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The value of $\int \frac{(1-{\cos }^{2}x)\tan x}{{\cos }^{4}x}dx$ is

Evaluate: ${\int }_0^{\frac{\pi }{4}}\sin 2xdx$

Construct
a 3 ×
4 matrix, whose elements are given by

(i) (ii)

For the given equation, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.​
$y=e^{2x}(a+bx)$

The eccentricity of the ellipse $9x^2+16y^2$ = 576 is

If with standard notations $t_1,t_2,t_3,t_4$ are four co–normal points on the hyperbola $xy=c^2$ then the orthocentre of the triangle formed by joining the points $t_1,t_2,t_3$ is given by

Find four numbers
forming a geometric progression in which third term is greater than
the first term by 9, and the second term is greater than the 4^th
by 18.

The value of ${\cos }^{-1}\left(\cos \frac{13\pi }{6}\right)$ is

The mean and variance of $7$ observations are $8$ and $16$ respectively. If two observations are $6$ and $8,$ then the variance of the remaining $5$ observations is

The value of $\underset{x\to 0}{lim}\left(\frac{x}{\sqrt[8]{81-\sin x}-\sqrt[8]{81+\sin x}}\right)$ is equal to

Two sets such that $n(B-A)=25;n(A-B)=15$ and $n(A\cap B)=10,$ then $n(A\cup B)=$

The ratio in which the line joining points (1,-2,3) and (4,2,-1) is divided by XOY plane is

Find coordinates of the centre of mass of a uniform $L-$shaped lamina (a thin flat plate) wrt origin, having dimensions as shown in figure. (Consider mass of lamina as $3mkg$)

The point of inflection for the function $f\left(x\right)=\frac{\ln x}{x}$ is:

At constant pressure, the volume of a fixed mass of a gas varies as a function of temperature as shown in the graph.

The volume of the gas at 300${}^{\circ }$C is larger than that at 0${}^{\circ }$C by a factor of

The sum of all values of $\theta \in (0,\frac{\pi }{2})$ satisfying ${\sin }^{2}2\theta +{\cos }^{4}2\theta =\frac{3}{4}$ is :

Let $f:R\to R$ be a function defined by $f(x+1)=\frac{f\left(x\right)-5}{f\left(x\right)-3}\forall x\in R$ then which of the following statements are true?

A bag contains 6 red and 4 blue balls (all are different). A fair die is rolled and number of balls equals to that appearing on the die is chosen from the bag at random. The probability that all the balls selected are red is

Find the 8th term from the end of the GP 3, 6, 12, 24, ..., 12288.

In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.

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:

Find the area of the circle 4x²
+ 4y² = 9 which is interior to the parabola x²
= 4y

Latus rectum of the parabola $y^2-4y-2x-8=0$ is

Which of
the following differential equation hasas
one of its particular solution?

A.

B.

C.

D.

The houses on one side of a road are numbered using consecutive even numbers. The sum of the numbers of all the houses in that row is 170. If there are at least 6 houses in that row and a is the number of the sixth house, then

The relation between keys of piano ($p$) and its notes ($n$) is given by: $p=13+7n$.



If the value of notes is $6$, the value of keys is

Find the equation of the chord of the ellipse $4x^2+25y^2=100$ whose middle point is (1,1).

The complete solution set of $4\cot 2\theta ={\cot }^2\theta -{\tan }^2\theta$ is