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If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three non coplanar vectors and a vector $\overrightarrow{r}=\frac{(\overrightarrow{r}.\overrightarrow{a})(\overrightarrow{b}\times \overrightarrow{c})+(\overrightarrow{r}.\overrightarrow{b})(\overrightarrow{c}\times \overrightarrow{a})+(\overrightarrow{r}.\overrightarrow{c})(\overrightarrow{a}\times \overrightarrow{b})}{\lambda },$ then the value of $\lambda$ is:

The range of values of $x$ for which the inequality $\frac{3x-2}{5x-3}\ge 4$ is satisfied is given by

$2.\stackrel{\cdots }{357}$=

If the sum of the binomial coefficients of the expansion ${(2x+\frac{1}{x})}^n$ is equal to 256, then the term independent of x is

The value of $\underset{x\to 0}{lim}\frac{{\log }_e(1+x)-x}{x^2}=$

The absolute difference between the greatest and the least possible values of the expression $3-\cos x+{\sin }^{2}x$ is

If a curve $y=f\left(x\right)$, passing through the point $(1,2)$ is the solution of the differential equation, $2x^{2}dy=(2xy+y^2)dx$, then $f\left(\frac{1}{2}\right)$ is equal to:

If θ
is the angle between two vectors
and
,
then
only
when

(A) (B)

(C) (D)

The number of ways in which we can get a sum of $11$ by throwing three dice is :

If in a $△ABC$, $\tan \frac{A}{2},\tan \frac{B}{2},\tan \frac{C}{2}$ are in H.P., then the minimum value of $\cot \frac{B}{2}$ is

If ${1690}^{2608}+{2608}^{1690}$ is divided by $7$, then the remainder is

If the real valued function $f\left(x\right)=\frac{a^x-1}{x^n(a^x+1)}$ is even, then $n$ is equal to

Number of goals scored by Ronaldo in champions league for past 10 years are as follows. {1, 3, 8, 4, 7, 6, 10, 12, 17, 10} Find the mean deviation about the mean.

In throwing a pair of dice, consider two events :

$E_1:$ coming up of $4$ on first dice.

$E_2:$ coming up of $5$ on second dice.

Which of the following(s) is/are true?

A line divides a plane into 2 regions. Two lines divide the plane into maximum 4 regions. If $L_n$ is the maximum number of regions divided by $n$ lines then the following is/are true?

For a parabola, if $L_1:x=y+1$ is the axis of symmetry, $L_2:x+y=5$ is tangent at vertex and $L_3:y=4$ is a tangent at a point $P,$ then the equation of circumcircle of the triangle formed by the tangent and normal at point $P$ and axis of parabola is

A variable straight line through $A(-1,-1)$ is drawn to cut the circle $x^2+y^2=1$ at the points $B,C.$ If $P$ is chosen on the line $ABC$ such that $AB,AP,AC$ are in $A.P$ then the locus of $P$ is

Let $k$ be the non-zero real number such that the quadratic equation $k{x}^2+2x+k=0$ has two distinct real roots $\alpha \text{and}\beta (\alpha <\beta ).$

If $\alpha <2$ and $\beta >5$, then

If two normals on the parabola $(y-2{)}^2=-12(x+3)$ intersect each other at right angle then the chord joining their feet passes through a fixed point whose coordinate are

If 1, w, $w^2$ are three cube roots of unity, then $(1-w+w^2)(1+w-w^2)$ is .............

${lim}_{x\to 0}\frac{2x}{\sqrt{a+x}-\sqrt{a-x}}$

The standard deviation of 17 numbers is zero. Then

Prove that the greatest integer function defined by f(x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.

$\frac{2x+3}{4}-3<\frac{x-4}{3}-2$

Consider the parabola $y=ax–b{x}^2$. If the least positive value of a for which there exist $\alpha ,\alpha ϵR–\left\{0\right\}$ such that both the point $(\alpha ,\beta )$ and $(\beta ,\alpha )$ lies on the given parabola is k then [k] is equal to ___