List of top Mathematics Questions

For the Function \(f(x)=\tan x-4x,\ x\in\left(0,\frac{\pi}{2}\right)\), which of the following statements are correct?
A. \(f(x)\) is increasing in \(\left(0,\frac{\pi}{3}\right)\)
B. \(f(x)\) is increasing in \(\left(\frac{\pi}{3},\frac{\pi}{2}\right)\)
C. \(f(x)\) is decreasing in \(\left(0,\frac{\pi}{3}\right)\)
D. \(f(x)\) is decreasing in \(\left(\frac{\pi}{3},\frac{\pi}{2}\right)\)
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The difference between the greatest and least values of the function \(f(x)=\sin 2x-x\) on \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\) is
In the given figure, area of the shaded portion is equal to
\(\int\frac{dx}{(x+1)\sqrt{x^2-1}}\) is equal to:
[\(c\) is an arbitrary constant]
\(\int_{-\pi/2}^{\pi/2}\frac{\cos x}{1+e^{x}}\,dx=\)
Match the LIST-I with LIST-II
LIST-I
Indefinite Integral		LIST-II
Solution (where \(c\) is an arbitrary constant)
A. \(\int\sqrt{16-x^2}\,dx\)I. \(\sin^{-1}\left(\frac{x}{4}\right)+c\)
B. \(\int\frac{dx}{\sqrt{16-x^2}}\)II. \(\frac{x}{2}\sqrt{16-x^2}+8\sin^{-1}\left(\frac{x}{4}\right)+c\)
C. \(\int\frac{dx}{\sqrt{x^2-16}}\)III. \(\frac{1}{8}\log\left|\frac{4+x}{4-x}\right|+c\)
D. \(\int\frac{dx}{16-x^2}\)IV. \(\log\left|x+\sqrt{x^2-16}\right|+c\)
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If \(x, y, z\) are all different from zero and \(\begin{vmatrix} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{vmatrix} = 0\), then the value of \(x^{-1} + y^{-1} + z^{-1}\) is
If \(y=x^x,\ x>0\) then \(y^{\prime\prime}(2)-2y^{\prime}(2)\) is equal to
If \(x=\sec\theta+\cos\theta,\ y=\sec^n\theta+\cos^n\theta\), then \(\frac{dy}{dx}\) is equal to
Function \(f(x)=\begin{cases} k\left(x^2-6x+13\right) & : x<1 \\ |x-3| & : x\ge 1 \end{cases}\) is differentiable at \(x=1\), then the value of \(k\) is
If \(A(x_1,y_1)\), \(B(x_2,y_2)\) and \(C(x_3,y_3)\) are the vertices of an equilateral triangle whose each side is equal to \(k\), then the value of \(\begin{vmatrix} x_1 & y_1 & 2 \\ x_2 & y_2 & 2 \\ x_3 & y_3 & 2 \end{vmatrix}^2\) is
If \(A=\begin{bmatrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{bmatrix}\), then
Match the LIST-I with LIST-II
LIST-ILIST-II
A. \(|\text{adj }A|\)I. 81
B. \(|A|\)II. \(-27\)
C. \(|A\ \text{adj }A|\)III. 9
D. \(|\text{adj (adj }A)|\)IV. \(-3\)

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If \(A\) is square matrix such that \(A^2 = A\), then \((2I + A)^4 - 65A\) is equal to (Where \(I\) is identity matrix)
If \(A = [a_{ij}]_{3\times 3}\), \(a_{ij} = \begin{cases} 0, & i \neq j \\ i+j, & i = j \end{cases}\), then which of the following statements are correct ?
A. \(A\) is scalar matrix
B. \(A\) is diagonal matrix
C. \(A\) is unit matrix
D. \(A\) is symmetric matrix
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A function \(f: N \to N\), (\(N\): set of natural numbers) defined by \(f(x) = x^2 + x + 1\) is
Match the LIST-I with LIST-II
LIST-I
Function		LIST-II
Range (Principal value)
A.\(\tan^{-1}x\)I.\([0,\pi]-\left\{\frac{\pi}{2}\right\}\)
B.\(\sec^{-1}x\)II.\(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\)
C.\(\operatorname{cosec}^{-1}x\)III.\(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]-\{0\}\)
D.\(\sin^{-1}x\)IV.\(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\)

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If \(A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}\), such that \(A^2 - 4A + \lambda I = 0\), where \(I\) is identity matrix of order 2, then '\(\lambda\)' is equal to
Match the LIST-I with LIST-II
LIST-I
Inequality		LIST-II
Solution Space
A.\(2x+3y \geq 6\)I.half plane contains the origin, also points on the line \(2x+3y=6\)
B.\(2x+3y \leq 6\)II.half plane contains the origin but not points on the line \(2x+3y=6\)
C.\(2x+3y > 6\)III.half plane not contains the origin but contains points on the line \(2x+3y=6\)
D.\(2x+3y < 6\)IV.half plane neither contains the origin nor contains points on the line \(2x+3y=6\)

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The solution of the differential equation \(\frac{dy}{dx}=\frac{x}{y}\) represents a family of
The solution of initial value problem: \(\frac{dy}{dx}=e^{3x+4y};\ y(0)=-\frac{1}{4}\) is
Three coins are tossed, then
Match the LIST-I with LIST-II
LIST-ILIST-II
A.Probability of occurrence of 3 heads or 3 tailsI.\(\frac{1}{8}\)
B.Probability of occurrence of at least two headsII.\(\frac{7}{8}\)
C.Probability of occurrence of at most two headsIII.\(\frac{1}{2}\)
D.Probability of occurrence of no headIV.\(\frac{1}{4}\)
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The order and degree of the differential equation \(\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}=\left(\frac{d^{2}y}{dx^{2}}\right)^{\frac{1}{3}}\) are
Area of the region bounded by the lines \(|x|+|y|=1,\ x\geq 0,\ y\geq 0\) is
If \(A\) is a square matrix of order \(3 \times 3\), and \(|\text{adj } A| = 25\), then the value of \(|2A|\) is
If \(y = \sqrt{\dfrac{1-x}{1+x}}\), then the value of \((1 - x^2)\dfrac{dy}{dx} + y\) is