List of top Mathematics Questions

The number of points, where the curve y = x⁵ – 20x³ + 50x + 2 crosses the x-axis, is _____.

Let a curve y = f(x), x ∈ (0, ∞) pass through the points  \(p ( 1\frac{3 }{2} )\) and  Q \( ( a,\frac{1 }{2} )\). If the tangent at any point R(b, f(b)) to the given curve cuts the y-axis at the point S (0, c) such that bc = 3, then (PQ)² is equal to ______.

Let f (x) = \(\frac{x}{ (1+x ^n )^{\frac{1}{n} }}\), x∈R-{-1}, n∈N. n>2. 
If fⁿ (x) = (fofof____upto n times) (x), then \(lim ∫^1_ {n→∞ 0} x^{n-2}\)( f ⁿ(x))dx is equal to ___.

Let the eccentricity of an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), is reciprocal to that of the hyperbola 2x²-2y²=1. if the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is___.

If the lines \(\frac{x−1}{2}=\frac{2−y }{−3} =\frac{ z−3 }{α}\)  and \( \frac{x−4 }{5} =\frac{y−1}{ 2}= \frac{z}{ β}\) intersect, then the magnitude of the minimum value of 8αβ is ___.

The value of tan 9°-tan27°-tan63°+tan81° is____.

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 _____

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 for x∈R, S₀(x)=x,Sk(x)=Ckx+k\(∫^x_0S_{k−1}\)(t)dt, where C₀=1,C_k≡1−\(∫^1_0S_{k−1}\)(x)dx,k==1,2,3,..... Then s₂(3)+6c₃ is equal to___.

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 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 ____.

Let A = {1, 2, 3, 4,.....10} and B = {0, 1, 2, 3, 4}. The number of elements in the relation R = {(a, b) ∈ A × A: 2(a – b)2 + 3(a – b) ∈ B} is______.

Let the point (p, p+1) lie inside the region \(E = \{(x,y):3-x≤y≤\sqrt{9-x^2}, 0 ≤ x ≤ 3\} \)If the set of all values of p is the interval (a,b). then b2 + b - a2 is equal to_____

Let a∈ Z and [t] be the greatest integer t. Then the number of points, where the function f(x)=[a+13 sin x], x ∈ (0, π) is not differentiable, is ____

If the area of the region \( S={(x,y) : 2y-y^2 ≤ x^2 ≤ 2y, x ≥ y}\) is equal to \(\frac{ n+2}{n+1}−\frac{π}{n−1}\), then the natural number n is equal to _____.

Let the tangent to the curve x² + 2x – 4y + 9 = 0 at the point P(1, 3) on it meet the y-axis at A. Let the line passing through P and parallel to the line x – 3y = 6 meet the parabola y² = 4x at B. If B lies on the line 2x – 3y = 8. then (AB)² is equal to___

Let y = y(x) be a solution of the differential equation (x cos x)dy +(xy sin x + y cos x – l)dx = 0, 0 < x < \(\frac{π}{2}\). If\(\frac{π}{3}y(\frac{π}{3})=\sqrt3\), then \(| \frac{π}{6 }y“( \frac{π}{6})+2 y'( \frac{π}{6})| \) is equal to______.

The number of relations, on the set {1,2,3} containing (1,2) and (2,3), which are reflexive and transitive but not symmetric, is______

Let D_k=\(\begin{vmatrix} 1 & 2k & 2k-1\\ n & n^2+n+2 & n^2 \\ n & n^2+n & n^2+n+2 \end{vmatrix} \) If \(∑^n_{k=1} D_k=96,\) The n is equal to

Let the digits a, b, c be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

Let [x] be the greatest integer ≤ x. Then the number of points in the interval (-2,1), where the function f(x)=|[x]|+√x−[x] is discontinuous is _____.

Let I(x) = \(∫√\frac{x+7}{x} dx\) and I (9) = 12 + 7 loge 7. If I (1) = α + 7 loge (1 + 2√2), then α4 is equal to_____.