Ncert Solutions Maths class 12th

Let f be any continuous function on [0, 2] and twice differentiable on (0, 2). If f(0) = 0, f(1) = 1 and f(2) = 2, then

<!-- [if gte mso 9]><xml> </xml><![endif]--> $Let{a}_1,a_2,a_3,.....beanA.P.If\frac{a_1+a_2+...+a_{10}}{a_1+a_2+...+a_p}=\frac{100}{p^2},p\ne 10,$ then $\frac{a_{11}}{a_{10}}$ <!-- [if gte mso 9]><xml> </xml><![endif]--> is equal to :

If α + β + <!-- [if gte mso 9]><xml> </xml><![endif]-->= 2π, then the system of equations

$x+\left(cos\gamma \right)y+\left(cos\beta \right)z=0$        <!-- [if gte mso 9]><xml> </xml><![endif]-->

$\left(cos\gamma \right)x+y+\left(cos\alpha \right)z=0$
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$\left(cos\beta \right)x+\left(cos\alpha \right)y+z=0$
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has :

Let f : R -> R be a continuous function. The $\underset{x\to \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{lim}\frac{\frac{\pi }{4}{\displaystyle \underset{2}{\overset{se{c}^{2}x}{\int }}f\left(x\right)dx}}{x^2-\frac{{\pi }^2}{16}}$ <!-- [if gte mso 9]><xml> </xml><![endif]-->is equal to :

The area, enclosed by the curves y = sin x + cos x and y = |cos x – sin x| and the lines x = 0,  $x=\frac{\pi }{2},$ is:

Let S = {1, 2, 3, 4, 5, 6}. Then the probability that a randomly chosen onto function g from S to S satisfies g(3) = 2g (1) is :

Consider the system of linear equations

-x + y + 2z = 0

3x – ay + 5z = 1

2x – 2y – az = 7

Let S₁ be the set of all a $\in$ <!-- [if gte mso 9]><xml> </xml><![endif]--> R for which the system is inconsistent and S₂ be the set of all $a\in R$ <!-- [if gte mso 9]><xml> </xml><![endif]-->for which the system has infinitely many solutions. If $n\left(S_1\right)andn\left(S_2\right)$ <!-- [if gte mso 9]><xml> </xml><![endif]-->denote the number of elements in S₁ and S₂ respectively, then

The distance of the point (-1, 2, -2) from the line of intersection of the planes 2x + 3y + 2z = 0 and x – 2y + z = 0 is :

Let θ be the acute angle between the tangents to the ellipse $\frac{x}{9}+\frac{y}{1}=1$ <!-- [if gte mso 9]><xml> </xml><![endif]-->and the circle x² + y² = 3 at their point of intersection in the first quadrant. The tanq is equal to :

Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ <!-- [if gte mso 9]><xml> </xml><![endif]-->be three vectors mutually perpendicular to each other and have same magnitude. If a vector $\overrightarrow{r}$ <!-- [if gte mso 9]><xml> </xml><![endif]-->satisfies $\overrightarrow{a}\times \left\{\left(\overrightarrow{r}-\overrightarrow{b}\right)\times \overrightarrow{a}\right\}+\overrightarrow{b}\times \left\{\left(\overrightarrow{r}-\overrightarrow{b}\right)\times \overrightarrow{b}\right\}+\overrightarrow{c}\times \left\{\left(\overrightarrow{r}-\overrightarrow{a}\right)\times \overrightarrow{c}\right\}=\overrightarrow{0},$ <!-- [if gte mso 9]><xml> </xml><![endif]-->then $\overrightarrow{r}$ <!-- [if gte mso 9]><xml> </xml><![endif]-->is equal to