Ncert Solutions Maths class 11th

Let g(t) = $\int_{- π / 2}^{π / 2} c o s (\frac{π}{4} t + f (x)) d x,$ where f(x) = log_e $(x + \sqrt[]{x^{2} + 1}), x \in R .$ Then which one of the following is correct?

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$? f (x) = l n (x + \sqrt[]{1 + x^{2}}) ∴ f (- x) = - f (x)$ .

Hence f(x) is an odd function.

$N o w g (t) = \int_{- \frac{π}{2}}^{\frac{π}{2}} c o s (\frac{π}{4} t + f (x)) d x$

Put t = 0, g(0) = $\int_{- \frac{π}{2}}^{\frac{π}{2}} c o s (f (x)) d x = 2 \int_{0}^{\frac{π}{2}} c o s (f (x)) d x . . . . . . . . . . . . . . . (i)$

where cos(f(x)) is an even function.

Now again put t = 1,

$∴ g (1) = \frac{1}{\sqrt[]{2}} * g (0), f r o m (i)$

$∴ g (0) = \sqrt[]{2} g (1)$

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Let r₁ and r₂ be the radii of the largest and smallest circles, respectively, which pass through the point (-4, 1) and having their centres on the circumference of the circle x² + y² + 2x + 4y – 4 = 0. If $\frac{r_{1}}{r_{2}}$ = a + $b \sqrt[]{2},$ then a + b is equal to:

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Centre of the smallest circle is A

Centre of the largest circle is B

$r_{1} = | C P + C B | = 3 \sqrt[]{2} + 3$ and

$r_{2} = | C P - C B | = 3 \sqrt[]{2} - 3$

$\frac{r_{1}}{r_{2}} = \frac{3 \sqrt[]{2} + 3}{3 \sqrt[]{2} - 3} = 3 + 2 \sqrt[]{2}$

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Let P be a variable point on the parabola y = 4x² + 1. Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn from the point P to the line y = x is:

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Let mid point of PQ is R (h, k)

$∴ h = \frac{α + \frac{4 α^{2} + α + 1}{2}}{2} a n d$

$k = \frac{4 α^{2} + 1 + \frac{4 α^{2} + α + 1}{2}}{2}$

Eliminate a from above these two, we get

2 (3x – y)² + (x – 3y) + 2 = 0

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The value of $t a n (2 t a n^{- 1} (\frac{3}{5}) + s i n^{- 1} (\frac{5}{13})) i s$ equal to:

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$t a n^{- 1} (\frac{2 * \frac{3}{5}}{1 - \frac{9}{25}}) + s i n^{- 1} \frac{5}{13} \Rightarrow t a n^{- 1} \frac{15}{8} + t a n^{- 1} \frac{5}{12} = t a n^{- 1} \frac{\frac{15}{8} + \frac{5}{12}}{1 - \frac{15}{8} \frac{5}{12}} = t a n^{- 1} \frac{220}{21}$

$∴ t a n (t a n^{- 1} \frac{220}{21}) = \frac{220}{21}$

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A man starts walking from the point P(-3, 4), touches the x-axis at R, and then turns to reach at the point Q(0, 2). The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then 50(PR)² + (RQ)² is equal to__________.

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$\Rightarrow \frac{6}{3} = \frac{2}{- t} \Rightarrow t = - 1$

->R (-1,0)

$P R^{2} + R Q^{2} = 20 + 5 = 25$

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All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is________.

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Let the points of intersection of the lines x – y + 1 = 0, x – 2y + 3 = 0 and 2x – 5y + 11 = 0 are the mid points of the sides of a triangle ABC. Then the area of the triangle ABC is_____.

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ar (ABC) = 4ar (DEF)

$= 4 * \frac{1}{2} * | 2 (2 - 5) + 1 (5 - 3) + 7 (3 - 2) | = 2 | - 6 + 2 + 7 | = 6$

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If for the complex numbers z satisfying $| z - 2 - 2 i | \leq 1,$ the maximum value of is attained at then a + b is equal to__________.

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Which of the following is equivalent to the Boolean expression $p \land \sim q ?$

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$\sim (\sim p \lor q) = p \land \sim q$

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The number of pairs (a, b) of real numbers, such that whenever a is a root of the equation x² + ax + b = 0, a² – 2 is also a root of this equation , is :

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