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If events $E_1,E_2,...,E_n$ represent a partition of sample space S.

Then which of the following is/are correct?

The solution of the differential equation $\frac{dy}{dx}=secx(secx+tanx)$ is

If the normal to a parabola $y^2=4ax$ at $P$ meets the curve again at $Q$ and if $PQ$ and the normal at $Q$ makes angle $\alpha$ and $\beta$, respectively with the $x$-axis then $\tan \alpha (\tan \alpha +\tan \beta )$ has the value equal to

Find the modulus and the arguments of each of the complex numbers

$z=-1-i\sqrt{3}$

If p, q, r are in A.P. and are positive, the roots of the quadratic

equation p $x^2$ + qx + r = 0 are all real for ___.

The value of $\frac{\cos {15}^{\circ }+\sin {15}^{\circ }}{\cos {15}^{\circ }-\sin {15}^{\circ }}$ is

If $a,b,c\in R$ and $a^2+b^2+c^2=1,$ then $ab+bc+ca$ lies in the interval

The range of $f\left(x\right)=\frac{1}{-x^2+4x+5}$,$x\in R-\{-1,5\}$ is

Define $x\%a$ as the remainder obtained when $x$ is divided by $a$.
Let $f:Z\to R$ be defined as $f\left(x\right)=x\%k$, where $kϵN$. If Y is the range of $f\left(x\right)$, then

The equation of the circle concentric with the circle $x^2+y^2-8x+6y-5=0$ and passing through the point $(-2,-7)$ is

Let $x_n,y_n,z_n,w_n$ denote $n^{th}$ terms of four different arithmetic progressions with positive terms. If $x_4+y_4+z_4+w_4=8$ and $x_{10}+y_{10}+z_{10}+w_{10}=20$ then the maximum value of $x_{20}\cdot y_{20}\cdot z_{20}\cdot w_{20}$ is

If R is a relation then which of the following is correct?

If $x+y=\pi +7$, then $si{n}^{2}x+si{n}^{2}y-si{n}^{2}7=$

Which of the following is logically equivalent to $\sim (\sim p\Rightarrow q)$

Fourth term of a G.P is 3 and the product of the $3^{\text{rd}}$ and $8^{\text{th}}$ term is 243. Find its $7^{\text{th}}$ term.

Find the number of integers satisfying the condition |x-3| $>$ 5 and |x + 1 | $\le$ 4

if a, b, c are in

An atom Y crystallizes in fcc unit cell having an edge length $a=200\sqrt{2}pm$ . What would be the radius of octahedral void present in the unit cell at the body centre?

1. 40

$(A-B)\cap (C-B)=$

$\frac{5+9+13+.......upton.terms}{If7+9+11+.....upto(n+1)terms}=\frac{17}{16}$ then n is equal to

If the slope of the curve $y=\frac{ax}{b-x}$ at the point (1, 1) is 2 then values of a and b respectively

The co-ordinates of a point which is equidistant from the points (0, 0, 0), (a, 0, 0), (0, b, 0) and (0, 0, c) are given by [MP PET 1993]

Complete the following sentences.

1. If nobody passes the ball in a basketball game, then you can’t______________________.

2. In a relay race, if no one passes the baton, then __________________________.

If $A=\{x:x\text{is a multiple of}3\}$ and $B=\{x:x\text{is a multiple of}5\}$, then $A-B=$

$sin\left(2si{n}^{-1}\sqrt{\frac{63}{65}}\right)=$

If $x^2$ + $y^2$ = $z^2$. Then ${\log }_{y+z}x$ + ${\log }_{z-y}x$ = ?

If coordinates of the points A and B are (2, 4) and (4, 2) respectively and point M is such that A-M-B also AB =

3 AM, then the coordinates of M are

Find the value of other five trigonometric functions if sin x = $\frac{3}{5}$, and x lies in second quadrant.

Match the following

1. $PC{l}_3$ (i). 2 lone pairs (I) 5 Ligands

2. $Cl{F}_3$ (ii). 3 lone pair (II) 3 Ligands

3. $Xe{F}_6$ (iii). 0 lone pair (III) 6 Ligands

4. $SbC{l}_5$ (iv). 1 lone pairs (IV) 4 Ligands

Find the minimum value of 10 $co{s}^{2}x$ - 6 sinxcos x + 2 $si{n}^{2}x$

If $Ax+By=1$ is a normal to the curve $ay=x^2$ , then

If $|z-i|=1$ and arg $\left(z\right)=\theta$ where $\theta \in (0,\frac{\pi }{2})$, then

$cot\theta -\frac{2}{z}$

What are the initial position vector ${\overrightarrow{r}}_i$ and final position vector ${\overrightarrow{r}}_f$, both in unit-vector notation? What is the x component of displacement $\Delta \overrightarrow{r}$?

(i) Position vector ${\overrightarrow{r}}_i$ (x) $5\hat{i}-3\hat{j}-1\hat{k}$
(ii) Postion vector ${\overrightarrow{r}}_f$ (y) $-2\hat{i}-4\hat{j}+1\hat{k}$
(iii)x-component of displacement $\Delta \overrightarrow{r}$ (z) $7\hat{i}+1\hat{j}-2\hat{k}$

Match the following. Graph of f(x) is given

Solve the inequalities in Exercieses 7 to 10 and represent the solution graphically on number line:

5x +1 > -24, 5x -1 < 24

Let $\left[x\right]$ denote the greatest integer less than or equal to $x$. Then, the values of $x\in R$ satisfying the equation $[e^x{]}^2+[e^x+1]-3=0$ lies in the interval:

Let the curve $y=f\left(x\right)$ pass through the origin. If the mid point of the line segment of its normal between any point on the curve and the $x-$axis lies on the parabola $y^2=4x$, then the equation of the curve $y=f\left(x\right)$ satisfies

The relation between phase difference (Df) and path difference (Dx) is

[MNR 1995; UPSEAT 1999, 2000]

Let $p,q$ be integers and let $\alpha ,\beta$ be the roots of the equation, $x^2-x-1=0,$ where $\alpha \ne \beta .$ For $n=0,1,2,....,$ let $a_n=p{\alpha }^n+q{\beta }^n.$



FACT: If $a$ and $b$ are rational numbers and $a+b\sqrt{5}=0.$ then $a=0=b.$



$a_{12}=$

Find the equation of the hyperbola, the length of whose latus rectum is $4$ and the eccentricity is $3$.

Derivative of $f\left(x\right)=4e^x-4^x+2\ln x$ is

In a G.P. the first, third and fifth terms may be considered as the first, fourth and sixteenth terms of an A.P. Then the fourth term of the A.P., knowing that its first term is 5 is

In a hyperbola the distance between the foci is 2 and the distance between directrices is 1 Its eccentricity is

Find X if $2X+3Y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]and3X+2Y=\left[\begin{array}{cc}7& -2\\ -1& 5\end{array}\right]$

Let $Z_1$=3+4i and |$Z_2$|=1, then

The range of the function f(x) defined by $f\left(x\right)=\frac{x}{1+x^2}$ is

If $[x{]}^2-7[x]+10>0$, then $x$ lies in the interval
(Here, $[.]$ denotes the greatest integer function)