List of top Mathematics Questions on types of differential equations

The population $ p(t) $ of a certain mouse species follows \[ \frac{dp}{dt} = 0.5p - 450. \] If $ p(0)=850 $, then the time at which population becomes zero is:}

If \[ x = \int_0^y \frac{1}{\sqrt{1+9t^2}} \, dt \quad \text{and} \quad \frac{d^2 y}{dx^2} = ay, \] then $ a $ is equal to:

Let the system of equations $x+2y+3z = 5$, $2x+3y+z = 9$, $4x+3y+λz = μ$ have an infinite number of solutions. Then $λ + 2μ$ is equal to

The area enclosed by the closed curve $C$ given by the differential equation $\frac{d y}{d x}+\frac{x+a}{y-2}=0, y(1)=0$ is $4 \pi$.

Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis If normals at $P$ and $Q$ on the curve $C$ intersect $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $R S$ is

Let $y=y(t)$ be a solution of the differential equation$\frac{d y}{d t}+\alpha y=\gamma e^{-\beta t}$ where, $\alpha > 0, \beta>0$ and $\gamma > 0$. Then $\displaystyle\lim _{t \rightarrow \infty} y(t)$

The negation of the expression $q \vee((\sim q) \wedge p)$ is equivalent to

Among the two statements
(S1) : (p ⇒ q)∧(q ∧(~q)) is a contradiction and
(S2) : (p∧q) v ((~p)∧q) v (p ∧ (~q)) v ((~p) ∧ (~q)) is a tautology

$lim _{n→∞}{(2^½-2^½)(2^½-2^½)….(2^½-^2{\frac{1}{2n+1}})}$ is equal to

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 s={$z=x+ry:\frac{2z-3i}{4z+2i}$ is a real number}. Then which of the following is not correct?

If the coefficient of x⁷ in ($ax^2 +\frac{ 1}{2bx}^{11}$) and x^-7 in ($ax - \frac{1}{3bx^2}^{11}$), are euual then

The statement ~[p v (~(p ∧ q))] is equivalent to

Let $y=y(x)$ be the solution of the differential equation $\left(3 y^2-5 x^2\right) y d x+2 x\left(x^2-y^2\right) d y=0$ such that $y(1)=1$ Then $\left|(y(2))^3-12 y(2)\right|$ is equal to :

the points P and Q are respectively the circumcentre and the orthocentre of a ΔABC, then $ \overrightarrow{PA}+ \overrightarrow{PB}+ \overrightarrow{PC}$ is

if $2x^y+3y^x=20$ then $\frac{dy}{dx}$ at (2,2) is equal to:

If the equation of the normal to the curve $𝑦= 𝑥−𝑎 (𝑥+𝑏)(𝑥−2)$ at the point $(1,−3)$ is $𝑥−4𝑦=13$, then the value of $𝑎+𝑏$ 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

$(S1) : lim_{n→∞}\frac{1}{ n ^2} ( 2 + 4 + 6 + . . . . + 2n) = 1$
$(S2): lim_n→∞\frac{1}{n^{16}}( 1^15 +2^15 +3^15 + . . . . + n^15 ) = \frac{1}{16} $

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

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

If the system of equations
x + y + az = b
2x + 5y + 2z = 6
x + 2y + 3z = 3
Has infinitely many solutions, then 2a + 3b is equal to

If the point $(α ,\frac{7√3}{3}) $lies on the curve traced by the mid-points of the line segments of the lines x cosθ + y sinθ = 7, θ ∈ (0 , $\frac{π}{2}$ ) between the coordinates axes, then α is equal to

Two dice A and B are rolled, Let the numbers obtained on A and B be α and β respectively. If the variance of α - β is $\frac{p}{q}$, where p and q are coprime, then the sum of the positive divisors of p is equal to

Let g(x) = f(x) + f(1 - x) and f''(x) > 0, x ∈ (0,1). If g is decreasing in the interval (0, α) and increasing in the interval (α, 1), then tan-1 (2α) + tan^-1 ($\frac{1}{α}$) + tan^-1$(\frac{α+1}{α})$ is equal to

For, α, β, γ, δ ∈ $\mathbb{N},$ if $∫\left((\frac{x}{e})^{2x}+(\frac{e}{x})^{2x}\right)log_e\,xdx=$ $\frac{1}{\alpha}(\frac{x}{e})^{βx}-\frac{1}{γ}(\frac{e}{x})^{δx}+c,$ Where $e=\displaystyle\sum_{n=10}^{∞}\frac{1}{n!}$ and C is constant of integration, then α + 2β + 3γ - 4δ is equal to