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If $\int e^{\sin x}\left[\frac{x{\cos }^{3}x-\sin x}{{\cos }^{2}x}\right]dx=e^{\sin x}f\left(x\right)+c$ where c is constant of integration, then $f\left(x\right)=$

Find the inverse of the given matrix
$\left[\begin{array}{ccc}2& 1& 3\\ 4& -1& 0\\ -7& 2& 1\end{array}\right]$

Let $f:R\to$ be defined as f(x) =10x +7. Find the function $g:R\to R$ such that gof =fog =$I_R$.

The equation $|\sqrt{(x-2{)}^2+(y-1{)}^2}-\sqrt{(x+2{)}^2+y^2}|=c$ will represent a hyperbola if

Q is a point on the auxiliary circle of an ellipse. P is the corresponding point on ellipse. N is the foot of perpendicular from focus S, to the tangent of auxillary circle at Q. Then

$1^2+3^2+5^2+7^2+9^2$ =___.

If the range of f(x) is $(-(\infty ,\infty )$, then the range of g(x) = $\frac{1}{\sqrt{f\left(x\right)}}$ is

What is the value of sin 18 degree.

If in a railway carriage $5$ persons can sit each side then number of ways in which a party of $4$ girls and $6$ boys can seat themselves so that the girls will always occupy the corner seats is

The angle between planes $2x+7y+11z-3=0$ & $5x+3y+9z+1=0$ is .

If three positive real numbers a, b, c are in A.P., with abc=4, then the minimum value of b is

If n(A) = 3 and n(B) =4 then find the number of elements in the power set of A × B

A unit vector perpendicular to the plane of $a=2\hat{i}-6\hat{j}-3\hat{k},b=4\hat{i}+3\hat{j}-\hat{k}$ is

The auxiliary circle of $\frac{x^2}{9}+\frac{y^2}{4}=1$ touches a parabola having axis of symmetry as $y-$axis at its vertex . If the latus rectum of parabola is equal to radius of director circle of auxiliary circle, then the equation of parabola can be

Consider the parabola $(x-1{)}^2+(y-2{)}^2=\frac{(12x-5y+3{)}^2}{169}$
$\begin{array}{cc}ColumnI& ColumnII\\ \text{a. Locus of point of intersection of perpendicular tangent}& \text{p.}(12x-5y-2=0)\\ \text{b. Locus of foot of perpendicular from focus upon any tangent}& \text{q.}(5x+12y-29=0)\\ \text{c. Line along which minimum length of focal chord occurs}& \text{r.}(12x-5y+3=0)\\ \text{d. Line about which parabola is symmetrical}& \text{s.}(24x-10y+1=0)\end{array}$

If $5^{th}$ and $6^{th}$ term of an $A.P.$ are $6$ and $5$ respectively, then the ${11}^{th}$ term is

If x, y, z are real and distinct, then $x^2+4y^2+9z^2-6yz-3zx-2xy$

Curve $16x^2+8xy+y^2-74x-78y+212=0$ represents

The equation of the chord of the circle $x^2+y^2=r^2$ passing through $(2,3)$ and farthest from the centre is

Two numbers x and y are chosen at random (without replacement) from amongst the numbers 1, 2, 3,……, 2004. The probability that $(x^3+y^3)$ is divisible by 3 is $\frac{p}{q}$ (p and q are co-prime). Find the value of (p + q).___

The number of numbers from 1 to 10 ¹⁰⁰ having the sum of their digits equal to 3 is

The expression $3[{\sin }^4(\frac{3\pi }{2}-\alpha )+{\sin }^4(3\pi +\alpha \left)\right]-2[{\sin }^6(\frac{\pi }{2}+\alpha )+{\sin }^6(5\pi +\alpha \left)\right]$ is equal to

Consider the lines represented by the equation $(x^2+xy-x)(x-y)=0$, forming a triangle, then
$\begin{array}{cc}Column1& Column2\\ \left(a\right)Orthocentreoftraingle& (\frac{1}{6},\frac{1}{2})\\ \left(b\right)Circumcenter& \left(\frac{1}{2+2\sqrt{2}}\right)\\ \left(c\right)Centroid& (0,\frac{1}{2})\\ \left(d\right)Incenter& (\frac{1}{2},\frac{1}{2})\end{array}$

Suppose $p,q,r$ are real numbers such that $q=p(4-p),r=q(4-q),p=r(4-r).$ The maximum possible value of $p+q+r$ is

Find the derivative of the following functions at the indicated points :

$\left(i\right)sinx\text{at}x=\frac{\pi }{2}$

$\left(ii\right)x\text{at}x=1$

$\left(iii\right)2cosx\text{at}x=\frac{\pi }{2}$

$\left(iv\right)sin2x\text{at}x=\frac{\pi }{2}$

y=f(x) be a real valued twice differentiable function defined on R, then $\frac{d^{2}y}{d{x}^2}{\left(\frac{dx}{dy}\right)}^3+\frac{d^{2}x}{d{y}^2}$ =

If $y=\sqrt{sinx+\sqrt{sinx+\sqrt{sinx+....\infty }}}$,then $\frac{dy}{dx}$ is equal to

If $\omega$ $(\ne 1)$ is a cube root of unity, and $(1+\omega {)}^7$ = A + B $\omega$, then (A, B) is equals:

The logical statement $[\sim (\sim p\vee q)\vee (p\wedge r\left)\right]\wedge (\sim q\wedge r)$ is equivalent to :

Let $f:[0,\infty ]\to R$ and $g:R\to R$ be defined by $f\left(x\right)=\sqrt{x}$ and g(x) = x. Find f + g, f - g, fg and $\frac{f}{g}$

If $(\lambda -2)x^2+(\lambda -1)x+3<0,\forall x\in R,$ then the range of $\lambda$ is

The value of $6+{\log }_{\frac{3}{2}}\left(\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\dots \dots }}}\right)$ is

The given truth table is:

$\begin{array}{ccc}A& B& Y\\ 0& 0& 0\\ 1& 0& 1\\ 0& 1& 1\\ 1& 1& 0\end{array}$

A die is thrown repeatedly until a sixcomes up. What is the sample space for this experiment.

The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1, find the first term and the common difference.

Let M be a 2 $\times$ 2 symmetric matrix with integer entries. Then M is invertible if

Point of intersection of the lines $\sqrt{3}x-y-4\sqrt{3}m=0$ and $\sqrt{3}mx+my-4\sqrt{3}=0$ describes-

The number of integers in the range of ‘c’ such that there exists a line which intersects the curve $y=x^4–6x^3+12x^2+cx+1$ at four distinct points is equal to ___

If x cos(a+y) = cos y, then prove that $\frac{dy}{dx}=\frac{co{s}^2(a+y)}{sina}$

Hence show that sin a $\frac{d^{2}y}{d{x}^2}+sin2(a+y)\frac{dy}{dx}=0$

OR Find $\frac{dy}{dx}$ if $y=si{n}^{-1}\left[\begin{array}{c}\frac{6x-4\sqrt{1-4x^2}}{5}\end{array}\right].$

Show that the relation R in the set R of real numbers, defined as $R=\left\{\right(a,b):a\le b^2\}$ is neither reflexive nor symmetric nor transitive.

Here, the result is disproved using some speicific examples. In order to prove a result. we must prove it in generlity and in order to disprove a result we can just provide one example. where the condition is false. It is important to pick up the examles suitably. Since there are certain ordered pairs like (1,1) for which the relation is reflexive.

The ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and hyperbola $\frac{x^2}{A^2}-\frac{y^2}{B^2}=1$ are having a same foci and length of minor axis of ellipse is same as the conjugate axis of the hyperbola. If $e_1$ & $e_2$ are the eccentricities of ellipse and hyperbola respectively, then the value of $\frac{1}{e_1^2}+\frac{1}{e_2^2}$ is

The equation of circle passing through the points $(4,1),(6,5)$ and having centre on the line $4x+y=16$ is

If the mean deviation of the numbers $1,1+d,1+2d,\dots ,1+100d$ from their mean is $255$, then $|d|$ is equal to