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9th term of the sequence 1,1,2,3,5... Is

If the roots of the equation $x^3-12x^2+39x-28=0$ are in A.P., then their common difference will be

Which of the following point(s) lies on the line passing through $-2\hat{i}+2\hat{j}-\hat{k}$ and $3\hat{i}+2\hat{k}$ is/are

A group of $r$ boys is to be formed from $9$ boys. The value of $r$ for which we get maximun number of different groups is

Eight players $P_1,P_2,\cdots ,P_8$ paly a knock - out tournament. It is known that whenever the players $P_{i}and{P}_j$ play, the player $P_i$ will win if i < j. Assuming that the players are paired at random in each round, what is the probability that the player $P_4$ reaches the final?

If $F\left(x\right)=\left[\begin{array}{ccc}\cos x& -\sin x& 0\\ \sin x& \cos x& 0\\ 0& 0& 1\end{array}\right]$ and $G\left(x\right)=\left[\begin{array}{ccc}\cos x& 0& \sin x\\ 0& 1& 0\\ -\sin x& 0& \cos x\end{array}\right],$ then $\left[F\right(x).G(x){]}^{-1}$ is equal to:

If $S_r$ denotes the sum of the infinite geometric series whose first term is $r$ and common ratio is $\frac{1}{1+r}$, where $r\in N$, then the value of $\sum _{r=1}^{10}{S}_r^2$ is

Let $P$ be a point on the parabola, $y^2=12x$ and $N$ be the foot of the perpendicular drawn from $P$ on the axis of the parabola. A line is now drawn through the mid-point $M$ of $PN$, parallel to its axis which meets the parabola at $Q$. If the y-intercept of the line $NQ$ is $\frac{4}{3}$, then

Let the circle $S:36x^2+36y^2-108x+120y+C=0$ be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, $x-2y=4$ and $2x-y=5$ lies inside the circle $S$, then:

If $p\equiv$ "It rains today", $q\equiv$ "I go to school", $r\equiv$ "I shall meet my friends"
What can be concluded from "I will go to school and meet my friends if it doesn't rain today"?

If the coefficient of $m^{th}$, $(m+1{)}^{th}$ and $(m+2{)}^{th}$ terms in the expansion of $(1+x{)}^n$ are in A.P., then:

The position vector of a point in which a line through the origin and perpendicular to the plane $2x-y-z=4,$ meets the plane $\overrightarrow{r}\cdot (3\hat{i}-5\hat{j}+2\hat{k})=6$ is

Prove that

I.Q. of a person is given by $I=\frac{M}{C}\times 100$, where $M$ is mental age and $C$ is chronological age. If $80\le I\le 140$ for a group of $12$ years old children, then the range of their mental age is

If $\left|\begin{array}{ccc}2a& x_1& y_1\\ 2b& x_2& y_2\\ 2c& x_3& y_3\end{array}\right|=\frac{abc}{2}\ne 0$,

then the area of the triangle whose vertices are $(\frac{x_1}{a},\frac{y_1}{a}),(\frac{x_2}{b},\frac{y_2}{b}),(\frac{x_3}{c},\frac{y_3}{c})$ is

Which of the following holds true in $(0,1)$ :

Let $A$ be a point on the line $\overrightarrow{r}=(1-3\mu )\hat{i}+(\mu -1)\hat{j}+(2+5\mu )\hat{k}$ and $B(3,2,6)$ be a point in the space. Then the value of $\mu$ for which the vector $\overrightarrow{AB}$ is parallel to the plane $x-4y+3z=1$ is :

The maximum value of the function f(x)=sin x + cos x is

If $\cos \alpha =\frac{2}{3},$ then the range of values of $\varphi$ on the ellipse
$x^2+4y^2=4$ falls inside the circle $x^2+y^2+4x+3=0$ is

a,b are positive integers. If 21ab² and 15ab are perfect squares, the minimum value of a + b is

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors, then the greatest value of $\sqrt{3}\left|\begin{array}{c}\overrightarrow{a}+\overrightarrow{b}\end{array}\right|+\left|\begin{array}{c}\overrightarrow{a}-\overrightarrow{b}\end{array}\right|$ is

Let $f:[0,1]\to R$ be such that $f\left(xy\right)=f\left(x\right)\cdot f\left(y\right)$, for all $x,y\in [0,1]$, and $f\left(0\right)\ne 0$. If $y=y\left(x\right)$ satisfies the differential equation, $\frac{dy}{dx}=f\left(x\right)$ with $y\left(0\right)=1,\text{then}y\left(\frac{1}{4}\right)+y\left(\frac{3}{4}\right)$ is equal to :

If A and B are mutually exclusive events such that P(A = 0.35 and P(B=0.45, find

(i) $P(A\cup B)$

(ii) $P(A\cap B)$

(iii) $P(A\cap \overline{B})$

(iv) $P(\overline{A}\cap \overline{B})$

Three coins are tossed once. Describe the following events associated with this random experiment :

A = Getting three heads,

B = Getting two heads and one tail,

C = Getting three tails,

D = Getting a head on the first coin.

(i) Which pairs of events are mutually exclusive?

(ii) Which events arc elementary events?

(iii) Which events are compound events?

Relation between Exradius ,semiperimeter and circumradius can be given by

($r_1$ is the radius of the circle opposite the angle A)

The image of point $P(1,-2,3)$ in the plane $2x+3y-4z+22=0$ measured parallel to the line $\frac{x}{1}=\frac{y}{4}=\frac{z}{5}$ is

If $\alpha ,\beta ,\gamma$ are roots of the equation $a{x}^3+b{x}^2+c=0$, the value of determinant $\left|\begin{array}{ccc}\alpha \beta & \beta \gamma & \gamma \alpha \\ \beta \gamma & \gamma \alpha & \alpha \beta \\ \gamma \alpha & \alpha \beta & \beta \gamma \end{array}\right|$ is

A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to that getting 9 heads, then the probability of getting 3 heads is