The value of a for which the function
Question: The value of a for which the function $f(x)=\left\{\begin{array}{ll}\frac{\left(4^{x}-1\right)^{3}}{\sin (x / a) \log \left\{\left(1+x^{2} / 3\right)\right\}}, x \neq 0 \\ 12(\log 4)^{3} , x=0\end{array}\right.$ may be continuous at $x=0$ is (a) 1 (b) 2 (c) 3 (d) none of these Solution: (d) none of these For $f(x)$ to be continuous at $x=0$, we must have $\lim _{x \rightarrow 0} f(x)=f(0)$ $\lim _{x \rightarrow 0}\left[\frac{\left(4^{x}-1\right)^{3}}{\sin \frac{x}{a} \log \left(1+\frac...
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Question: Express $\frac{-48}{60}$ as a rational number with denominator $5 .$ Solution: If $\frac{a}{b}$ is a rational integer and $m$ is a common divisor of $a$ and $b$, then $\frac{a}{b}=\frac{a \div m}{b \div m}$. $\therefore \frac{-48}{60}=\frac{-48 \div 12}{60 \div 12}=\frac{-4}{5}$...
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Question: Express $\frac{-42}{98}$ as a rational number with denominator 7 . Solution: If $\frac{a}{b}$ is a rational number and $m$ is a common divisor of $a$ and $b$, then $\frac{a}{b}=\frac{a \div m}{b \div m}$. $\therefore \frac{-42}{98}=\frac{-42 \div 14}{98 \div 14}=\frac{-3}{7}$...
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Question: Express $\frac{-3}{5}$ as a rational number with denominator (i) 20 (ii) 30 (iii) 35 (iv) 40 Solution: If $\frac{a}{b}$ is a fraction and $m$ is a non-zero integer, then $\frac{a}{b}=\frac{a \times m}{b \times m}$. Now, (i) $\frac{-3}{5}=\frac{-3 \times 4}{5 \times 4}=\frac{-12}{20}$ (ii) $\frac{-3}{5}=\frac{-3 \times-6}{5 \times-6}=\frac{18}{-30}$ (iii) $\frac{-3}{5}=\frac{-3 \times 7}{5 \times 7}=\frac{-21}{35}$ (iv) $\frac{-3}{5}=\frac{-3 \times-8}{5 \times-8}=\frac{24}{-40}$...
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Question: If $f(x)=\frac{1-\sin x}{(\pi-2 x)^{2}}$, when $x \neq \pi / 2$ and $f(\pi / 2)=\lambda$, then $f(x)$ will be continuous function at $x=\pi / 2$, where $\lambda=$ (a) $1 / 8$ (b) $1 / 4$ (c) $1 / 2$ (d) none of these Solution: (a) $\frac{1}{8}$ If $f(x)$ is continuous at $x=\frac{\pi}{2}$, then $\lim _{x \rightarrow \frac{\pi}{2}} f(x)=f\left(\frac{\pi}{2}\right)$ $\lim _{x \rightarrow \frac{\pi}{2}} \frac{1-\sin x}{(\pi-2 x)^{2}}=f\left(\frac{\pi}{2}\right)$ Suppose $\left(\frac{\pi}{...
Read More →A school awarded 42 medals in hockey, 18 in basketball and 23 in cricket.
Question: A school awarded 42 medals in hockey, 18 in basketball and 23 in cricket. if these medals were bagged by a total of 65 students and only 4 students got medals in all the three sports, how many students received medals in exactly two of the three sports? Solution: Given: - Total number of students $=65$ - Medals awarded in Hockey $=42$ - Medals awarded $\mathrm{n}$ Basketball $=18$ - Medals awarded in Cricket $=23$ - 4 students got medals in all the three sports. To Find: Number of stud...
Read More →The function
Question: The function $f(x)=\left\{\begin{array}{cc}x^{2} / a , \quad 0 \leq x1 \\ a , \quad 1 \leq x\sqrt{2} \\ \frac{2 b^{2}-4 b}{r^{2}}, \sqrt{2} \leq x\infty\end{array}\right.$ is continuous for $0 \leq x\infty$, then the most suitable values of $a$ and $b$ are (a) $a=1, b=-1$ (b) $a=-1, b=1+\sqrt{2}$ (c) $a=-1, b=1$ (d) none of these Solution: (c) $a=-1, b=1$ Given: $f(x)$ is continuous for $0 \leq x\infty$. This means that $f(x)$ is continuous for $x=1, \sqrt{2}$. Now, If $f(x)$ is contin...
Read More →In a group of 65 people, 40 like cricket and 10 like both cricket and tennis.
Question: In a group of 65 people, 40 like cricket and 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis? Solution: Given: In a group of 65 people: - 40 people like cricket - 10 like both cricket and tennis To Find: - Number of people like tennis only - Number of people like tennis Let us consider, Number of people who like cricket $=n(C)=40$ Number of people who like tennis $=n(T)$ Number of people who like cricket or tennis $=n(C \cup T)=65$ Numbe...
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Question: $f(x)= \begin{cases}\frac{\sqrt{1+p x}-\sqrt{1-p x}}{x}, -1 \leq x0 \\ \frac{2 x+1}{x-2} , 0 \leq x \leq 1\end{cases}$ is continuous in the interval $[-1,1]$, then $p$ is equal to (a) $-1$ (b) $-1 / 2$ (c) $1 / 2$ (d) 1 Solution: (b) $-\frac{1}{2}$ Given: $f(x)=\left\{\begin{array}{c}\frac{\sqrt{1+p x}-\sqrt{1-p x}}{x}, \text { if }-1 \leq x0 \\ \frac{2 x+1}{x-2}, \text { if } 0 \leq x \leq 1\end{array}\right.$ If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0^{-}} f(x)=\l...
Read More →In an examination, 56% of the candidates failed in English and 48% failed
Question: In an examination, 56% of the candidates failed in English and 48% failed in science. If 18% failed in both English and science, find the percentage of those who passed in both the subjects. Solution: Given: In an examination: - $56 \%$ of candidates failed in English $-48 \%$ of candidates failed in science - $18 \%$ of candidates failed in both English and Science To Find; Percentage of students who passed in both subjects. Let us consider, Percentage of candidates who failed in Engl...
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Question: $f(x)=\frac{2-(256-7 x)^{1 / 8}}{(5 x+32)^{1 / 5}-2}, x \neq 0$ is continuous everywhere, is given by (a) $-1$ (b) 1 (c) 26 (d) none of these Solution: (d) none of these Given: $f(x)=\frac{2-(256-7 x)^{\frac{1}{8}}}{(5 x+32)^{\frac{1}{5}}-2}$ For $f(x)$ to be continuous at $x=0$, we must have $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow f(0)=\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0} \frac{2-(256-7 x)^{\frac{1}{8}}}{(5 x+32)^{\frac{1}{5}}-2}$ $\Rightarrow f(0)=\lim _{x \r...
Read More →In a class of a certain school, 50 students, offered mathematics
Question: In a class of a certain school, 50 students, offered mathematics, 42 offered biology and 24 offered both the subjects. Find the number of students offering (i) mathematics only, (ii) biology only, (iii) any of the two subjects. Solution: Given: Number of students offered Mathematics $=50$ Number of students offered Biology $=42$ Number of students offered both Mathematics and Biology $=24$ To Find: (i) Number of students offered Mathematics only Let us consider, Number of students offe...
Read More →The value of f(0), so that the function
Question: The value of $f(0)$, so that the function $f(x)=\frac{(27-2 x)^{1 / 3}-3}{9-3(243+5 x)^{1 / 5}}(x \neq 0)$ is continuous, is given by (a) $\frac{2}{3}$ (b) 6 (c) 2 (d) 4 Solution: (c) 2 Forf(x) to be continuous atx= 0, we must have $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow f(0)=\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0} \frac{(27-2 x)^{\frac{1}{3}-3}}{9-3(243+5 x)^{\frac{1}{5}}}$ $\Rightarrow f(0)=\lim _{x \rightarrow 0} \frac{(27-2 x)^{\frac{1}{3}}-27^{\frac{1}{3}}}{3...
Read More →There are 200 individuals with a skin disorder,
Question: There are 200 individuals with a skin disorder, 120 had been exposed to the chemical C1, 50 to chemical C2, and 30 to both the chemicals C1 and C2. Find the number of individuals exposed to (i) Chemical $\mathrm{C}_{1}$ but not chemical $\mathrm{C}_{2}$ (ii) Chemical $\mathrm{C}_{2}$ but not chemical $\mathrm{C}_{1}$ (iii) Chemical $C_{1}$ or chemical $C_{2}$ Solution: Given: Total number of individuals with skin disorder $=200$ Individuals exposed to chemical $C_{1}=120$ Individuals e...
Read More →If the points A(1, 2), B(0, 0)
Question: If the points A(1, 2), B(0, 0) and C(a, b) are collinear, then (a) a = b (b) a = 2b (c) 2a = b (d) a = b Solution: (c)Let the given points areB = (x1,y1) = (1,2), B = (x2,y2) = (0,0) and C3 = (x3,y3)= (a, b). $\because$ Area of $\triangle A B C \Delta=\frac{1}{2}\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right]$ $\therefore \quad \Delta=\frac{1}{2}[1(0-b)+0(b-2)+a(2-0)]$ $=\frac{1}{2}(-b+0+2 a)=\frac{1}{2}(2 a-b)$ Since, the points $...
Read More →The function
Question: The function $f(x)=\left\{\begin{array}{cc}1, |x| \geq 1 \\ \frac{1}{n^{2}}, \frac{1}{n}|x| \\ 0, x=0\end{array}\right.$$\frac{1}{n-1}, n=2,3, \ldots$ (a) is discontinuous at finitely many points (b) is continuous everywhere (c) is discontinuous only at $x=\pm \frac{1}{n}, n \in Z-\{0\}$ and $x=0$ (d) none of these Solution: Given: $f(x)=\left\{\begin{array}{cc}1, |x| \geq 1 \\ \frac{1}{n^{2}}, \frac{1}{n}|x|\frac{1}{n-1} \\ 0, x=0\end{array}\right.$ $\Rightarrow f(x)=\left\{\begin{arr...
Read More →The function
Question: The function $f(x)=\left\{\begin{array}{cc}1, |x| \geq 1 \\ \frac{1}{n^{2}}, \frac{1}{n}|x| \\ 0, x=0\end{array}\right.$$\frac{1}{n-1}, n=2,3, \ldots$ (a) is discontinuous at finitely many points (b) is continuous everywhere (c) is discontinuous only at $x=\pm \frac{1}{n}, n \in Z-\{0\}$ and $x=0$ (d) none of these Solution: Given: $f(x)=\left\{\begin{array}{cc}1, |x| \geq 1 \\ \frac{1}{n^{2}}, \frac{1}{n}|x|\frac{1}{n-1} \\ 0, x=0\end{array}\right.$ $\Rightarrow f(x)=\left\{\begin{arr...
Read More →The function
Question: The function $f(x)=\left\{\begin{array}{cc}1, |x| \geq 1 \\ \frac{1}{n^{2}}, \frac{1}{n}|x| \\ 0, x=0\end{array}\right.$$\frac{1}{n-1}, n=2,3, \ldots$ (a) is discontinuous at finitely many points (b) is continuous everywhere (c) is discontinuous only at $x=\pm \frac{1}{n}, n \in Z-\{0\}$ and $x=0$ (d) none of these Solution: Given: $f(x)=\left\{\begin{array}{cc}1, |x| \geq 1 \\ \frac{1}{n^{2}}, \frac{1}{n}|x|\frac{1}{n-1} \\ 0, x=0\end{array}\right.$ $\Rightarrow f(x)=\left\{\begin{arr...
Read More →The function
Question: The function $f(x)=\left\{\begin{array}{cc}1, |x| \geq 1 \\ \frac{1}{n^{2}}, \frac{1}{n}|x| \\ 0, x=0\end{array}\right.$$\frac{1}{n-1}, n=2,3, \ldots$ (a) is discontinuous at finitely many points (b) is continuous everywhere (c) is discontinuous only at $x=\pm \frac{1}{n}, n \in Z-\{0\}$ and $x=0$ (d) none of these Solution: Given: $f(x)=\left\{\begin{array}{cc}1, |x| \geq 1 \\ \frac{1}{n^{2}}, \frac{1}{n}|x|\frac{1}{n-1} \\ 0, x=0\end{array}\right.$ $\Rightarrow f(x)=\left\{\begin{arr...
Read More →If the distance between the points
Question: If the distance between the points (4, p) and (1, 0) is 5, then the value of pis (a) 4 only (b) 4 (c) 4 only (d) 0 Solution: (b)According to the question, the distance between the points (4, p) and (1, 0) = 5 i.e., $\quad \sqrt{(1-4)^{2}+(0-p)^{2}}=5$ $\left[\because\right.$ distance between the points $\left(x_{1}, y_{1}\right)$ and $\left.\left(x_{2}, y_{2}\right), d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\right]$ $\Rightarrow \quad \sqrt{(-3)^{2}+p^{2}}=5$ $...
Read More →In a group of 50 persons, 30 like tea, 25 like coffee and 16 like both.
Question: In a group of 50 persons, 30 like tea, 25 like coffee and 16 like both. How many like (i) either tea or coffee? (ii) neither tea nor coffee? Solution: Given: In a group of 50 persons, -30 like tea -25 like coffee -16 like both tea and coffee To find: (i) People who like either tea or coffee. Let us consider Total number of people = n(X) = 50 People who like tea = n(T) = 30 People who like coffee = n(C) = 25 People who like both tea and coffee $=n(T \cap C)=16$ People who like either te...
Read More →The area of a triangle with
Question: The area of a triangle with vertices (a, b + c) , (b, c + a) and (c, a + b) is (a) (a + b + c) (b) 0 (c) (a + b + c) (d) abc Solution: (b)Let the vertices of a triangle are, A (x1, y1) (a, b + c) B (x2, y2) (b,c + a) and C = (x3, y3) (c, a + b) $\because$ Area of $\triangle A B C=\Delta=\frac{1}{2}\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right]$ $\therefore \quad \Delta=\frac{1}{2}[a(c+a-a-b)+b(a+b-b-c)+c(b+c-c-a)]$ $=\frac{1}{2}[a...
Read More →A line intersects the y-axis and X-axis at the points P and Q,
Question: A line intersects the y-axis and X-axis at the points P and Q, respectively. If (2, 5) is the mid-point of PQ, then the coordinates of P and Q are, respectively (a) (0,-5) and (2, 0) (b) (0, 10) and (- 4, 0) (c) (0, 4) and (- 10, 0) (d) (0, 10) and (4, 0) Solution: (d)Let the coordinates of P and 0 (0, y) and (x, 0), respectively. So, the mid-point of $P(0, y)$ and $Q(x, 0)$ is $M\left(\frac{0+x}{2}, \frac{y+0}{2}\right)$ $\left[\because\right.$ mid-point of a line segment having point...
Read More →If a circle drawn with origin as the centre passes
Question: If a circle drawn with origin as the centre passes through $\left(\frac{13}{2}, 0\right)$, then the point which does not lie in the interior of the circle is (a) $\left(\frac{-3}{4}, 1\right)$ (b) $\left(2, \frac{7}{3}\right)$ (c) $\left(5, \frac{-1}{2}\right)$ (d) $\left(-6, \frac{5}{2}\right)$ Solution: (d) it is given that, centre of circle in $(0,0)$ and passes through the point $\left(\frac{13}{2}, 0\right)$. $\therefore$ Radius of circle $=$ Distance between $(0,0)$ and $\left(\f...
Read More →In a committee, 50 people speak Hindi, 20 speak English and 10 speak
Question: In a committee, 50 people speak Hindi, 20 speak English and 10 speak both Hindi and English. How many speak at least one of these two languages? Solution: Given: People who speak Hindi = 50 People who speak English = 20 People who speak both English and Hindi = 10 To Find: People who speak at least one of these two languages Let us consider, People who speak Hindi = n(H) = 50 People who speak English = n(E) = 20 People who speak both Hindi and English $=n(H \cap E)=10$ People who speak...
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