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Let $f:\left(0,1\right]\to R$ be a continuous function such that ${\int }_0^{\pi }f\left(\sin x\right)\mathrm{dx}=2018,\mathrm{then}{\int }_0^{\pi }xf\left(\sin x\right)\mathrm{dx}$ is equal to

A function $y=f\left(x\right)$ satisfies $x{f}^{\prime }\left(x\right)-2f\left(x\right)=x^4\left(f\right(x){)}^2$ for all $x>0$ and $f\left(1\right)=-6.$ Then the value of $f\left(2\right)$ is

The value of the expression $1-\frac{si{n}^{2}y}{1+cosy}+\frac{1+cosy}{siny}-\frac{siny}{1-cosy}$

If ${\sin }^4\alpha +4{\cos }^4\beta +2=4\sqrt{2}\sin \alpha \cos \beta$;

$\alpha ,\beta \in [0,\pi ]$, then $\cos (\alpha +\beta )-\cos (\alpha -\beta )$ is equal to :

$li{m}_{x\to \frac{\pi }{4}}\frac{1-co{t}^{3}x}{2-cotx-co{t}^{3}x}=$___

If $\cos \theta -\sin \theta =\frac{1}{5}$, where $0<\theta <\frac{\pi }{2}$



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(1)}& \frac{(\cos \theta +\sin \theta )}{2}& \left(p\right)& \frac{4}{5}\\ \text{(2)}& \sin 2\theta & \left(q\right)& \frac{7}{10}\\ \text{(3)}& \cos 2\theta & \left(r\right)& \frac{24}{25}\\ \text{(4)}& \cos \theta & \left(s\right)& \frac{7}{25}\end{array}$



Which of the following is the correct combination?

For a set $K,\text{if}n\left(K\right)=m;$ then the number of proper subsets of $K=$

The equation of the lines represented by $4x^2+24xy+11y^2=0$ is/are

Value of $\lambda$ for which the function $f\left(x\right)=2x^3-3(\lambda +2)x^2+12\lambda x$ has one local maxima and one local minima in $R$, can not be

The points of intersection of the curves whose parametric equations are $x=t^2+1,y=2tandx=2s,y=\frac{2}{s}$ Is given by

If $lo{g}_{7}2=m,thenlo{g}_{49}28$ is equal to

Let $x_1,x_2$ be the roots of $a{x}^2+bx+c=0$ and $x_1.x_2<0$, $x_1+x_2$ is non - zero. Roots of $x_1(x-x_2{)}^2+x_2(x-x_1{)}^2=0$ are

The lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ and $\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$ are coplanar if

If f : D $\to$R $f\left(x\right)=\frac{x^2+bx+c}{x^2+b_{1}x+c_1},where\alpha ,\beta$ are th roots of the equation $x^2+bx+c=0$ and ${\alpha }_1,{\beta }_1$ are the roots of $x^2+b_{1}x+c_1=0$. Now, answer the following question for f(x). A combination of graphical and analytical approach may be helpful in solving these problems. If ${\alpha }_{1}and{\beta }_1$ are real, then f(x) has vertical asymptote at $x=({\alpha }_1,{\beta }_1$). Then if ${\alpha }_1<\alpha <{\beta }_1<\beta$, then

Kanchan has $10$ friends among whom two are married to each other. She wishes to invite five of them for a party. If the married couples refuse to attend separately, then the number of different ways in which she can invite five friends is

If the coefficient of the middle term in the expansion of $(1+x{)}^{2n+2}$ is p and the coefficients of middle terms in the expansion of $(1+x{)}^{2n+1}$ are q and r, then

The symmetric form of the equation of the line x + y – z = 1, 2x – 3y + z = 2 is

If $\omega$ is a complex cube root of unity, then the equation whose roots are $2\omega$ and $2{\omega }^2$ is

If tan(A - B)=1, sec (A + B)= $\frac{2}{\sqrt{3}}$, then the smallest positive value of B is

A line L passes through the points (1, 1) and (2, 0) and another line L’ passes through $[\frac{1}{2},0]$ and perpendicular to L. Then the area of the triangle formed by the lines L, L’ and y –axis, is

Answer the following by appropriately matching the lists based on the information in Column I and Column II​​​​​​

$\begin{array}{cc}ColumnI& ColumnII\\ \text{a}.y=f\left(x\right)\text{is given by}& \\ x=t^5-5t^3-20t+7& \\ \text{and}y=4t^3-3t^2-18t+3.& \\ \text{Then}-5\times \frac{dy}{dx}\text{at}t=1& \text{p.}0\\ \text{b}.\text{Let}P\left(x\right)\text{be a polynomial of degree}4,& \\ \text{with}P\left(2\right)=-1,P^{\prime }\left(2\right)=0,& \\ P^{\prime \prime }\left(2\right)=2,P^{\prime \prime \prime }\left(2\right)=-12\text{and}& \\ P^{iv}\left(2\right)=24,\text{then}{P}^{\prime \prime }\left(3\right)\text{is}& \text{q.}-2\\ \text{c}.y=\frac{1}{x},\text{then}\frac{\frac{dy}{\sqrt{1+y^4}}}{\frac{dx}{\sqrt{1+x^4}}}& r.2\\ \text{d}.f\left(\frac{2x+3y}{5}\right)=\frac{2f\left(x\right)+3f\left(y\right)}{5}\text{and}& \\ f^{\prime }\left(0\right)=p\text{and}f\left(0\right)=q.\text{Then},f^{\prime \prime }\left(0\right)\text{is}& \text{s.}-1\end{array}$

The incentre of the triangle formed by (0, 0), (5,12), (16, 12) is

Which of the following is/are true, For $f\left(x\right)=\frac{ln\left(lnx\right)}{lnx}$

If $\alpha ,\beta$ are the roots of the equation $\tan x+\sec x=2\cos x$ where $x\in [0,2\pi )$, then the value of $|\alpha -\beta |$ is

If $A,$ $B$ and $C$ are the angles of a non-right angled triangle $ABC,$ then the value of $\left|\begin{array}{ccc}\tan A& 1& 1\\ 1& \tan B& 1\\ 1& 1& \tan C\end{array}\right|$ is equal to

The domain and range of the function ${\text{cosec}}^{-1}\sqrt{{\log }_{\left(\frac{3-4\sec x}{1-2\sec x}\right)}2}$ are respectively

If a ray of light passing through $(2,2)$ reflects on the $x-$axis at a point $P$ and the reflected ray passes through the point $(6,5)$, then the co-ordinates $P$ is

If a focal chord to $y^2=16x$ is tangent to $(x-6{)}^2+y^2=2$, then the possible value(s) of the slope of this chord is/are

The maximum number of points of intersection of five lines and four circles in a plane is

If $\alpha +\beta =\frac{\pi }{2}$ and $\beta +\gamma =\alpha ,$ then tan $\alpha$ equal to

The coefficient of $x^4$ in the expansion of $(x^2-x-2{)}^6$ is

If sum of infinite terms of a G.P. is 3 and sum of squares of its terms is 3, then its first term and common ratio are

[RPET 1999]

If $x_1$ and $x_2$ are the roots of the equation $e^2{x}^{\ln x}=x^3$ with $x_1>x_2$, then

A variable circle whose centre lies on $y^2-36=0$ cuts rectangular hyperbola $xy=16$ at $(4t_i,\frac{4}{t_i}),i=1,2,3,4$ then $\sum _{i=1}^4\frac{1}{t_i}$ can be

An experiment is called random experiment if

The product of the lengths of the perpendiculars from any point on the hyperbola $x^2-2y^2=2$ to its asymptotes is

The value of $\frac{\sin {70}^{\circ }-\cos {40}^{\circ }}{\cos {70}^{\circ }-\sin {40}^{\circ }}$ is equal to



$\frac{\sin {70}^{\circ }-\cos {40}^{\circ }}{\cos {70}^{\circ }-\sin {40}^{\circ }}$ का मान बराबर है

An artillery target may be either at point I with probability $\frac{8}{9}$ or at point II with probability $\frac{1}{9}$. We have 55 shells, each of which can be fired either at point I or II. Each shell may hit the target, independent of the other shells, with probability $\frac{1}{2}$. Maximum number of shells that must be fired at point I to have maximum probability is

If (a + 3b)(3a + b) = $4h^2$, then the angle between the lines represented by $a{x}^2+2hxy+b{y}^2=0$ is

If $y=f\left(x\right)$ is quadratic polynomial having vertex at $(6,8),$ as shown in the figure below, then $f\left(x\right)$ is