Mathematics Test-3

Directions For Questions

Let \(\psi_{1}:[0, \infty) \rightarrow \mathbb{R}, \psi_{2}:[0, \infty) \rightarrow \mathbb{R}, f:[0, \infty) \rightarrow \mathbb{R}\) and \(g:[0, \infty) \rightarrow \mathbb{R}\) be functions such that \(f(0)=g(0)=0\),

\(\psi_{1}(x)=e^{-x}+x, x \geq 0 \)

\(\psi_{2}(x)=x^{2}-2 x-2 e^{-x}+2, x \geq 0 \)

\(f(x)=\int_{-x}^{x}\left(|t|-t^{2}\right) e^{-t^{2}} d t, x>0 \)

and \(g(x)=\int_{0}^{x^{2}} \sqrt{t} e^{-t} d t, x>0\)

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Which of the following statements is TRUE?

Mathematics Tes...

Question 1

\(\frac{(\sin x-x \times \cos x \times \log x)}{x}\)

\(\frac{(\sin x+x \times \cos x \times \log x)}{x}\)

\(\frac{x^{\sin x}(\sin x+x \times \cos x \times \log x)}{x}\)

\(\frac{x^{\sin x}(\sin x-x \times \cos x \times \log x)}{x}\)

Question 2

\(A\) is a finite set but \(B\) is an infinite set

\(A\) is an infinite set but \(B\) is a finite set

Both \(A\) and \(B\) are infinite sets

Both \(A\) and \(B\) are finite sets

Question 3

\(n \in N\)

\(n \in R\)

\(n \in I\)

\(n \in Z\)

Question 4

\({ }^{10} \mathrm{C}_{5}\)

\({ }^{10} \mathrm{C}_{4}\)

\({ }^{10} \mathrm{C}_{3}\)

\({ }^{10} \mathrm{C}_{2}\)

Question 5

\(\frac{x^{2}}{8}-\frac{y^{2}}{6}=1\)

\(\frac{x^{2}}{4}-\frac{y^{2}}{5}=1\)

\(\frac{x^{2}}{5}-\frac{y^{2}}{4}=1\)

None of the above

Question 6

\(\frac{\pi}{2}\)

0

\(\pi\)

\(\frac{\pi}{4}\)

Question 7

1680

7500

5880

7560

Question 8

\(\frac{7}{12}\)

\(\frac{7}{16}\)

\(\frac{16}{7}\)

\(\frac{12}{7}\)

Question 9

3

4

5

9

Question 10

\(x+2 y+2 z=1\)

\(x+2 y+2 z=3\)

\(x+2 y+2 z=\frac{1 }{ 3}\)

\(x+2 y+2 z=0\)

Question 11

\(f(\sqrt{\ln 3})+g(\sqrt{\ln 3})=\frac{1}{3}\)

For every \(x>1\), there exists an \(\alpha \in(1, x)\) such that \(\Psi_{1}(x)=1+\alpha x\)

For every \(x>0\), there exists a \(\beta \in(0, x)\) such that \(\Psi_{2}(x)=2 x\left(\Psi_{1}(\beta)-1\right)\)

\(f\) is an increasing function on the interval \(\left[0, \frac{3}{2}\right]\)

Question 12

\(a=3, b=-1\)

\(a=3, b=1\)

\(\ a=-3, b=1\)

\(a=-3, b=-1\)

Question 13

\(-\frac{2 p q}{a b}\)

\(\frac{2 p q}{a b}\)

\(-\frac{p q}{a b}\)

\(\frac{p q}{a b}\)

Question 14

\(\left(2+e^{2}\right)^{2}\)

\(\left(2-e^{2}\right)^{\frac{1}{2}}\)

\(\left(2+e^{2}\right)^{\frac{1}{2}}\)

\(\left(2-e^{2}\right)^{\frac{3}{2}}\)

Question 15

\(30^{\circ}\)

\(45^{\circ}\)

\(60^{\circ}\)

\(90^{\circ}\)

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