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Fundamental period of the function $f\left(x\right)=|\sin \pi x|$ is

The equation of the base of an equilateral triangle is $x+y=2$ and its vertex is (2, -1). Find the length and equations of its sides.

If $\int \frac{dx}{(x-4)\sqrt{x+5}}=\frac{1}{3}\ln \left|\frac{\sqrt{f\left(x\right)}-\alpha }{\sqrt{f\left(x\right)}+\beta }\right|+C$ where $(\alpha ,\beta ,C)$ are constants, then which of the following is/are CORRECT?

If the function $f\left(x\right)=\{\begin{array}{cc}\frac{1}{x}{\log }_e\left(\frac{1+\frac{x}{a}}{1-\frac{x}{b}}\right),& x<0\\ k,& x=0\\ \frac{{\cos }^{2}x-{\sin }^{2}x-1}{\sqrt{x^2+1}-1},& x>0\end{array}$

is continuous at $x=0,$ then $\frac{1}{a}+\frac{1}{b}+\frac{4}{k}$ is equal to:

Find the
equations of the tangent and normal to the given curves at the
indicated points:

(i) y = x⁴ − 6x³ +
13x² − 10x + 5 at (0, 5)

(ii) y
= x⁴ − 6x³ + 13x²
− 10x + 5 at (1, 3)

(iii) y = x³ at (1, 1)

(iv) y = x² at (0, 0)

(v) x = cos t, y = sin t at

If $0\le x<\frac{\pi }{2}$, then the number of values of $x$ for which $\sin x-\sin 2x+\sin 3x=0$, is :

Find the area of the region bounded by the parabola $y=x^2$ and y = |x|.

Area of a
rectangle having vertices A, B, C, and D with position vectors
and

respectively is

(A) (B) 1

(C) 2 (D)

The sum of coefficients of intergral power of $x$ in the binomial expansion ${(1-2\sqrt{x})}^{50}$is:

If $6{\sin }^4\theta +3{\cos }^4\theta =2,$ where $\theta \in R,$ then the value of $8{\sec }^6\theta +3{\text{cosec}}^6\theta$ is equal to

The area of the region bounded by the curve $y=x-x^2$ and the line $y=mx$ equals $\frac{9}{2}\text{sq.units}$. Then the possible values of $m$ is/are:

If the roots of $x^2-5x+4=0$ are $p,q$, then which of the following CANNOT be the equation with the roots $\sqrt{p},\sqrt{q}$ ?

The equation of the line, passing through the centre and bisecting the chord $7x+y-1=0$ of the ellipse $\frac{x^2}{1}+\frac{y^2}{7}=1,$ is

For every natural number n, n$(n^2-1)$ is divisible by

If $f\left(x\right)$ is a quadratic polynomial such that graph of $y=f\left(x\right)$ touches at $(4,0)$ and intersects the positive $y-$axis at $4$, then which of the following is/are correct?

The value of $\underset{0}{\overset{\pi /4}{\int }}\frac{\sin 2x}{({\sin }^{4}x+{\cos }^{4}x)}dx$ is

The value of $\underset{x\to \infty }{lim}{2}^x\sin \left(\frac{a}{2^x}\right)$ is

If a line in the space makes angles $\alpha ,\beta$ and $\gamma$ with the coordinate axes, then $\cos 2\alpha +\cos 2\beta +\cos 2\gamma +{\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =$

Prove that

Find the equation of the straight line drawn through the point of intersection of the lines $x+y=4$ and $2x-3y=1$ and perpendicular to the line cutting of intercepts 5,6 on the axes.

If $p,q,r$ are in A.P. and $p^2,q^2,r^2$ are in G.P, where $p<q<r$ and $p+q+r=\frac{3}{2},$ then the value of $p$ is

If $ABCD$ (in order) is a quadrilateral inscribed in a circle, then which of the following is/are always true?

How many words can be formed out of the letters of the word 'ARTICLE', so that vowels occupy even places?

While throwing a pair of dice, an event A is defined as 'sum of faces will be at least 10'. Find the total number of favorable outcomes.

$7^{2n}+2^{3n-3}{.3}^{n-1}$ is divisible by 25 for all $nϵ$ N.

The equation of parabola, whose axis is parallel to $y-$axis and which passes through points $(0,2),(-1,0)$ and $(1,6)$ is

If the standard deviation of $X$ is $\sigma$, then s.d. of the variable $U=\frac{aX+b}{c}$ where $a,b,c$ are constants is

If the lines represented by the equation $2x^2-3xy+y^2=0$ make angles $\alpha$ and $\beta$ with x - axis, then $co{t}^2\alpha +co{t}^2\beta =$