List of top Mathematics Questions

If the tangents drawn at the points O(0, 0) and P(1 + √5, 2) on the circle x2 + y2 – 2x – 4y = 0 intersect at the point Q, then the area of the triangle OPQ 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 the plane
\(P : \stackrel{→}{r} . \stackrel{→}{a} = d\)
contain the line of intersection of two planes
\(\stackrel{→}{r} . ( \hat{i} + 3\hat{j} - \hat{k} ) = 6\)
and
\(\stackrel{→}{r} . ( -6\hat{i} + 5\hat{j} - \hat{k} ) = 7\)
. If the plane P passes through the point (2, 3, 1/2), 
then the value of \(\frac{| 13a→|² }{d²}\) is equal to

Let A = {1, a1, a2…a18, 77} be a set of integers with 1 <a1<a2<….<a18< 77. Let the set A + A = {x + y :x, y∈A} contain exactly 39 elements. Then, the value of a1 + a2 +…+a18 is equal to _____.
Let r ∈ {p, q, ~p, ~q} be such that the logical statement r ∨ (~p) ⇒ (p ∧ q) ∨ r is a tautology. Then r is equal to :
The number of positive integers k such that the constant term in the binomial expansion of \(( 2x^3 + \frac {3}{x^k} )^{12}, x ≠ 0\) is \(2^8\) . l, where l is an odd integer, is______.
The number of elements in the set { \(z = a + ib ∈ C : a,b ∈ Z\) and \(1 < | z - 3 + 2i | < 4\) } is _______ .
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 _______.
Let R1 and R2 be relations on the set {1, 2, ….., 50} such that R1 ={(p, pn) : p is a prime and n≥ 0 is an integer} and R2 = {(p, pn) : p is a prime and n = 0 or 1}. Then, the number of elements in R1 – R2 is ______.
The number of real solutions of the equation e4x + 4e3x - 58e2x + 4ex + 1 = 0 is _____.
The mean and standard deviation of 15 observations are found to be 8 and 3, respectively. On rechecking, it was found that, in the observations, 20 was misread as 5. Then, the correct variance is equal to _______.
If \(\vec a= 2\hat i +\hat j + 3\hat k\), \(\vec b = 3\hat i + 3\hat j + \hat k\) and \(\vec c = c_1\hat i + c_2\hat j + c_3\hat k\) are coplanar vectors and \(\vec a.\vec c = 5\), \(\vec b⊥\vec c\)  then \(122( c_1 + c_2 + c_3 )\) is equal to _____ .
Let AB and PQ be two vertical poles, 160 m apart from each other. Let C be the middle point of B and Q, which are feet of these two poles. Let \(\frac \pi 8\) and \(θ\) be the angles of elevation from C to P and A, respectively. If the height of pole PQ is twice the height of pole AB, then \(tan^2θ\) is equal to
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?
If the system of linear equations.
\(8x + y + 4z = –2\)
\(x + y + z = 0\)
\(λx–3y=μ\)
has infinitely many solutions, then the distance of the point \((λ, μ, -1/2)\) from the plane \(8x + y + 4z + 2 = 0\) is
The odd natural number a, such that the area of the region bounded by \(y = 1, y = 3, x = 0, x = y^a\) is \(\frac {364}{3}\), is equal to
Consider two G.Ps. \(2, 2^2, 2^3, ….\) and \(4, 4^2, 4^3, …\) of \(60\) and n terms respectively. If the geometric mean of all the \(60 + n\) terms is \((2)^{\frac {225}{8}}\) then \(\sum_{k=1}^{n}k(n−k)\) is equal to
If the function
\(f(x) =   \begin{cases}    \frac {log_e(1−x+x^2)+log_e(1+x+x_2)}{sec⁡\ x−cos⁡\ x},      & \quad x∈(−\frac \pi2,\frac \pi2)−{0}\\     k,  & \quad x=0   \end{cases}\)
is continuous at \(x = 0\), then k is equal to

Let
A=\(\begin{bmatrix} 2 & -1 \\ 0 & 2 \end{bmatrix}\)
If B = I – 5C1(adjA) + 5C2(adjA)2 – …. – 5C5(adjA)5, then the sum of all elements of the matrix B is

The sum of the infinite series
\(1 + \frac{5}{6} + \frac{12}{6^2} + \frac{22}{6^3} + \frac{35}{6^4} +\frac{51}{6^5} + \frac{70}{6^6}+…..\)
is equal to

Let f : R → R be a function defined by:
\(ƒ(x) = (x-3)^{n_1} (x-5)^{n_2} , n_1, n_2 ∈ N\)
Then, which of the following is NOT true?

Let f be a real valued continuous function on [0, 1] and
\(f(x) = x + \int_{0}^{1} (x - t) f(t) \,dt\)
Then, which of the following points (x, y) lies on the curve y = f(x)?

If
\(\int_{0}^{2} (\sqrt{2x} - \sqrt{2x - x^2}) \,dx = \int_{0}^{1} \left(1 - \sqrt{1 - y^2} - \frac{y^2}{2}\right) \,dy + \int_{1}^{2} (2 - \frac{y^2}{2}) \,dy + I\)
then I equal is

Let P : y2 = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of π/4 with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are collinear is

Let a triangle ABC be inscribed in the circle
\(x² - \sqrt2(x+y)+y² = 0\)
such that ∠BAC= π/2. If the length of side AB is √2, then the area of the ΔABC is equal to :