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The value of $(\cot {18}^{\circ }\cot {12}^{\circ }-\cot {45}^{\circ })({\sin }^2{15}^{\circ }-{\sin }^2{3}^{\circ })$ is

Find the modulus and argument of the following complex numbers and hence express each of them in the poloar form :

$\left(i\right)1+i\left(ii\right)\sqrt{3}+i\left(iii\right)1-i\left(iv\right)\frac{1-i}{1+i}\left(v\right)\frac{1}{1+i}\left(vi\right)\frac{1+2i}{1-3i}\left(vii\right)sin{120}^0-icos{120}^0\left(viii\right)\frac{-16}{1+i\sqrt{3}}$

Tangents are drawn from a point $P$ to the parabola $y^2=4ax$. If the chord of contact of the parabola is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then the locus of $P$ is

how many ap with 10 terms are there whose first term is in the set {1,2,3} and whose common difference is in the set {1,2,3,4,5}

The maximum distance between the points $(a\cos \alpha ,a\sin \alpha )$ and $(a\cos \beta ,a\sin \beta )$ is

Appropriately matching the information given in the three columns of the following table.

$\begin{array}{cccccc}& \text{Column 1}& & \text{Column 2}& & \text{Column 3}\\ \text{(I)}& \text{If}a,b,cϵR-\left\{0\right\}\text{such that}a\ne b\ne c& & & & \\ & \text{and}\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\text{and}A=\left[\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right],\text{then}& \left(i\right)& \text{A is singular matrix}& \text{(P)}& |adjA|=|A{|}^2\\ \text{(II)}& \text{If}\alpha ,\beta ,\gamma ϵR,\text{and}& & & & \\ & A=\left[\begin{array}{ccc}1& \cos (\alpha -\beta )& \cos (\alpha -\gamma )\\ \cos (\beta -\alpha )& 1& \cos (\beta -\gamma )\\ \cos (\gamma -\alpha )& \cos (\gamma -\beta )& 1\end{array}\right],\text{then}& \left(ii\right)& \text{A is singular matrix}& \left(Q\right)& adj\left(adjA\right)=|A|A\\ \left(III\right)& \text{If}\omega \ne 1\text{be cube root of unity and}& \left(iii\right)& \text{A is non-singular matrix}& \left(R\right)& |A|\text{is equal to}\\ & A=\left[\begin{array}{ccc}1+2{\omega }^{100}+{\omega }^{200}& {\omega }^2& 1\\ 1& 1+{\omega }^{101}+2{\omega }^{202}& \omega \\ \omega & {\omega }^2& 2+{\omega }^{100}+2{\omega }^{200}\end{array}\right]& & & & \text{is equal to minimum value of}\\ & ,then& & & & {\cos }^{-1}(x-\frac{1}{x})\\ & & & & & +{\cos }^{-1}\left(\frac{y^2}{y+1}\right)\\ & & & & & +{\cos }^{-1}(z^2+z+1)\\ & & & & & \text{(where}x,y,z\text{are real numbers)}\\ \text{(IV)}& \text{If}a,b,cϵR-\left\{0\right\}\text{such that}a\ne b\ne c,\text{and}& & & & \\ & A=\left[\begin{array}{ccc}0& (a-b{)}^3& (a-c{)}^3\\ (b-a{)}^3& 0& (b-c{)}^3\\ (c-a{)}^3& (c-b{)}^3& 0\end{array}\right],\text{then}& \left(iv\right)& \text{Invertible}& \left(S\right)& |A^{-1}|=\frac{1}{|A|}\end{array}$

Which of the following is only correct combination?

Find the principal values of the following questions:

$co{s}^{-1}\left(\frac{\sqrt{3}}{2}\right)$

The equation of the circle passing through three points $(1,0),$ $(-1,0)$ and $(0,1),$ is

Which of the following setences are statements? Give reasons for your answer.

(i) There are 35 days in a month.

(ii) Mathematics is difficult.

(iii) The sum of 5 and 7 is greater than 10.

(iv) The square of a number is an even number.

(v) The sides of a quadrilateral have equal length.

(vi) Answer this question.

(vii) The product of (-1) and 8 is 8.

(viii) The sum of all interior angles of a triangle is ${180}^{\circ }$.

(ix) Today is a windy day.

(x) All real numbers are complex numbers.

Common tangents are drawn to the parabola $y^2=4x$ and the ellipse $3x^2+8y^2=48$ touching the parabola $A$ and $B$ and the ellipse at $C$ and $D$. Area of quadrilateral $ABCD$ (in sq.units) is:

${\int }_0^{\pi }\frac{xtanx}{secx+tanx}dx.$

Show that the function defined by f(x) = cos $x^2$ is a continuous function.

Determine the number of 5-card combinations out of a deck of 52 cards if there is exactly one ace in each combination.

Find $\frac{dy}{dx}$in the following questions:

$y=ta{n}^{-1}\frac{3x-x^3}{1-3x^2},-\frac{1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}}.$

Draw the graph of {y} = $e^x$

Convert the vector rvector is equal to 3 ICAP + 2 J cap into the unit vector

The probability that A speaks is 4/5and probability for B is 3/4. the probability that they contradict each other when asked to speak on a fact is

7/20 . Aieee2004

Pls ans in detail ☺ pls put it on paper

The vertices of a triangle are $A(0,-6),$ $B(-6,0)$ and $C(1,1)$ respectively. Then coordinates of the excentre opposite to vertex $A$ are

Integration of (ax+b)^2

Domain of $f\left(x\right)=\frac{\sqrt{9-x^2}}{\sqrt{\left[x\right]+3}}$ is

(99)² is equivalent to

If $y\left(x\right)$ is the solution of the differential equation $(x+2)\frac{dy}{dx}=x^2+4x-9,x\ne 2$ and $y\left(0\right)=0$, then $y(-4)$ is equal to

One ticket is selected at random from 50 tickets numbered 00, 01, 02, ...., 49. Then the probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is zero, equals

The equation of the diameter of the circle $x^2+y^2+2x-4y-4=0$ which is parallel to $3x+5y-4=0$ is

${lim}_{x\to a}\frac{asinx-xsina}{a{x}^2-x{a}^2}$

In the following series, you will be looking at both the letter pattern and the number pattern. Complete the Series using appropriate option given below

निम्नलिखित श्रृंखला में, आप अक्षर पैटर्न और संख्या पैटर्न दोनों देख रहे होंगे। नीचे दिए गए उचित विकल्प का उपयोग करके श्रृंखला को पूरा करें

Q. FAG, GAF, HAI, IAH, ____

If ω is a complex cube root of unity, then the value of the

$Rangeofthefunctiony=\frac{2^x-2^{-x}}{2^x+2^{-x}}is$

The equivalent conductance of NaCl at concentration C and at infinite dilution are ${\lambda }_C$ and ${\lambda }_{\infty }$ respectively. The correct relationship between ${\lambda }_{C}and{\lambda }_{\infty }$ is given as (where, the constant B is positive)

The point equidistant from the points (a, 0, 0), (0, b, 0), (0, 0, c) and (0, 0, 0) is

The area bounded by the x-axis and the curve $y=4x–x^2–3$ is

$6(si{n}^6\theta +co{s}^6\theta )-9(si{n}^4\theta +co{s}^4\theta )$ is equal to

If ${\int }_0^{\frac{\pi }{2}}\frac{sinx}{1+sinx+cosx}dx=k$, then ${\int }_0^{\frac{\pi }{2}}\frac{dx}{1+sinx+cosx}$ is-

The number of words that can be made by arranging the letter of the word ROORKEE that neither begin with r nor end with e

If x + y + z = xyz, then tan^-1x + tan^-1y + tan^-1z =

The probability that a teacher will give an unannounced test during any class meeting is $\frac{1}{5}$. If a student is absent twice, then the probability that the student will miss atleast one test is

The acute angle between the line x + y = 3 and the line joining the points (1, 1) and (–3, 4) is

If $\alpha \in [\frac{\pi }{2},\pi ],$ then the value of $\sqrt{1+\sin \alpha }-\sqrt{1-\sin \alpha }$ is

$\text{The value of}{lim}_{x\to \infty }{\left(\frac{x^2-1}{x^2+1}\right)}^{x^2}is$

Two parabolas with a common vertex and with axes along $x-$ axis and $y-$ axis, respectively, intersect each other in the first quadrant. If the length of the latus rectum of each parabola is $3$, then the equation of the common tangent to the two parabolas is :