Board Paper of Class 12-Commerce 2017 Maths Delhi(SET 1) - Solutions

Show that all the diagonal elements of a skew symmetric matrix are zero. VIEW SOLUTION

Find $\frac{dy}{dx} \mathrm{at} x=1, y=\frac{\pi }{4} \mathrm{if} {\sin }^{2}y+\cos xy = K$. VIEW SOLUTION

The volume of a sphere is increasing at the rate of 3 cubic centimeter per second. Find the rate of increase of its surface area, when the radius is 2 cm. VIEW SOLUTION

Show that the function $f\left(x\right)=4x^3-18x^2+27x-7$ is always increasing on $\mathrm{ℝ}$. VIEW SOLUTION

Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7y = 35z. VIEW SOLUTION

Prove that if E and F are independent events, then the events E and F' are also independent. VIEW SOLUTION

A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is Rs 100 and that on a bracelet is Rs 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit?
It is being given that at least one of each must be produced. VIEW SOLUTION

Find $\int \frac{\mathrm{d}x}{x^2+4x+8}$ VIEW SOLUTION

Prove that $\tan \left\{\frac{\pi }{4}+\frac{1}{2}{\cos }^{-1}\frac{a}{b}\right\}+\tan \left\{\frac{\pi }{4}-\frac{1}{2}{\cos }^{-1}\frac{a}{b}\right\}=\frac{2b}{a}$ VIEW SOLUTION

Using properties of determinants, prove that $\left|\begin{array}{ccc}x& x+y& x+2y\\ x+2y& x& x+y\\ x+y& x+2y& x\end{array}\right|=9y^2\left(x+y\right).$
 
Let $A=\left(\begin{array}{cc}2& -1\\ 3& 4\end{array}\right), B=\left(\begin{array}{cc}5& 2\\ 7& 4\end{array}\right), C=\left(\begin{array}{cc}2& 5\\ 3& 8\end{array}\right)$, find a matrix D such that CD − AB = O. VIEW SOLUTION

Differentiate the function ${\left(\sin x\right)}^x+{\sin }^{-1}\sqrt{x}$ with respect to x.

OR

If $x^m{y}^n={\left(x+y\right)}^{m+n}$, prove that $\frac{d^{2}y}{d{x}^2}=0$. VIEW SOLUTION

Find $\int \frac{2x}{\left(x^2+1\right){\left(x^2+2\right)}^2}dx$ VIEW SOLUTION

Evaluate : $\underset{0}{\overset{\mathrm{\pi }}{\int }}\frac{x \sin x}{1+{\cos }^{2}x}\mathrm{d}x$
 
Evaluate : $\underset{0}{\overset{3/2}{\int }}\left|x \sin \pi x\right|\mathrm{d}x$ VIEW SOLUTION

Prove that x² – y² = c(x² + y²)² is the general solution of the differential equation (x³ – 3xy²)dx = (y³ – 3x²y)dy, where C is parameter. VIEW SOLUTION

Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}},\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}$ and $\overrightarrow{\mathrm{c}}={\mathrm{c}}_1\hat{\mathrm{i}}+{\mathrm{c}}_2\hat{\mathrm{j}}+{\mathrm{c}}_3\hat{\mathrm{k}},$ then

(a) Let c₁ = 1 and c₂ = 2, find c₃ which makes $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ coplanar.

(b) If c₂ = –1 and c₃ = 1, show that no value of c₁ can make $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ coplanar. VIEW SOLUTION

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\overrightarrow{a}+ \overrightarrow{b}+ \overrightarrow{c}$ is equally inclined to $\overrightarrow{a}, \overrightarrow{b} \mathrm{and} \overrightarrow{c}$.  Also, find the angle which $\overrightarrow{a}+ \overrightarrow{b}+ \overrightarrow{c}$ makes with $\overrightarrow{a} \mathrm{or} \overrightarrow{b} \mathrm{or} \overrightarrow{c}$. VIEW SOLUTION

The random variable X can take only the values 0, 1, 2, 3. Give that P(X = 0) = P(X = 1) = p and P(X = 2) = P(X = 3) such that $\mathrm{\Sigma }{p}_{\mathit{i}}{x}_i^2=2\mathrm{\Sigma }{p}_{\mathit{i}}{x}_{\mathit{i}}$, find the value of p. VIEW SOLUTION

Often it is taken that a truthful person commands, more respect in the society. A man is known to speak the truth 4 out of 5 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.
Do you also agree that the value of truthfulness leads to more respect in the society? VIEW SOLUTION

Solve the following L.P.P. graphically:
 

Minimise	Z = 5x + 10y
Subject to	x + 2y ≤ 120
Constraints	x + y ≥ 60
	x – 2y ≥ 0
and	x, y ≥ 0

VIEW SOLUTION

Use product $\left[\begin{array}{ccc}1& -1& 2\\ 0& 2& -3\\ 3& -2& 4\end{array}\right]\left[\begin{array}{ccc}-2& 0& 1\\ 9& 2& -3\\ 6& 1& -2\end{array}\right]$ to solve the system of equations x + 3z = 9, −x + 2y − 2z = 4, 2x − 3y + 4z = −3. VIEW SOLUTION

Consider f : R₊ → [−5, ∞), given by f(x) = 9x² + 6x − 5. Show that f is invertible with ${\mathrm{f}}^{-1}\left(y\right)\left(\frac{\sqrt{y+6}-1}{3}\right)$.

Hence Find
(i) f⁻¹(10)
(ii) y if ${\mathrm{f}}^{-1}\left(y\right)=\frac{4}{3},$

where R₊ is the set of all non-negative real numbers.
Discuss the commutativity and associativity of binary operation '*' defined on A = Q − {1} by the rule a * b = a − b + ab for all, a, b ∊ A. Also find the identity element of * in A and hence find the invertible elements of A. VIEW SOLUTION

If the sum of lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum, when the angle between them is $\frac{\mathrm{\pi }}{3}.$ VIEW SOLUTION

Using integration, find the area of region bounded by the triangle whose vertices are (–2, 1), (0, 4) and (2, 3).
 
Find the area bounded by the circle x² + y² = 16 and the line $\sqrt{3}\mathrm{y}=x$ in the first quadrant, using integration. VIEW SOLUTION

Solve the differential equation $x\frac{\mathrm{dy}}{\mathrm{d}x}+y=x \cos x + \sin x,$ given that y = 1 when $x=\frac{\mathrm{\pi }}{2}$ VIEW SOLUTION

Find the equation of the plane through the line of intersection of $\underset{r}{\to }·\left(2\hat{i}-3\hat{j}+4\hat{k}\right)=1$ and $\underset{r}{\to }·\left(\hat{i}-\hat{j}\right)+4=0$ and perpendicular to the plane $\underset{r}{\to }·\left(2\hat{i}-\hat{j}+\hat{k}\right)+8=0$. Hence find whether the plane thus obtained contains the line x − 1 = 2y − 4 = 3z − 12.
  Find the vector and Cartesian equations of a line passing through (1, 2, –4) and perpendicular to the two lines $\frac{x-8}{3}=\frac{\mathrm{y}+19}{-16}=\frac{\mathrm{z}-10}{7}$ and $\frac{x-15}{3}=\frac{\mathrm{y}-29}{8}=\frac{\mathrm{z}-5}{-5}$. VIEW SOLUTION