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12th Class Mathematics Sample Paper

Mathematics Sample Paper-12 Total Questions - 29 [Download]

1)

If A is an invertible matrix of order \[3\times 3\] such that | A| = 5, find the value of \[|{{A}^{-1}}|.\]
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2)

If A, B and C are the vertices of \[\Delta ABC,\]what is the value of \[\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}?\]
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3)

The sides of an equilateral triangle are increasing at the rate of 2 cm/s. Find the rate at which the area increases, when the side is 10 cm.
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4)

Evaluate \[\int{\frac{1}{x{{(\log \,x)}^{n}}}\,dx.}\]
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5)

Find the integrating factor of differential equation \[\frac{dy}{dx}+y\tan x-\sec x=0.\]
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6)

Find the projection of vector \[2\hat{i}+\hat{j}\] on the vector \[\hat{i}+2\hat{j}.\]
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7)

Let A and B be square matrices of the order \[3\times 3.\]Is \[{{(AB)}^{2}}={{A}^{2}}{{B}^{2}}?\] Give reason.
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8)

If \[{{x}^{16}}{{y}^{9}}={{({{x}^{2}}+y)}^{17}},\] then prove that \[x\frac{dy}{dx}=2y.\]
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9)

Find the value of m and n, where m and n are order and degree of differential equation \[\frac{4{{\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{3}}}{\frac{{{d}^{3}}y}{d{{x}^{3}}}}+\frac{{{d}^{3}}y}{d{{x}^{3}}}={{x}^{2}}-1.\]
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10)

Find the approximate value of \[\frac{1}{\sqrt{251}}.\]
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11)

If the slope of the curve \[2{{y}^{2}}=a{{x}^{2}}+b\] at \[(1,\,\,-\,1)\] is \[-\,1,\] find the values of a and b.
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12)

There are 5 white and 8 red balls in bag A, 7 white and 6 red balls in bag B, 6 white and 5 red balls in bag C. One ball is taken out at random from each bag. Find the probability that all the three balls are of the same colour.
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13)

A total amount of Rs. 7000 is deposited in three different saving bank accounts with annual interest rates of 5%, 8% and \[8\frac{1}{2}%,\] respectively. The total annual interest from these three accounts is Rs. 550. Equal amounts have been deposited in the 5% and 8% saving accounts. Find the amount deposited in each of the three accounts, with the help of matrix multiplication. Keeping nation's growth in mind, justify the value of saving in individual life.
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14)

Find the intervals in which \[f(x)=\frac{3}{10}{{x}^{4}}-\frac{4}{5}{{x}^{3}}-3{{x}^{2}}+\frac{36}{5}x+11\] is (a) Strictly increasing (b) strictiy decreasing.
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15)

Differentiate \[{{\tan }^{-1}}\left( \frac{a\cos x-b\sin x}{b\cos x+a\sin x} \right),\] \[\frac{-\,\pi }{2}<x<\frac{\pi }{2}\] and \[\frac{a}{b}\tan x>-\,1\,\,w.r.t.x.\]
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16)

\[\] Prove that \[{{\cot }^{-1}}7+{{\cot }^{-1}}8+{{\cot }^{-1}}18={{\cot }^{-1}}3.\]
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17)

Evaluate \[\int{{{x}^{2x}}(1+\log \,x)dx.}\]
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18)

Evaluate \[\int_{0}^{\pi /2}{\log (\sin \,x)\,dx.}\]

OR

Evaluate \[\int{\frac{{{\sin }^{6}}x+{{\cos }^{6}}x}{{{\sin }^{2}}{{\cos }^{2}}x}}\,dx.\]

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19)

Find a unit vector perpendicular to each of the vectors \[\vec{a}+\vec{b}\] and \[\vec{a}-\vec{b},\] where \[\vec{a}=3\hat{i}+2\hat{j}+2\hat{k}\] and \[\vec{b}=\hat{i}+2\hat{j}-2\hat{k}.\]
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20)

Find the foot of perpendicular drawn from the point A (1, 8, 4) to the line joining the points \[B(0,\,\,-\,1,\,\,3)\] and \[C(2,\,\,-3,\,\,1).\]
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For 6 trials of an experiment, let X be a binomial variate which satisfies the relation 9P(X = 4) = P(X = 2). Find the probability of success.

A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 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.

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22)

Two groups are competing for the position on the board of directors of a corporation. The probability that the first and second groups will win are 0.6 and 0.4, respectively. Further, if the first group wins the probability of introducing a new product is 0.7 and the corresponding probability is 0.3, if the second group wins. Find the probability that the new product introduce was by the second group.
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Show that the differential equation \[\left[ x\,{{\sin }^{2}}\left( \frac{y}{x} \right)-y \right]\,dx+x\,dy=0\]

Is homogeneous. Find the particular solution of this differential equation, given that \[y=\frac{\pi }{4},\] when x = 1.

Find the solution of differential equation

\[{{x}^{2}}dy+y(x+y)dx=0,\] if x = 1 and y = 1.

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Consider the function \[f:{{R}^{+}}\to [4,\,\,\infty )\] defined by \[f(x)={{x}^{2}}+4,\] where \[{{R}^{+}}\] is the set of all non-negative real numbers. Show that f is invertible. Also, find the inverse of f.

Show that the relation S in the set

\[A=\{x\in Z:0\,\,\le x\le 12\}\] given by

\[S=\{(a,\,\,b):a,\,\,b\in Z,\,\,|a-b|\]is divisible by 4} is an equivalence relation. Find the set of all elements related to 4.

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Find the equation of plane determined

by points \[A(3,\,\,-\,1,\,\,2),\] B(5, 2, 4), \[C(-\,1,\,\,-\,1,\,\,6)\] and hence find the distance between plane and point P(6, 5, 9).

Show that the lines

\[\vec{r}=3\hat{i}+2\hat{j}-4\hat{k}+\lambda (\hat{i}+2\hat{j}+2\hat{k})\] and

\[\vec{r}=5\hat{i}-2\hat{j}+\mu (3\hat{i}+2\hat{j}+6\hat{k})\] are intersecting.

Hence, find their point of intersection.

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