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Question: Let $f(x)=\left\{\begin{array}{ll}1, x \leq-1 \\ |x|, -1x1 \\ 0, x \geq 1\end{array}\right.$ Then, $f$ is (a) continuous atx= 1(b) differentiable atx= 1(c) everywhere continuous(d) everywhere differentiable Solution: (b) differentiable at $x=-1$ $f(x)= \begin{cases}1, x \leq-1 \\ |x|, -1x1 \\ 0, x \geq 1\end{cases}$ Differentiabilty at $x=-1$ $(\mathrm{LHD} x=-1)$ $\lim _{x \rightarrow-1^{-}} \frac{f(x)-f(-1)}{x+1}$ $=\lim _{x \rightarrow-1} \frac{f(x)-f(-1)}{x+1}$ $=\lim _{x \rightarr...
Read More →If from an external point B of a circle with centre 0,
Question: If from an external point B of a circle with centre 0, two tangents BC and BD are drawn such that DBC = 120, prove that BC + BD = B0 i.e., BO = 2 BC. Solution: Two tangents BD and BC are drawn from an external point B. To prove $\quad B O=2 B C$ Given, $\angle D B C=120^{\circ}$ Join $O C, O D$ and $B O$. Since, $B C$ and $B D$ are tangents. $\therefore \quad O C \perp B C$ and $O D \perp B D$ We know, $O B$ is a angle bisector of $\angle D B C$. $\therefore$ $\angle O B C=\angle D B O...
Read More →The set of points where the function f (x) given by
Question: The set of points where the function $f(x)$ given by $f(x)=|x-3| \cos x$ is differentiable, is (a) $R$ (b) $R-\{3\}$ (c) $(0, \infty)$ (d) none of these Solution: (b) $R-(3)$ $(\mathrm{LHD}$ at $x=3)=\lim _{x \rightarrow 3^{-}} \frac{f(x)-f(3)}{x-3}$ $(\mathrm{LHD}$ at $x=3)=\lim _{h \rightarrow 0} \frac{f(3-h)-f(3)}{3-h-3}$ $(\mathrm{LHD}$ at $x=3)=\lim _{h \rightarrow 0} \frac{f(3-h)-f(3)}{-h}$ $(\mathrm{LHD}$ at $x=3)=\lim _{h \rightarrow 0} \frac{|3-h-3| \cos (3-h)-f(3)}{-h}$ $(\ma...
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Question: If $f(x)= \begin{cases}\frac{1-\cos x}{x \sin x}, x \neq 0 \\ \frac{1}{2} , x=0\end{cases}$ then atx= 0,f(x) is(a) continuous and differentiable(b) differentiable but not continuous(c) continuous but not differentiable(d) neither continuous nor differentiable Solution: (a) continuous and differentiable we have, $f(x)= \begin{cases}\frac{1-\cos x}{x \sin x}, x \neq 0 \\ \frac{1}{2} , x=0\end{cases}$ $f(x)= \begin{cases}\frac{1-\cos x}{x \sin x}, x \neq 0 \\ \frac{1}{2} , x=0\end{cases}$...
Read More →Show that each of the following numbers is a perfect square.
Question: Show that each of the following numbers is a perfect square. In each case, find the number whose square is the given number: (i) 1225 (ii) 2601 (iii) 5929 (iv) 7056 (v) 8281 Solution: A perfect square is a product of two perfectly equal numbers.(i) Resolving into prime factors: $1225=25 \times 49=5 \times 5 \times 7 \times 7=5 \times 7 \times 5 \times 7=35 \times 35=(35)^{2}$ Thus, 1225 is the perfect square of 35.(ii) Resolving into prime factors: $2601=9 \times 289=3 \times 3 \times ...
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Question: If $f(x)= \begin{cases}\frac{1-\cos x}{x \sin x}, x \neq 0 \\ \frac{1}{2} , x=0\end{cases}$ then atx= 0,f(x) is(a) continuous and differentiable(b) differentiable but not continuous(c) continuous but not differentiable(d) neither continuous nor differentiable Solution: (a) continuous and differentiable we have, $f(x)= \begin{cases}\frac{1-\cos x}{x \sin x}, x \neq 0 \\ \frac{1}{2} , x=0\end{cases}$ $f(x)= \begin{cases}\frac{1-\cos x}{x \sin x}, x \neq 0 \\ \frac{1}{2} , x=0\end{cases}$...
Read More →Prove that the centre of a circle
Question: Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines. Solution: Given Two tangents PQ and PR are drawn from an external point P to a circle with centre 0. To prove Centre of a circle touching two intersecting lines lies on the angle bisector of the lines.In RPQ. Construction Join $\mathrm{OR}$, and $\mathrm{OQ}$. In $\triangle \mathrm{POP}$ and $\triangle \mathrm{POO}$ $\angle P R O=\angle P Q O=90^{\circ}$ [tangent at any point of a...
Read More →Using the prime factorisation method, find which of the following numbers are perfect squares:
Question: Using the prime factorisation method, find which of the following numbers are perfect squares: (i) 441 (ii) 576 (iii) 11025 (iv) 1176 (v) 5625 (vi) 9075 (vii) 4225 (viii) 1089 Solution: A perfect square can always be expressed as a product of equal factors.(i) Resolving into prime factors: $441=49 \times 9=7 \times 7 \times 3 \times 3=7 \times 3 \times 7 \times 3=21 \times 21=(21)^{2}$ Thus, 441 is a perfect square.(ii) Resolving into prime factors: $576=64 \times 9=8 \times 8 \times 3...
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Question: If $f(x)=\left\{\begin{array}{ll}\frac{1}{1+e^{1 / x}}, x \neq 0 \\ 0, x=0\end{array}\right.$ then $f(x)$ is (a) continuous as well as differentiable atx= 0(b) continuous but not differentiable atx= 0(c) differentiable but not continuous atx= 0(d) none of these Solution: (d) none of these we have, $(\mathrm{LHL}$ at $x=0)$ $=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(-h)$ $=\lim _{h \rightarrow 0} \frac{1}{1+e^{1 /-h}}$ $=\lim _{h \rightar...
Read More →Two tangents PQ and PR are drawn
Question: Two tangents PQ and PR are drawn from an external point to a circle with centre 0. Prove that QORP is a cyclic quadrilateral. Solution: Given Two tangents PQ and PR are drawn from an external point to a circle with centre 0. To prove QORP is a cyclic quadrilateral. proof Since, $P R$ and $P Q$ are tangents. So, $O R \perp P R$ and $O Q \perp P Q$ [since, if we drawn a line from centre of a circle to its tangent line. Then, the line always perpendicular to the tangent line] $\therefore$...
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Question: If $f(x)=\left\{\begin{array}{ll}\frac{1}{1+e^{1 / x}}, x \neq 0 \\ 0, x=0\end{array}\right.$ then $f(x)$ is (a) continuous as well as differentiable atx= 0(b) continuous but not differentiable atx= 0(c) differentiable but not continuous atx= 0(d) none of these Solution: (d) none of these we have, $(\mathrm{LHL}$ at $x=0)$ $=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(-h)$ $=\lim _{h \rightarrow 0} \frac{1}{1+e^{1 /-h}}$ $=\lim _{h \rightar...
Read More →Out of the two concentric circles,
Question: Out of the two concentric circles, the radius of the outer circle is 5 cm and the chord AC of length 8 cm is a tangent to the inner circle. Find the radius of the inner circle. Solution: Let C1and C2be the two circles having same centre O. AC is a chord which touches the C1at point D. Join $O D$. Also, $O D \perp A C$ $\therefore \quad A D=D C=4 \mathrm{~cm} \quad$ [perpendicular line OD bisects the chord] In right angled $\triangle A O D, \quad O A^{2}=A D^{2}+D O^{2}$ [by Pythagoras ...
Read More →AB is a diameter of a circle and AC
Question: AB is a diameter of a circle and AC is its chord such that BAC 30. If the tangent at C intersects AB extended at D, then BC = BD. Solution: True To Prove, BC = BD Join $B C$ and $O C$. Given, $\angle B A C=30^{\circ}$ $\Rightarrow$$\angle B C D=30^{\circ}$ [angle between tangent and chord is equal to angle made by chord in the alternate segment] $\therefore$$\angle A C D=\angle A C O+\angle O C D=30^{\circ}+90^{\circ}=120^{\circ}$ $\left[\because O C \perp C D\right.$ and $O A=O C=$ ra...
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Question: If $f(x)=|3-x|+(3+x)$, where $(x)$ denotes the least integer greater than or equal to $x$, then $f(x)$ is (a) continuous and differentiable atx= 3(b) continuous but not differentiable atx= 3(c) differentiable nut not continuous atx= 3(d) neither differentiable nor continuous atx= 3 Solution: (d) neither differentiable nor continuous atx= 3 We have, $f(x)=|3-x|+(3+x)$, where $(x)$ denotes the least integer greater than or equal to $x$. $f(x)=\left\{\begin{array}{lr}3-x+3+3, 2x3 \\ -3+x+...
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Question: If $f(x)=|3-x|+(3+x)$, where $(x)$ denotes the least integer greater than or equal to $x$, then $f(x)$ is (a) continuous and differentiable atx= 3(b) continuous but not differentiable atx= 3(c) differentiable nut not continuous atx= 3(d) neither differentiable nor continuous atx= 3 Solution: (d) neither differentiable nor continuous atx= 3 We have, $f(x)=|3-x|+(3+x)$, where $(x)$ denotes the least integer greater than or equal to $x$. $f(x)=\left\{\begin{array}{lr}3-x+3+3, 2x3 \\ -3+x+...
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Question: Let $f(x)=a+b|x|+c|x|^{4}$, where $a, b$, and $c$ are real constants. Then, $f(x)$ is differentiable at $x=0$, if (a) $a=0$ (b) $b=0$ (c) $c=0$ (d) none of these Solution: (b) $b=0$ We have, $f(x)=a+b|x|+c|x|^{4}$ $f(x)= \begin{cases}a+b x+c x^{4} x \geq 0 \\ a-b x+c x^{4} x0\end{cases}$ Here, $f(x)$ is differentiable at $x=0$ $\therefore(\mathrm{LHD}$ at $x=0)=(\mathrm{RHD}$ at $x=0)$ $\Rightarrow \lim _{x \rightarrow 0^{-}} \frac{f(x)-f(0)}{x-0}=\lim _{x \rightarrow 0^{+}} \frac{f(x)...
Read More →The function
Question: The function $f(x)=\frac{\sin (\pi[x-\pi])}{4+[x]^{2}}$, where [.] denotes the greatest integer function, is (a) continuous as well as differentiable for allx R(b) continuous for allxbut not differentiable at somex(c) differentiable for allxbut not continuous at somex.(d) none of these Solution: (a) continuous as well as differentiable for allx R Here, $f(x)=\frac{\sin (\pi[x-\pi])}{4+[x]^{2}}$ Since, we know that $\pi[(x-\pi)]=n \pi$ and $\sin n \pi=0$. $\because 4+[x]^{2} \neq 0$ $\t...
Read More →Fill in the blanks.
Question: Fill in the blanks. (i) 360000 written in standard form is ......... (ii) 0.0000123 written in standard form is ......... (iii) $\left(\frac{-2}{3}\right)^{-2}=\ldots \ldots \ldots$ (iv) 3 103in usual form is ......... (v) 5.32 104in usual form is ......... Solution: (i) The standard form of 36000 is $3.6 \times 10^{5}$. $360000=36 \times 10^{4}=3.6 \times 10 \times 10^{4}=3.6 \times 10^{(1+4)}=3.6 \times 10^{5}$ (ii) The standard form of $0.0000123$ is $1.23 \times 10^{-5}$ $0.0000123...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer $\left(\frac{-1}{3}\right)^{3}=?$ (a) $\frac{-1}{9}$ (b) $\frac{1}{9}$ (c) $\frac{-1}{27}$ (d) $\frac{1}{27}$ Solution: (c) $\frac{-1}{27}$ $\left(\frac{-1}{3}\right)^{3}=\frac{-1^{3}}{3^{3}}=\frac{-1}{27}$...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer $\left(\frac{-6}{5}\right)^{-1}=?$ (a) $\frac{6}{5}$ (b) $\frac{-6}{5}$ (c) $\frac{5}{6}$ (d) $\frac{-5}{6}$ Solution: (d) $\frac{-5}{6}$ $\left(\frac{-6}{5}\right)^{-1}=\left(\frac{5}{-6}\right)^{1}=\frac{5}{-6}=\frac{5 \times-1}{-6 \times-1}=\frac{-5}{6}$...
Read More →If a number of circles pass through the end
Question: If a number of circles pass through the end points P and Q of a line segment PQ, then their centres lie on the perpendicular bisector of PQ. Solution: True We draw two circle with centres $C_{1}$ and $C_{2}$ passing through the end points $P$ and $Q$ of a line segment $P Q$. We know, that perpendicular bisectors of a chord of a circle always passes through the centre of circle. Thus, perpendicular bisector of $P Q$ passes through $C_{1}$ and $C_{2}$. Similarly, all the circle passing t...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer $\left(\frac{3}{5}\right)^{0}=?$ (a) $\frac{5}{3}$ (b) $\frac{3}{5}$ (c) 1 (d) 0 Solution: (c) 1 Using the law of exponents, which says $\left(\frac{a}{b}\right)^{0}=1$, we get: $\left(\frac{3}{5}\right)^{0}=1$...
Read More →If a number of circles touch a given
Question: If a number of circles touch a given line segment PQ at a point A, then their centres lie on the perpendicular bisector of PQ. Solution: False Given that PQ is any line segment and S1,S2, S3, S4, circles are touch a line segment PQ at a point A. Let the centres of the circlesS1,S2, S3, S4, be C1C2, C3, C4, respectively. To prove centres of these circles lie on the perpendicular bisector PQ Now, joining each centre of the circles to the point A on the line segment PQ by a line segment i...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer If $\left(\frac{5}{12}\right)^{-4} \times\left(\frac{5}{12}\right)^{3 x}=\left(\frac{5}{12}\right)^{5}$, then $x=?$ (a) $-1$ (b) 1 (c) 2 (d) 3 Solution: (d) 3 $\left(\frac{5}{12}\right)^{-4} \times\left(\frac{5}{12}\right)^{3 x}=\left(\frac{5}{12}\right)^{5}$ $\Rightarrow\left(\frac{5}{12}\right)^{-4+3 x}=\left(\frac{5}{12}\right)^{5}$ $\Rightarrow-4+3 x=5$ $\Rightarrow 3 x=5+4=9$ $\Rightarrow x=\frac{9}{3}=3$...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer (36 34) = ? (a) 32 (b) 32 (c) 310 (d) 310 Solution: (c) 310 $\left(3^{-6} \div 3^{4}\right)$ $=\left(\frac{1}{3^{6}} \div 3^{4}\right)$ $=\frac{1}{3^{6}} \times \frac{1}{3^{4}}$ $=\frac{1}{3^{(6+4)}}$ $=\frac{1}{3^{10}}$...
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