List of top Mathematics Questions

Find the absolute maximum value of \( f(x) = \cos x + \sin^2 x \) in the closed interval \( x \in [0, \pi] \).

If \(\vec{AB} = \hat{j} + \hat{k}\) and \(\vec{AC} = 3\hat{i} - \hat{j} + 4\hat{k}\) represent two vectors along the sides \(AB\) and \(AC\) of \(\Delta ABC\), prove that the median vector satisfies \(\vec{AB} + \vec{AC} = 2\vec{AD}\), where \(D\) is the midpoint of \(BC\). Hence, find the length of the median \(AD\).

Assertion (A): $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2|\vec{b}|^2$.
Reason (R): $|\vec{a} \times \vec{b}| = (\vec{a} \cdot \vec{b})\tan\theta$, where $\theta$ is the angle between vectors $\vec{a}$ and $\vec{b}$ ($\theta \neq \frac{\pi}{2}$).

Assertion (A): A line can have direction cosines $\langle 1, 1, 1 \rangle$.
Reason (R): $\cos\theta = 1$ is possible for $\theta = 0$.

If $e^{x + y} = 3x$, then $\frac{dy}{dx}$ is equal to:

If $A$ and $B$ are square matrices of the same order, then which of the following statements is/are always true?
(i) $(A + B)(A - B) = A^2 - B^2$
(ii) $AB = BA$
(iii) $(A + B)^2 = A^2 + AB + BA + B^2$
(iv) $AB = 0 \implies A = 0$ or $B = 0$

For a square matrix $A$ of order $n$, the inverse matrix $(3A)^{-1}$ is equal to:

For two events $A$ and $B$ such that $P(A) \neq 0$ and $P(B) \neq 1$, the conditional probability $P(A'/B')$ is equal to:

A relation $R$ on the set $A = \{1, 2, 3\}$ defined as $R = \{(1, 2), (2, 1), (2, 2)\}$ is:

For a feasible region bounded by the corner points $(0,3)$, $(3,2)$, and $(5,0)$ in the first quadrant, the non-trivial constraints of the linear programming problem are:

If $|\vec{a}| = 5$ and $-2 \le \lambda \le 1$, then the sum of the greatest and the smallest value of $|\lambda \vec{a}|$ is:

The direction cosines of the line given by $x = y = 1 - z$ are:

In a linear programming problem, the linear function which has to be maximized or minimized is called:

A vector of magnitude 3 making equal angles with the $x$ and $y$ axes and perpendicular to the $z$ axis is:

The integrating factor of the differential equation $2x \frac{dy}{dx} - y = 3$ is:

Which of the following expressions will give the area of the region bounded by the curve $y = x^2$ and the line $y = 16$?

The indefinite integral $\int \frac{1}{1 + \cos x} \, dx$ is equal to:

For the function $f(x) = x + \frac{1}{x}$ ($x \neq 0$), which of the following statements is true?

The general solution of the differential equation $x \, dy - y \, dx = 0$ is:

$\frac{dy}{dx} = xy + x + y + 1$ has the solution}

The equation of the curve passing through the origin and satisfying the differential equation $\frac{dy}{dx} = (x-y)^2;$

Which of the following is an integrating factor for the differential equation $x \cos x \frac{dy}{dx} + (x \sin x + \cos x) y = 1$?

If the solution $y(x)$ of the given differential equation $(e^y+1) \cos x dx + e^y \sin x dy = 0$ passes through the point $(\pi/2, 0)$ then the value of $y(\pi/6)$ is

The general solution of $\frac{ydx - xdy}{y^2} = 0$ represents a family of}

If $k \int_{0}^{1} x f(3x) dx = \int_{0}^{3} t f(t) dt$ then $k =$