VITEEE 2023

The maximum slope of the curve \(y=\frac{1}{2} x^4-5 x^3+18 x^2-19 x+7\) occurs at the point

The area enclosed by \(y=x^3+1\) and \(y=x+2\) in first quadrant, is

The area of the region containing the points \((x, y)\) satisfying \(4 \leq x^2+y^2 \leq 2|x|+|y|\), is

Last three digits in \((9)^{50}\) be

If \(x^n=a_0+a_1(1+x)+a_2(1+x)^2+\ldots \ldots \ldots+ a_n(1+x)^n=b_0+b_1(1-x)+b_2(1-x)^2+\ldots . .+ b_n(1-x)^n\), then for \(n=201,\left(a_{101}, b_{101}\right)\) is equal to

The line \(a x+b y+c=0\) will be a tangent to the circle \(x^2+y^2=r^2\), then

If \(z_1, z_2\) and \(z_3\) are the vertices \(A, B\) and \(C\) respectively of an isosceles right angled triangle with right angled at \(C\), then \(\left(z_1-z_3^{\prime}\right)\left(z_2-z_3\right)\) equals to

If \(\alpha\) is a non -real fifth root of unity, then the value of \(3^{\left|1+\alpha+\alpha^2+\alpha^{-2}-\alpha^{-1 \mid}\right|}\), is

If \(\alpha, \beta\) and \(\gamma\) are the cube roots of \(P,(P<0)\), then for any \(x, y\) and \(z\) which does not make denominator zero, the expression \(\frac{x \alpha+y \beta+z \gamma}{x \beta+y \gamma+z \alpha}\) equals to

If \(x+\frac{1}{x}=1\) and \(p=x^{4000}+\frac{1}{x^{4000}}\) and \(q\) is the digit at unit place in the number \(2^{2 n}+1\), then the value of \((p+q)\) is equal to

Let \(z_k=\cos \left(\frac{2 k \pi}{10}\right)+i \sin \left(\frac{2 k \pi}{10}\right) ; k=1,2, \ldots \ldots \ldots\) 9, then \(\frac{1}{10}\left\{\left|1-z_1\right|\left|1-z_2\right| \ldots .\left|1-z_a\right|\right\}\) equals to

The value of \(\int_\lambda^{\lambda+\pi / 2}\left(\cos ^4 x+\sin ^4 x\right) d x\) is

\(\lim _\limits{n \rightarrow \infty}\left(\frac{(n+1)(n+2) \ldots 3 n}{n^{2 n}}\right)^{1 / n}\) is equal to

If \(m\) and \(n\) are order and degree of the question \(\left(\frac{d^2 y}{d x^2}\right)^4+8 \frac{\left(d^2 y / d x^2\right)^3}{\left(d^4 y / d x^4\right)^5}+\left(\frac{d^4 y}{d x^4}\right)=x^2+4\), then \(m-n\) is equal to

The solution of \(d y / d x=1+x+y+x y\) is

If \(x d y / d x=x^2+y-2, y(1)=1\), then \(y(2)\) is equal to

The solution of differential equation \(y y^{\prime}=x\left(\frac{y^2}{x^2}+\frac{f\left(y^2 / x^2\right)}{f^{\prime}\left(y^2 / x^2\right)}\right)\) is

If \(\left(\sin ^{-1} x\right)^2-\left(\cos ^{-1} x\right)^2=a \pi^2\), then the range of \(a\) is

If \(\cot ^{-1}(y)=\cot ^{-1}(x)+\cot ^{-1}\left(\frac{x^2-1}{2 x}\right)\), then the value of \(y\) is

\(\tan ^{-1}\left(\frac{1}{5}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{8}\right)\)
equals to

The equation \(3 \cos ^{-1} x-\pi x-\pi / 2=0\) has

For \(x \in R, f(x)=|\log 2-\sin x|\) and \(g(x)=f(f(x))\), then

If matrix $$A=\left[\begin{array}{ccc}0 & 2 b & -2 \\ 3 & 1 & 3 \\ 3 a & 3 & -1\end{array}\right]$$ is given to be symmetric, then the value of \(a b\) is

The determinant of the matrix $$\left[\begin{array}{ccc}1 & 4 & 8 \\ 1 & 9 & 27 \\ 1 & 16 & 64\end{array}\right]$$ is

Suppose, $$A=\left[\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right]$$ is an adjoint of the matrix $$\left[\begin{array}{rrr}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{array}\right]$$. The value of $$\fr...

If the 4th term in the expansion of \(\left(p x+\frac{1}{x}\right)^n, n \in N\) is \(\frac{5}{2}\) and three normals to the parabola \(y^2=x\) are drawn through a point \((q, 0)\), then

Out of 9 consonants and 4 vowels, how many words of 4 consonants and 3 vowels can be formed?

The sum of all the numbers of four different digits that can be made using the digits 0, 1, 2 and 3 is

Let \(A\) and \(B\) be two independent events such that the odds in favour of \(A\) and \(B\) are \(1: 1\) and \(3: 2\), respectively. Then, the probability that only one of the two occurs is

Let \(a_n\) be a sequence of numbers which is defined by relation \(a_1=2, \frac{a_n}{a_{n+1}}=3^{-n}\), then \(\log _2\left(a_{50}\right)\) is equal to (take \(\log _2 3=1.6\) )

The value of \(\frac{1}{2}\left(\frac{1}{5}\right)^2+\frac{2}{3}\left(\frac{1}{5}\right)^3+\frac{3}{4}\left(\frac{1}{5}\right)^4+\ldots . . \infty\) is

If \(R\) is a relation from a finite set A having \(m\) elements to finite set \(B\) having \(n\) elements, then the number of relation from \(A\) to \(B\) is

The image of the point \((2,3,7)\) in the plane \(2 x+5 y-3 z-19=0\), is

The distance between the point \((7,2,4)\) and the plane determined by the points \((2,5,-3),(-2,-3,5)\) and \((5,3,-3)\) is

The plane is perpendicular to the planes \(x-y+2 z-4=0\) and \(2 x-2 y+z=0\) and passes through \((1,-2,1)\) is

If \(\theta_1, \theta_2\) and \(\theta_3\) are the angles made by a line with the positive direction of \(X, Y\) and \(Z\)-axes, then \(\cos 2 \theta_1+\cos 2 \theta_2+\cos 2 \theta_3\) is equal to

The minium value of \(\left[2-\cos \theta+\sin ^2 \theta\right]\) is

If \(\sin A, \sin B\) and \(\cos A\) are in GP, then the roots of \(x^2+2 x \cot B+1=0\) are always

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

Let \(a, b\) and \(c\) be three unit vectors such that \(a \times(b \times c)=\frac{\sqrt{3}}{2}(b+c)\). If \(b\) is not parallel to \(c\), then the angle between \(a\) and \(b\) is