfrac{x^2}{4}+\dfrac{y^2}{12}=1 \) is __________.
The angle between the tangents drawn from the point \( (1,4) \) to the parabola \( y^2=4x \) is __________.
The point \( P(7, 3, 4) \) lie in the octant_____
In a square \( ABCD, O(0,0,0) \) is the intersecting point of two diagonals, where \( A (a,7,10) \) and \( C (5,b,10) \) are opposite vertices, then find \( a \) and \( b \).
The locus of a point for which \( x=0 \) is __________.
If the distance between the points \( (5,a,1) \) and \( (1,1,1) \) is \( 5 \), then \( a \) is __________.
If \( 4\sin\theta+3\cos\theta=5 \), find \( \tan\theta \).
Solve: \( \dfrac{\sin40^o}{\sin140^o} \)
If \( \tan\theta=\dfrac{b}{a} \), then \( a\cos2\theta+b\sin2\theta \) is __________.
If \( \sin\theta+\cos\theta=0 \), then find \( \sin2\theta \).
Find the maximum value of \( 12+\sin^2\theta \).
Solve: \( \sin10^o\sin50^o\cdot \sin70^o \).
The third term of the geometric progression is \( 2 \). The product of the first five terms is __________.
The sum of \( n \) terms of the series \( \dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+… \) is _________.
If \( 2, k, \dfrac{25}{4} \) are in arithmetic progression, then find the value of \( k \) is __________.
The sum of first \( 5 \) terms of the geomteric progression \( 2,6,18,54,… \) is __________.
\( \lim_{x \to 0} \dfrac{\tan x^o}{x} \) is equal to __________.
Solve: \( \lim_{x \to 0} \dfrac{5^x1}{x} \)
Solve the value of \( n \), where \( \lim_{x \to 3} \dfrac{x^n3^n}{x3}=108, n\in N \).
\( \lim_{x \to \pi} \dfrac{\sin x}{x\pi} \) is equal to __________.
Evaluate:\( \lim_{x \to \frac{\pi}{6}} \dfrac{\sqrt{3}\sin x\cos x}{x \dfrac{\pi}{6}} \)
If \( \begin{vmatrix}
6i & 4 & 20 \\
3i & 3i & 3 \\
1 & 1 & i
\end{vmatrix} = x+iy \), then ___________.
For the smallest positive integer \( n\left( \dfrac{i+1}{i1} \right)^n=1 \), is ___________.
If \( \left z_1 \right=\left z_2 \right= \left z_3 \right=\left \dfrac{1}{z_1} +\dfrac{1}{z_2}+\dfrac{1}{z_3}\right=1 \), where \( z_1, z_2, z_3\neq 0 \), then \( \left z_1+z_2+z_3 \right \) is __________.
For any two complex numbers \( z_1 \) and \( z_2 \) and any two real numbers \( a \) and \( b \), \( \left az_1+bz_2 \right^2+\left bz_1az_2 \right^2 \) is equal to __________.
If \( x+iy=\sqrt{\dfrac{p+iq}{r+i5}} \), then \( (x^2+y^2)^2 \) is __________.
A jar contains \( 36 \) marbles. Some are red and others are white. If a marble is drawn at random from the jar, the probability that it is white is \( \dfrac{3}{4} \), then number of red marbles in the jar is __________.
If number \( x \) is chosen at random from the numbers \( 2,2,0,1,2 \). Then the probability that \( x^2 \)< \( 1 \) is __________.
A card is drawn from a wellshuffled deck of \( 52 \) cards. Find the probability of getting a Ace of black suit.
Two dice are thrown simultaneously. Find the probability of getting even entries only.
If \( P(A)=\dfrac{1}{6},\ P(B)=\dfrac{2}{3} \) and \( P\left( \dfrac{B}{A} \right)=\dfrac{1}{3} \) then \( P(A\cup B) \) is equal to __________.
The range of the following set of observations \( 3,4,7,6,5,7,9,8,11,13 \) is __________.
The variance of the data \( 3,7,5,9 \) is ___________.
Mean of \( 100 \) observations is \( 46 \). It was later found that two observations \( 17 \) and \( 33 \) were incorrectly recorded as \( 93 \) and \( 57 \). The corrrect mean is __________.
In a frequency distribution, the mean and median are \( 42 \) and \( 53 \) respectively, then find mode.
The mean of a set of observations is \( \overline{x} \). If each observation is divided by \( a \) and then is increased by \( 5 \), then the new mean is __________.