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

The equation of tangent at (2, 3) on the curve y² = ax³ + b is y = 4x – 5. Find the values of a and b. VIEW SOLUTION

Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane. VIEW SOLUTION

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

The two adjacent sides of a parallelogram are $2\hat{i}-4\hat{j}-5\hat{k} \mathrm{and} 2\hat{i}+2\hat{j}+3\hat{k}$. Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram. VIEW SOLUTION

Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes. VIEW SOLUTION

In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, when a fair die is thrown. The man decided to thrown a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses.
A bag contains 4 balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white? VIEW SOLUTION

A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received Rs 2,800 as interest. However, if trust had interchanged money in bonds, they would have got Rs 100 less as interest. Using matrix method, find the amount invested by the trust. Interest received on this amount will be given to Helpage India as donation. Which value is reflected in this question? VIEW SOLUTION

Differentiate $x^{\sin x}+{\left(\sin x\right)}^{\cos x}$ with respect to x.
  If $y=2 \cos \left(\log x\right)+3 \sin \left(\log x\right)$, prove that $x^2\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}+x\frac{\mathrm{d}y}{\mathrm{d}x}+y=0$. VIEW SOLUTION

Solve the equation for $x:{\sin }^{-1}x+{\sin }^{-1}\left(1-x\right)={\cos }^{-1}x$ If ${\cos }^{-1}\frac{x}{a}+{\cos }^{-1}\frac{y}{\mathrm{b}}=\alpha$, prove that $\frac{x^2}{a^2}-2\frac{xy}{\mathrm{ab}}\cos \alpha +\frac{y^2}{{\mathrm{b}}^2}={\sin }^2\alpha$ VIEW SOLUTION

If $x=a \sin 2\mathrm{t}\left(1+\cos 2\mathrm{t}\right) \mathrm{and} y=\mathrm{b} \cos 2\mathrm{t}\left(1-\cos 2\mathrm{t}\right), \mathrm{find} \frac{\mathrm{d}y}{\mathrm{d}x}\mathrm{at} \mathrm{t}=\frac{\mathrm{\pi }}{4}$. VIEW SOLUTION

Solve the differential equation :

$y+x\frac{\mathrm{d}y}{\mathrm{d}x}=x-y\frac{\mathrm{d}y}{\mathrm{d}x}$ VIEW SOLUTION

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

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

Using properties of determinants, show that ΔABC is isosceles if:

$\left|\begin{array}{ccc}1& 1& 1\\ 1+\mathrm{cosA}& 1+\mathrm{cosB}& 1+\mathrm{cosC}\\ {\cos }^2\mathrm{A}+\mathrm{cosA}& {\cos }^2\mathrm{B}+\mathrm{cosB}& {\cos }^2\mathrm{C}+\mathrm{cosC}\end{array}\right|=0$

                                                                      OR

A shopkeeper has 3 varieties of pens 'A', 'B' and 'C'. Meenu purchased 1 pen of each variety for a total of Rs 21. Jeevan purchased 4 pens of 'A' variety 3 pens of 'B' variety and 2 pens of 'C' variety for Rs 60. While Shikha purchased 6 pens of 'A' variety, 2 pens of 'B' variety and 3 pens of 'C' variety for Rs 70. Using matrix method, find cost of each variety of pen. VIEW SOLUTION

There are two types of fertilisers 'A' and 'B'. 'A' consists of 12% nitrogen and 5% phosphoric acid whereas 'B' consists of 4% nitrogen and 5% phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If 'A' costs Rs 10 per kg and 'B' cost Rs 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. VIEW SOLUTION

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is $6\sqrt{3}$r.
 
If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is $\frac{\mathrm{\pi }}{3}.$ VIEW SOLUTION

Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution. VIEW SOLUTION

Prove that the curves y² = 4x and x² = 4y divide the area of square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts. VIEW SOLUTION

Show that the binary operation * on A = R – { – 1} defined as a*b = a + b + ab for all a, b ∈ A is commutative and associative on A. Also find the identity element of * in A and prove that every element of A is invertible. VIEW SOLUTION

Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector $2\hat{i}+3\hat{j}+4\hat{k}$ to the plane $\overrightarrow{r}.\left(2\hat{i}+\hat{j}+3\hat{k}\right)-26=0$. Also find image of P in the plane. VIEW SOLUTION