List of practice Questions

Let f and g be twice differentiable functions on R such that
f''(x)=g''(x)+6x
f''(1)=4g'(1)-3=9
f(2)=3g(2)=12.
Then which of the following is NOT true?

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

Let f be a continuous function satisfying $∫^{t ^2}_ 0 ( f ( x ) + x ^2 ) d x = \frac{4}{3} t^3 , ∀ t > 0.$ Then $f(\frac{π^2}{4})$ is equal to

Let $f(x)=\frac{sinx+cosx-\sqrt2}{sinx-cosx},x∈[0.\pi]-\left[\frac{\pi}{4}\right]. Then f\left(\frac{7\pi}{12}\right)f^n\left(\frac{7\pi}{12}\right)$is equal to

Let a, b, c be three distinct real numbers, none equal to one. If the vectors $a\^i+\^j+\^k , \^i+b\^j+\^k$ and $ \^i+\^j+c\^k $ are coplanar, then $\frac{1}{ 1 − a }+ \frac{1}{ 1 − b} + \frac{1 }{1 − c }$ is equal to

Let A₁ and A₂ be two arithmetic means and G₁, G₂, G₃ be three geometric means of two distinct positive numbers. Then $ G_{1}^{4} + G_{2}^{4}+ G_{3}^{4} +G_{1}^{2}G_{3}^{2} $ is equal to

Let A = {2, 3, 4} and B = {8, 9, 12}. Then the number of elements in the relation R = {((a₁, b₁), (a₂, b₂)) ∈ (A × B, A × B) : a₁ divides b₂ and a₂ divides b₁} is

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

The domain of the function f(x) = $\frac{1}{\sqrt{[x]^2-3[x]-10}}$ is (where [x] denotes the greatest integer less than or equal to x)

The sum of the coefficients of three consecutive terms in the binomial expansion of (1 + x)ⁿ⁺², which are in the ratio 1 : 3 : 5, is equal to

The random valuable X follows binomial distribution B (n, p) for which the difference of the mean and the variance is 1. If 2P(X = 2) = 3P(X = 1), then n²P(X > 1) is equal to

The straight lines l₁ and l₂ pass through the origin and trisect the line segment of the line L: 9x + 5y = 45 between the axes. If m₁ and m₂ are the slopes of the lines l₁ and l_2, then the point of intersection of the line y = (m₁ + m₂)x with L lies on

Let S = {$ x∈(−\frac{ π}{ 2} , \frac{π}{2} )$ : 9^1−tan2 x + 9^tan2 x=10 } and $β=∑_{ x∈s} t a n ^2 (\frac{ x}{ 3} )$ , then $\frac{1}{6}$ ( β − 14 )² is equal to

Let
$A=\{(x,y) ∈R^2:y≥0,2x≤y≤\sqrt{4-(x-1)^2} \}$
and
$B=\{(x,y) ∈R\times R:0≤y≤min \{2x,\sqrt{4-(x-1)^2}\}\}$
Then the ratio of the area of A to the area of B is

A line segment AB of length λ moves such that the points A and B remain on the periphery of a circle of radius λ. Then the locus of the point, that divides the line segment AB in the ratio 2 : 3, is a circle of radius

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 f be a differentiable function such that x² f(x) - x = $4 \int^x_0tf(t)dt,f(1)=\frac{2}{3}. \;\text{Then 18f(3)is equal to}$

Let N be the foot of perpendicular from the point P (1, –2, 3) on the line passing through the points (4, 5, 8) and (1, –7, 5). Then the distance of N from the plane 2x – 2y + z + 5 = 0 is

Let $λ ≠ 0$ be a real number. Let α, β be the roots of the equation $14x^2 – 31x + 3λ = 0$ and α, γ be the roots of the equation $35x^2 – 53x + 4λ = 0$. Then $\frac{3\alpha}{\beta}$ and $\frac{4\alpha}{\lambda}$ are the roots of the equation

Let A =$\left[\begin{matrix} 2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2 \end{matrix} \right]$. If |adj(adj(adj 2A)) | = (16)ⁿ, then n is equal to

If $ A = \begin{bmatrix} 1 & 5 \\ \lambda & 10 \end{bmatrix} $, $ A^{-1} = \alpha A + \beta I $ and $ \alpha + \beta = -2 $, then $ 4\alpha^2 + \beta^2 + \lambda^2 $ is equal to:

Let (a+bx+cx²)¹⁰ = $ \sum_{i=0}^{20} $ p_ixⁱ, a,b,c∈N. If p₁=20 and P₂ = 210, then 2(a+b+c) is equal to

If a plane passes through the points $(-1, k, 0),(2, k,-1),(1,1,2)$ and is parallel to the line $\frac{x-1}{1}=\frac{2 y+1}{2}=\frac{z+1}{-1}$, then the value of $\frac{k^2+1}{(k-1)(k-2)}$ is

Let the plane P : 4x – y + z = 10 be rotated by an angle $\frac{π}{2}$ about its line of intersection with the plane x + y – z = 4. If α is the distance of the point (2, 3, -4) from the new position of the plane P, then 35α is

Let$ |\vec{a}|=2, |\vec{b}|=3$ and the angle between the vectors $\vec{a}$ and $\vec{b}$ be $\frac{π}{4}$. Then $|(\vec{a}+2\vec{b})×(2\vec{a}-3\vec{b})|^2$ is equal to