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The function $f\left(x\right)=\frac{2x^2-1}{x^4},x>0$, decreases in the interval

Consider the letters of the word $MATHEMATICS$.

Possible number of words in which no two vowels are together is

The point of inflection for the function $f\left(x\right)={\sin }^{-1}x$ is:

Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be reciprocal to that of the ellipse $x^2+9y^2=9,$ then the ratio $a^2:b^2$ equals

Find the derivative of cot² x³

Let $f\left(x\right)=(x+1{)}^2-1,(x\ge -1)$. Then the set $S=\{x:f(x)=f^{-1}(x\left)\right\}$. is

Prove that $\frac{1-\cos A+\cos B-\cos (A+B)}{1+\cos A-\cos B-\cos (A+B)}=\tan \frac{A}{2}\cot \left(\frac{B}{2}\right)$.

What is the equation of polar of the point (1, 2) with respect to the circle $x^2+y^2+8x+2y+1=0$

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse

If $y=3^{{\log }_9(1+{\tan }^{2}x)},x\in (0,\frac{\pi }{2}),$ then $\frac{dy}{dx}=$

The area of the triangle formed by the intersection of a line parallel to x-axis and passing through P(h,k) with the lines y=x and x+y=2 is $4h^2$. Find the locus of the point P.

Question 2(a)

Change the following statements using expressions into statements in ordinary language.(For example, given Salim scores r run in a cricket match, Nalin scores (r + 15) runs. In ordinary language – Nalin scores 15 runs more than Salim.)

A notebook costs ₹ p. A book costs ₹ 3p.

If R be a relation $<$ from A={1,2,3,4} to B={1,3,5} i.e., (a,b) $\in R\iff a<b,thenRo{R}^{-1}$ is

If $y=y\left(x\right)$ is the solution of the differential equation $x^{2}dy+(y-\frac{1}{x})dx=0;x>0$ and $y\left(1\right)=1,$ then the value of $y\left(\frac{1}{2}\right)$ is

Let $S$ be the set of all non-zero real number $\alpha$ such that the quadratic equation $\alpha x^2-x+\alpha =0$ has two distinct real roots $x_1$ and $x_2$ satisfying the inequality $|x_1-x_2|<1.$ Which of the following interval is(are) a subset of $S?$

The number of hyperconjugative structures in the following molecule will be:

For any angle $x^{\circ },x^c=$

The mean deviation is least when taken from the central value

If $\cos x\frac{dy}{dx}-y\sin x=6x,(0<x<\frac{\pi }{2})$ and $y\left(\frac{\pi }{3}\right)=0,$ then $y\left(\frac{\pi }{6}\right)$ is equal to :

If the co-ordinates of the points P,Q,R,S be (1, 2, 3), (4, 5, 7), (– 4, 3, – 6) and (2, 0, 2) respectively, then

When and why do we take π value sometimes 22/7 and sometimes 180°?

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sinⁿ x

A line makes angles $\alpha ,\beta ,\gamma ,\delta$ with the four diagonals of a cube. Then $co{s}^2\alpha +co{s}^2\beta +co{s}^2\gamma +co{s}^2\delta$ is

From mean value theorem $f\left(b\right)-f\left(a\right)=(b-a)f^{\prime }\left(x_1\right);a<x_1<b$ if $f\left(x\right)=\frac{1}{x}$, then $x_1$

If A=$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, then $A^{-1}$ = [MP PET 1988]

The probability of getting a total of 10 in a single throw of the dice is

$\underset{n\to \infty }{lim}\frac{1-2+3-4+5-6+.......2n}{\sqrt{n^2+1}+\sqrt{4n^2-1}}\text{is equal to:}$

If $A=\{1,2,4\},B=\{2,4,5\},C=\{2,5\}$ then $(A-C)\times (B-C)$ is