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Let A = {1, 2} and B = {3, 4}, write $A\times B.$ How many sub sets will $A\times B$ have? List them.

Find $\underset{x\to 5}{\text{lim}}$, where f(x) = |x| - 5,

$\frac{d}{dx}\left\{\right(1+x{)}^{\frac{1}{2}}-(1-x{)}^{\frac{1}{2}}\}$

Find the $8^{th}$ term of the series 1 + 5 + 18 + 58 + 179 + ..........

Find the value of $\theta$, if the equation $cos\theta x^2-2sin\theta x-cos\theta =0$ real root.

The number of integral value of K for which the equation 7cosx+5sinx=2K+1 has a solution is

If A + B + C = ${180}^{\circ }$, then sin (B + 2C) + sin (C + 2A) + sin (A + 2B)

A ray of light passing through the point A and the reflected ray passes through the point (1,2) reflects on the x-axis at point A and the reflected ray passes through the point (5,3). Find the coordinates of A.

The sum of an infinite number of terms of a G.P. is $20$, and the sum of their squares is $100$, then the common ratio is

If a line, OA(O is origin) makes an angle of ${30}^{\circ }$ with the negative x-axis, the angle it makes with the positive direction of x-axis is .

Convert the complex number z=-1-i in to the polar form.

If sum of n terms of a sequence is given by $S_n=2n^2+3n$, Find it's ${10}^{th}$ term.

The centre of the circle $z\overline{z}-(2+3i)z-(2-3i)\overline{z}+9$ = 0 is

Find the sum to n terms of the sequences 8, 88, 888, 8888 ....

Superman is going on a vacation to his home planet krypton from earth via. Sun. (Let's assume for a moment that Krypton is not destroyed). His energy v/s distance travelled graph is as shown below:

At which point is superman's (dE/ds) positive

The tangent at any point P on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ with centre C,

meets the asymptotes in Q and R and cut off triangle CQR. Find the area of the triangle CQR.

In a $△$ ABC, if a cos A = b cos B, show that the triangle is either isosceles or right angled.

Find the set of real values of x for which ${\log }_{0.2}$ $\frac{x+2}{x}$ $\le$ 1, is ________

The sum of an infinite G.P. is 23 and the sum of squares of the series is 69, then first term is :

If tan A and tan B are the roots of the quadratic equation $x^2$ -ax+b=0, then the value of $si{n}^2$ (A+B) is

If $\tan {20}^{\circ }=p,then\frac{\tan {160}^{\circ }-\tan {110}^{\circ }}{1+\tan {160}^{\circ }\tan {110}^{\circ }}=$

Find out range and coefficient of range of the following series.

$\begin{array}{cccccc}\text{Size}& 5-10& 10-15& 15-20& 20-25& 25-30\\ \text{Frequency}& 4& 9& 15& 30& 40\end{array}$

If P = sin a .sin 2a . sin 3a . . . . . . sin 999a &

Q = sin 2a .sin 4a . sin 8a . . . . . . sin 1998a

Find the value of $\frac{Q}{P}$ Where a = $\frac{2\pi }{1999}$

Let A, B and C be three events, if the probability of occuring exactly one event out of A and B is (1-x), out of B and C is (1-2x), out of C and A is (1-x) and that of occuring three events simultaneously is x^2, then prove that the probability that atleast one out of A, B and C will occut is greater than $\frac{1}{2}$

If cos$\alpha$+cos$\beta$+cos$\gamma$=0=sin$\alpha$+sin$\beta$+sin$\gamma$ then

cos2$\alpha$+cos2$\beta$+cos2$\gamma$ equals

Find the value of $5^{{\log }_5{9}^2}$

The maximum value of $5+(sinx-4{)}^2$ is

If 8 arithmetic means are placed between 2 and 17 then the 5th arithmetic mean will be:

Use quanlifiers to convert each of the following open sentences defined on N, into a true statement:

(i) $x+5=8$ (ii) $x^2>0$ (iii) $x+2<4$

Mr. Akshat keeps his books on incomplete records. Following information is given below

$\begin{array}{ccc}\text{Items}& \text{April 1, 2004}& \text{April 1, 2004}\\ \text{Cash in hand}& 1,000& 1,500\\ \text{Cash at bank}& 15,000& 10,000\\ \text{Stock}& 1,00,000& 95,000\\ \text{Debtors}& 42,500& 70,000\\ \text{Business premises}& 75,000& 1,35,000\\ \text{Furniture}& 9,000& 7,500\\ \text{Creditors}& 66,000& 87,000\\ \text{Bills payable}& 44,000& 58,000\end{array}$

During the year, he withdrew Rs. 45,000 and introduced Rs. 25,000 as further capital in the business.
Compute the profit or loss of the business.

The quadratic equation tan$\theta x^2+2(sec\theta +cos\theta )x+(tan\theta +3\sqrt{2}cot\theta )$ always has

If $\alpha$ is the fifth root of unity then which of the following is false

Origin is shifted to the point (-1,2) to form a new co-ordinate system. What will be the co-ordinate of (3,-5) in the new co-ordinate system.

If $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+.......+C_n{x}^n$, then

$\frac{C_1}{C_0}$ + $\frac{2C_2}{C_1}$ + $\frac{3C_3}{C_2}$ + ........+ $\frac{n{C}_n}{C_{n-1}}$ =

If A = [(x,y): $x^2+y^2=25$]

And B = [(x,y): $x^2+9y^2=144],thenA\cap B$ contains

$co{s}^2{48}^0-si{n}^2{12}^0$ =

[MNR 1977]

There are 16 points in a plane out of which 6 are collinear, then how many lines can be drawn by joining these points

A point P on the ellipse is at a distance of 6 units from the focus. If the eccentricity of the ellipse is 0.6, then the distance of P from the corresponding directrix is ___

Prove that:

${\text{sin}}^2\frac{\pi }{6}+{\text{cos}}^2\frac{\pi }{3}-{\text{tan}}^2\frac{\pi }{4}$ =- $\frac{1}{2}$

State whether each of the following statements is true or false. If the statement is false, rewrite the given statement correctly.

(i) If P = {m, n} and Q = {n, m}, then $P\times Q=$ {(m, n), (n, m)}

(ii) If A and B are non-empty sets, then $A\times B$ is a non-empty set of ordered pairs (x, y) such that $xϵA$ and $yϵB.$

(iii) If A = {1, 2}, B = {3, 4}, then $A\times (B\cap \varphi )=\varphi$

A physical quantity is measured and its value is found to be equal to $nu$, where $n$ is the numerical value and $u$ is the unit. Which of the following relations is true?

The solution set of $16-x^2\ge 0$ is

With the help of mathematical induction find for all $n\ge 1$ the sum of the series $\frac{1}{1.2}+\frac{1}{2.3}...\frac{1}{n.n+1}$ is equal to

What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many ways these

(i) four cards are of the same suit?

(ii) four cards are belongs to four different suits?

(iii) four cards are of the same colour?

Prove the following:
cos 4x = 1 - 8 $si{n}^2$x $co{s}^2$x