STD 12 - 10. vector algebra — Mathematics STD 12 Science — Question

$1.$ The correct statement$(s)$ is(are)

$(A)$ $f^{\prime}(1) < 0$

$(B)$ $f(2) < 0$

$(C)$ $f^{\prime}(x) \neq 0$ for any $x \in(1,3)$

$(D)$ $f^{\prime}(x)=0$ for some $x \in(1,3)$

$2.$ If $\int_1^3 x^2 F^{\prime}(x) d x=-12$ and $\int_1^3 x^3 F^{\prime \prime}(x) d x=40$, then the correct expression$(s)$ is(are)

$(A)$ $9 f^{\prime}(3)+f^{\prime}(1)-32=0$

$(B)$ $\int_1^3 f(x) d x=12$

$(C)$ $9 f^{\prime}(3)-f^{\prime}(1)+32=0$

$(D)$ $\int_1^3 f(x) d x=-12$

Give the answer question $1$ and $2.$

$1.$  If $\mathrm{f}(-10 \sqrt{2})=2 \sqrt{2}$, then $\mathrm{f}^{\prime \prime}(-10 \sqrt{2})=$

$(A)$ $\frac{4 \sqrt{2}}{7^3 3^2}$ $(B)$ $-\frac{4 \sqrt{2}}{7^3 3^2}$ $(C)$ $\frac{4 \sqrt{2}}{7^3 3}$ $(D)$ $-\frac{4 \sqrt{2}}{7^3 3}$

$2.$ The area of the region bounded by the curves $y=f(x)$, the $x$-axis, and the lines $x=a$ and $x=b$, where $-\infty < \mathrm{a} < \mathrm{b} < -2$, is

$(A)$ $\int_a^b \frac{x}{3\left((f(x))^2-1\right)} d x+b f(b)-a f(a)$

$(B)$ $-\int_a^b \frac{x}{3\left((f(x))^2-1\right)} d x+b f(b)-a f(a)$

$(C)$ $\int_a^b \frac{x}{3\left((f(x))^2-1\right)} d x-b f(b)+a f(a)$

$(D)$ $-\int_a^b \frac{x}{3\left((f(x))^2-1\right)} d x-b f(b)+a f(a)$

$3.$ $\int_{-1}^1 g^{\prime}(x) d x=$

$(A)$ $2 g(-1)$ $(B)$ 0 $(C)$ $-2 g(1)$ $(D)$ $2 \mathrm{~g}(1)$

Give the answer question $1,2$ and $3.$

Statement $-1 :$ ${\rm{tr}}\left( A \right) = 0$

Statement $-2 :$  $\det \left( A \right) = 1$