List of top Mathematics Questions

Let the image of the point \(\left(\frac{5}{3},\frac{5}{3},\frac{8}{3}\right)\) in the plane x - 2y + z - 2 = 0 be P. If the distance of the point Q(6,-2,α), α > 0, from P is 13, then α is equal to___.

Let \(\overrightarrow{a}=3\^i+\^j −\^k\)  and \(\overrightarrow{c}=2\^i+3\^j −3\^k\)  if \(\overrightarrow{b}\) is a vector such that  \(\overrightarrow{a}=\overrightarrow{b}\times\overrightarrow{c}\) and \(|\overrightarrow{b}|=50\) , then \(|72|\overrightarrow{b}+\overrightarrow{c}|^2|\)| |  is equal to _______. 

If S={ x ∈ R : sin⁻¹\(( \frac{x + 1}{ √ x^ 2 + 2 x + 2} ) \)−-sin⁻¹\((\frac{x}{√x ^2+1 }) = \frac{π}{ 4} \)}
then \(∑_{x∈R }(sin( (x^2+x+5)\frac{π}{2})−cos( ( x^2+x+5)π ) ) \)is equal to ____.

Find area bounded by the curves \(y\) = \(max\{sin x, cos x\}\) and \(x-axis\) between \(x = -  \pi  \)and \(x =   \pi\)

Let m1 and m2 be the slopes of the tangents drawn from the point P(4,1) to the hyperbola H : \(\frac{y^2}{25} − \frac{x^2}{16 }\)= 1. If Q is the point from which the tangents drawn to H have slopes |m₁| and |m₂| and they make positive intercepts α and β on the x-axis, then  \(\frac{( P Q ) ^2}{ α β}\) α β is equal to _____.

Let α be the constant term in the binomial expansion of \( (√x−\frac{6}{x^{\frac{3}{2}}} )^n ,n≤15.\)  If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of x^-n is λα, then λ is equal to _______.

Let ω = z\(\bar{z}\) + k₁z + k₂iz + λ(1+i), k₁ , k₂ ∈ \(\mathbb{R}\). Let Re(ω) = 0 be the circle C of radius 1 in the first quadrant touching the line y =1 and the y-axis. If the curve Im (ω) = 0 intersects C at A and B, then 30 (AB)² is equal to _____

The negation of the statement \(((A\land(B\lor C)) ⇒ (A\lor B)) ⇒ A\) is

Let O be the origin and the position vector of the point P be \(\hat i-2\hat j+3\hat k\). If the position vectors of the points A, B and C are \(-2\hat i+\hat j-3\hat k,2\hat i+4\hat j-2\hat k\) and \(-4\hat i+2\hat j-\hat k\) respectively then the projection of the vector \(\overrightarrow{OP}\) on a vector perpendicular to the vectors \(\overrightarrow{ AB}\) and \(\overrightarrow {AC}\) is

A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If X denotes the number of tosses of the coin, then the mean of X is

The distance of the point ( -1,2,3) from the plane \(\overrightarrow{r}( \^i−2 \^j+3 \^k )=10\) parallel to the line of the shortest distance between the lines \(\overrightarrow{r}=(\^i=\^j)+λ(2\^i+\^k)\) and  \(\overrightarrow{r}=(2\^i=\^j)+μ(\^i+\^j+\^k)\)  is 

Let PQ be a focal chord of the parabola y² = 36x of length 100, making an acute angle with the positive x-axis. Let the ordinate of P be positive and M be the point on the line segment PQ such that PM:MQ = 3:1. Then which of the following points does \(\underline{NOT}\) lie on the line passing through M and perpendicular to the line PQ? 

Let the tangent and normal at the point \((3\sqrt{3},1)\) on the ellipse \(\frac{ x^2}{36}+\frac{y^2}{4} =1\) meet the y-axis at the points A and B respectively. Let the circle C be drawn taking AB as a diameter and the line x = \(2\sqrt{5}\) intersect C at the points P and Q. If the tangents at the points P and Q on the circle intersect at the point (α,β) , then α² - β² is equal to 

For differentiable function f: R-{0}⟶ R, let 3f(x)+2f(\(\frac{1}{x}\))-\(\frac{1}{x}\)-10, then |f(3)+f(\(\frac{1}{4}\))| is equal to

The integral \(∫^∞_0  \frac{6}{(e^{3x} + 6e^{2x} + 11e{x} + 6)}dx\)

\(max\\{0≤x≤π}\) \(\{x−2\,\, sin. x \,\,cos+\frac{1}{3} sin\,\, 3x \}=\)

Let a =\(\^i+4\^j​+2\^k,b=3\^i−2\^j​+7\^k \,\,\,\,and\,\,\.\ c=2\^i−\^j​+4\^k\). Find a vector d satisfies \(\overrightarrow{d}\times{\overrightarrow{b}}=\overrightarrow{c}\times{\overrightarrow{b}}\) and \(\overrightarrow{d}.\overrightarrow{a}=24\), then \(|\overrightarrow{d}|^2\) is equal to

Let s₁, s₂, s₃,…..,s₁₀ respectively be the sum to 12 terms of 10 A.P.s whose first terms are 1,2,3, …. ,10 and the common differences are 1, 3, 5, …. , 19 respectively. Then  \(10∑^{10}_{i=1} s_i\) is

For the system of linear equations  
2x+4y+2az=b
x+2y+3z=4 
2x-5y+2z=8 
which of the following is NOT correct ?  

The set of all a∈ R for which the equation x|x-1|+|x-2|+a=0 has exactly one real root is