List of top Mathematics Questions

For differentiable function f: R-{0}⟶ R, let 3f(x)+2f(\(\frac{1}{x}\))-\(\frac{1}{x}\)-10, then |f(3)+f(\(\frac{1}{4}\))| is equal to
Consider the following statements : $P$ : Ramu is intelligent $Q: $ Ramu is rich $R$ : Ramu is not honest The negation of the statement "Ramu is intelligent and honest if and only if Ramu is not rich" can be expressed as :
The points of intersection of the line $a x+b y=0,(a \neq b)$ and the circle $x^2+y^2-2 x=0$ are $A (\alpha, 0)$ and $B (1, \beta)$ The image of the circle with $AB$ as a diameter in the line $x+y+2=0$ is :
Let a, b, c be three distinct real numbers, none equal to one. If the vectors \(a\^i+\^j+\^k , \^i+b\^j+\^k\) and \( \^i+\^j+c\^k \) are coplanar, then  \(\frac{1}{ 1 − a }+ \frac{1}{ 1 − b} + \frac{1 }{1 − c }\) is equal to 
If $\tan 15^{\circ}+\frac{1}{\tan 75^{\circ}}+\frac{1}{\tan 105^{\circ}}+\tan 195^{\circ}=2 a$, then the value of $\left(a+\frac{1}{a}\right)$ is :

Let \(f(x)=\frac{sinx+cosx-\sqrt2}{sinx-cosx},x∈[0.\pi]-\left[\frac{\pi}{4}\right]. Then f\left(\frac{7\pi}{12}\right)f^n\left(\frac{7\pi}{12}\right)\)is equal to

A vector $\vec{v}$ in the first octant is inclined to the $x$-axis at $60^{\prime}$, to the $y$-axis at $45$ and to the $z$-axis at an acute angle If a plane passing through the points $(\sqrt{2},-1,1)$ and $(a, b, c)$, is normal to $\vec{v}$, then
Let A = {2, 3, 4} and B = {8, 9, 12}. Then the number of elements in the relation R = {((a1, b1), (a2, b2)) ∈ (A × B, A × B) : a1 divides b2 and a2 divides b1} is

Let \(A=\) [\(a_{ij}\)]\(_{2\times2}\) be a matrix and \(A^2 = I\) where \(a_{ij} \neq0\). If a sum of diagonal elements and b=det(A), then \(3a^2+4b^2\) is

Let A₁ and A₂ be two arithmetic means and G1, G2, G3 be three geometric means of two distinct positive numbers. Then $ G_{1}^{4} + G_{2}^{4}+ G_{3}^{4} +G_{1}^{2}G_{3}^{2} $ is equal to
Let a smooth curve $y=f(x)$ be such that the slope of the tangent at any point $(x, y)$ on it is directly proportional to $\left(\frac{-y}{x}\right)$ If the curve passes through the point $(1,2)$ and $(8,1)$, then $\left|y\left(\frac{1}{8}\right)\right|$ is equal to

Let $\vec{a}=2 \hat{i}+\hat{j}+\hat{k}$, and $\vec{b}$ and $\vec{c}$ be two nonzero vectors such that $|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|$ and $\vec{b} \cdot \vec{c}=0$. Consider the following two statements: 

(A) $|\vec{a}+\lambda \vec{c}| \geq|\vec{a}|$ for all $\lambda \in R$ 

(B) $\vec{a}$ and $\vec{c}$ are always parallel. Then. is

Let $y=f(x)=\sin ^3\left(\frac{\pi}{3}\left(\cos \left(\frac{\pi}{3 \sqrt{2}}\left(-4 x^3+5 x^2+1\right)^{\frac{3}{2}}\right)\right)\right)$ Then, at $x=1$

The number of points on the curve \(y=54 x^5-135 x^4-70 x^3+180 x^2+210 x\) at which the normal lines are parallel \(to x+90 y+2=0\) is 

Let the tangent and normal at the point \((3\sqrt{3},1)\) on the ellipse \(\frac{ x^2}{36}+\frac{y^2}{4} =1\) meet the y-axis at the points A and B respectively. Let the circle C be drawn taking AB as a diameter and the line x = \(2\sqrt{5}\) intersect C at the points P and Q. If the tangents at the points P and Q on the circle intersect at the point (α,β) , then α2 - β2 is equal to 
The domain of the function f(x) = \(\frac{1}{\sqrt{[x]^2-3[x]-10}}\) is (where [x] denotes the greatest integer less than or equal to x)
The straight lines l1 and l2 pass through the origin and trisect the line segment of the line L: 9x + 5y = 45 between the axes. If m1 and m2 are the slopes of the lines l1 and l2, then the point of intersection of the line y = (m1 + m2)x with L lies on
The random valuable X follows binomial distribution B (n, p) for which the difference of the mean and the variance is 1. If 2P(X = 2) = 3P(X = 1), then n2P(X > 1) is equal to
If the coefficient of $x^{15}$ in the expansion of $\left(a x^3+\frac{1}{b x^{1 / 3}}\right)^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of $\left(a x^{1 / 3}-\frac{1}{b x^3}\right)^{15}$, where a and $b$ are positive real numbers, then for each such ordered pair $(a, b)$ :
The sum of the coefficients of three consecutive terms in the binomial expansion of (1 + x)n+2, which are in the ratio 1 : 3 : 5, is equal to
Let $f(x)=2 x+\tan ^{-1} x$ and $g(x)=\log _e\left(\sqrt{1+x^2}+x\right), x \in[0,3]$ Then
Let S = {\( x∈(−\frac{ π}{ 2} , \frac{π}{2} )\) : 91−tan2 x + 9tan2 x=10 } and \(β=∑_{ x∈s} t a n ^2 (\frac{ x}{ 3} )\) , then \(\frac{1}{6}\) ( β − 14 )2  is equal to
Let $z_1=2+3 i$ and $z_2=3+4 i$ The set $S=\left\{z \in C:\left|z-z_1\right|^2-\left|z-z_2\right|^2=\left|z_1-z_2\right|^2\right\}$ represents a
If $[ t ]$ denotes the greatest integer $\leq t$, then the value of $\frac{3( e -1)}{ e } \int\limits_1^2 x^2 e^{[x]+\left[x^3\right]} dx$ is :

Let $\alpha \in(0,1)$ and $\beta=\log _e(1-\alpha)$ Let $P_n(x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\ldots+\frac{x^n}{n}, x \in(0,1)$ Then the integral $\int\limits_0^\alpha \frac{t^{50}}{1-t} d t$ is equal to