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If polynomial $P\left(x\right)=x^2+ax+b$ has factors $(x-a)\text{and}(x-b),\text{where}a,b\in R,$ then the value of $P\left(2\right)$ is

If in a $\Delta ABC,A\equiv (1,10),$ circumcentre $\equiv (-\frac{1}{3},\frac{2}{3})$ and orthocentre $\equiv (-\frac{11}{3},\frac{4}{3}),$ then the equation of side $BC$ is

The graph of $f\left(x\right)=-|\log (x-3)|+3$ will be

The altitude of a cone is 20 cm and its semi-vertical angle is ${30}^{\circ }$. If the semi-vertical angle is increasing at the rate of $2^{\circ }$ per second, then the radius of the base is increasing at the rate of

If a sequence is given by $9,12,15,18,\cdots$, then the value of ${16}^{th}$ term is

The equation of common tangent(s) to the hyperbola $9x^2-16y^2=144$ and circle $x^2+y^2=9$ is/are

The value of $x$ in the interval $[0,2\pi ]$ for which $4{\sin }^{2}x-8\sin x+3\le 0$ is

The square root of 3 - 4i is

$\underset{x\to \infty }{lim}\frac{\sqrt{x^2+1}-\sqrt[3]{3x^2+1}}{\sqrt[4]{4x^4+1}-\sqrt[5]{5x^4-1}}isequalto$

If $\mu$ is the mean of a distribution, then $\sum f_i(y_i-\mu )$ is equal to

Distance of the point (1, 2, 3) from the co-ordinate axes are

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

If PS is the median of the triangle with vertices P(2, 2), Q(6, -1) and R(7, 3) then equation of the line passing through (1, -1) and parallel to PS is

Let A, B and C are the angles of a plane triangle and $tan\frac{A}{2}=\frac{1}{3},tan\frac{B}{2}=\frac{2}{3}$.Then $tan\frac{C}{2}$ is equal to

The equation of the director circle of the circle $(x-2{)}^2+(y+3{)}^2=16$ is .

The area of the region bounded by the curve $y=\tan x$, the tangent to the curve at $x=\frac{\pi }{4}$ and the $x$-axis is

1 . The sum of the series 1 + 3x + 6$x^2$ + 10 $x^3$ + ....... ∞ will be

If a₁, a₂, a₃, . . . . . . , a_n . . . . . . .are in GP, then the value of the determinant

The length of minor axis (along $y$-axis) of an ellipse of the standard form is $\frac{4}{\sqrt{3}}$. If this ellipse touches the line $x+6y=8$, then its eccentricity is:

The coefficient of $a^8{b}^4{c}^9{d}^9$ in $(abc+abd+acd+bcd{)}^{10}$ is

Number of ways in which $10$ different diamonds can be arranged to make a necklace is

$$\begin{array}{cc}coloumn1& coloumn2\\ a& p)\frac{1}{x}\\ b& q)\frac{1}{x^2}\\ c& r)\frac{1}{x^3}\\ d& s)\frac{1}{x^4}\end{array}$$

Out of a pack of 52 cards one is lost, from the remainder of the pack , two cards are drawn and are found to be spades .Find the chance that the missing card is a spade ?

The range of the observations in 3, 8, 4, 9, 16, 19 is

If $x=\sum _{n=0}^{\infty }(-1{)}^n{\tan }^{2n}\theta$ and $y=\sum _{n=0}^{\infty }{\cos }^{2n}\theta ,$ where $0<\theta <\frac{\pi }{4},$ then:

Find the equation of radical axis of two circle $x^2+y^2-4x-2y=4$ and $x^2+y^2-12x-8y=12.$

Which of the following equation has exactly one root as $0$?

$(a,b,c>0)$

If $\sin (\theta +\alpha )=\cos (\theta +\alpha )$, then which of the following is/are correct?

where $\theta ,\alpha \in (0,\frac{\pi }{2})-\left\{\frac{\pi }{4}\right\}$

Let a and b be two unit vectors. If the vectors c=a+2b and d=5a-4b are perpendicular to each other, then the angle between a and b is

A hat contains a number of cards with $30\%$ white on both sides, $50\%$ black on one side and white on the other side, $20\%$ black on both sides. The cards are mixed up, and a single card is drawn at random and placed on the table. Its upper side shows up black. The probability that its other side is also back is

The number of ways in which ten candidates $A_1,A_2,A_3......A_{10}$ can be ranked if $A_1$ and $A_2$ are next to each other is

The value of the expression $\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\sin \frac{\pi }{30}$ is

$1+cos{56}^{\circ }+cos{58}^{\circ }-cos{66}^{\circ }=$

Twelve persons are to be arranged around two round tables such that one table can accommodate seven persons and another table can accommodate five persons only.



The total number of possible arrangements if two particular persons $A$ and $B$ do not want to be on the same table is

[IIT Screening 2001]

If $|z|=1$, then ${\left(\frac{1+z}{1+\overline{z}}\right)}^n+{\left(\frac{1+\overline{z}}{1+z}\right)}^n$ is equal to

If ${\log }_{10}\left[\frac{1}{2^x+x-1}\right]=x[{\log }_{10}5-1]$, then $x$ equals to

A rectangle with side lengths as $2m-1$ and $2n-1$ units is divided into squares of unit length by drawing parallel lines as shown in diagram, then the number of rectangles possible with odd side length is

Let $d\in R$, and

$A=\left[\begin{array}{ccc}-2& 4+d& \left(\sin \theta \right)-2\\ 1& \left(\sin \theta \right)+2& d\\ 5& \left(2\sin \theta \right)-d& (-\sin \theta )+2+2d\end{array}\right]$,

$\theta \in [0,2\pi ].$ If the minimum value of $det\left(A\right)$ is $8$, then a value of $d$ is:

The value of $\frac{\tan {53}^{\circ }}{\cot {37}^{\circ }}-\frac{\cot {40}^{\circ }}{\tan {50}^{\circ }}+2$ is

If $[\overrightarrow{a}\times \overrightarrow{b}\overrightarrow{b}\times \overrightarrow{c}\overrightarrow{c}\times \overrightarrow{a}]=\lambda {\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}^2$ then $\lambda$ is equal to

If $\alpha ,\beta$ are the root of a quadratic equation $x^2-3x+5=0$, then the equation whose roots are $({\alpha }^2-3\alpha +7)\text{and}({\beta }^2-3\beta +7)$ is

If $\sum _{k=1}^{n}k(k+1)(k-1)=p{n}^4+q{n}^3+t{n}^2+sn$, where $p,q,t,s$ are constant, then the correct option(s) is/are

If $|cos\theta +\{sin\theta +\sqrt{si{n}^2\theta +si{n}^2\alpha }\}|\le k$, then the value of k is