VITEEE 2022

The maximum slope of the curve \(y=\frac{1}{2} x^4-5 x^3+18 x^2-19 x\) occurs at the point

\(f\) and \(g\) are differentiable function in \((0,1)\) satisfying \(f(0)=2=g(\mathrm{l}), g(0)=0\) and \(f(l)=6\), then for some \(c \in] 0,1[\)

Find the area enclosed by \(y=x^2\) and \(y=x+2\)

The area between \(y=x^2\) and \(y=8-x^2\)

If the 2nd, 3rd and 4th terms in the expansion of \((a+b)^n\) be \(240,720\) and 1080 respectively, then the value of \((n, b, a)\) is

If a circle of constant radius '\(r\)' passes through the origin and meets the coordinate axes at points \(A\) and \(B\) respectively, then the locus of the centroid of triangle \(O A B\), '\(O\)' being the origin, is

If the tangent at point \(P\) on the circle \(x^2+y^2+6 x+6 y-2=0\) meets the straight line \(5 x-2 y+6=0\) at a point \(Q\) on \(Y\)-axis, the length of \(P Q\) is

The image of the centre of the circle \(x^2+y^2=a^2\) with respect to the mirror \(x+y=1\) is

The condition in order that \(Z_1, Z_2, Z_3\) are vertices of an isosceles triangle right angled at \(z_2\), is

If \(X \phi(x)=\int_\limits5^x 3 t^2-2 \phi(t) d t, x>-2\) and \(\phi(0)=4\), then \(\phi(2)\) is

The solution of \(\frac{d y}{d x}=1+x+y+x y\) is

If \(p\) and \(q\) are order and degree of the question \(\left(\frac{d^2 y}{d x^2}\right)^4+4 \frac{\left(\frac{d^2 y}{d x^2}\right)^2}{\left(\frac{d^3 y}{d x^3}\right)^3}+\frac{d^3 y}{d x^3}=x^2-1\), then

The solution of the equation \(\frac{d y}{d x}+x(x+y)=x^3(x+y)^3-1\) is

Find the solution of \(\frac{d y}{d x}=\frac{1}{\cos (x-y)}\)

Find the solution of equation \(\frac{d y}{d x}=\frac{1}{\cos (x+y)}\)

If a man running around a race-course notes that the sum of the distances of two flag-posts from him is always \(10 \mathrm{~m}\) and the distance between the flag-posts is \(8 \mathrm{~m}\), then the area of the path he encloses in square ...

The domain of
\(f(x)=\sqrt{\log \frac{1}{4}\left(\frac{5 x-x^2}{4}\right)}+{ }^{10} C_x\) is

Through a fixed point \(P(\alpha, \beta), a\), variable line is drawn to cut the coordinate axes at \(A\) and \(B\). The locus of the mid-point of \(A B\) is

The integral \(\int \frac{d x}{x^2\left(x^4+1\right)^{3 / 4}}\) equals

If \(\tan ^{-1} y=\tan ^{-1} x+\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)\), where \(|x|< \frac{1}{\sqrt{3}}\) then value of \(y\) is

If \(\cos ^{-1} x-\cos ^{-1} \frac{y}{2}=\alpha\) where \(-1 \leq x \leq 1, -2 \leq y \leq 2, x \leq \frac{y}{2}\), then for all \(x, y, 4 x^2-4 x y \cos \alpha+y^2\) is equal to

The value of \(f(x) = \mathop {\lim }\limits_{x \to 2} {{{x^3} - 3{x^2} + 4} \over {{x^4} - 7x - 2}}\)

The value of
\(\lim _\limits{x \rightarrow \infty}\left[\frac{p^{1 / x}+q^{1 / x}+r^{1 / x}+s^{1 / x}}{4}\right]^{3 x}, p, q, r, s>0 \text {, }\) is

The value of \(\lim _\limits{t \rightarrow \infty} \frac{\ln \left(\frac{3}{2} t\right)}{t^2}\)

If $$f(x)=\left\{\begin{array}{cc}(\sin x+\cos x)^{\operatorname{cosec} x} & ,-\frac{\pi}{2}< x<0 \\ a & ,x=0 \\ \frac{e^{1 / x}+e^{2 / x}+e^{3 / x}}{a e^{-2+\frac{1}{x}}+b e^{-1+\frac{3}{x}}} & , 0< x<\frac{\pi}{2}\end{array}\right.$$ is c...

For all values of \(\lambda\), rank of matrix
$$A=\left[\begin{array}{ccc} { }^h C_0 & { }^4 C_3 & { }^5 C_4 \\ \lambda & 8 & 8 \lambda-6 \\ 1+\lambda^2 & 8 \lambda+4 & 2 \lambda+21 \end{array}\right]$$

If the tangent at \(P\) on \(y^2=4 a x\) meets the tangent at the vertex in \(Q\) and \(S\) is the focus of the parabola, then \(\angle S Q P\) is equal to

An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that at the most three of them are red is

There are 50 intermediate stations on a railway line from one terminus to another. The number of ways a train can stop at 3 of these intermediate stations if no two of these stopping stations are to be consecutive, are

Two dices are rolled. If both dices have six faces numbered \(1,2,3,5,7\) and \(11\) then the probability that the sum of the number on the top faces is less than or equal to 8 is

The value of \(\frac{4}{1 !}+\frac{11}{2 !}+\frac{22}{3 !}+\frac{37}{4 !}+\frac{56}{5 !}+\ldots \infty\) is

If the real numbers \(x, y, z, t\) be in GP then the value of \((x^2+y^2+z^2)(y^2+z^2+t^2)\) is euqal to

A ray of light is sent along the line \(x-2 y+5=0\). Upon reaching the line \(3 x-2 y+7=0\), the ray is reflected from it. The equation of the line containing the reflected ray, is

A force of magnitude \(\sqrt{6}\) acting along, the line joining points \(A(2,-1,1)\) and \(B(3,1,2)\) displaces a particle from \(A\) to \(B\). The work done by the force is

If a variable plane cuts the coordinate axes in \(A, B, C\) and is at constant distance $p$ from the origin, then the locus of the centroid of the tetrahedron \(A B C\) is equal to

The image of the point \((2,-3,4)\) with respect to the plane \(4 x+2 y-4 z+3=0\), is

The roots of the equation \(\cos x+\sqrt{3} \sin x=2 \cos 2 x\), are

If \(0< a<5,0< b<5\) and \(\frac{x^2+5}{2}=x-2[\cos (a+b x)]\) is satisfied for atleast one real \(x\), then the least value of \(\frac{a+b}{\pi}\) is equal to

\(\cos (x+y), \cos x, \cos (x-y)\) are in HP, then \(\cos x \sec \frac{y}{2}\) is

A unit vector perpendicular to both the vectors \(\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\hat{\mathbf{j}}+\hat{\mathbf{k}}\) is