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Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola

Let $F_1$ be the set of all parallelograms, $F_2$ the set of all reactangles, $F_3$ the set of all rhombuses, $F_4$ the set of all sqauares and $F_5$ the set of trapeziums in a plane. Then $F_1$ may be equal to

Which of the following sets are finite and which are infinite ?

(i) Set of concentric circles in a plane.

(ii) Set of letter of the English Alphabets.

(iii) {$xϵN;x>5$}

If $\sin B=\frac{1}{5}\sin (2A+B),$ then $\frac{tan(A+B)}{tanA}$ is equal to

Let any tangent plane to the sphere $(x-a{)}^2+(y-b{)}^2+(z-c{)}^2=r^2$ makes intercepts $a,b,c$ with the coordinate axes at $A,B,C$ respectively. If $P$ is the centre of the sphere, then

$($$ar.$ and $vol.$ denote the area and volume respectively$)$

If the straight line y = mx + c touches the circle $x^2+y^2-4y=0,$ then the value of c will be

Find the equation of a line for which

(i) p = 5, $\alpha ={60}^{\circ }$

(ii) p = 4, $\alpha ={150}^{\circ }$

(iii) p = 8, $\alpha ={225}^{\circ }$

(iv) p = 8, $\alpha ={300}^{\circ }$

In a class of $40$ students, $12$ enrolled for both English and German. $22$ enrolled for German. If the students of the class enrolled for at least one of the two subjects, then how many students enrolled for English but not German ?

The value of $\underset{2}{\overset{5/2}{\int }}\sqrt{\frac{x-2}{3-x}}dx$ is

The condition to be imposed on $\beta$ so that $(0,\beta )$ lies on or inside the triangle having sides y + 3x + 2 = 0, 3y – 2x – 5 = 0 and 4y + x – 14 = 0 is

In how many ways can 3 distinct rings be worn on five fingers of right hand?

Let $C(m,n)$ be a point on the curve $y=x^3,$ where the tangent is parallel to the chord connecting the points $O(0,0)$ and $A(2,8)$. Then the value of $m\cdot n$ is:

If $X=\{x:x=8^n-7n-1,n\in N\}$ and $Y=\{y:y=49n-49,n\in N\},$ then which of the following is (are) TRUE?

If x = -5 and y = -7, what is 222 - (x + y)?

The polar coordinates of the point whose Cartesian coordinates are $(-\frac{1}{\sqrt{2}},\sqrt{2})$, are

In a multiple choice question there are four alternative answers, of which one or more are correct. A candidate will get marks in the question only if he ticks all the correct answers. The candidate decides to tick answers at random. If he is allowed upto three chances to answer the question, find the probability that he will get marks in the question.___

Question 1 (iv)

In which of the following situations does the list of numbers involved make an arithmetic progression and why?

(iv) The amount of money in the account every year, when Rs. 10000 is deposited at compound interest at $8\%$ per annum.

If the lengths of the chords intercepted by the circle $x^2+y^2+2gx+2fy=0$ from the co-ordinate axes be 10 and 24 respectively, then the radius of the circle is

If $f\left(0\right)=0,f^{\prime }\left(0\right)=2$ then the derivative of $y=f\left(f\right(f\left(f\right(x\left)\right))$ at x = 0 is

The sum of three positive numbers $\alpha ,\beta ,\gamma$ is equal to $\frac{\pi }{2}.$ If $e^{\cot \alpha },e^{\cot \beta }$ and $e^{\cot \gamma }$ form a geometric progression, then which of the following hold(s) good?

If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x < -1 or x > 3.

The solution of the equation $8\sin x=\frac{\sqrt{3}}{\cos x}+\frac{1}{\sin x}$ is/are given by

For which of the following functions the second derivative test for finding extremum fails at x = 0 ?

A square of side length $a$ units lies above the $x-$axis and has one vertex at the origin. The side passing through the origin makes an angle $\alpha (0<\alpha <\pi /4)$ with the positive direction of $x-$axis. The equation of its diagonal not passing through the origin is

$sin\left(\frac{B-C}{2}\right)=\frac{b-c}{a}cos\frac{A}{2}$

[(sec A + tan A) (1 - sin A)] on simplification gives: