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List of top Mathematics Questions

Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi}{4}<\beta<0<\alpha<\frac{\pi}{4}$ If $\sin (\alpha+\beta)=\frac{1}{3}$ and $\cos (\alpha-\beta)=\frac{2}{3}$, then the greatest integer less than or equal to $\left(\frac{\sin \alpha}{\cos \beta}+\frac{\cos \beta}{\sin \alpha}+\frac{\cos \alpha}{\sin \beta}+\frac{\sin \beta}{\cos \alpha}\right)^2$ is ____.

Let A = {1, 2, 3, 4, 5, 6, 7} and B = {3, 6, 7, 9}. Then the number of elements in the set {C ⊆ A :Cߏ∩B ≠ φ} is _____________.

Let the matrix $A=\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{bmatrix}$ and the matrix $B_0=A^{49}+2 A^{98}$ If $B_n=\text{Adj}\left(B_{n-1}\right)$ for all $n \geq 1$, then $\text{det}\left( B _4\right)$ is equal to :

Let $z$ be a complex number with non-zero imaginary part If $\frac{2+3 z+4 z^2}{2-3 z+4 z^2}$ is a real number, then the value of $|z|^2$ is ___ .

The set of values of $k$, for which the circle $C : 4x^2 + 4y^2 – 12x + 8y + k = 0$ lies inside the fourth quadrant and the point $\bigg(1, -\frac{1}{3}\bigg)$ lies on or inside the circle $C$, is

Let $x * y$ = $x^2 + y^3$ and $(x * 1) * 1$ = $x * (1 * 1)$. Then a value of $2 sin^{-1}\bigg(\frac{x^4+x^2-2}{x^4+x^2+2}\bigg)$ is

The probability that a relation R from {x, y} to {x, y} is both symmetric and transitive, is equal to

Let

p : Ramesh listens to music.

q : Ramesh is out of his village.

r : It is Sunday.

s : It is Saturday.

Then the statement “Ramesh listens to music only if he is in his village and it is Sunday or Saturday” can be expressed as

The inner and outer radius of a ring-shaped iron sheet of thickness 1 cm is 8 cm and 10 cm respectively. When it is melted to make a solid iron ball, its diameter will be 1 cm

Let a, b and c be the length of sides of a triangle ABC such that:
$\frac {a+b}{7}=\frac {b+c}{8}=\frac {c+a}{9}$
If $r$ and $R$ are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of $\frac Rr$ is equal to :

Let $ƒ : R→R$ be defined as $ƒ(x) = x^3 + x – 5$ . If g(x) is a function such that $ƒ(g(x)) = x$, ∀‘x‘∈R. Then $g^′(63)$ is equal to_____.

A tower $PQ$ stands on a horizontal ground with base $Q$ on the ground The point $R$ divides the tower in two parts such that $QR =15 \,m$ If from a point $A$ on the ground the angle of elevation of $R$ is $60^{\circ}$ and the part $PR$ of the tower subtends an angle of $15^{\circ}$ at $A$, then the height of the tower is :

If three times of a number is, 3 less than three times of its square, then the number will be

The number of elements in the set $S = \{ \theta \in [-4\pi, 4\pi] : 3 \cos^2(2\theta) + 6 \cos^2(\theta) - 10\cos(2\theta) + 5 = 0 \}$ is______.

Let C be a circle passing through the points A(2, –1) and B (3, 4). The line segment AB is not a diameter of C. If r is the radius of C and its centre lies on the circle
$(x−5)^2+(y−1)^2=\frac{13}{2}$
then r² is equal to

The value of $\frac{ cot 45° }{sin 30° + cos 60°}$ is equal to

$cos^{-1}(\frac{2sin^{-1}(\frac{1}{4x^2-1})}{π})$ is:

If $cos^{-1}(\frac{y}{2})=log_e(\frac{x}{5})^5, |y| < 2$ then:

Let $|M|$ denote the determinant of a square matrix $M$. Let $g:\left[0, \frac{\pi}{2}\right] \rightarrow R$ be the function defined by
$g (\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}$ where
$f(\theta)=\frac{1}{2}\begin{vmatrix}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{vmatrix}+\begin{vmatrix}\sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _{ e }\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _{ e }\left(\frac{\pi}{4}\right) & \tan \pi\end{vmatrix}$
Let $p (x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g (\theta)$, and $p (2)=2-\sqrt{2}$. Then, which of the following is/are TRUE?

The distance between the two points A and A′ which lie on y = 2 such that both the line segments AB and A′B (where B is the point (2, 3)) subtend angle π/4 at the origin, is equal to

The sum of all the real roots of the equation $(e^{2x} – 4)(6e^{2x} – 5e^x + 1) = 0$ is

Let $S$ be the reflection of a point $Q$ with respect to the plane given by
$\vec{r}=-(t+p) \hat{ i }+t \hat{ j }+(1+p) \hat{ k }$
where $t, p$ are real parameters and $\hat{ i }, \hat{ j }, \hat{ k }$ are the unit vectors along the three positive coordinate axes. If the position vectors of $Q$ and $S$ are $10 \hat{ i }+15 \hat{ j }+20 \hat{ k }$ and $\alpha \hat{ i }+\beta \hat{ j }+\gamma \hat{ k }$ respectively, then which of the following is/are TRUE ?

The number of 4-digit integers in the closed interval $[2022,4482]$ formed by using the digits $0,2,3,4,6,7$ is ___

The greatest integer less than or equal to $\int\limits_1^2 \log _2\left(x^3+1\right) d x+\int\limits_1^{\log _2 9}\left(2^x-1\right)^{\frac{1}{3}} dx$ is _____.

If $\beta=\displaystyle\lim _{x \rightarrow 0} \frac{e^{x^3}-\left(1-x^3\right)^{\frac{1}{3}}+\left(\left(1-x^2\right)^{\frac{1}{2}}-1\right) \sin x}{x \sin ^2 x}$ then the value of $6 \beta$ is ____.