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Let S= {$x\in R:x\geq0$ and $2\mid \sqrt{x}-3\mid +\sqrt{x}(\sqrt{x}-6)+6=0$. Then , S

A) is an empty set

B) contains exactly one element

C) contains exactly two element

D) contains exactly four element

If α,β $\in$ C are the distinct roots of the equation $x^{2}-x+1=0, $ , then $\alpha^{101}+\beta^{107}$ is equal to

A) -1

B) 0

C) 1

D) 2

If the tangent at ( 1,7) to the curve $x^{2}=y-6$ touches the circle $x^{2}+y^{2}+16x+12y+c=0$, then the value of c is

A) 195

B) 185

C) 85

D) 95

Let $g(x)=\cos x ^{2}, f(x)=\sqrt{x}$ and $\alpha ,\beta (\alpha <\beta)$ be the roots of the quadratic equation

$18x^{2}-9\pi x +\pi^{2}=0$ .Then , the area (in sq units) bounded by the curve $y= (gof) (x)$ and the lines x=α, x=β

and y=0, is

A) $\frac{1}{2}(\sqrt{3}-1)$

B) $\frac{1}{2}(\sqrt{3}+1)$

C) $\frac{1}{2}(\sqrt{3}-\sqrt{2})$

D) $\frac{1}{2}(\sqrt{2}-1)$

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\sin ^{2}x}{1+2^{x}}$ is

A) $\frac{\pi}{8}$

B) $\frac{\pi}{2}$

C) $\frac{\pi}{4}$

D) $4\pi$

The length of the projection of the line segment joining the points (5,-1,4) and (4,-1,3) on the plane , x+y+z=7 is

A) $\frac{2}{\sqrt{3}}$

B) $\frac{2}{3}$

C) $\frac{1}{3}$

D) $\sqrt{\frac{2}{3}}$

The Integral

$\int_{}^{} \frac{\sin^{2}x cos^{2}x}{(\sin^{5}x+\cos^{3}x\sin^{2}x+\sin^{3}x\cos^{2}x+\cos^{5}x)^{2}}dx$

is equal to (where C is constant of integration)

A) $\frac{1}{3( 1+tan^{3}x)}+C$

B) $\frac{-1}{3( 1+tan^{3}x)}+C$

C) $\frac{1}{ 1+\cot^{3}x}+C$

D) $\frac{-1}{ 1+\cot^{3}x}+C$

If sum of all the solutions of the equation

$8\cos x.( \cos ( \frac{\pi}{6}+x).\cos ( \frac{\pi}{6}-x)-\frac{1}{2})=1$ in [0,π ] is kπ , then k is equal to

A) $\frac{2}{3}$

B) $\frac{13}{9}$

C) $\frac{8}{9}$

D) $\frac{20}{9}$

Let A be the sum of the first 20 terms and B be the sum of first 40 terms of the series

1²+2.2²+3²+2.4²+5²+2.6²+........ If B-2A=100λ then λ is equal to

A) 232

B) 248

C) 464

D) 496

If $\begin{bmatrix}x-4 & 2x & 2x\\2x & x-4 & 2x\\ 2x & 2x &x-4 \end{bmatrix}=\left(A+Bx\right)\left(x-A\right)^{2}$ then the ordered pair ( A,B) is equal to

A) ( -4, -5)

B) ( -4, 3)

C) ( -4,5)

D) ( 4,5)

Let a,b,c be three non-zero real numbers such that the equation $\sqrt{3} a \cos x+2b \sin x= c$ , $x\in[-\frac{\pi}{2},\frac{\pi}{2}]$, has two distinct real roots $\alpha$ and $\beta$ with $\alpha$ + $\beta$= $\frac{\pi}{3}$ , Then , the value of $\frac{b}{a}$ is.....

A) 0.5

B) 0.75

C) 1.5

D) 1

A farmer F₁ has a land in the shape of a triangle with vertices at P(0,0), Q(1,1), and R (2,0). From this land, a neighbouring farmer F₂ takes away the region which lies between the sides PQ and a curve of the form y=xⁿ(n>1). If the area of the region taken away by the farmer F₂ is exactly 30% of the area Δ PQR , then the value of n is........

A) 5

B) 4

C) 2

D) 1

Let $f:R\rightarrow R $ and $g :R\rightarrow R $ e two non-constant differentiable functions. If $f'(x)=(e^{(f(x)-g(x)})g'(x)$ for all $x \in R$ and f(1)=g(2)=1, then which of the following statement(s) is (are) TRUE ?

A) $f(2)<1-\log_{e}{2}$

B) $f(2)>1-\log_{e}{2}$

C) $g(1)>1-\log_{e}{2}$

D) $g(1)<1-\log_{e}{2}$

The value of integral $\int_{0}^{1/2} \frac{1+\sqrt{3}}{((x+1)^{2}(1-x)^{6})^{1/4}}dx$ is ............

A) 3

B) 4

C) 1

D) 2

Let $f:R\rightarrow R$ be a differentiable function with f(0)=0, y=f(x) satisfies the differential equation

$\frac{\text{d}y}{\text{d}x}= (2+5y)(5y-2)$ , then the value of $\lim_{x \rightarrow \infty} f(x) $ is .............

A) 0.6

B) 0.4

C) 0.8

D) 0.1

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