VITEEE 2021

The interval in which the function \(f(x)=\sin x-\cos x, 0 \leq x \leq 2 \pi\) is strictly decreasing, is

The slope of normal to the curve \(y=x^3+2 x+6\) which is parallel to line \(x+14 y+4=0\) is

The area bounded by the circle \(x^2+y^2=16\) and the line \(y=x\) in the first quadrant is

Which of the following is the correct principle of Mathematical induction?

The coefficient of the term independent of \(x\) in the expansion \(\left(\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right)^{10}\) is

The radius of the circle \((x \cos \theta+y \sin \theta-a)^2+(x \sin \theta-y \cos \theta-b)^2=k^2\) is

If \((1+i)(2 i+1)(1+3 i) \ldots(1+n i)=x+i y\), then \(2 \cdot 5 \cdot 10 \ldots\left(1+n^2\right)\) is equal to

The non-zero solutions of the equation \(z^2+|z|=0\), where \(z\) is a complex number, are

The value of \(\int\limits_0^{\pi / 2} \frac{d x}{1+\tan x}\) is

The particular solution of the differential equation \(\frac{d y}{d x}+y \cot x=2 x+x^2 \cot x\), such that \(y(\pi / 2)=0\) is

The general solution of the linear differential equation \(\frac{d y}{d x}+\sec x \cdot y=\tan x\left(0 \leq x \leq \frac{\pi}{2}\right)\) is

On solving the differential equation \(x^2 y d x-\left(x^3+y^3\right) d y=0\), the value of \(\log y\) is

The value of \(\frac{d}{d x}\left(x^n \log _a x e^x\right)\) is

If \(y=1+\frac{x}{1 !}+\frac{x^2}{2 !}+\frac{x^3}{3 !} \ldots\), then \(\frac{d y}{d x}\) is equal to

The focal distance of the point \((x, y)\) from the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b\) is

The eccentricity of the ellipse \(25 x^2+9 y^2-150 x-90 y+225=0\) is

The quotient of the identity function by the reciprocal function is given by

Which of the following is not a function?

Evaluate \(\int \frac{3 x-2}{(x+3)(x+1)^2} d x\).

Using the principal values, the value of \(\sin ^{-1}\left\{\sin \frac{5 \pi}{6}\right\}+\tan ^{-1}\left\{\tan \frac{\pi}{6}\right\}\) is equal to

Find the value of \(\cos ^{-1}\left(\frac{4}{5}\right)+\tan ^{-1}\left(\frac{3}{5}\right)\).

If $$f(x)=\left\{\begin{array}{cc}\frac{(1-\cos 4 x)}{x^2}, & \text { if } x < 0 \\ a, & \text { if } x=0, \\ \frac{\sqrt{x}}{\sqrt{(16+\sqrt{x})}-4}, & \text { if } x > 0\end{array}\right.$$ then \(f(x)\) is continuous at \(x=0\), for $$a$...

The value of \(\lim _\limits{n \rightarrow \infty}\left\{\frac{1+2+3+\ldots+n}{n+2}-\frac{n}{2}\right\}\) is

\(\lim _\limits{x \rightarrow 0}\left\{\tan \left(\frac{\pi}{4}+x\right)\right\}^{1 / x}\) is equal to

The negation of \(\sim s \vee(\sim r \wedge s)\) is equivalent to

If $$A^{-1}=\left[\begin{array}{rr}5 & -2 \\ -7 & 3\end{array}\right]$$ and $$B^{-1}=\frac{1}{2}\left[\begin{array}{rr}9 & -7 \\ -8 & 6\end{array}\right]$$, then \((A B)^{-1}\) is equal to

If $$A=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$$, then \(\left(A-A^{\prime}\right)\) is equal to (where, \(A^{\prime}\) is transpose of matrix \(A\) )

The locus of a point which moves in a plane such that its distance from a fixed point in the plane is always equal to its distance from a fixed straight line in the same plane represents

Five persons entered the lift cabin on the ground floor of an eight floor house. Suppose that each of them independently and with equal probability can leave the cabin at any floor beginning with the first, then the probability of all 5 per...

A bag contains 50 tickets numbered \(1,2,3, ..., 50\) of which five are drawn at random and arranged in ascending order of magnitude \(\left(x_1 < x_2 < x_3 < x_4< x_5\right)\), then the probability that $x_3=30$ is

If \(S=\{(a, b): b=|a-1|, a \in Z\) and \(|a|<3\}\), where \(Z\) denotes the set of integers. Then, the range set of \(S\) is

The cartesian product \(A \times A\) has 9 elements among which are found \((-1,0)\) and \((0,1)\), then set \(A\) is equal to

The equation of a straight line upon which the length of the perpendicular from the origin is 5 and slope of this perpendicular is 3/4 is

The equation of a straight line which cuts off intercept on \(X\)-axis which is twice that on \(Y\)-axis and is at a unit distance from origin is given by

If \((3,4,-1)\) and \((-1,2,3)\) be end points of the diameter of a sphere, then the radius of the sphere is

The following lines are
$$\begin{aligned}
\mathbf{r} & =(\hat{\mathbf{i}}+\hat{\mathbf{j}})+\lambda^{\prime}(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}), \\
\text { and } \quad \mathbf{r} & =(\hat{\mathbf{i}}+\hat{\mathbf{j}})+\mu...

The angle between the lines \(\frac{x-5}{-3}=\frac{y+3}{-4}=\frac{z-7}{0}, \frac{x}{1}=\frac{y-1}{-2}=\frac{z-6}{2}\) is

The position vector of a point \(R\) which divides the line joining \(P(6,3,-2)\) and \(Q(3,1,-4)\) in the ratio \(2 : 1\) externally is

If \(\theta=\frac{\pi}{2^n+1}\), then the value of \(2^n \cos \theta \cos 2 \theta \cos 2^2 \theta \ldots \cos 2^{n-1} \theta\) is

The value of \(\cos \left(\frac{3 \pi}{2}+x\right) \cos (2 \pi+x)\left\{\cot \left(\frac{3 \pi}{2}-x\right)+\cot (2 \pi+x)\right\}\) is