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A hat contains a number of cards with $30\%$ white on both sides, $50\%$ black on one side and white on the other side, $20\%$ black on both sides. The cards are mixed up, and a single card is drawn at random and placed on the table. Its upper side shows up black. The probability that its other side is also back is

If the eccentricity of a hyperbola is $\sqrt{2}$ and if the distance between the foci is 16, then its equation is

Solve $|x-2|\ge 5$

If $lo{g}_{3}x=a$, find the value of ${81}^{a-1}$ in terms of x.

In a $\Delta$ ABC, $2a{­}^2+4b^2$ + $c^2=4ab+2ac,$then cos B is equal to

Find the distance between parallel lines

(i) 15x + 8y - 34 = 0 and 15x + 8y + 31 = 0

(ii) l(x + y) + p = 0 and l(x + y) - r = 0

The line segment joining (5,0) and (10 cos t, 10 sin t) is divided internally in the ratio 2:3 at P. If t varies, then the locus of P is

Find centre and radius for the given equation of the circle.

$x^2+y^2-4x-8y-45=0$

$\underset{x\to \infty }{lim}{\left(\frac{3x^2+2x+1}{x^2+x+2}\right)}^{\frac{6x+1}{3x+2}}\text{is equal to}$

Show that the following statement is true by the method of contrapositive p : "If x is an integer and $x^2$ is even, then x is also even"

If $\frac{2}{11}$ is the probability of an event, what is
the probability of the event not A'

Find the equation of the hyperbola whose foci are $(\pm 5,0)$ and the transverse axis is of length 8.

If 4-digit numbers greater than 5,000 are randomly formed from the digits 0, 1, 3, 5, and 7, what is the probability of forming a number divisible by 5 when,
(i) the digits are repeated? (ii) the repetition of digits is not allowed?

Which of the following equations represents a circle with centre $(g,f)$ and radius $\sqrt{g^2+f^2+c}$ ?

What is the conjugate hyperbola of the hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

You can remove a removable discontinuity by

Co-efficient of $x^5$ in the expansion of $(1+x^2{)}^5(1+x{)}^4$ is ___________.

Find the coefficient of $x^8{y}^{12}$ in the expansion of $(x+2y{)}^{20}$

If f(x) = x² - x + 1, g(x) = 7x - 3, be two real functions then (f - g)(8) is

How many of following statements are true?

(1)Period of sin $\theta$ is $\pi$, because sin 0 and sin $\pi$ = 0

(2) Period of cos $\theta$ is $\pi$

(3) Period of tan $\theta$ is $\pi$

Express the following in the form a + ib where a, b Є R.

The expression $tan(ilog(\frac{a-ib}{a+ib}))$ reduces to:

If $(a,a^2)$ falls inside the angle made by the lines $y=\frac{x}{2},x>0,$ and $y=3x,x>0,$ then a belongs to

If x is real, then the maximum and minimum values of expression $\frac{x^2+14x+9}{x^2+2x+3}$will be

How many of following statements are true?
(1)Period of $sin\theta$is $pi$, because sin 0 = 0 and $sin\pi =0$,
(2) Period of $cos\theta$ is $\pi$
(3) Period of $tan\theta$ is $\pi$

The temperature at which RMS velocity of $S{O}_2$ molecules is half that of He molecules at 300 K is

The equation of the line passing through the points $(3,-2)$ and $(-3,2)$ is

The intercept of the line drawn for log P (P in atm) and $log\frac{1}{V}$ (V in the litre) for 1 mole of an ideal gas at ${27}^{\circ }$ C is equal to

If $x_1,x_2,\cdots ,x_n$ and $\frac{1}{h_1},\frac{1}{h_2},\cdots ,\frac{1}{h_n}$ are two $A.P.$s such that $x_3=h_2=8$ and $x_8=h_7=20,$ then $x_5\cdot h_{10}$ equals

Which of the following intervals belong(s) to the domain of the function $f\left(x\right)=\frac{\sqrt{-{\log }_{0.4}(x^4-1)}}{x^2+2x+8}$

$Thedomainofthefunctionf\left(x\right)=lo{g}_e(x^2+x+1)+sin\sqrt{x-1}is$

Which of the following relations are functions? Give reasons. If it is a function determine its domain and range.

(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

(iii) {(1, 3), (1, 5), (2, 5)}

What is the value of $\sqrt{25}$

Find the value of $sin{210}^{\circ }+cos{120}^{\circ }+tan{225}^{\circ }+cot{315}^{\circ }+sec{300}^{\circ }+cosec{150}^{\circ }$.

If $\tan \theta =\frac{-4}{3}$ then $\sin \theta$ is

Let z be any non-zero complex number, then the value of arg z + arg($\overline{z})$ is __

If A(1, 2, -1) and B(-1, 0, 1) are given, then the co-ordinates of P which divides AB externally in the ratio 1:2, are [MP PET 1989]

The coefficient of $ab{c}^{3}d{e}^2$ in the expansion of $(a+b+c+d+e{)}^8$ is equal to

The value of ${\sum }_{n=1}^{45}{i}^n+i^{n+1}$ is ____________

Hint: $\sum t_n+t_m$ = $\sum t_n+\sum t_m$

If $(a+bx{)}^{-2}$ = $\frac{1}{4}$ - 3x + ......, then (a,b) =

A large tank filled with water to a height ‘h’ is to be emptied through a small hole at the bottom. The ratio of times taken for the level of water to fall from h to $\frac{h}{2}$ and from $\frac{h}{2}$ to zero is

If $x=1+y+y^2+\cdots \cdots \infty$, then y is

If $C_k$ is the coefficient of in the expansion of ${(1+x)}^{2005}$ and if a, d are real numbers then ${\sum }_{k=0}^{2005}(a+kd).C_k=$

Evaluate $\int x^2{e}^{x}dx$

Range of the function $f\left(x\right)=x^2+\frac{1}{x^2+1}$, is

Three critics review a book. Odds in favour of th ebook are $5:2,4:3$ and $3:4$, respectively, for the three critics. The probability that majority are in favor off the book is