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Let $f:R\to R$ and $g:R\to R$ be a continous functions. Then the value of the integral

$\underset{-\pi /2}{\overset{\pi /2}{\int }}\left[f\right(x)+f(-x\left)\right]\left[g\right(x)-g(-x\left)\right]dx$ is

The parametric equation of parabola $(y-2{)}^2=12(x-4)$ is

If $R+r=r_3$ then $\angle C=$

For different values of k, the circle $x^2+y^2$ + (8 + k)x + (8 + k)y + (16 + 12k) = 0, always passes through two fixed point P and Q. For $k=k_1$, the tangents at P and Q intersect at the origin. Which of the following is/are correct?

The greatest positive integer $k$, for which ${49}^k+1$ is a factor of the sum ${49}^{125}+{49}^{124}+\cdots +{49}^2+49+1,$ is

Let $A$ be a set of all $4-$digit natural numbers whose exactly one digit is $7.$ Then the probability that a randomly chosen element of $A$ leaves remainder $2$ when divided by $5$ is:

A function $f:[-3,7)\to \text{R}$ is defined as follows:

$f\left(x\right)=$$(\begin{array}{c}4x^2-1;-3\le x<2\\ 3x-2;2\le x\le 4\\ 2x-3;4<x<7\end{array}$



Find: $\frac{f\left(3\right)+f(-1)}{2f\left(6\right)-f\left(1\right)}$

The equation $x^{\frac{3}{4}(lo{g}_{2}x{)}^2+lo{g}_{2}x-\frac{5}{4}}=\sqrt{2}$ has

The order and degree of the differential equation $y=x\frac{dy}{dx}+\sqrt{a^2{\left(\frac{dy}{dx}\right)}^2+b^2}$ are

Let the observations $x_i(1\le i\le 10)$ satisfy the equations, $\sum _{i=1}^{10}(x_i-5)=10$ and $\sum _{i=1}^{10}(x_i-5{)}^2=40$. If $\mu$ and $\lambda$ are the mean and the variance of observations, $(x_1-3),(x_2-3),...,(x_{10}-3)$, then the ordered pair $(\mu ,\lambda )$ is equal to :

If the equation of a straight line $3x+4y+12=0$ is transformed into normal form $x\cos \alpha +y\sin \alpha =p$, then the value of $\tan \alpha +\cot \alpha$ is equal to

Match the possible scenarios for system of linear equations in 3 variables.

Choose the correct answer in the following question:

The normal at the point (1, 1) on the curve $2y+x^2=3$ is

(a) x + y = 0 (b) x - y = 0 (c) x + y + 1 = 0 (d) x - y + 1 = 0

Which of the following statements are not well defined?​



$\text{I}.$ Set of best cricketers in the Indian cricket team

$\text{II}.$ Set of odd natural numbers divisible by $2$

$\text{III}.$ Set of letters in the word $\text{BYJUS}$

$\text{IV}.$ Set of large numbers

$x^2-y^2+5x+8y-4=0$ represents

The equation of the curve that passes through the point (1,2) and satisfies the differential equation $\frac{dy}{dx}=\frac{-2xy}{(x^2+1)}$ is

If $a,b>0$ and $\underset{0}{\overset{\infty }{\int }}\frac{\ln \left(bx\right)}{x^2+a^2}dx=\frac{\pi }{2a},$ then the value of $ab$ is

If the function $f\left(x\right)=\frac{k\sin x+2\cos x}{\sin x+\cos x}$ is strictly increasing for all values of $x$ in its domain, then

Prove that

The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.

If $P$ is the set of all Irrational Numbers,

$Q$ is the set of all Rational Numbers,

$Z$ is the set of all Integers,

then which of the following is correct?