List of top Mathematics Questions

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ and $|\vec{a}| = 3, |\vec{b}| = 5, |\vec{c}| = 7$, then the angle between $\vec{a}$ and $\vec{b}$ is
The integrating factor of the differential equation $(x \log_e x) \frac{dy}{dx} + y = 2\log_e x$ is
Consider the differential equation, $x \frac{dy}{dx} = y(\log_e y - \log_e x + 1)$, then which of the following are true?
(A) It is a linear differential equation
(B) It is a homogenous differential equation
(C) Its general solution is $\log_e(\frac{y}{x}) = Cx$, where C is constant of integration
(D) Its general solution is $\log_e(\frac{x}{y}) = Cy$, where C is constant of integration
(E) If y(1) = 1, then its particular solution is y = x
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The area (in sq. units) of the region bounded by the curve \( y = x^5 \), the x-axis and the ordinates x = -1 and x = 1 is equal to

Match List-I with List-II

List-I (Definite integral)List-II (Value)
(A) \( \int_{0}^{1} \frac{2x}{1+x^2}\, dx \)(I) 2
(B) \( \int_{-1}^{1} \sin^3x \cos^4x\, dx \)(II) \(\log_e\!\left(\tfrac{3}{2}\right)\)
(C) \( \int_{0}^{\pi} \sin x\, dx \)(III) \(\log_e 2\)
(D) \( \int_{2}^{3} \frac{2}{x^2 - 1}\, dx \)(IV) 0


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The area (in sq. units) of the region bounded by y = $2\sqrt{1-x^2}$, x $\in$ [0,1] and x-axis is equal to
The rate of change of area of a circle with respect to its circumference when radius is 4cm, is
The function f(x) = tanx - x
$\int_{\pi/6}^{\pi/3} \frac{\tan x}{\tan x + \cot x} dx$ is equal to
The integral I = $\int e^x (\frac{x-1}{3x^2}) dx$ is equal to
Let AX = B be a system of three linear equations in three variables. Then the system has
(A) a unique solution if |A| = 0
(B) a unique solution if |A| $\neq$ 0
(C) no solutions if |A| = 0 and (adj A) B $\neq$ 0
(D) infinitely many solutions if |A| = 0 and (adj A)B = 0
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Let A = [aij]2x3 and B = [bij]3x2, then |5AB| is equal to
Let \( y=\sin(\cos(x^2)) \). Find \( \frac{dy}{dx} \) at \( x=\frac{\sqrt{\pi}}{2} \).
If the function f(x) = $\begin{cases}\frac{k\cos x}{\pi - 2x} & ; x \neq \frac{\pi}{2} \\ 3 & ; x = \frac{\pi}{2} \end{cases}$ is continuous at x = $\frac{\pi}{2}$, then k is equal to

Match List-I with List-II

List-IList-II
(A) \( f(x) = |x| \)(I) Not differentiable at \( x = -2 \) only
(B) \( f(x) = |x + 2| \)(II) Not differentiable at \( x = 0 \) only
(C) \( f(x) = |x^2 - 4| \)(III) Not differentiable at \( x = 2 \) only
(D) \( f(x) = |x - 2| \)(IV) Not differentiable at \( x = 2, -2 \) only


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Match List-I with List-II

List-IList-II
(A) The minimum value of \( f(x) = (2x - 1)^2 + 3 \)(I) 4
(B) The maximum value of \( f(x) = -|x + 1| + 4 \)(II) 10
(C) The minimum value of \( f(x) = \sin(2x) + 6 \)(III) 3
(D) The maximum value of \( f(x) = -(x - 1)^2 + 10 \)(IV) 5


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If A and B are invertible matrices then which of the following statement is NOT correct?
Let A = $\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$ and I = $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. If AT + A = I, then
Let A = \{1, 2, 3\}. Then, the number of relations containing (1, 2) and (1, 3), which are reflexive and symmetric but not transitive, is
If A and B are skew-symmetric matrices, then which one of the following is NOT true?
for $|x| < 1$, sin(tan-1x) equal to
The area (in sq. units) of the region bounded by the parabola y2 = 4x and the line x = 1 is
The solution of the differential equation $\log_e(\frac{dy}{dx}) = 3x + 4y$ is given by
Consider the LPP: Minimize Z = x + 2y subject to 2x + y $\ge$ 3, x + 2y $\ge$ 6, x, y $\ge$ 0. The optimal feasible solution occurs at
Let f: R $\rightarrow$ R be defined as f(x) = 10x. Then (Where R is the set of real numbers)