List of top Mathematics Questions

Let the function $f: R \rightarrow R$ be defined by $f(x)=x^{3}-x^{2}+(x-1) \sin x$ and let $g: R \rightarrow R$ be an arbitrary function. Let $f g: R \rightarrow R$ be the product function defined by $(f, g)(x)=f(x) g(x)$. Then which of the following statements is/are TRUE?

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^{2}+20 x-2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^{2}-20 x+2020$. Then the value of $a c(a-c)+a d(a-d)+b c(b-c)+b d(b-d)$ is

The number of integral values of x satisfying the inequation |x| < 4 is

If 2^x = 7^y = 14^z, then find the value of z in terms of x and y.

If the price of cach book goes up by ₹ 5, then Asha can buy 20 books less for ₹ 1200. Find the original price and the number of books Asha could buy at the original price.

Bhiku borrowed some money from Dhanraj to admit his daughter in a reputed Law School. He agreed to pay the interest-free loan of ₹ 60, 000 in a monthly installments which increased by a constant amount. After the 20^th installment, he found that he has paid 3/4 of the loan. If the entire loan was cleared this way in 25 installments, find out the value of the first installment.

City A is connected to City B by four highways and City B is connected to City C by three highways. Kesari wants to travel from City A to City C via City B. In how many ways, Kesari can do it ?

If the third and the tenth term of an arithmetic progression are 14 and 56, respectively, then the arithmetic mean of the first 15 terms of the arithmetic progression is

Ronnic has a bag containing 7 red and 9 green balls. From it, he draws out 6 balls simultaneously at random. The probability that 4 of them are red and the rest are green is

The right hand and left hand limit of the function are respectively

Let \(m\) be the minimum possible value of \(\log _{3}\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}\right)\), where \(y_{1}, y_{2}, y_{3}\) are real numbers for which \(y_{1}+y_{2}+y_{3}=9\) .Let \(M\) be the maximum possible value of \(\left(\log _{3} x_{1}+\log _{3} x_{2}+\log _{3} x_{3}\right)\), where \(x_{1}, x_{2}, x_{3}\) are positive real numbers for which \(x_{1}+x_{2}+x_{3}=9\) .Then the value of \(\log _{2}\left(m^{3}\right)+\log _{3}\left(M^{2}\right)\) is_______

If $2^x+2^y = 2^{x+y}$, then $\frac {dy}{dx}$ is

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z^{2}+z+1\right|=1$. Then which of the following statements is/are TRUE?

For a complex number $z$, let $Re(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^{4}-|z|^{4}=4 i z^{2}$, where $i=\sqrt{-1}$. Then the minimum possible value of $\left|z_{1}-z_{2}\right|^{2}$, where $z_{1}, z_{2} \in S$ with \(Re\left( z _{1}\right)\)\(>\)0 and \(Re\left( z _{2}\right)\) \(<\) 0, is ______

The value of $cos\left(\sin^{-1}\frac{\pi}{3}+\cos^{-1}\frac{\pi}{3}\right)$ is

Let the function $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by
$f(x)=e^{x-1}-e^{-|x-1|}$ and $g(x)=\frac{1}{2}\left(e^{x-1}+e^{1-x}\right)$.
Then the area of the region in the first quadrant bounded by the curves $y=f(x), y=g(x)$ and $x=0$ is

In a triangle \(P Q R\), let \(\vec{a}=\overrightarrow{Q R}, \vec{b}=\overrightarrow{R P}\) and \(\vec{c}=\overrightarrow{P Q}\) If
 \(|\vec{a}|=3,|\vec{b}|=4\) and \(\frac{\vec{a} \cdot(\vec{c}-\vec{b})}{\vec{c} \cdot(\vec{a}-\vec{b})}=\frac{|\vec{a}|}{|\vec{a}|+|\vec{b}|}\) then the value of \(|\vec{a} \times \vec{b}|^{2}\) is

Let $x, y$ and $z$ be positive real numbers Suppose $x, y$ and $z$ are the lengths of the sides of a triangle opposite to its angles $X, Y$ and $Z$, respectively If $\tan \frac{x}{2}+\tan \frac{z}{2}=\frac{2 y}{x+y+z}$ then which of the following statements is/are TRUE?

The order of the differential equation obtained by eliminating arbitrary constants in the family of curves $c_1y = (c_2 +c_3 )e^{x+c_4}$ is

Let \(f:[0,2] \rightarrow R\) be the function defined by
\(f(x)=(3-\sin (2 \pi x)) \sin \left(\pi x-\frac{\pi}{4}\right)-\sin \left(3 \pi x+\frac{\pi}{4}\right)\) If \(\alpha, \beta \in[0,2]\) are such that \(\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]\), then the value of \(\beta-\alpha\) is ______

Corner points of the feasible region determined by the system of linear constraints are $(0, 3), (1, 1)$ and $(3, 0)$. Let $z = px = qy$, where $p, q > 0$. Condition on $p$ and $q$ so that the minimum of $z$ occurs at $(3, 0)$ and $(1, 1)$ is

If a line makes an angle of $\pi/3$ with each of $x$ and and $y$-axis, then the acute angle made by $z$-axis is

If the curves $2x = y^2$ and $2xy = K$ intersect perpendicularly, then the value of $K^2$ is

The standard deviation of the data $6, 7, 8, 9, 10$ is