Board Paper of Class 12-Science 2015 Maths (SET 1) - Solutions

• Question 1
  If $\overrightarrow{\mathrm{a}}=7\hat{\mathrm{i}}+\hat{\mathrm{j}}-4 \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}} + 6 \hat{\mathrm{j}} + 3\hat{\mathrm{k}},$then find the projection of $\overrightarrow{\mathrm{a}} \mathrm{on}\overrightarrow{\mathrm{b}}$. VIEW SOLUTION

• Question 2
  Find λ, if the vectors $\overrightarrow{a}=\hat{i}+3\hat{j}+\hat{k}, \overrightarrow{b}=2\hat{i}-\hat{j}-\hat{k} \mathrm{and} \overrightarrow{c}=\lambda \hat{j}+3\hat{k}$ are coplanar. VIEW SOLUTION

• Question 3
  If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ. VIEW SOLUTION

• Question 4
  Write the element a₂₃ of a 3 ✕ 3 matrix A = (a_ij) whose elements a_ij are given by $a_{ij}=\frac{\left|i-j\right|}{2}.$ VIEW SOLUTION

• Question 5
  Find the differential equation representing the family of curves $\mathrm{v}=\frac{\mathrm{A}}{\mathrm{r}}$+ B, where A and B are arbitrary constants. VIEW SOLUTION

• Question 6
  Find the integrating factor of the differential equation
  $\left(\frac{{e^{-2}}^{\sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right)\frac{dx}{dy}=1$. VIEW SOLUTION

• Question 7
  If $\mathrm{A}=\left(\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right)$ find ${\mathrm{A}}^2-5\mathrm{A}+4\mathrm{I}$ and hence find a matrix X such that ${\mathrm{A}}^2-5\mathrm{A}+4\mathrm{I}+\mathrm{X}=\mathrm{O}$
  
  OR
  
  If $\mathrm{A}=\left[\begin{array}{ccc}1& -2& 3\\ 0& -1& 4\\ -2& 2& 1\end{array}\right], \mathrm{find} {\left(\mathrm{A}\text{'}\right)}^{-1}.$ VIEW SOLUTION

• Question 8
  If $f\left(x\right)=\left|\begin{array}{ccc}a& -1& 0\\ ax& a& -1\\ a{x}^2& ax& a\end{array}\right|$, using properties of determinants find the value of f(2x) − f(x). VIEW SOLUTION

• Question 9
  Find : $\int \frac{dx}{\sin x+\sin 2x}$
   

  OR

  
  Integrate the following w.r.t. x
  $\frac{x^2-3x+1}{\sqrt{1-x^2}}$ VIEW SOLUTION

• Question 10
  Evaluate : $\underset{-x}{\overset{x}{\int }} {\left(\cos ax-\sin bx\right)}^2 dx$ VIEW SOLUTION

• Question 11
  A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B, If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.

  OR

  An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained. VIEW SOLUTION

• Question 12
  $\mathrm{If} \stackrel{\to }{\mathrm{r}}=\mathrm{x}\hat{\mathrm{i}}+\mathrm{y}\hat{\mathrm{j}}+\mathrm{z}\hat{\mathrm{k}}, \mathrm{find} \left(\stackrel{\to }{\mathrm{r}}\times \hat{\mathrm{i}}\right)·\left(\stackrel{\to }{\mathrm{r}}\times \mathrm{j}\right)+\mathrm{xy}$ VIEW SOLUTION

• Question 13
  Find the distance between the point (−1, −5, −10) and the point of intersection of the line $\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}$ and the plane x − y + z = 5. VIEW SOLUTION

• Question 14
  If sin [cot⁻¹ (x+1)] = cos(tan⁻¹x), then find x.

  
  OR

  
  If (tan⁻¹x)² + (cot⁻¹x)² = $\frac{5{\mathrm{\pi }}^2}{8}$, then find x. VIEW SOLUTION

• Question 15
  If $y={\tan }^{-1} \left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right), x^2\le 1,$ then find $\frac{dy}{dx}$. VIEW SOLUTION

• Question 16
  If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that $y^2\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+y=0.$ VIEW SOLUTION

• Question 17
  The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ? VIEW SOLUTION

• Question 18
  Find : $\int \left(x+3\right)\sqrt{3-4x-x^2} \mathrm{dx}$ VIEW SOLUTION

• Question 19
  Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The number of articles sold are given below:
   

  School
  Article	A	B	C
  Hand-fans	40	25	35
  Mats	50	40	50
  Plates	20	30	40

  
  Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose.
  
  Write one value generated by the above situation. VIEW SOLUTION

• Question 20
  Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation. VIEW SOLUTION

• Question 21

  Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle $x^2+y^2=4 \mathrm{at} \left(1, \sqrt{3}\right)$

  OR

  Evaluate $\underset{1}{\overset{3}{\int }}\left(e^{2-3x}+x^2+1\right) dx$ as a limit of a sum.
   

  VIEW SOLUTION

• Question 22

  Solve the differential equation :

  $\left({\tan }^{-1}y-x\right)dy=\left(1+y^2\right)dx.$

  OR

  Find the particular solution of the differential equation $\frac{dy}{dx}=\frac{xy}{x^2+y^2}$ given that y = 1, when x = 0. VIEW SOLUTION

• Question 23
  If lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4} \mathrm{and} \frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect, then find the value of k and hence find the equation of the plane containing these lines. VIEW SOLUTION

• Question 24
  If A and B are two independent events such that $\mathrm{P}\left(\overline{\mathrm{A}} \cap \mathrm{B}\right) =\frac{2}{15} \mathrm{and} \mathrm{P}\left(\mathrm{A} \cap \overline{\mathrm{B}}\right) = \frac{1}{6},$ then find P(A) and P(B). VIEW SOLUTION

• Question 25
  Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π. VIEW SOLUTION

• Question 26
  Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below :
  
  $$\begin{gathered} 2x + 4y \le 8 \\ 3x + y \le 6 \\ x + y \le 4 \\ x \ge 0, y\ge 0 \end{gathered}$$ VIEW SOLUTION