Board Paper of Class 12-Humanities 2016 Maths Abroad(SET 2) - Solutions

Find the values of a and b, if the function f defined by $f\left(x\right)=\left\{\begin{array}{ccc}{x}^2+3x+a& ,& x⩽1\\ bx+2& ,& x>1\end{array}\right.$ is differentiable at x = 1. VIEW SOLUTION

Differentiate ${\tan }^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right) \mathrm{w}.\mathrm{r}.\mathrm{t}. {\sin }^{-1}\frac{2x}{1+x^2},$if x ∈ (–1, 1)

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If x = sin t and y = sin pt, prove that $\left(1-x^2\right)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+p^{2}y=0$ VIEW SOLUTION

Find the angle of intersection of the curves $y^2=4ax \mathrm{and} x^2=4by$. VIEW SOLUTION

Evaluate :$\underset{0}{\overset{\pi }{\int }}\frac{x}{1+\sin \alpha \sin x}dx$ VIEW SOLUTION

Find : $\int \left(2x+5\right)\sqrt{10-4x-3x^2}dx$
Find : $\int \frac{\left(x^2+1\right)\left(x^2+4\right)}{\left(x^2+3\right)\left(x^2-5\right)}dx$ VIEW SOLUTION

Find : $\int \frac{x {\sin }^{-1}x}{\sqrt{1-x^2}}dx$ VIEW SOLUTION

Solve the following differential equation :

$y^{2}dx+\left(x^2-xy+y^2\right)dy=0$ VIEW SOLUTION

Solve the following differential equation:

$\left({\cot }^{-1}y+x\right) dy=\left(1+y^2\right) dx$ VIEW SOLUTION

If $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{d} \mathrm{and} \overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{d}$, show that $\overrightarrow{a}-\overrightarrow{d}$  is parallel to $\overrightarrow{b}-\overrightarrow{c}$, where $\overrightarrow{a}\ne \overrightarrow{d} \mathrm{and} \overrightarrow{b}\ne \overrightarrow{c}$. VIEW SOLUTION

Prove that the line through A(0, –1, –1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(–4, 4, 4). VIEW SOLUTION

A box has 20 pens of which 2 are defective. Calculate the probability that out of 5 pens drawn one by one with replacement, at most 2 are defective.
Let, X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that
$\mathrm{P}\left(\mathrm{X} = x\right) = \left\{\begin{array}{lll}kx& ,& \mathrm{if} x = 0 \mathrm{or} 1\\ 2 kx& ,& \mathrm{if} x = 2\\ k\left(5-x\right)& ,& \mathrm{if} x = 3 \mathrm{or} 4\\ 0& ,& \mathrm{if} x > 4\end{array}\right.$
where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges. VIEW SOLUTION

Prove that :

${\cot }^{-1}\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}=\frac{\mathrm{x}}{2}, 0<x<\frac{\mathrm{\pi }}{2}$
 
Solve for x :

${\tan }^{-1} \left(\frac{x-2}{x-1}\right)+{\tan }^{-1} \left(\frac{x+2}{x+1}\right)=\frac{\mathrm{\pi }}{4}$ VIEW SOLUTION

A coaching institute of English (subject) conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich children and total monthly collection is Rs 9,000, whereas in batch II, it has 5 poor and 25 rich children and total monthly collection is Rs 26,000. Using matrix method, find monthly fees paid by each child of two types. What values the coaching institute is inculcating in the society? VIEW SOLUTION

Using integration find the area of the region bounded by the curves $y=\sqrt{4-x^2}, x^2+y^2-4x=0$ and the x-axis. VIEW SOLUTION

Find the equation of the plane which contains the line of intersection of the planes $x+2y+3z-4=0 \mathrm{and} 2x+y-z+5=0$ and whose x-intercept is twice its z-intercept. VIEW SOLUTION

Bag A contains 3 red and 5 black balls, while bag B contains 4 red and 4 black balls. Two balls are transferred at random from bag A to bag B and then a ball is drawn from bag B at random. If the ball drawn from bag B is found to be red find the probability that two red balls were transferred from A to B. VIEW SOLUTION

In order to supplement daily diet, a person wishes to take X and Y tablets. The contents (in milligrams per tablet) of iron, calcium and vitamins in X and Y are given as below :
 

Tablets	Iron	Calcium	Vitamin
X	6	3	2
Y	2	3	4

The person needs to supplement at least 18 milligrams of iron, 21 milligrams of calcium and 16 milligrams of vitamins. The price of each tablet of X and Y is Rs 2 and Rs 1 respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an LPP and solve graphically. VIEW SOLUTION

If f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x| –$x,\forall x\in \mathrm{R}$. Then find fog and gof. Hence find fog(–3), fog(5) and gof (–2). VIEW SOLUTION

If a, b and c are all non-zero and $\left|\begin{array}{ccc}1+\mathrm{a}& 1& 1\\ 1& 1+\mathrm{b}& 1\\ 1& 1& 1+\mathrm{c}\end{array}\right|=0,$ then prove that $\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}+1=0$
 
If $\mathrm{A}=\left(\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right),$ find adj·A and verify that A(adj·A) = (adj·A)A = |A| I₃. VIEW SOLUTION

The sum of the surface areas of a cuboid with sides x, 2x and $\frac{x}{3}$ and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of sphere. Also find the minimum value of  the sum of their volumes.

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Find the equation of tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π that are parallel to the line x + 2y = 0. VIEW SOLUTION