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List of top Mathematics Questions

If
$\sum\limits_{k=1}^{31}$ $(^{31}C_k) (^{31}C_{k-1})$ $-\sum\limits_{k=1}^{30}$ $(^{30}C_k) (^{30}C_{k-1})$ $= \frac{α (60!)} {(30!) (31!)}$
where $α ∈ R$, then the value of 16α is equal to

If
$(^{40}C_0) + (^{41}C_1) + (^{42}C_2) + ...... + (^{60}C_{20}) \frac{m}{n} ^{60}C_{20}$
m and n are coprime, then m + n is equal to _____.

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is _____.

The total number of four digit numbers such that each of first three digits is divisible by the last digit, is equal to _______.

There are ten boys B₁, B₂, …, B₁₀ and five girls G₁, G₂,…, G₅ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both B₁ and B₂ together should not be the members of a group, is ________

The number of ways to distribute 30 identical candies among four children C₁, C₂, C₃ and C₄ so that C₂ receives atleast 4 and atmost 7 candies, C₃ receives atleast 2 and atmost 6 candies, is equal to:

Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits 1, 2, 3, 4, 5 and 6 without repetition of digits. Then the total number of such numbers is ______________.

A line, with the slope greater than one, passes through the point A (4,3) and intersects the line x-y-2=0 at the point B If the length of the line segment AB is $\frac{\sqrt{29}}{3},$ then B also lies on the line :

Let $P$ be the plane containing the straight line $\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5}$ and perpendicular to the plane containing the straight lines $\frac{ x }{2}=\frac{ y }{3}=\frac{ z }{5}$ and $\frac{ x }{3}=\frac{ y }{7}=\frac{ z }{8}$ If $d$ is the distance of $P$ from the point $(2,-5,11)$, then $d ^2$ is equal to :

If $y = m_1x + c_1 $and $y = m_2x + c_2$, $m_1≠m_2$ are two common tangents of circle $x_2 + y_2 = 2$ and parabola $y^2 = x$, then the value of $8|m_1m_2|$ is equal to :

Let $Q$ be the mirror image of the point $P(1, 0, 1)$ with respect to the plane $S$: $x + y + z = 5$. If a line L passing through $(1, –1, –1)$, parallel to the line $PQ$ meets the plane $S$ at $R$, then $QR_2$ is equal to :

Let $x=2t,\ y=\frac {t^2}{3}$ be a conic. Let S be the focus and B be the point on the axis of the conic such that SA⊥BA, where A is any point on the conic. If k is the ordinate of the centroid of the ΔSAB, then $\lim\limits_{t \to 1} k$ is equal to

A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2:1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be (α, β). Then, the value of $7α + 3β$ is equal to ________ .

Let l be a line which is normal to the curve $y = 2x^2 + x + 2$ at a point P on the curve. If the point Q(6, 4) lies on the line l and O is origin, then the area of the triangle OPQ is equal to _________ .

Let the lines
$y + 2x = \sqrt {11} + 7\sqrt 7$ and $2y + x = 2\sqrt {11} + 6\sqrt 7$
be normal to a circle $C : (x – h)^2 + (y – k)^2 = r^2$. If the line
$\sqrt {11}y - 3x =\frac { 5\sqrt {17}}{3} + 11$
is tangent to the circle C, then the value of $(5h – 8k)^2 + 5r^2$ is equal to _______.

A circle of radius 2 unit passes through the vertex and the focus of the parabola y² = 2x and touches the parabola $y=\left (x−\frac{1}{4}\right)^2+α,$ where $α > 0$. Then $(4α – 8)^2$ is equal to ___________.

Let the common tangents to the curves 4(x² + y²) = 9 and y² = 4x intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and I respectively denote the eccentricity and the length of the latus rectum of this ellipse, then
$\frac{1}{e^2}$ is equal to

Let the tangent drawn to the parabola y² = 24x at the point (α, β) is perpendicular to the line 2x + 2y = 5. Then the normal to the hyperbola
$\frac{x^2}{α^2}−\frac{y^2}{β^2}=1$
at the point (α + 4, β + 4) does NOT pass through the point

An ellipse
$E:\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
passes through the vertices of the hyperbola
$H:\frac{x^2}{49} - \frac{y^2}{64} = -1$
Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H, respectively. Let the product of the eccentricities of E and H be 1/2. If the length of the latus rectum of the ellipse E, then the value of 113l is equal to _____.

Let $O$ be the origin and $A$ be the point $z _1=1+2 i$. If $B$ is the point $z _2, \operatorname{Re}\left( z _2\right)<0$, such that $OAB$ is a right angled isosceles triangle with $OB$ as hypotenuse, then which of the following is NOT true ?

Let z₁ and z₂ be two complex numbers such that

$z_1=iz_2 \,and \,arg(\frac{z_1}{z_2})=π.$

If z² + z + 1 = 0,
$z ∈ C$, then $\left| \sum_{n=1}^{15} \left( z_n + (-1)^n \frac{1}{z_n} \right)^2 \right|$
is equal to ________.

Let $S = z ∈ C: |z-3| <= 1$ and $z (4+3i)+z(4-3)≤24.$
If α + iβ is the point in S which is closest to 4i, then 25(α + β) is equal to ______.

The number of elements in the set { $z = a + ib ∈ C : a,b ∈ Z$ and $1 < | z - 3 + 2i | < 4$ } is _______ .

Let O be the origin and A be the point $z_1 = 1 + 2i$. If B is the point $z_2, Re(z_2) < 0$, such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT true?