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# Board Paper of Class 12-Science Term-I 2021 Applied Math Delhi(Set 4) - Solutions

General Instructions:
(i) This question paper contains 50 questions out of which 40 questions are to be attempted as per instructions. All questions carry equal marks.
(ii) The question paper consists three Sections – Section A, B and C.
(iii) Section A contains of 20 questions. Attempt any 16 questions from Q.No. 1 to 20.
(iv) Section B also consists of 20 questions. Attempt any 16 questions from Q.No. 21 to 40.
(v) Section C consists of a Case Studies containing 5 questions (Q.No. 46 – 50). Attempt any 8 from Q.No. 41 to 50.
(vi) There is only one correct option for every multiple choice question (MCQ). Marks will not be awarded for answering more than one option.
(vii) There is no negative marking.

• Question 1
If , then the least positive value of x is :
(a) 2
(b) 3
(c) 6
(d) 4 VIEW SOLUTION

• Question 2
If $\tau \left(n\right)$ denotes the number of divisors of n, then the value of $\tau \left(15\right)$ is :
(a) 3
(b) 4
(c) 5
(d) 7

VIEW SOLUTION

• Question 3
If a man rows 32 km downstream and 14 km upstream in 6 hours each, then the speed of the stream is:
(a) 2 km/h
(b) 1.5 km/h
(c) 2.5 km/h
(d) 2.25 km/h VIEW SOLUTION

• Question 4
In a 2 km race, P can give Q a start of 200 m and R a start of 560 m. Then, in the same race, Q can give R a start of :
(a) 360 m
(b) 380 m
(c) 400 m
(d) 430 m VIEW SOLUTION

• Question 5
Pipe A and B can fill a tank in 5 hours and 6 hours respectively. Pipe C can empty it in 12 hours. If all the three pipes are opened together, then the time taken to fill the tank is :
(a) 2 hours

(b) $2\frac{3}{4}$ hours

(c) 3 hours

(d) $3\frac{9}{17}$ hours VIEW SOLUTION

• Question 6
The solution of  is
(a) x > 3
(b) x < –5
(c) x < –5 or x > 3
(d) no solution VIEW SOLUTION

• Question 7
If matrix A is given by A = [aij]2×2, where aij = i + j, then A is equal to:

(a) $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$

(b) $\left[\begin{array}{cc}2& 3\\ 3& 4\end{array}\right]$

(c) $\left[\begin{array}{cc}1& 1\\ 2& 2\end{array}\right]$

(d) $\left[\begin{array}{cc}1& 2\\ 1& 2\end{array}\right]$ VIEW SOLUTION

• Question 8
If A is a square matrix such that A2 = A, then (I + A)2 – 3A is equal to:
(a) I
(b) 2A
(c) 3I
(d) A VIEW SOLUTION

• Question 9
If A is a square matrix of order 3 × 3 such that |A| = 4, then |3A | is equal to:
(a) 27
(b) 81
(c) 108
(d) 256 VIEW SOLUTION

• Question 10
The function f(x) = ax is increasing on R, if:
(a) a > 0
(b) a < 0
(c) 0 < a < 1
(d) a > 1 VIEW SOLUTION

• Question 11
If C(x) and R(x) are respectively Cost function and Revenue function, then the Profit function P(x) is given by :
(a) P(x) = R(x
(b) P(x) = C(x) + R(x)
(c) P(x) = R(x) – C(x)
(d) P(x) = R(x) . C(x) VIEW SOLUTION

• Question 12
If 'm' is the mean of Poisson distribution, then its standard deviation is given by:
(a) $\sqrt{m}$
(b) m2
(c) m
(d) $\frac{m}{2}$  VIEW SOLUTION

• Question 13
The normal distribution curve is symmetrical about :
(a) X = μ
(b) X = σ
(c) X = $\frac{\mathrm{\mu }}{\mathrm{\sigma }}$
(d) X = $\frac{\mathrm{\sigma }}{\mathrm{\mu }}$ VIEW SOLUTION

• Question 14
Let X be a discrete random variable whose probability distribution is given below:
 X = xi : 0 1 2 3 4 5 6 7 P(X = xi) : 0 2K 2K 3K K2 2K2 7K2 2K
​​The value of K is :
(a) $\frac{1}{10}$

(b) –1
(c) –$\frac{1}{10}$
(d) $\frac{1}{5}$

​​ VIEW SOLUTION

• Question 15
In a box of 100 bulbs, 10 are defective. What is the probability that out of a sample of 5 bulbs, none is defective?
(a) ${\left(\frac{9}{10}\right)}^{5}$
(b) $\frac{9}{10}$
(c) 10–5
(d) ${\left(\frac{1}{2}\right)}^{2}$ VIEW SOLUTION

• Question 16
If X is a normal variate with mean $\mathrm{\mu }$ and standard deviation $\mathrm{\sigma }>0,$ then the new random variate Z = $\frac{\mathrm{X}-\mathrm{\mu }}{\mathrm{\sigma }}$ is a variate with :
(a) Mean = 1, Standard deviation = 0.
(b) Mean = 1, Standard deviation = 1.
(c) Mean = 2, Standard deviation = 1.
(d) Mean = 0, Standard deviation = 1. VIEW SOLUTION

• Question 17
The mean E(x) of the number obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is :
(a) 1
(b) 2
(c) 5
(d) $\frac{8}{3}$ VIEW SOLUTION

• Question 18
Which of the following index number satisfies the "time reversal test"?
(a) Fisher's ideal index number
(b) Laspeyres' index number
(c) Paasche's index number
(d) None of these VIEW SOLUTION

• Question 19
To calculate Paasche's price index, the weights are taken as :
(a) p0
(b) p1
(c) q0
(d) q1 VIEW SOLUTION

• Question 20
Given that  where subscript 0 and 1 are used for base year and current year respectively. The Laspeyres' index number is :
(a) 117.81
(b) 119.5
(c) 121.15
(d) 123.35 VIEW SOLUTION

• Question 21
The remainder when 561 is divided by 7 is:
(a) 1
(b) 2
(c) 4
(d) 5 VIEW SOLUTION

• Question 22
20 litres of a mixture contains milk and water in the ratio 3 : 1. The amount of milk, in litres, to be added to the mixture so as to have milk and water in the ratio 4 : 1, is:
(a) 7
(b) 4
(c) 5
(d) 6 VIEW SOLUTION

• Question 23
Pipe A can fill a tank 6 times faster than a pipe B. If B can fill a tank in 21 minutes, then the time taken by both the pipes together to fill the tank is:

(a) 3 minutes

(b) $4\frac{1}{2}$minutes

(c) 7 minutes

(d) 9 minutes VIEW SOLUTION

• Question 24
The ratio of investments of two partners A and B is 11 : 12 and the ratio of their profits is 2 : 3. If A invested the money for 8 months, then for how much time did B invest his money?
(a) 11 months
(b) 10 months
(c) 9 months
(d) 5 months VIEW SOLUTION

• Question 25
The solution set of the inequation |x + 2| ≤ 5 is:
(a) (–7, 5)
(b) [–7, 3]
(c) [–5, 5]
(d) (–7, 3) VIEW SOLUTION

• Question 26
If   and AB = I3, then (x + y) equals:
(a) 0
(b) –1
(c) 2
(d) –2 VIEW SOLUTION

• Question 27
If $\mathrm{A}=\left[\begin{array}{ccc}1& 0& 1\\ 0& 0& 1\\ a& b& 2\end{array}\right]$ , then aI + bA + 2A2 equals:
(a) A
(b) –A
(c) 2abA
(d) None of these VIEW SOLUTION

• Question 28
If A2 A + I = 0, then the inverse of matrix A is:
(a) A2
(b) A + I
(c) IA
(d) AI VIEW SOLUTION

• Question 29
If the points (1, 3), (x, 5) and (2, 7) are collinear, then the value of x is:

(a) 2

(b) $\frac{3}{2}$

(c) 1

(d) $\frac{3}{4}$ VIEW SOLUTION

• Question 30
If y = Ae5x + Be–5x, then $\frac{{d}^{2}y}{d{x}^{2}}$ is:
(a) 25y
(b) 5y
(c) –25y
(d) 15y VIEW SOLUTION

• Question 31
The point on the curve x= 2y which is nearest to the point (0, 5) is:

(a)

(b)

(c) (0, 0)

(d) (2, 2) VIEW SOLUTION

• Question 32
If the total revenue (₹) received from the sale of x units of a product is given by:
R(x) = 3x2 + 36x + 5,
then the marginal revenue, when x = 15, is
(a) 116
(b) 96
(c) 90
(d) 126 VIEW SOLUTION

• Question 33
The equation of normal at the point (1, 1) to the curve 2y + x= 3 is:
(a) x + y = 0
(b) xy = 0
(c) x + y = 1
(d) xy = 1 VIEW SOLUTION

• Question 34
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. Then the possible values of X are:
(a) 0, 1, 3, 5
(b) 0, 2, 4, 6
(c) 0, 2, 5, 6
(d) 1, 3, 4, 5 VIEW SOLUTION

• Question 35
If the mean of a binomial distribution is 81, then the standard deviation lies in the interval:
(a) [0, 9)
(b) (0, 9]
(c) [0, 3]
(d) (0, 3] VIEW SOLUTION

• Question 36
If a random variable X has the Poisson distributions with mean 2. Then, P(X > 1.5) is:
(a) 2e–2
(b) 3e2
(c) 1 – 2e2
(d) 1 – 3e–2 VIEW SOLUTION

• Question 37
There are 50 telephone lines in an exchange. The probability that any one of them will be busy is 0.1. The probability that all the lines are busy is:
(a) $\frac{{5}^{0}{e}^{–5}}{0!}$
(b) $1–\frac{{5}^{0}{e}^{–5}}{0!}$
(c) $\frac{{5}^{50}{e}^{–5}}{50!}$
(d) $1–\frac{{5}^{50}{e}^{–5}}{50!}$ VIEW SOLUTION

• Question 38
Price relative of sugar is 135 in the year 2020 compared to the year 2019. If the sugar cost ₹30 per kg in 2019, then cost in 2020 is:
(a) ₹15 per kg
(b) ₹40.50 per kg
(c) ₹45.20 per kg
(d) ₹65 per kg VIEW SOLUTION

• Question 39
If ∑W log p = 199.50 and ∑W = 100, then the weighted index number is:
(a) 120.86
(b) 88.86
(c) 98.86
(d) 78.86 VIEW SOLUTION

• Question 40
The condition for the time reversal test to hold good with usual notation is:
(a) Pbc × Pcb = 1
(b) Pbc × Pcb = 0
(c) Pbc + Pcb = 1
(d) $\frac{{\mathrm{P}}_{\mathrm{cb}}}{{\mathrm{P}}_{\mathrm{bc}}}=1$
VIEW SOLUTION

• Question 41
In this section, attempt any 8 questions out of questions no. 41-50. Each question is of 1 mark.
(Questions no. 46-50 are based on a Case-Study)

The least value of 'a' such that the function f(x) = x2 + ax + 1 is increasing on (1, 2) is :
(a) 0
(b) –1
(c) –2
(d) –4 VIEW SOLUTION

• Question 42
In this section, attempt any 8 questions out of questions no. 41-50. Each question is of 1 mark.
(Questions no. 46-50 are based on a Case-Study)

The demand function of a commodity is given by :
x = 82 – p
and its total cost function is given by :
TC = 100 + 60x
For maximum profit, the value of x is :
(a) 15
(b) 14
(c) 13
(d) 11 VIEW SOLUTION

• Question 43
The mean of the probability distribution of the number of doublets in 4 throws of a pair of dice, is :

(a) 1

(b) $\frac{2}{3}$

(c) $1\frac{3}{5}$

(d) $2\frac{2}{3}$ VIEW SOLUTION

• Question 44
In this section, attempt any 8 questions out of questions no. 41-50. Each question is of 1 mark.
(Questions no. 46-50 are based on a Case-Study)

It is known from the past experience that the number of telephone calls made daily in a certain community between 3 p.m. and 4 p.m. has a mean of 352 and a standard deviation of 31. What percentage of the time will there be more than 400 telephone calls made in the community between 3 p.m. to 4 p.m. ? [Use : P (0 ≤  Z ≤ 1.5) = 0.4394]
(a) 11.4%
(b) 9.6%
(c) 7.08%
(d) 6.06% VIEW SOLUTION

• Question 45
In this section, attempt any 8 questions out of questions no. 41-50. Each question is of 1 mark.
(Questions no. 46-50 are based on a Case-Study)

The index number of the following data :
 Relative Index 181 116 110 157 Weight 4 12 3 7

is :
(a) 118.74
(b) 136.34
(c) 142.04
(d) 146.14 VIEW SOLUTION

• Question 46
Two products P and Q are produced such that 0.4 tonne of P and 0.7 tonne of Q are required to produce one tonne of P. Similarly, 0.1 tonne of P and 0.6 tonne of Q are required to produce one tonne of Q. The economy needs 68 tonnes of P and 102 tonnes of Q.
Based on the above information, answer the following questions:

The technology matrix A is:
(a) $\left[\begin{array}{cc}0.4& 0.1\\ 0.7& 0.6\end{array}\right]$

​(b) $\left[\begin{array}{cc}0.4& 0.6\\ 0.7& 0.1\end{array}\right]$

​(c) $\left[\begin{array}{cc}0.6& 0.1\\ 0.7& 0.4\end{array}\right]$

​(d) $\left[\begin{array}{cc}0.4& 0.7\\ 0.1& 0.6\end{array}\right]$ VIEW SOLUTION

• Question 47
Two products P and Q are produced such that 0.4 tonne of P and 0.7 tonne of Q are required to produce one tonne of P. Similarly, 0.1 tonne of P and 0.6 tonne of Q are required to produce one tonne of Q. The economy needs 68 tonnes of P and 102 tonnes of Q.
Based on the above information, answer the following questions:

The demand matrix is:
(a) $\left[\begin{array}{c}68\\ 102\end{array}\right]$

​(b) $\left[\begin{array}{c}68\\ 34\end{array}\right]$

​(c) $\left[\begin{array}{c}102\\ 68\end{array}\right]$

​(d) $\left[\begin{array}{c}34\\ 68\end{array}\right]$ VIEW SOLUTION

• Question 48
Two products P and Q are produced such that 0.4 tonne of P and 0.7 tonne of Q are required to produce one tonne of P. Similarly, 0.1 tonne of P and 0.6 tonne of Q are required to produce one tonne of Q. The economy needs 68 tonnes of P and 102 tonnes of Q.
Based on the above information, answer the following questions:

(I – A) is:

(a) $\left[\begin{array}{cc}0.6& -0.6\\ -0.7& 0.9\end{array}\right]$

​(b) $\left[\begin{array}{cc}0.4& -0.1\\ -0.7& 0.6\end{array}\right]$

​(c) $\left[\begin{array}{cc}0.6& -0.7\\ -0.1& 0.4\end{array}\right]$

​(d) $\left[\begin{array}{cc}0.6& -0.1\\ -0.7& 0.4\end{array}\right]$ VIEW SOLUTION

• Question 49
Two products P and Q are produced such that 0.4 tonne of P and 0.7 tonne of Q are required to produce one tonne of P. Similarly, 0.1 tonne of P and 0.6 tonne of Q are required to produce one tonne of Q. The economy needs 68 tonnes of P and 102 tonnes of Q.
Based on the above information, answer the following questions:

(I – A)–1 is:

(a) $\frac{1}{0.17}\left[\begin{array}{cc}0.6& 0.1\\ 0.7& 0.4\end{array}\right]$

​(b) $\frac{1}{0.17}\left[\begin{array}{cc}0.4& 0.1\\ 0.7& 0.6\end{array}\right]$

​(c) $\frac{1}{0.17}\left[\begin{array}{cc}0.4& 0.1\\ 0.7& 0.6\end{array}\right]$

​(d) $\frac{1}{0.17}\left[\begin{array}{cc}0.9& 0.6\\ 0.7& 0.6\end{array}\right]$ VIEW SOLUTION

• Question 50
Two products P and Q are produced such that 0.4 tonne of P and 0.7 tonne of Q are required to produce one tonne of P. Similarly, 0.1 tonne of P and 0.6 tonne of Q are required to produce one tonne of Q. The economy needs 68 tonnes of P and 102 tonnes of Q.
Based on the above information, answer the following questions:

The gross outputs of P and Q are:
(a) P = 260; Q = 360
​(b) P = 220; Q = 640
​(c) P = 520; Q = 300
​(d) P = 420; Q = 433 VIEW SOLUTION
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