Chapter 7

Let $\mathrm{a},\mathrm{b},\mathrm{c}$ are three Arithmatic means between two numbers such that $a+b+c=15$ and p,q,r be Harmonic means between same numbers such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{5}{3}$, then the numbers are (1) 3,10 (2) 3,12 (3) 3,15 (4) 9,1

If $(1-p)(1+3x+9x^2+27x^3+81x^4+243x^5)$ $=1-{\mathrm{p}}^6,\mathrm{p}\ne 1$, then the value of $\frac{\mathrm{p}}{\mathrm{x}}$ may be equal to (1) $\frac{1}{3}$ (2) 3 (3) $\frac{1}{2}$ (4) 2

$\int \frac{\sec x}{a+b\tan x}dx$ is equal to (1) $\frac{1}{\sqrt{a^2+b^2}}$ \elln $\tan \frac{x}{2}+C$ (2) $\frac{1}{\sqrt{a^2+b^2}}\ell n(\tan \left(\frac{x+{\tan }^{-1}\left(\frac{a}{b}\right)}{2}\right))+C$ (3) $\frac{1}{\sqrt{a^2+b^2}}\ell n(\tan \left(\frac{x+{\tan }^{-1}\left(\frac{b}{a}\right)}{2}\right))+C$ (4) $\frac{1}{\sqrt{a^2+b^2}}$ \elln $\sin \frac{x}{2}+C$

If the roots of the equation $x^2+px+c=0$ are $2,-2$ and the roots of the equation $x^2+bx+q=0$ are $-1,-2$, then the roots of the equation $x^2+bx+c=0$ are $(1)-3,-2$ $(2)-1,4$ (3) $1,-4$ (4) $-2,3$

At the foot of a mountain the elevation of its peak is found to be $\frac{\pi }{4}$. After ascending 'h' m toward the mountain up a slope of $\frac{\pi }{6}$ inclination, the elevation is found to be $\frac{\pi }{3}$. Height of the mountain is (1) $\frac{h}{2}(\sqrt{3}+1)m$ (2) $\mathrm{h}(\sqrt{3}+1)\mathrm{m}$ (3) $\frac{\mathrm{h}}{2}(\sqrt{3}-1)\mathrm{m}$ (4) $\mathrm{h}(\sqrt{3}-1)\mathrm{m}$

In how many ways can five different examination papers of maths, physics, chemistry, computer and physical education be arranged so that physics and chemistry papers never come together (1) 31 (2) 48 (3) 60 (4) 72

Let $\mathrm{P}$ and $\mathrm{Q}$ be two statements, then $\sim (\sim \mathrm{P}\wedge \mathrm{Q})\wedge (\mathrm{P}\vee \mathrm{Q})$ is logically equivalent to (1) $Q$ (2) $\mathrm{P}$ (3) $P\vee Q$ (4) $\mathrm{P}\wedge \mathrm{Q}$

The probabilities of different faces of biased dice to appear are as follows Face number $12$ $\begin{array}{lll}3& 4& 5\end{array}$ Probability $\begin{array}{llllll}0.1& 0.32& 0.21& 0.15& 0.05& 0.17\end{array}$ The dice is thrown and it is known that either the face number 1 or 2 will appear. Then, the probability of the face number 1 to appear is (1) $5/21$ (2) $5/12$ (3) $7/23$ (4) $3/10$

The set of all values of a for which equation $\frac{x^2}{8-a}+\frac{y^2}{a-2}=1$ will represent an ellipse, is (1) $(1,4)$ (2) $(-\mathrm{\infty },2)\cup (8,\mathrm{\infty })$ (3) $$\begin{gathered} (2,8)- \\ 5 \end{gathered}$$ (4) $(2,8)$

Consider the following statements : ${\mathrm{S}}_1$ : For an odd function $\mathrm{f}(\mathrm{x})$, graph of $y=f(x)$ always passes through origin. ${\mathbf{S}}_2$ : If $f$ and $g$ are two bijective function then $\mathrm{f}(\mathrm{g}(\mathrm{x}))$ is also bijective. ${\mathbf{s}}_3:$ All points of intersection of $y=f(x)$ and $y=f^{-1}(x)$ lies on $y=x$ only. State, in order, whether ${\mathrm{S}}_1,{\mathrm{S}}_2,{\mathrm{S}}_3,{\mathrm{S}}_4$ are true or false (1) T F T (2) T T F (3) $\mathrm{FTT}$ (4) F FF