List of top Mathematics Questions

Let the line $L_1$ be parallel to the vector $-3\hat{i }+ 2\hat{j} + 4\hat{k}$ and pass through the point $(2, 6, 7)$, and the line $L_2$ be parallel to the vector $2\hat{i} + \hat{j} + 3\hat{k}$ and pass through the point $(4, 3, 5)$. If the line $L_3$ is parallel to the vector $-3\hat{i} + 5\hat{j} + 16\hat{k}$ and intersects the lines $L_1$ and $L_2$ at the points $C$ and $D$, respectively, then $|\vec{CD}|^2$ is equal to :

Let the maximum value of \( (\sin^{-1}x)^2 + (\cos^{-1}x)^2 \) for \( x \in \left[ -\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}} \right] \) be \( \frac{m}{n}\pi^2 \), where \( \gcd(m, n) = 1 \). Then \( m + n \) is equal to _________.

Let $f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f'''(3), x \in \mathbb{R}$. Then the value of $f'(5)$ is :

Let \( [ \cdot ] \) denote the greatest integer function and \( f(x) = \lim_{n} \to \infty \frac{1}{n^3} \sum_{k=1}^{n} \left[ \frac{k^2}{3^x} \right] \). Then \( 12 \sum_{j=1}^{\infty} f(j) \) is equal to _________.

If the line $ax + 4y = \sqrt{7}$, where $a \in \mathbb{R}$, touches the ellipse $3x^2 + 4y^2 = 1$ at the point $P$ in the first quadrant, then one of the focal distances of $P$ is :

If \( P \) is a point on the circle \( x^2 + y^2 = 4 \), \( Q \) is a point on the straight line \( 5x + y + 2 = 0 \) and \( x - y + 1 = 0 \) is the perpendicular bisector of \( PQ \), then 13 times the sum of abscissa of all such points \( P \) is _________.

If \( \int_0^1 4 \cot^{-1}(1 - 2x + 4x^2) dx = a \tan^{-1}(2) - b \ln(5) \), where \( a, b \in \mathbb{N} \), then \( (2a + b) \) is equal to _________.

Let $\alpha$ and $\beta$ be the roots of the equation $x^2 + 2ax + (3a + 10) = 0$ such that $\alpha<1<\beta$. Then the set of all possible values of $a$ is :

Consider two identical metallic spheres of radius $R$ each having charge $Q$ and mass $m$. Their centers have an initial separation of $4R$. Both the spheres are given an initial speed of $u$ towards each other. The minimum value of $u$, so that they can just touch each other is :
(Take $k = \frac{1}{4 \pi \epsilon_0}$ and assume $kQ^2>Gm^2$ where $G$ is the Gravitational constant)

Let $z$ be the complex number satisfying $|z - 5| \le 3$ and having maximum positive principal argument. Then $34 \left| \frac{5z - 12}{5iz + 16} \right|^2$ is equal to :

The sum of all the roots of the equation \((x-1)^2 - 5|x-1| + 6 = 0\), is:

Let the foci of a hyperbola coincide with the foci of the ellipse \(\frac{x^2}{36} + \frac{y^2}{16} = 1\). If the eccentricity of the hyperbola is 5, then the length of its latus rectum is:

Let \((\alpha, \beta, \gamma)\) be the co-ordinates of the foot of the perpendicular drawn from the point (5, 4, 2) on the line \(\vec{r}=(-\hat{i}+3\hat{j}+\hat{k})+\lambda(2\hat{i}+3\hat{j}-\hat{k})\). Then the length of the projection of the vector \(\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}\) on the vector \(6\hat{i}+2\hat{j}+3\hat{k}\) is:

The area of the region, inside the ellipse \(x^2+4y^2=4\) and outside the region bounded by the curves \(y=x-1\) and \(y=1-x\), is:

Let PQ and MN be two straight lines touching the circle \(x^2+y^2-4x-6y-3=0\) at the points A and B respectively. Let O be the centre of the circle and \(\angle AOB = \pi/3\). Then the locus of the point of intersection of the lines PQ and MN is:

The value of \(\text{cosec}10^\circ - \sqrt{3} \sec10^\circ\) is equal to:

Let \(\vec{a} = -\hat{i} + 2\hat{j} + 2\hat{k}\), \(\vec{b} = 8\hat{i} + 7\hat{j} - 3\hat{k}\) and \(\vec{c}\) be a vector such that \(\vec{a} \times \vec{c} = \vec{b}\). If \(\vec{c} \cdot (\hat{i}+\hat{j}+\hat{k}) = 4\), then \(|\vec{a}+\vec{c}|^2\) is equal to:

If the domain of the function \(f(x) = \cos^{-1}\left(\frac{2x-5}{11-3x}\right) + \sin^{-1}(2x^2-3x+1)\) is the interval \([\alpha, \beta]\), then \(\alpha + 2\beta\) is equal to:}

Let \(a_1, a_2, a_3, \dots\) be a G.P. of increasing positive terms such that \(a_2 \cdot a_3 \cdot a_4=64\) and \(a_1 + a_3 + a_5 = \frac{813}{7}\). Then \(a_3 + a_5 + a_7\) is equal to:

Let \(y=y(x)\) be the solution curve of the differential equation \((1+x^2)dy+(y-\tan^{-1}x) dx=0\), \(y(0) = 1\). Then the value of \(y(1)\) is:

Let \(f: R \to R\) be a twice differentiable function such that the quadratic equation \(f(x)m^2-2f'(x) m+f''(x) = 0\) in \(m\), has two equal roots for every \(x \in R\). If \(f(0)=1, f'(0) = 2\), and \((\alpha, \beta)\) is the largest interval in which the function \(g(x) = f(\log_e x - x)\) is increasing, then \(\alpha+\beta\) is equal to:

Let O be the vertex of the parabola \(x^2=4y\) and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio 2:3 be the conic C. Then the equation of the chord of C, which is bisected at the point (1, 2), is:

If $f(a)$ is the area bounded in the first quadrant by $x=0$, $x=1$, $y=x^2$ and $y=|ax-5|-|1-ax|+ax^2$, then find $f(0)+f(1)$.