List of top Mathematics Questions

The number of ordered tuples \((p,q,r)\) in truth table for which the statement \((\neg p \lor q)\Rightarrow r\) is true is:

Let \(R\subset \mathbb N\times \mathbb N\). Which of the following statements is necessarily true?

If \[ f(x)= \begin{cases} 1,& |x|\le 1\\ 0,& |x|>1 \end{cases} \] \[ g(x)= \begin{cases} 2-x^2,& |x|\le 2\\ 2,& |x|>2 \end{cases} \] and \(h(x)=f(g(x))\), then the interval in which \(h(x)=1\) for all values of \(x\) is:

The number of values of \(\theta\) in \([0,2\pi]\) for which the following homogeneous system has a non-trivial solution is: \[ x+(\sin\theta)y+(\cos\theta)z=0 \] \[ x+(\cos\theta)y+(\sin\theta)z=0 \] \[ x-(\sin\theta)y-(\cos\theta)z=0 \]

Let \(f:\mathbb{R}\to\mathbb{R}\), \(f(x)=|x+1|e^{-x^2}\), then which of the following option is true?

Number of triplets of sets \((A,B,C)\) with \(A,B,C\subseteq \{1,2,3,\ldots,n\}\) such that \[ (A\cap B)\subseteq C\subseteq (A\cup B) \] is:

Find the value of \[ \lim_{x\to\infty} \frac{\sqrt{x}} {\sqrt{x+\sqrt{x+\sqrt{x}}}} \]

Let \(f:[0,\infty)\to\mathbb{R}\) be defined by \[ f(x)=\frac{3x^2+4x+1}{x^2+3x+2}. \] Then the value of \((f^{-1})'(2)\) is:

Find the value of determinant at \(x=2026\): \[ \begin{vmatrix} x & x+1 & x+3\\ x+1 & x+3 & x+6\\ x+3 & x+6 & x+10 \end{vmatrix} \]

From top of view point at height of \(80\,\text{m}\), the angles of depression of the top and bottom of a flag standing on the same plane are observed to be \(30^\circ\) and \(45^\circ\). Find the height of the flag.

Let \(a, b, c\) be non-zero number such that \(a+b+c\neq 0\) and \(4a-2b+c\neq 0\). If \(\alpha\) and \(\beta\) are the roots of quadratic equation \(ax^2+bx+c=0\), then which of the following equation has roots \(\dfrac{\alpha+2}{\alpha-1}\) and \(\dfrac{\beta+2}{\beta-1}\)?

A triangle has a vertex at \((1,2)\) and the midpoints of two sides through it are \((-1,1)\) and \((2,3)\). Find its area.

If \(x,y\) are real numbers such that \[ 2^{x+\frac12}\times4^{y-\frac56} = 3^{x-\frac12}\times9^{y-\frac13}, \] then which of the following is true?

The eccentricity of ellipse whose centre at origin is \(\frac{1}{2}\). If one of its directrices is \(x=-4\), then equation of normal to it at \(\left(1,\frac{3}{2}\right)\) is:

Evaluate \[ \int_{0}^{\frac{\pi}{4}}\frac{dx}{\cos^4x}. \]

If \(f:[0,\infty)\to\mathbb R\), \[ f(x)=\frac{x^2-1}{x^2+1}, \] then find \[ \int_{-1}^{1}f^{-1}(y)\,dy. \]

A die is rolled twice independently. Probability that either first die shows number no less than \(4\) or second die shows at least \(4\)?

Given \[ \cos6x=(a)\cos^6x+(b)\cos^4x+(c)\cos^2x+(d) \] for any real number \(x\), where \(a,b,c,d\) are constants. Find the value of \(a+b+c\).

Find the area of the triangle in the right half plane formed by the lines \(x-y=0\) and \(x+y=0\), and which is tangent to the hyperbola \[ x^2-y^2=a^2. \]

Find the value of \[ \lim_{x\to 0}\frac{x^2+2\cos x-2}{x\sin 3x} \]

The value of \(\cos^{-1}\left(\cos\left(-\frac{\pi}{6}\right)\right)+\sin^{-1}\left(\sin\left(\frac{5\pi}{6}\right)\right)\) is:

Let \(A_k\) be the arithmetic mean of the squares of \(k\) natural numbers. \[ \sum_{k=1}^{n}(6A_k-3k)=31 \] Find the value of \(n\):

If the HCF of 210 and 55 is expressed as \(210 \times 5 + 55m\), then find the value of \(m\).

Let $f : \mathbb{R} \to \mathbb{R}$ be the function given by \[ f(x) = |x - 2| + 3|x - 1| + ||x - 2| - 1| . \] What is the number of points where $f$ is NOT differentiable?

Suppose there are two boxes $B_1$ and $B_2$, each having 3 red and 4 black balls. One ball is drawn at random from $B_1$. If it is red, 4 red balls are put into $B_2$, otherwise 3 black balls are put into $B_2$. Then one ball is randomly drawn from $B_2$. If this ball is red, what is the conditional probability that the ball drawn from $B_1$ was also red?