ISC Board Exam 2024 Solution Mathematics Class 12

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begin{center}
textbf{Section A }
end{center}
 
textbf{Question 1}
 
begin{enumerate}[label=roman*]
 
 
 
item Let L be a set of all straight lines in a plane. The relation R on L defined as “perpendicular to ” is : 
 
begin{enumerate}
item Symmetric and Transitive 
item Transitive 
item Symmetric 
item Equivalence 
end{enumerate}
 
item The order and degrees of differential equation $1 + ( frac{dy}{dx})^2 = frac{d^2y}{dx^2}$
 
begin{enumerate}
item 2 and 3/2 item 2 and 3 item 3 and 4 item 2 and 1 
end{enumerate}
 
item Let A be a non empty set 
Statement 1 : Identity relation on A is Reflexive  \
 
 
Statement 2 : Every reflexive relation on A is an identity relation 
 
begin{enumerate}
item Both statements are true 
item bot statements are false 
item Statement 1 is true and Statement 2 is false 
item Statement 1 is false and statement 2 is true  
end{enumerate}
 
item The graph of the function F is shown below : 
 
 
 
 
of the following on what all points the function f is not differentiable. 
 
begin{enumerate}
item at x = 0 and x = 2 item at x = 1 and x = 3 item at x = -1 and x = 1 item at x = -1.5 and x = 1.5 
end{enumerate}
 
item The value of $cosec (sin^{-1}(frac{-1}{2}))- sec(cos^{-1}(frac{-1}{2}))$ is equal to 
 
begin{enumerate}
item -4 item 0 item -1 item 4
end{enumerate}
 
item The value of $int^{sqrt{3}}_1 frac{dx}{1+x^2}$ is begin{enumerate}
item $frac{pi}{2}$ item $frac{2pi}{3}$item $frac{pi}{6}$item $frac{pi}{12}$
end{enumerate}
 
 
item textbf{Assertion:} Let the matrices A = $begin{pmatrix}
-3 & 2 \ -5 & 4 \
end{pmatrix}$ and $B = begin{pmatrix}
4 & -2 \ 5 & -3 \
end{pmatrix}$ be such that $A^{100} B = B A ^{100}$
 
textbf{Reason}: AB = BA implies $A^nB = BA^m$ for all positive integers n. 
 
 
begin{enumerate}
item Both assertion and reason are true and Reason is the correct explanation of assertion 
 
item Both Assertion and Reason are true but Reason is not the correct explanation of assertion. 
 
item Assertion is true and Reason is false 
item Assertion is false and Reason is true. 
end{enumerate}
 
 
item If $int (cotx – cosec^2x)e^x dx = e^x f(x) + c $ then f(x) will be 
begin{enumerate}
item cotx + cosec x item $cot^2x$ item cotx item cosec x 
end{enumerate}
 
 
item In which one of the following intervals is the function $f(x  ) = x^3-12x$ increasing ? 
 
begin{enumerate}
item $(-2,2)$ item $(-infty, -2) cup (2,infty)$ item $(-2, infty)$ item $(-infty, 2)$
end{enumerate}
 
item If A and B are symmetric matrices of the same order, then AB – BA is : 
 
begin{enumerate}
item skew symmetric item symmetric matrix item diagonal matrix item identity matrix 
end{enumerate}
 
item Find the derivative of $y = logx + frac{1}{x}$ 
item Teena is practising for an upcoming Rifle Shooting tournament. The probability of her shooting the target in the 1st , 2nd , 3rd and 4th shots are 0.4, 0.3 , 0.2 amd 0.1 respectively. Find the probability of at least one shot of Teen hitting the target. 
 
item Which of the following graphs is a function of x ? 
 
 
item Evaluate $int^4_0 |x+3|dx$ 
 
item Given that $frac{1}{y}+ frac{1}{x} = frac{1}{12}$ and y decreases at a rate of $1cm/s^{-1}$ find the rate of change of x when x = 5 cm and y = 1 cm. 
 
textbf{Question 2 } 
 
begin{enumerate}[label=roman*]
item Let $f:R-(frac{-1}{3}) rightarrow R – {0}$ be defined as $f(x) = frac{5}{3x+1}$ is invertible . Find $f^{-1}(x)$ 
 
begin{center}
textbf{OR}
end{center}
 
item If $f:R rightarrow R $ be defined as $f(x) = frac{2x-7}{4}$ . Show that f(x) is one and onto. 
end{enumerate}
 
 
textbf{Question 3 } 
 
begin{enumerate}[label=roman*]
 
item Find the value of the determinant given below, without expanding it at any stage 
 
 
$begin{vmatrix}
beta gamma & 1 & alpha (beta + gamma) \
gamma alpha & 1 & beta (gamma + alpha) \
alpha beta & 1 & gamma (alpha + beta) \
end{vmatrix}$
 
 
end{enumerate}
 
 
 
textbf{Question 4 } 
 
begin{enumerate}[label=roman*]
 
item Determine the value of k for which the following function is continuous at x = 3 
 
f(x) =
$begin{cases}
frac{(x+3)^2 -36}{x-3}; x neq 3 \ 
k; ~~~~~~~~x = 3 \ 
end{cases}$
 
item Find a point on the curve $y =(x-2)^2$ at which the tangent is parallel to the line joining the chord through the points (2,0) and (4,4)
 
 
end{enumerate}
 
 
 
 
textbf{Question 5 } 
 
begin{enumerate}[label=roman*]
 
item Evaluate : $$int^{2pi}_0 frac{1}{1+e^{sinx}}dx$$ 
 
end{enumerate}
 
 
 
textbf{Question 6 } 
 
begin{enumerate}[label=roman*]
 
item Evaluate : $  P(A cup B)$ if $2P(A) = P(B) =frac{5}{13}$ and $P(A|B) = frac{2}{5}$
 
end{enumerate}
 
 
 
 
 
textbf{Question 7 } 
 
begin{enumerate}[label=roman*]
 
item If $y= 3cos(logx) + 4sin(logx)$ then show that $$x^2 frac{d^2y}{dx^2}+x frac{dy}{dx}+y=0$$
 
end{enumerate}
 
 
 
 
 
 
textbf{Question 8 } 
 
begin{enumerate}[label=roman*]
 
item Solve for x : $$sin^{-1}(frac{x}{2})+ cos^{-1}(x) = frac{pi}{6}$$
 
begin{center}
textbf{Large  OR}
end{center}
 
item If $sin^{-1}x + sin^{-1}y+ sin^{-1}z =pi$ Show that $$x^2-y^2-x^2+ 2yz sqrt{1-x^2}=0$$
 
end{enumerate}
 
 
 
 
textbf{Question 9 } 
 
begin{enumerate}[label=roman*]
 
item Evaluate ;$int x^2cosx dx$ 
 
begin{center}
textbf{Large OR}
end{center}
 
item Evaluate $int frac{x+7}{x^2+4x+7}dx$
 
 
end{enumerate}
 
 
 
 
 
 
 
textbf{Question 10 } 
 
begin{enumerate}[label=roman*]
 
item A jewellery seller has precious gems in white and red colour which he has put in three boxes. The distribution of these gemes is shown in the table given below : 
 
 
 
begin{table}[ht]
centering
 
begin{tabular}{|c|c|c|}
hline
Box & multicolumn{2}{c|}{Numbers of Gems} \ cline{2-3}
& White & Red \ hline
I   & 1     & 2   \
II  & 2     & 3   \
III & 3     & 1   \ hline
end{tabular}
label{tab:gem_numbers}
end{table}
 
 
He wants to gift two gems to his mother. So, he asks here to select one box at random and pick out any two gems one after the other without replacement from the selected box. The mother selects one white and one red gem. Calculate the probability that the gems drawn are from Box II. 
 
end{enumerate}
 
 
 
 
 
textbf{Question 11 } 
 
begin{enumerate}[label=roman*]
 
item A furniture factory uses three types of wood namely , teakwood, rosewood and satinwood for manufacturing three types of furniture , that are, table chair and cot. 
The wood requirements (in tonnes) for each type of furniture are given below : 
 
 
 
 
begin{table}[ht]
centering
 
begin{tabular}{|l|c|c|c|}
hline
Material  & Table & Chair & Cot \ hline
Teakwood  & 2     & 3     & 4   \
Rosewood  & 1     & 1     & 2   \
Satinwood & 3     & 2     & 1   \ hline
end{tabular}
label{tab:furniture_materials}
end{table}
 
 
It is found that 29 tonnes of teakwood, 13 tonnes of rosewood and 16 tonnes of satinwood are available to make all three types of furniture. Using the above information, answer the following questions : 
 
begin{enumerate}
item Express the data given in the table above in the form of a set of simultaneous equation item Solve the set of simultaneous equations formed in subpart (above) by matrix method item Hence, find the number of table(s), chair(s) and cot(s) produced. 
end{enumerate}
 
end{enumerate}
 
 
 
 
 
 
 
 
 
 
begin{comment}
begin{enumerate}[label=Roman*]
item One
item Two
item Three
end{enumerate}
 
 
end{comment}
 
 
end{enumerate}
 
begin{center}
textbf{Section (C)}
end{center}
 
begin{enumerate}
 
item A company sells hand towels at Rs  100 per unit. The fixed cost for the company to manufacture hand towels is Rs. 35000 and variable cost is estimated to be Rs. 30% of total revenue. What will be the total cost function for manufacturing the hand towels ? 
 
textbf{Solution}
To find the total cost function for manufacturing the hand towels, we need to understand and incorporate both the fixed costs and the variable costs into the equation. Let’s break it down step by step.
 
Fixed costs are costs that do not change with the level of output. In this case, the fixed cost for the company is given as Rs. 35,000. This means no matter how many hand towels are manufactured, the company will always have to pay Rs. 35,000.
 
Variable costs change with the level of output. It’s given that the variable cost is 30% of the total revenue. The total revenue can be calculated as the selling price per unit times the number of units sold. If we let (x) represent the number of hand towels manufactured and sold, and given that each hand towel is sold at Rs. 100, the total revenue ((R)) would be (100x).
 
Therefore, the variable cost ((VC)) would be 30% of the total revenue, which can be written as (0.30 times 100x = 30x).
 
The total cost ((TC)) function is the sum of the fixed costs and the variable costs. Therefore, the total cost function can be represented as:
[
TC = text{Fixed Costs} + text{Variable Costs}
]
 
Substituting the given values into the equation:
[
TC = 35000 + 30x
]
 
So, the total cost function for manufacturing the hand towels is:
[
TC(x) = 35000 + 30x
]
 
This equation tells us the total cost of manufacturing (x) hand towels, incorporating both the fixed costs and the costs that vary with production level.
 
 
item If the correlation coefficient of two sets of variables (X,Y)is $frac{-3}{4}$ which one of the following statement is true for the same set of variables ? 
 
begin{enumerate}
item Only one of the two regression lines has a negative coefficient
item Both regression coefficients are positive
item Both regression coefficients are negative 
item one of the lines of regression is parallel to the x axis.
end{enumerate}
 
 
textbf{Solution:}
 
Given that the correlation coefficient of two sets of variables (X, Y) is (r = frac{-3}{4}), we need to determine the truthfulness of the following statements regarding the regression coefficients of these variables.
 
subsection*{Understanding the Correlation Coefficient}
The correlation coefficient ((r)) measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where:
begin{itemize}
item A value of 1 indicates a perfect positive linear relationship,
item A value of -1 indicates a perfect negative linear relationship, and
item A value of 0 indicates no linear relationship.
end{itemize}
 
Given (r = frac{-3}{4}), this indicates a strong negative linear relationship between (X) and (Y).
 
 
textit{Both regression coefficients are negative.} \
This statement is correct. Given the negative correlation coefficient, both the slope of the regression line of (Y) on (X) and the slope of the regression line of (X) on (Y) are negative, indicating that both variables move in opposite directions.
 
 
 
end{enumerate}
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