\displaystyle \text{ASSERTION - REASON QUESTIONS - 1 Mark Each}

\displaystyle \textbf{Question 36: }\text{Assertion (A): The relation }f:\{m,n,p,q\}\to\{11,12,13,14\}
\displaystyle \text{ defined by} \ f=\{(m,11),(n,12),(p,13)\}\text{ is a bijective function.}
\displaystyle \text{Reason (R): The function }f:\{m,n,p\}\to\{11,12,13,14\}\text{ such that}
\displaystyle f=\{(m,11),(n,12),(p,13)\}\text{ is one-one.}
\displaystyle \text{(a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(b) Both Assertion (A) and Reason (R) are true, and Reason (R) is not the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(c) Assertion (A) is true; Reason (R) is false.}
\displaystyle \text{(d) Assertion (A) is false; Reason (R) is true.}
\displaystyle \text{Answer:}
\displaystyle \text{For Assertion (A), the domain is }\{m,n,p,q\}.
\displaystyle \text{But }f=\{(m,11),(n,12),(p,13)\}\text{ gives no image for }q.
\displaystyle \therefore f\text{ is not a function from }\{m,n,p,q\}\text{ to }\{11,12,13,14\}.
\displaystyle \therefore f\text{ cannot be bijective.}
\displaystyle \therefore \text{Assertion (A) is false.}
\displaystyle \text{For Reason (R), the domain is }\{m,n,p\}.
\displaystyle f(m)=11,\qquad f(n)=12,\qquad f(p)=13.
\displaystyle \text{Distinct elements of the domain have distinct images.}
\displaystyle \therefore f\text{ is one-one.}
\displaystyle \therefore \text{Reason (R) is true.}
\displaystyle \therefore \text{Assertion (A) is false and Reason (R) is true.}
\displaystyle \therefore \text{Option (d) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 37: }\text{Assertion (A): Let }A=\begin{bmatrix}d_1&0&0\\0&d_2&0\\0&0&d_3\end{bmatrix},\text{ where }d_1,d_2,d_3\ne0,
\displaystyle \text{then }A^{-1}=\begin{bmatrix}d_1^{-1}&0&0\\0&d_2^{-1}&0\\0&0&d_3^{-1}\end{bmatrix}.
\displaystyle \text{Reason (R): If }A\text{ is a non-singular diagonal matrix, then }A^{-1}\text{ exists and it is also}
\displaystyle \text{a diagonal matrix.}
\displaystyle \text{(a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(b) Both Assertion (A) and Reason (R) are true, and Reason (R) is not the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(c) Assertion (A) is true; Reason (R) is false.}
\displaystyle \text{(d) Assertion (A) is false; Reason (R) is true.}
\displaystyle \text{Answer:}
\displaystyle \text{Given, }A=\begin{bmatrix}d_1&0&0\\0&d_2&0\\0&0&d_3\end{bmatrix},\text{ where }d_1,d_2,d_3\ne0.
\displaystyle |A|=d_1d_2d_3\ne0.
\displaystyle \therefore A^{-1}\text{ exists.}
\displaystyle \text{Let }B=\begin{bmatrix}d_1^{-1}&0&0\\0&d_2^{-1}&0\\0&0&d_3^{-1}\end{bmatrix}.
\displaystyle AB=\begin{bmatrix}d_1d_1^{-1}&0&0\\0&d_2d_2^{-1}&0\\0&0&d_3d_3^{-1}\end{bmatrix}.
\displaystyle =\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}=I.
\displaystyle \therefore B=A^{-1}.
\displaystyle \therefore A^{-1}=\begin{bmatrix}d_1^{-1}&0&0\\0&d_2^{-1}&0\\0&0&d_3^{-1}\end{bmatrix}.
\displaystyle \therefore \text{Assertion (A) is true.}
\displaystyle \text{Also, the inverse of a non-singular diagonal matrix is again a diagonal matrix.}
\displaystyle \therefore \text{Reason (R) is true and correctly explains Assertion (A).}
\displaystyle \therefore \text{Option (a) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 38: }\text{If }E_1\text{ and }E_2\text{ are two mutually exclusive events associated with a random}
\displaystyle \text{experiment and }E\text{ is an event such that }P(E)\ne0.
\displaystyle \text{Assertion (A): }P\left(\frac{E_1\cup E_2}{E}\right)=P\left(\frac{E_1}{E}\right)+P\left(\frac{E_2}{E}\right).
\displaystyle \text{Reason (R): For two mutually exclusive events }E_1\text{ and }E_2,\ P(E_1\cap E_2)=0.
\displaystyle \text{(a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(b) Both Assertion (A) and Reason (R) are true, and Reason (R) is not the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(c) Assertion (A) is true; Reason (R) is false.}
\displaystyle \text{(d) Assertion (A) is false; Reason (R) is true.}
\displaystyle \text{Answer:}
\displaystyle \text{Since }E_1\text{ and }E_2\text{ are mutually exclusive,}
\displaystyle E_1\cap E_2=\phi.
\displaystyle \therefore (E_1\cap E)\cap(E_2\cap E)=\phi.
\displaystyle \text{Hence, }E_1\cap E\text{ and }E_2\cap E\text{ are mutually exclusive.}
\displaystyle P\left(\frac{E_1\cup E_2}{E}\right)=\frac{P\{(E_1\cup E_2)\cap E\}}{P(E)}.
\displaystyle =\frac{P\{(E_1\cap E)\cup(E_2\cap E)\}}{P(E)}.
\displaystyle =\frac{P(E_1\cap E)+P(E_2\cap E)}{P(E)}.
\displaystyle =P\left(\frac{E_1}{E}\right)+P\left(\frac{E_2}{E}\right).
\displaystyle \therefore \text{Assertion (A) is true.}
\displaystyle \text{Also, }P(E_1\cap E_2)=0\text{ for mutually exclusive events.}
\displaystyle \therefore \text{Reason (R) is true and correctly explains Assertion (A).}
\displaystyle \therefore \text{Option (a) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 39: }\text{The vectors }\overrightarrow{PQ},\overrightarrow{QR},\overrightarrow{RS},\overrightarrow{ST},\overrightarrow{TU}\text{ and }\overrightarrow{UP}\text{ represent}
\displaystyle \text{the sides of a regular hexagon.} \displaystyle \text{Assertion (A): }\overrightarrow{PQ}\times(\overrightarrow{RS}+\overrightarrow{ST})\ne\overrightarrow{0}.
\displaystyle \text{Reason (R): }\overrightarrow{PQ}\times\overrightarrow{RS}=\overrightarrow{0}\text{ and }\overrightarrow{PQ}\times\overrightarrow{ST}\ne\overrightarrow{0}.
\displaystyle \text{(a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(b) Both Assertion (A) and Reason (R) are true, and Reason (R) is not the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(c) Assertion (A) is true; Reason (R) is false.}
\displaystyle \text{(d) Assertion (A) is false; Reason (R) is true.}
\displaystyle \text{Answer:}
\displaystyle \overrightarrow{RS}+\overrightarrow{ST}=\overrightarrow{RT}.
\displaystyle \text{Since }\overrightarrow{PQ}\text{ and }\overrightarrow{RT}\text{ are not parallel,}
\displaystyle \overrightarrow{PQ}\times\overrightarrow{RT}\ne\overrightarrow{0}.
\displaystyle \therefore \overrightarrow{PQ}\times(\overrightarrow{RS}+\overrightarrow{ST})\ne\overrightarrow{0}.
\displaystyle \therefore \text{Assertion (A) is true.}
\displaystyle \text{Now, }\overrightarrow{PQ}\text{ and }\overrightarrow{RS}\text{ are not parallel.}
\displaystyle \therefore \overrightarrow{PQ}\times\overrightarrow{RS}\ne\overrightarrow{0}.
\displaystyle \text{Also, }\overrightarrow{PQ}\text{ and }\overrightarrow{ST}\text{ are parallel.}
\displaystyle \therefore \overrightarrow{PQ}\times\overrightarrow{ST}=\overrightarrow{0}.
\displaystyle \therefore \text{Reason (R) is false.}
\displaystyle \therefore \text{Assertion (A) is true and Reason (R) is false.}
\displaystyle \therefore \text{Option (c) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 40: }\text{Assertion (A): A company uses a demand function }
\displaystyle p=\frac{a}{x+b}-c,\text{ where }a,b,c\in R \ \text{and }x=\text{number of units.} \\ \text{The Marginal Revenue decreases with the increase of }x.
\displaystyle \text{Reason (R): }\frac{d}{dx}(MR)<0,\text{ when }0<a<b.
\displaystyle \text{Which one of the following options is correct?}
\displaystyle \text{(a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(b) Both Assertion (A) and Reason (R) are true, and Reason (R) is not the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(c) Assertion (A) is true; Reason (R) is false.}
\displaystyle \text{(d) Assertion (A) is false; Reason (R) is true.}
\displaystyle \text{Answer:}
\displaystyle \text{Given, }p=\frac{a}{x+b}-c.
\displaystyle \text{Revenue }R=xp=\frac{ax}{x+b}-cx.
\displaystyle \therefore MR=\frac{dR}{dx}.
\displaystyle =\frac{a(x+b)-ax}{(x+b)^2}-c.
\displaystyle =\frac{ab}{(x+b)^2}-c.
\displaystyle \therefore \frac{d}{dx}(MR)=-\frac{2ab}{(x+b)^3}.
\displaystyle \text{Since }0<a<b,\text{ we have }a>0\text{ and }b>0.
\displaystyle \text{Also, }x\geq0\text{ as }x\text{ represents the number of units.}
\displaystyle \therefore x+b>0.
\displaystyle \therefore -\frac{2ab}{(x+b)^3}<0.
\displaystyle \therefore \frac{d}{dx}(MR)<0.
\displaystyle \therefore \text{Marginal Revenue decreases as }x\text{ increases.}
\displaystyle \therefore \text{Assertion (A) is true and Reason (R) is true.}
\displaystyle \text{Reason (R) correctly explains Assertion (A).}
\displaystyle \therefore \text{Option (a) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 41: }\text{Assertion (A): The curve in the graph below is not a one-one function.}
\displaystyle \text{Reason (R): If any straight line parallel to }y\text{-axis does not cut the curve at more than}
\displaystyle \text{one point, then that curve represents a function.}
\displaystyle \text{(a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(b) Both Assertion (A) and Reason (R) are true, and Reason (R) is not the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(c) Assertion (A) is true; Reason (R) is false.}
\displaystyle \text{(d) Assertion (A) is false; Reason (R) is true.}
\displaystyle \text{Answer:}
\displaystyle \text{From the graph, a horizontal line can intersect the curve at more than one point.}
\displaystyle \text{Thus, there exist distinct values }x_1\text{ and }x_2\text{ such that }f(x_1)=f(x_2).
\displaystyle \therefore f\text{ is not a one-one function.}
\displaystyle \therefore \text{Assertion (A) is true.}
\displaystyle \text{Reason (R) describes the vertical line test for determining whether a curve represents a function.}
\displaystyle \text{The given curve passes the vertical line test and hence represents a function.}
\displaystyle \therefore \text{Reason (R) is true.}
\displaystyle \text{However, one-one nature is determined by the horizontal line test, not the vertical line test.}
\displaystyle \therefore \text{Reason (R) is not the correct explanation of Assertion (A).}
\displaystyle \therefore \text{Option (b) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 42: }\text{Assertion (A): Let }f(x)\text{ be a polynomial function of degree }7\text{ such that}
\displaystyle \frac{d}{dx}\{f(x)\}=(x-2)^3(x+1)^2(7x-2)\text{ has a local minimum at }x=-1.
\displaystyle \text{Reason (R): Let }f\text{ have first derivative at }c\text{ such that }f'(c)=0\text{ and }f'(x)>0
\displaystyle \text{for }x\in(c-\delta,c),\ f'(x)<0\text{ for }x\in(c,c+\delta),\text{ then }c\text{ is a point of local minimum.}
\displaystyle \text{(a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(b) Both Assertion (A) and Reason (R) are true, and Reason (R) is not the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(c) Assertion (A) is true; Reason (R) is false.}
\displaystyle \text{(d) Assertion (A) is false; Reason (R) is true.}
\displaystyle \text{Answer:}
\displaystyle f'(x)=(x-2)^3(x+1)^2(7x-2).
\displaystyle f'(-1)=0.
\displaystyle \text{For }x\text{ close to }-1,\ (x-2)^3<0,\ (x+1)^2>0\text{ and }7x-2<0.
\displaystyle \therefore f'(x)>0\text{ on both sides of }x=-1.
\displaystyle \therefore f'(x)\text{ does not change sign at }x=-1.
\displaystyle \therefore x=-1\text{ is not a point of local minimum.}
\displaystyle \therefore \text{Assertion (A) is false.}
\displaystyle \text{For a local minimum, }f'(x)\text{ must change from negative to positive at }x=c.
\displaystyle \text{But Reason (R) states that }f'(x)>0\text{ to the left of }c\text{ and }f'(x)<0\text{ to the right.}
\displaystyle \text{This sign change gives a local maximum, not a local minimum.}
\displaystyle \therefore \text{Reason (R) is false.}
\displaystyle \therefore \text{both Assertion (A) and Reason (R) are false.}
\displaystyle \therefore \text{none of the given options is correct.}
\displaystyle \\

\displaystyle \textbf{Question 43: }\text{Assertion (A): If }y=\sin^{-1}(x\sqrt{x}),\text{ then }\frac{dy}{dx}=\frac{3\sqrt{x}}{2\sqrt{1-x^3}}.
\displaystyle \text{Reason (R): }\frac{d}{dx}(\sin^{-1}x)=\frac{1}{\sqrt{1-x^2}},\quad |x|\leq1.
\displaystyle \text{(a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(b) Both Assertion (A) and Reason (R) are true, and Reason (R) is not the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(c) Assertion (A) is true; Reason (R) is false.}
\displaystyle \text{(d) Assertion (A) is false; Reason (R) is true.}
\displaystyle \text{Answer:}
\displaystyle \text{Given, }y=\sin^{-1}(x\sqrt{x})=\sin^{-1}(x^{3/2}).
\displaystyle \text{Using }\frac{d}{dx}(\sin^{-1}u)=\frac{1}{\sqrt{1-u^2}}\frac{du}{dx},
\displaystyle \frac{dy}{dx}=\frac{1}{\sqrt{1-(x^{3/2})^2}}\frac{d}{dx}(x^{3/2}).
\displaystyle =\frac{1}{\sqrt{1-x^3}}\times\frac32x^{1/2}.
\displaystyle =\frac{3\sqrt{x}}{2\sqrt{1-x^3}}.
\displaystyle \therefore \text{Assertion (A) is true.}
\displaystyle \text{Reason (R) gives the derivative formula used to obtain the result in Assertion (A).}
\displaystyle \therefore \text{Reason (R) is true and correctly explains Assertion (A).}
\displaystyle \therefore \text{Option (a) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 44: }\text{Assertion (A): Degree of the differential equation }
\displaystyle a\left(\frac{dy}{dx}\right)^2+b\frac{dx}{dy}=c \ \text{cannot be determined.}
\displaystyle \text{Reason (R): If each term involving derivatives of a differential equation is a polynomial}
\displaystyle \text{or can be expressed as a polynomial, then the highest exponent of the highest order}
\displaystyle \text{derivative is called the degree of the differential equation.}
\displaystyle \text{(a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(b) Both Assertion (A) and Reason (R) are true, and Reason (R) is not the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(c) Assertion (A) is true; Reason (R) is false.}
\displaystyle \text{(d) Assertion (A) is false; Reason (R) is true.}
\displaystyle \text{Answer:}
\displaystyle \text{Let }p=\frac{dy}{dx}.
\displaystyle \therefore \frac{dx}{dy}=\frac{1}{p}.
\displaystyle \text{The given differential equation becomes}
\displaystyle ap^2+\frac{b}{p}=c.
\displaystyle \text{Multiplying throughout by }p,
\displaystyle ap^3+b=cp.
\displaystyle \therefore ap^3-cp+b=0.
\displaystyle \text{This is a polynomial in the first-order derivative }p=\frac{dy}{dx}.
\displaystyle \text{The highest power of }\frac{dy}{dx}\text{ is }3.
\displaystyle \therefore \text{the degree of the differential equation is }3.
\displaystyle \therefore \text{Assertion (A) is false.}
\displaystyle \text{Reason (R) correctly states the definition of degree of a differential equation.}
\displaystyle \therefore \text{Reason (R) is true.}
\displaystyle \therefore \text{Assertion (A) is false and Reason (R) is true.}
\displaystyle \therefore \text{Option (d) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 45: }\text{Shown below is the graph of }f(x)=-2|x-3|. \displaystyle \text{Assertion (A): Maximum value of the function is }0.
\displaystyle \text{Reason (R): Minimum value of the function approaches }\infty.
\displaystyle \text{(a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(b) Both Assertion (A) and Reason (R) are true, and Reason (R) is not the correct }
\displaystyle \text{explanation of Assertion (A).}
\displaystyle \text{(c) Assertion (A) is true; Reason (R) is false.}
\displaystyle \text{(d) Assertion (A) is false; Reason (R) is true.}
\displaystyle \text{Answer:}
\displaystyle \text{Given, }f(x)=-2|x-3|.
\displaystyle \text{Since }|x-3|\geq0,
\displaystyle -2|x-3|\leq0.
\displaystyle \text{Equality holds when }|x-3|=0.
\displaystyle \therefore x=3.
\displaystyle f(3)=-2|3-3|=0.
\displaystyle \therefore \text{the maximum value of }f(x)\text{ is }0.
\displaystyle \therefore \text{Assertion (A) is true.}
\displaystyle \text{As }x\to\pm\infty,\ |x-3|\to\infty.
\displaystyle \therefore f(x)=-2|x-3|\to-\infty.
\displaystyle \therefore \text{the function has no minimum value and is unbounded below.}
\displaystyle \therefore \text{Reason (R) is false.}
\displaystyle \therefore \text{Assertion (A) is true and Reason (R) is false.}
\displaystyle \therefore \text{Option (c) is correct.}
\displaystyle \\

Discover more from ICSE / ISC / CBSE Mathematics Portal for K12 Students

Subscribe to get the latest posts sent to your email.