Mathematics Karnataka PUC II 2023 Model Question Paper solutions

Karnataka PUC II Mathematics Model Question Paper 2023 Solutions

PART-A

Q.1 The identity element for the binary operation $*$ if $a*b = \dfrac{ab}{4}, \forall a, b \in Q$

not exist

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Q.2 If $cot^{-1}x = y$, then

$0 \le y \le \pi$

$0 \lt y \lt \pi$

$-\dfrac{\pi}{2} \le y \le \dfrac{\pi}{2}$

$-\dfrac{\pi}{2} \lt y \lt \dfrac{\pi}{2}$

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Q.3 For $2 \times 2$ matrix $A = [a_{ij}]$ whose elements are given by $a_{ij} = \dfrac{i}{j}$, then A is equal to

$\left[ \begin{matrix}
2 & \phantom{-} 3 \\
\dfrac{1}{2} & \dfrac{9}{2} \\
\end{matrix} \right]$

$\left[ \begin{matrix}
\dfrac{1}{2} & \phantom{-} 1 \\
2 & \dfrac{1}{2} \\
\end{matrix} \right]$

$\left[ \begin{matrix}
1 & \phantom{-} \dfrac{1}{2} \\
2 & 1 \\
\end{matrix} \right]$

$\left[ \begin{matrix}
1 & \phantom{-} 2 \\
2 & 1 \\
\end{matrix} \right]$

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Q.4 If $\left| \begin{matrix}
3 & \phantom{-}x \\
x & 1 \\
\end{matrix} \right|$ $=$ $\left| \begin{matrix}
3 & \phantom{-}2 \\
4 & 1 \\
\end{matrix} \right|$ then the value of $x$ is equa to

$\pm 4$

$\pm 2$

$\pm \sqrt{2}$

$\pm 2 \sqrt{2}$

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Q.5 Left hand derivative of $f(x) = |x|$ at $x=0$ is

-1

does not exist

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Q.6 $\int e^x \left( \dfrac{1}{x} - \dfrac{1}{x^2} \right)$ $dx$

$ e^x$ + $C$

$ \dfrac{e^x}{x}$ + $C$

$ \dfrac{e^x}{x^2}$ + $C$

$ \dfrac{-e^x}{x}$ + $C$

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Q.7 The projection of vector $\overrightarrow{AB}$ on the directed line $l$, if angle $\theta = \pi$ will be

Zero vector

$\overrightarrow{AB}$

$\overrightarrow{BA}$

Unit vector

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Q.8 The equation of $xy$ plane is

$x=0$

$y=0$

$x=0$ and $y=0$

$z=0$

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Q.9 The corner points of the feasible region determined by the following system of linear inequalities: $2x+y \le 10$, $x+3y \le 15$, $x,y \ge 0$ are (0,0), (5,0), (3,4) and (0,5). Let $Z$ = $ax + by$, where $a, b> 0$ Condition on $a$ and $b$ so that the maximum of $Z$ occurs at both (3,4) and (0,5) is

$a=b$

$a=2b$

$a=3b$

$b=3a$

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Q.10 If $P(A) = \dfrac{1}{2}$, $P(B) = 0$, then $P(A|B)$ is

$\dfrac{1}{2}$

not defined

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Q.11

For a square matrix $A$ in matrix equation $AX = B$, if $|A| = 0$ and $(adj A) B \ne 0$, then there exists solution

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Q.12

The order of the differential equation $2x^2 \left( \dfrac{d^2y}{dx^2} \right) - 3 \left( \dfrac{dy}{dx} \right) + y$ is

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Q.13 Sum of the intercepts cut off by the plane $2x+y-z=5$ is !*n1*!

$-\dfrac{5}{2}$

$\dfrac{5}{2}$

$5$

$-\dfrac{2}{5}$

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Q.14 The slope of the normal to the curve $y=2x^2-3sinx$ at $x=0$ is

$3$

$-3$

$-\dfrac{1}{3}$

$\dfrac{1}{3}$

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Q.15 The probability of obtaining an even prime number on each die, when a pair of dice is rolled is

$\dfrac{1}{12}$

$\dfrac{1}{18}$

$\dfrac{1}{9}$

$\dfrac{1}{36}$

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Q.16

Define a bijective function.

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Q.17 Find the derivative of the function $sec(tan \sqrt{x})$ with respect to $x$

$\dfrac{1}{2 \sqrt{x}}$ $tan(tan \sqrt{x})$ $sec^2(\sqrt{x})$

$\dfrac{1}{\sqrt{x}}$ $sec(tan \sqrt{x})$ $tan(tan \sqrt{x})$ $sec^2(\sqrt{x})$

$tan(tan \sqrt{x})$ $sec^2(\sqrt{x})$

$\dfrac{1}{2 \sqrt{x}}$ $sec(tan \sqrt{x})$ $tan(tan \sqrt{x})$ $sec^2(\sqrt{x})$

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Q.18

Define feasible solutions in a linear programming problem.

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Q.19 $\int \dfrac{1-sinx}{cos^2x}$ $dx$

$tanx$ - $secx$ + $C$

$tanx$ + $secx$ + $C$

$secx$ $tanx$ + $C$

$sec^2x$ + $C$

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Q.20

Define Negative of a Vector

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PART-B

Q.21 Find $gof$ and $fog$, if $f: R \to R$ and $g: R \to R$ are given by $g(x) = x^{\dfrac{1}{3}}$ and $f(x) = 8x^3$

$gof(x) = 8x$ and $fog(x) = 2x$

$gof(x) = 8x$ and $fog(x) = 8x$

$gof(x) = 2x$ and $fog(x) = 2x$

$gof(x) = 2x$ and $fog(x) = 8x$

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Q.22

Prove that $tan^{-1}x + cot^{-1}x$ = $\dfrac{\pi}{2}$

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Q.23 If $sin \left( sin^{-1} \dfrac{1}{5} + cos^{-1} x \right) = 1$, find $x$

$5$

$\dfrac{1}{5}$

$\dfrac{1}{\pi}$

$\dfrac{\pi}{5}$

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Q.24 Find the area of the triangle whose vertices are (2,7), (1,1) and (10,8) using determinants.

$-\dfrac{47}{2}$ sq. units

$\dfrac{2}{47}$ sq. units

$47$ sq. units

$\dfrac{47}{2}$ sq. units

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Q.25 Find $\dfrac{dy}{dx}$, if $2x + 3y = sin x$

$\dfrac{cos x - 2}{3}$

$\dfrac{cos x + 2}{3}$

$-\dfrac{cos x + 2}{3}$

$cos x - \dfrac{2}{3}$

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Q.26 If $y = x^{sin x}$, $x > 0$, find $\dfrac{dy}{dx}$

$\dfrac{x^{sin x}}{x} + logx cosx$

$ \dfrac{sinx}{x} + logx cosx $

$x^{sin x}$ $\dfrac{sinx}{x} $

$x^{sin x}$ $\left[ \dfrac{sinx}{x} + logx cosx \right] $

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Q.27 Find the local maximum value of the function $g(x) = x^3 - 3x$

-1

-6

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Q.28 Evaluate $\int sin3x$ $cos4x$ $dx$

$ \dfrac{cosx}{2} - \dfrac{cos7x}{14} $ + $C$

$ \dfrac{sinx}{2} - \dfrac{sin7x}{14} $ + $C$

$ \dfrac{tanx}{2} - \dfrac{cot7x}{14} $ + $C$

$ -\dfrac{cosx}{2} + \dfrac{cos7x}{14} $ + $C$

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Q.29 Evaluate $\int_{0}^{\frac{\pi}{2}}$ $cos2x$ $dx$

$\dfrac{1}{2}$

$\dfrac{\pi}{2}$

$0$

$1$

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Q.30 Form the differential equation representing the family of curves $y=mx$, where $m$ is arbitrary constant.

$y=x$

$\dfrac{dy}{dx}$ = $\dfrac{x}{y}$

$\dfrac{dy}{dx}$ = $xy$

$\dfrac{dy}{dx}$ = $\dfrac{y}{x}$

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Q.31 Find the area of a triangle having the points A(1,1,1), B(1,2,3) and C(2,3,1) as its vertices.

$\dfrac{\sqrt{21}}{2}$

$\sqrt{21}$

$2$

$\dfrac{2}{\sqrt{21}}$

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Q.32 Find a vector in the direction of the $\overrightarrow{a}$ = $\hat{i} - 2 \hat{j}$ that has magnitude 7 units.

$ 7 \hat{i} - 14 \hat{j}$

$ \dfrac{7}{\sqrt5} \hat{i} - \dfrac{14}{\sqrt5} \hat{j}$

$ \dfrac{7}{\sqrt5} \hat{i} + \dfrac{14}{\sqrt5} \hat{j}$

$ 7 \hat{i} + 14 \hat{j}$

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Q.33 Find the angle between the line $\dfrac{x+1}{2}$=$\dfrac{y}{3}$=$\dfrac{z-3}{6}$ and the plane $10x+2y-11z=3$

$sin^{-1}\dfrac{4}{21}$

$\pi$

$sin^{-1}\dfrac{8}{21}$

$\dfrac{\pi}{2}$

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Q.34 Find the probability distribution of number of heads in two tosses of a coin.

X	0	1	2
P(X)	$\dfrac14$	$\dfrac34$	$\dfrac14$

X	0	1	2
P(X)	$\dfrac14$	$\dfrac24$	$\dfrac14$

X	0	1	2
P(X)	$0$	$\dfrac34$	$\dfrac14$

X	0	1	2
P(X)	$\dfrac14$	$\dfrac34$	$0$

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PART-C

Q.35

Show that the relation R in R defined as R = {(a,b) : a$\le$b}, is reflexive and transitive but not symmetric.

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Q.36 Solve: $tan^{-1} \left( \dfrac{x-1}{x-2}\right)$ + $tan^{-1} \left( \dfrac{x+1}{x+2}\right)$ = $\dfrac{\pi}{4}$

$x= \pm {\dfrac {1}{\sqrt3}}$

$x= \pm 2$

$x= \pm {\dfrac {1}{2}}$

$x= \pm {\dfrac {1}{\sqrt2}}$

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Q.37

Express $\left[ \begin{matrix}

3 & \phantom{-}5 \\

1 & -1 \\

\end{matrix} \right]$ as the sum of a symmetric and skew symmetric matrices.

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Q.38 If $y = cos^{-1} \left( \dfrac{1-x^2}{1+x^2} \right)$, 0 < $x$ < 1, then find $\dfrac{dy}{dx}$

$\dfrac{dy}{dx}$ = $\dfrac{2}{1+x}$

$\dfrac{dy}{dx}$ = $\dfrac{2}{1+x^2}$

$\dfrac{dy}{dx}$ = $1+x^2$

$\dfrac{dy}{dx}$ = $1+x$

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Q.39

Verify Rolle's theorem for the function $f(x) = x^2 + 2x - 8$, $x \in [-4, 2]$.

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Q.40 Find the intervals in which the function $f$ given by $f(x)=4x^3−6x^2-72x+30$ is (i) strictly increasing (ii) strictly decreasing.

During $(−∞, -2)$ and $(3, ∞)$, the function is strictly increasing.
During $(-2, 3)$, the function is strictly decreasing.

During $(−∞, -2)$, the function is strictly increasing.
During $(3, ∞)$, the function is strictly decreasing.

During $(−∞, 0)$, the function is strictly increasing.
During $(0, ∞)$, the function is strictly decreasing.

During $(−∞, -2)$ and $(3, ∞)$, the function is strictly decreasing.
During $(-2, 3)$, the function is strictly increasing.

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Q.41 Find $\int x cosx$ $dx$

$x$ $cosx$ + $sinx$ + $C$

$\dfrac{x^2}{2}$ $sinx$ + $cosx$ + $C$

$x$ $sinx$ - $cosx$ + $C$

$x$ $sinx$ + $cosx$ + $C$

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Q.42 Find $\int \dfrac{x}{(x+1)(x+2)}$ $dx$

$ log \left| \dfrac{2(x+2)}{x+1} \right| + C$

$ log \left| \dfrac{x+2}{(x+1)^2} \right| + C$

$ log \left| \dfrac{(x+2)^2}{x+1} \right| + C$

$ log \left| \dfrac{x+2}{x+1} \right| + C$

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Q.43 Find the area of the region bounded by the curve $y^2 = x$ and the lines $x=1$, $x=4$ and the x-axis in the first quadrant.

$\dfrac{4}{3}$

$\dfrac{16}{3}$

$\dfrac{2}{3}$

$\dfrac{14}{3}$

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Q.44 Find the equation of the curve passing through the point (-2, 3), given that the slope of the tangent to the curve at any point $(x, y)$ is $\dfrac{2x}{y^2}$

$y^3 = 3x^2 + 15$

$y^3 = 3x^2 - 19$

$y^3 = 3x^2$

$y^2 = 3x + 15$

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Q.45

For any three vectors $\overrightarrow{a}$, $\overrightarrow{b}$ and $\overrightarrow{c}$, prove that [$\overrightarrow{a}$ + $\overrightarrow{b}$ $\overrightarrow{b}$ + $\overrightarrow{c}$ $\overrightarrow{c}$ + $\overrightarrow{a}$] = 2[$\overrightarrow{a}$ $\overrightarrow{b}$ $\overrightarrow{c}$]

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Q.46 If $\overrightarrow{a}$, $\overrightarrow{b}$ and $\overrightarrow{c}$ are three unit vectors such that $\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = \overrightarrow{0}$ find the value of $\overrightarrow{a} . \overrightarrow{b}$ + $\overrightarrow{b} . \overrightarrow{c}$ + $\overrightarrow{c} . \overrightarrow{a}$

$\dfrac{3}{2}$

$-\dfrac{3}{2}$

$-\dfrac{2}{3}$

$\dfrac{3}{2}$

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Q.47 Find the vector equation of the plane passing through the intersection of the planes $3x-y+2z-4=0$ and $x+y+z-3=0$ and the point $(2,2,1)$.

$\overrightarrow{r} . (2\hat{i} - 2 \hat{j} + \hat{k}) =0$

$\overrightarrow{r} . (\hat{i} - \hat{j} + \hat{k}) =1$

$\overrightarrow{r} . (2\hat{i} + 2 \hat{j} + \hat{k}) =1$

$\overrightarrow{r} . (2\hat{i} - 2 \hat{j} + \hat{k}) =1$

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Q.48 A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.

$\dfrac{3}{8}$

$\dfrac{3}{7}$

$\dfrac{3}{4}$

$\dfrac{1}{6}$

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PART-D

Q.49 Let $f: N \rightarrow R$ be a function defined as $f(x) = 4x^2 + 12x + 15$. Show that $f: N \rightarrow S$, where $S$ is the range of function $f$, is invertible. Find the inverse of $f$.

$\dfrac{\sqrt{x-6} - 3}{2}$

$\dfrac{\sqrt{y-6} - 3}{2}$

$\sqrt{x-6} - 3$

$(2x + 3)^2 + 6$

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Q.50

If $A$ = $\left[ \begin{matrix}

0 & \phantom{-}6 & \phantom{-}7 \\

-6 & \phantom{-}0 & \phantom{-}8 \\

7 & \phantom{-}-8 & \phantom{-}0 \\

\end{matrix} \right]$,
$B$ = $\left[ \begin{matrix}

0 & \phantom{-}1 & \phantom{-}1 \\

1 & \phantom{-}0 & \phantom{-}2 \\

1 & \phantom{-}2 & \phantom{-}0 \\

\end{matrix} \right]$ and
$C$ = $\left[ \begin{matrix}

2 \\

-2 \\

3 \\

\end{matrix} \right]$, calculate $AB$, $AC$ and $A(B+C)$. Verify that $(A+B)C$ = $AC$+$BC$

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Q.51

Solve the following system of equations by matrix method:

$x-y+z = 4$; $x+y+z=2$ and $2x+y-3z=0$

$x=2$, $y=1$ and $z=2$

$x=2$, $y=-1$ and $z=1$

$x=2$, $y=-2$ and $z=2$

$x=1$, $y=-1$ and $z=1$

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Karnataka PUC II Mathematics Model Question Paper 2023 Solutions

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