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Applied Maths MCQ

Model Paper 7 · CBSE STD 12 Science (CBSE - English Medium). 18 questions.

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18 questions · timed · auto-graded

MCQ 11 Mark
The most commonly use mathematical method for measuring the trend is:
  • A
    Semi average method
  • B
    Time series method
  • ✓
    Method of least squares
  • D
    Moving average method
Answer
Correct option: C.
Method of least squares
(c) Method of least squares
Explanation: Method of least squares
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MCQ 21 Mark
$\int e^x\left(\frac{1-x}{1+x^2}\right)^2 d x$ is equal to
  • A
    $-\frac{e^x}{1+x^2}+C$
  • B
    $-\frac{e^x}{\left(1+x^2\right)^2}+C$
  • ✓
    $\frac{e^x}{1+x^2}+C$
  • D
    $\frac{e^x}{\left(1+x^2\right)^2}+C$
Answer
Correct option: C.
$\frac{e^x}{1+x^2}+C$
(c) $\frac{e^x}{1+x^2}+C$
$\begin{array}{l}\text { Explanation: Given } \int e^x\left(\frac{1-x}{1+x^2}\right)^2 d x \\ \Rightarrow \int e^x\left(\frac{1-x}{1+x^2}\right)^2 d x=\int e^x\left(\frac{1+x^2-2 x}{\left(1+x^2\right)^2}\right) d x \\ \Rightarrow \int e^x\left(\frac{1+x^2-2 x}{\left(1+x^2\right)^2}\right) d x=\int e^x\left\{\left(\frac{1+x^2}{\left(1+x^2\right)^2}\right)+\left(\frac{-2 x}{\left(1+x^2\right)^2}\right)\right\} d x \\ =\int e^x\left\{\left(\frac{1}{\left(1+x^2\right)}\right)+\left(\frac{-2 x}{\left(1+x^2\right)^2}\right)\right\} d x\end{array}$
Now using the property: $\int e ^{ x }\left( f ( x )+ f ^{\prime}( x )\right) dx = e ^{ x } f ( x )$
Now in $\int e^x\left\{\left(\frac{1}{\left(1+x^2\right)}\right)+\left(\frac{-2 x}{\left(1+x^2\right)^2}\right)\right\} d x$
$\begin{array}{l}\Rightarrow f ( x )=\frac{1}{\left(1+x^2\right)} \\ \Rightarrow f ^{\prime}( x )=\frac{-2 x}{\left(1+x^2\right)^2} \\ \Rightarrow \int e ^{ x }\left\{\left(\frac{1}{\left(1+ x ^2\right)}\right)+\left(\frac{-2 x }{\left(1+ x ^2\right)^2}\right)\right\} dx =\frac{ e ^{ x }}{1+ x ^2}+ C \\ \Rightarrow \int e^x\left(\frac{1-x}{1+x^2}\right)^2 d x=\frac{e^x}{1+x^2}+C\end{array}$
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MCQ 31 Mark
A sample of 50 bulbs is taken at random. Out of 50 we found 15 bulbs are of Bajaj, 17 are of Surya and 18 are of Crompton. What is the point estimate of population proportion of Surya?
  • A
    0.3
  • ✓
    0.34
  • C
    0.36
  • D
    0.4
Answer
Correct option: B.
0.34
(b) 0.34
Explanation: 0.34
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MCQ 41 Mark
The necessary condition for third quadrant region in xy plane is:
  • A
    x > 0, y < 0
  • ✓
    x < 0, y < 0
  • C
    x < 0, y = 0
  • D
    x < 0, y > 0
Answer
Correct option: B.
x < 0, y < 0
(b) x < 0, y < 0
Explanation: In $3^{\text {rd }}$ quadrant $x <0$ and $y <0$.
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MCQ 51 Mark
If the objective function for an L.P.P. is Z = 3x + 4y and the comer points for unbounded feasible region are (9, 0), (4, 3), (2, 5) and (0, 8), then the minimum value of Z occurs at
  • ✓
    (4, 3)
  • B
    (9, 0)
  • C
    (2, 5)
  • D
    (0, 8)
Answer
Correct option: A.
(4, 3)
(a) (4, 3)
Explanation: (4, 3)
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MCQ 61 Mark
If a man goes 18 km downstream in 4 hours and returns against the stream in 12 hours, then the speed of the stream in km/hr is
  • ✓
    $\frac{3}{2}$
  • B
    3
  • C
    1
  • D
    $\frac{7}{4}$
Answer
Correct option: A.
$\frac{3}{2}$
(a) $\frac{3}{2}$
Explanation: Rate downstream $=\frac{18}{4}=\frac{9}{2} km / hr$
Rate upstream $=\frac{18}{12}=\frac{3}{2} km / hr$
Now, the speed of the stream $=\frac{\text { Rate Downstream-Rate Upstream }}{2}$
$\Rightarrow \frac{\frac{9}{2}-\frac{3}{2}}{2}=\frac{6}{4}=\frac{3}{2}$
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MCQ 71 Mark
The solution of $\frac{x-3}{x+5}>0, \neq-5, x \in R$ is
  • A
    x < -5
  • ✓
    x < -5 or x > 3
  • C
    no solution
  • D
    x > 3
Answer
Correct option: B.
x < -5 or x > 3
(b) x < -5 or x > 3
Explanation: $\frac{x-3}{x+5}>0, x \neq-5$,
$\begin{array}{l}\Rightarrow(x-3)>0 \text { and }(x+5)>0 \text { or }(x-3)<0 \text { and }(x+5)<0 \\ \Rightarrow x>3 \text { and } x>-5 \text { or } x<3 \text { and } x<-5 \\ \Rightarrow x>3 \text { or } x<-5\end{array}$
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MCQ 81 Mark
A milkman mixed some water with milk to gain 25% by selling the mixture at the cost price. The ratio of water and milk respectively is:
  • A
    1 : 5
  • B
    5 : 4
  • C
    4 : 5
  • ✓
    1 : 4
Answer
Correct option: D.
1 : 4
(d) 1 : 4
Explanation: Cost price of 1 litres of milk = ₹ 100
$\therefore$ Mixture sold for ₹ 125
$=\frac{125}{100}=\frac{5}{4}$ litres
$\therefore$ Quantity of mixture $=\frac{5}{4}$ litres
$\therefore$ Quantity of milk $=1$ litre
$\therefore$ Quantity of water $=\frac{5}{4}-1=\frac{1}{4}$ litre
$\therefore$ Required ratio $=\frac{1}{4}: 1=1: 4$
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MCQ 91 Mark
Integrating factor of $x \frac{d y}{d x}-y= x ^4-3 x$ is
  • A
    log x
  • B
    x
  • C
    -x
  • ✓
    $\frac{1}{x}$
Answer
Correct option: D.
$\frac{1}{x}$
(d) $\frac{1}{x}$
Explanation: $\frac{d y}{d x}-\frac{y}{x}= x ^3-3$
I.F. $=e^{\int-\frac{1}{x} d x}=e^{-\log x}=e^{\log x^{-1}}=\frac{1}{x}$
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MCQ 101 Mark
In a 1000 m race, A can beat B by 100 m. In a race of 400 m, B can beat C by 40 m. By how many meters will A beat C in a race of 500 m?
  • A
    105 m
  • B
    85 m
  • C
    115 m
  • D
    95 m
Answer
(d) 95 m
Explanation: When A runs 1000 m, B runs 900 m
Hence, when A runs 500 m , B runs 450 m
Again, when $B$ runs $400 m, C$ runs 360 m
And, when $B$ runs $450 m, C$ runs $=360 \times \frac{450}{400}=405 m$
Required distance $=500-405=95$ meter
That means when A runs 500 meter then $B$ can run 450 m then $C$ runs 405 m
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MCQ 111 Mark
$y = ae m ^{ mx }+b e^{- mx }$, where $a , b$ are arbitrary constants satisfies which of the following differential equation?
  • A
    $\frac{d y}{d x}+ my =0$
  • B
    $\frac{d y}{d x}- my =0$
  • ✓
    $\frac{d^2 y}{d x^2}- m ^2 y =0$
  • D
    $\frac{d^2 y}{d x^2}+ m ^2 y =0$
Answer
Correct option: C.
$\frac{d^2 y}{d x^2}- m ^2 y =0$
(C) $\frac{d^2 y}{d x^2}- m ^2 y =0$
Explanation: $y = a e ^{ mx }+b e ^{- mx }$
$\begin{array}{l}\Rightarrow \frac{d y}{d x}= mae ^{ mx }- bme ^{- mx } \\ \text { and } \frac{d^2 y}{d x^2}= m ^2 ae ^{ mx }+ m ^2 be ^{- mx }= m ^2\left( ae ^{ mx }+ be ^{- mx }\right) \\ \Rightarrow \frac{d^2 y}{d x^2}= m ^2 y \text { i.e. } \frac{d^2 y}{d x^2}- m ^2 y =0\end{array}$
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MCQ 121 Mark
Two dice are thrown n times in succession. The probability of obtaining a double six atleast once is
  • A
    $\left(\frac{1}{12}\right)^n$
  • B
    $\left(\frac{1}{36}\right)^n$
  • C
    $1+\left(\frac{35}{36}\right)^n$
  • ✓
    $1-\left(\frac{35}{36}\right)^n$
Answer
Correct option: D.
$1-\left(\frac{35}{36}\right)^n$
(D) $1-\left(\frac{35}{36}\right)^n$
Explanation: $1-\left(\frac{35}{36}\right)^n$
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MCQ 131 Mark
If X is normally distributed with mean 20 and standard deviation 4, then standard normal variable Z corresponding to X = 21 is
  • A
    -1.25
  • B
    1.25
  • ✓
    0.25
  • D
    -0.25
Answer
Correct option: C.
0.25
(c) 0.25
Explanation: Given $\mu=20, \sigma=4$
$Z=\frac{X-\mu}{\sigma} \Rightarrow Z=\frac{21-20}{4}=0.25$
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MCQ 141 Mark
If $y =e^x+e^{x+\ldots \infty}$ then $\frac{d y}{d x}=$ ?
  • ✓
    $\frac{y}{(1-y)}$
  • B
    $\frac{1}{(y-1)}$
  • C
    $\frac{1}{(1-y)}$
  • D
    $\frac{y}{(y-1)}$
Answer
Correct option: A.
$\frac{y}{(1-y)}$
(a) $\frac{y}{(1-y)}$
Explanation: We can write it as
$\Rightarrow y=e^{x+y}$
log y = (x + y) log e
Differentiating with respect to x,we get
$\begin{array}{l}\Rightarrow \frac{1}{y} \frac{d y}{d x}=1+\frac{d y}{d x} \\ \Rightarrow\left(\frac{1}{y}-1\right) \frac{d y}{d x}=1 \\ \Rightarrow \frac{d y}{d x}=1\left(\frac{y}{1-y}\right)\end{array}$
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MCQ 151 Mark
Which of the term is not used in a linear programming problem:
  • ✓
    Concave region
  • B
    Objective function
  • C
    Slack in equation
  • D
    Feasible Region
Answer
Correct option: A.
Concave region
(a) Concave region
Explanation: Concave region
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MCQ 161 Mark
An 8 year annuity due has a present value of ₹ 1000. If the interest rate is 5% p.a., the amount of each annuity is closest to [Use $\left.(1.05)^{-8}=0.676\right]$
  • A
    ₹ 154.73
  • B
    ₹ 109.39
  • C
    ₹ 104.72
  • ✓
    ₹ 147.36
Answer
Correct option: D.
₹ 147.36
(d) ₹ 147.36
Explanation: Present value = ₹ 1000, Let each annuity be ₹ x,$n =8$ years, $r =5 \% \Rightarrow i =\frac{5}{100}=0.05$
$\begin{array}{l} n =8 \text { years, } r =5 \% \Rightarrow i =\frac{5}{100}=0.05 \\ \therefore 1000=\frac{x(1+0.05)}{0.05}\left[1-(1+0.05)^{-8}\right]\end{array}$
$\begin{array}{l}=\frac{x(1.05)}{0.05}\left[\left(1-(1.05)^{-8}\right]\right. \\ =21 x [1-0.676] \\ =21 x \times 0.324 \\ \Rightarrow x =\frac{1000}{21 \times 0.324}= ₹146.97\end{array}$
Closest to ₹ 147.36
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MCQ 171 Mark
Since α = probability of Type-I error, then 1 - α
  • ✓
    Probability of not rejecting $H _0$ when $H _0$ is true.
  • B
    Probability of rejecting $H _0$ when $H _0$ is true.
  • C
    Probability of rejecting $H _0$ when $H _0$ is true.
  • D
    Probability of not rejecting $H _0$ when $H _0$ is true.
Answer
Correct option: A.
Probability of not rejecting $H _0$ when $H _0$ is true.
(a) Probability of not rejecting $H _0$ when $H _0$ is true.
Explanation: Probability of not rejecting $H _0$ when $H _0$ is true.
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MCQ 181 Mark
If A is a square matrix of order 3 and |A| = 5, then the value of |2A'| is
  • A
    -10
  • B
    -40
  • ✓
    40
  • D
    10
Answer
Correct option: C.
40
(c) 40
Explanation: Given |A| = 5, order of A = 3.
So, $\left|2 A^{\prime}\right|=2^3\left|A^{\prime}\right|=8|A|=8 \times 5=40$
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MCQ - Applied Maths STD 12 Science Questions - Vidyadip