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Find the area of triangle whose vectors are (1,1,1), (2,1,3)and (0,0,1) using vectors .

The component vector of $\overline{b}$ Perpendicular to $\overline{a}$ is

$Thedomainofdefinitionofthefunction,f\left(x\right)givenbytheequation{2}^x+2^y=2is$

If $\omega$ is the imaginary cube root of unity, then find the member of ordered pairs of integers (a, b) such that $|a\omega +b|=1.$___

If $n$ be a positive integer,and $(7+4\sqrt{3}{)}^n=p+\beta$ where $p$ is a positive integer and $\beta$ is a proper fraction, then value of $(1-\beta )(p+\beta )$ is

Find the value of $\underset{x\to 0}{lim}\frac{a^x-1}{x}$

Suppose $z_1,z_2,z_3$ are the vertices of an equilateral triangle inscribed in the circle $|z|=2$. If $z_1=1+i\sqrt{3}$. then values of $z_{3}and{z}_2$ are respectively

If in the expansion of $(1+x{)}^n$, the coefficients of pth and qth terms are equal, prove that p+q =n+2, where $p\ne q.$

${\sum }_{r=1}^n\left({\sum }_{k=0}^{r-1}{}^n{C}_r{}^r{C}_k{2}^k\right)$ is equal to

If the vertices of a hyperbola be at $(-2,0)$ and $(2,0)$ and one of its foci be at $(-3,0),$ then which one of the following points does not lie on this hyperbola:

If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation,
$x^2+(2-\lambda )x+(10-\lambda )=0$ is minimum, then the magnitude of the difference of the roots of this equation is :

Find $\int \frac{cos\theta }{(4+si{n}^2\theta )(5-4co{s}^2\theta )}d\theta .$

The normal line to a given curve at each point $(x,y)$ on the curve passes through the point $(3,0)$. If the curve contains the point $(3,4)$, then the equation of the curve is

The least value of n for which $(n-2{)}^2+8x+n+4>si{n}^{-1}(sin12)+co{s}^{-1}(cos12)\forall xϵR$ and $nϵN$ is

Number of real solutions of $2^{|sinx|}=|cosx|$ in $[0,2\pi ]$ is ____________

Q67. Find the missing letter in the given figure series:

. निम्नलिखित आकृति श्रृंखला में लुप्त अक्षर पता कीजिएः

The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms ?

The equation of the circle having one of its diameter as end points of foci of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$, is

PQ. Show that the two formulae for the standard deviation of ungrouped data $\delta =\sqrt{\frac{1}{n}\sum (x_i-\overline{X}{)}^2}$ and $\delta =\sqrt{\frac{1}{n}\sum (x_i-\overline{X}{)}^2}$ are equivalent, wehere $\overline{X}=\frac{1}{n}\sum x_i$.

If we convert the denominator of the integral into a perfect square, $\int \frac{1}{x^2-x+1}dx$ then the correct integral will be

Negation of the compound statement ‘if the examination is difficult, then I shall pass if I study hard’ is ___.

A line meets the co-ordinate axes in A & B, A circle is circumscribed about the triangle OAB. If $d_{1}and{d}_2$ are the distances of the tangent to the circle at the origin O from the points A and B respectively the diameter of the circle is:

If ‘m’ and ‘n’ are the order and degree of the differential equation $(y^{\prime \prime }{)}^5+\frac{4(y^{\prime \prime }{)}^3}{y^{\prime \prime \prime }}+y^{\prime \prime \prime }=sinx$, then

Point charges of $+50\mu C,-250\mu C,+200\mu C$ are placed on the circumference of a circle of radius 0.5m to form the vertices of an equilateral triangle.The electric potential at the center of the circle is

What is probability that you spell " ERADICATE" correctly when you hit keyboard alphabet keys randomly?

Let $f\left(x\right)=\frac{x}{1+x^2}$. Then range of $f$ is

If a point in argand plane $A(2,3)$ rotated through origin about $\frac{\pi }{4}$ in anticlockwise, then new coordinates of the point will be

The acute angles between the curves $y=|x^2-3|$ at their points of intersection is

What is the rate of change of volume of a cube with respect to an edge when the diagonal of the cube is $6\sqrt{3}$ ?

The area bounded by the curve $=sin\left(\frac{x}{3}\right)$, x-axis and lines x = 0 and x = 3$\pi$ is

The probability of a bomb hitting a bridge is $\frac{1}{2}$ and two direct hits are needed to destroy it. The least number of bombs required so that the probability of the bridge being destroyed is greater then 0.9, is

State whether the following are true or false. Give reasons:

(a) At zero level of income, savings are positive.

(b) The relationship between multiplier and MPC is inverse and that with MPS is direct.

(c) The value of APC varies from 1 to infinity.

(d) The value of MPS can be negative.

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

The set of all real numbers $x$ for which $x^2-|x+2|+x>0$, is

What is the necessary condition for a matrix A to be invertible?

If $A,B,C$ are the angles of triangle such that $0<A\le \frac{\pi }{3}$, then the range of $\frac{\tan B+\tan C}{\tan B\tan C-1}$ is

In how many ways can the letters of the word 'FAILURE' be arranged so that the consonants may occupy only odd positions?

The principal value of ${\sin }^{-1}\left(\cos \frac{5\pi }{4}\right)$ is

Write the negation of the following statements:

(i) p = For every positive real number x, the number (x - 1) is also positive.

(ii) q : For every real number x, either x > 1 or x< 1.

(iii) r : There exists a number x such that 0 < x < 1.

Write the solution set of the inequation $\frac{x^2}{x-2}>0.$

A coin is tossed three times and the teomes are recorded, How many possible, twines are there ? How many possible outcomes if the coin is tossed four times ? Five times ? n times ?