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List of top Mathematics Questions

Let three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ be such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}, \vec{a} \cdot \vec{c} = 7$ and $\vec{b}$ is perpendicular to $\vec{c}$, where $\vec{a} = -\hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} + \hat{k}$, then the value of $|2\vec{a} + \vec{b} + \vec{c}|^2$ is __________

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a} = \vec{b} \times (\vec{b} \times \vec{c})$. If magnitudes of the vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ are $\sqrt{2}, 1$ and 2 respectively and the angle between $\vec{b}$ and $\vec{c}$ is $\theta$ ($0<\theta<\frac{\pi}{2}$), then the value of $1+\tan\theta$ is equal to :

Let $\vec{a}=\hat{i}-\alpha\hat{j}+\beta\hat{k}$, $\vec{b}=3\hat{i}+\beta\hat{j}-\alpha\hat{k}$ and $\vec{c}=-\alpha\hat{i}-2\hat{j}+\hat{k}$, where $\alpha$ and $\beta$ are integers. If $\vec{a} \cdot \vec{b} = -1$ and $\vec{b} \cdot \vec{c} = 10$, then $(\vec{a} \times \vec{b}) \cdot \vec{c}$ is equal to ________.

Let the vectors $(2 + a + b)\hat{i} + (a + 2b + c)\hat{j} - (b + c)\hat{k}$, $(1 + b)\hat{i} + 2\hat{j} - b\hat{k}$ and $(2 + b)\hat{i} + 2\hat{j} + (1 - b)\hat{k}$, $a, b, c \in \mathbb{R}$ be co-planar. Then which of the following is true ?

Rotation of axes: Vector ā (3p, 1) becomes (p+1, √10) in new system. Find p.

The sum of all 3-digit numbers less than or equal to 500, that are formed without using the digit "1" and they all are multiple of 11, is ___________.

The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is _________.

If ${}^1P_1 + 2 \cdot {}^2P_2 + 3 \cdot {}^3P_3 + \dots + 15 \cdot {}^{15}P_{15} = {}^qP_r - s$, where $0 \leq s \leq 1$, then $q + s + {}^qC_{r - s}$ is equal to _________.

A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is _________.

The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is ___________

The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is _________.

Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions from the set A to the set A $\times$ B. Then :

Let S = 1, 2, 3, 4, 5, 6, 7. Then the number of possible functions $f: S \rightarrow S$ such that $f(m \cdot n) = f(m) \cdot f(n)$ for every $m, n \in S$ and $m \cdot n \in S$ is equal to _________.

The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is _________

There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of class 10 and 11 is $100k$, then k is equal to __________.

The sum of all the 4-digit distinct numbers that can be formed with the digits 1, 2, 2 and 3 is :

A circle C touches the line $x=2y$ at the point (2, 1) and intersects the circle C$_1$ : $x^2 + y^2 + 2y - 5 = 0$ at two points P and Q such that PQ is a diameter of C$_1$. Then the diameter of C is :

The point P$(-2\sqrt{6}, \sqrt{3})$ lies on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ having eccentricity $\frac{\sqrt{5}}{2}$. If the tangent and normal at P to the hyperbola intersect its conjugate axis at the points Q and R respectively, then QR is equal to :

The locus of the mid points of the chords of the hyperbola $x^2-y^2=4$, which touch the parabola $y^2 = 8x$, is :

If a line along a chord of the circle $4x^2 + 4y^2 + 120x + 675 = 0$, passes through the point $(-30, 0)$ and is tangent to the parabola $y^2 = 30x$, then the length of this chord is :

On the ellipse $\frac{x^2}{8} + \frac{y^2}{4} = 1$ let $P$ be a point in the second quadrant such that the tangent at $P$ to the ellipse is perpendicular to the line $x + 2y = 0$. Let $S$ and $S'$ be the foci of the ellipse and $e$ be its eccentricity. If $A$ is the area of the triangle $SPS'$ then, the value of $(5 - e^2) \cdot A$ is :

If two tangents drawn from a point P to the parabola $y^2 = 16(x-3)$ are at right angles, then the locus of point P is :

Let A(sec$\theta$, 2tan$\theta$) and B(sec$\phi$, 2tan$\phi$), where $\theta+\phi=\pi/2$, be two points on the hyperbola $2x^2-y^2=2$. If ($\alpha$, $\beta$) is the point of the intersection of the normals to the hyperbola at A and B, then $(2\beta)^2$ is equal to _________.

A tangent line L is drawn at the point $ (2, -4) $ on the parabola $ y^2 = 8x $. If the line L is also tangent to the circle $ x^2 + y^2 = a $, then 'a' is equal to _________.

If the curve $x^2+2y^2 = 2$ intersects the line $x+y=1$ at two points P and Q, then the angle subtended by the line segment PQ at the origin is :