Show that
Question: $e^{x} \tan y d x+\left(1-e^{x}\right) \sec ^{2} y d y=0$ Solution: The given differential equation is: $e^{x} \tan y d x+\left(1-e^{x}\right) \sec ^{2} y d y=0$ $\left(1-e^{x}\right) \sec ^{2} y d y=-e^{x} \tan y d x$ Separating the variables, we get: $\frac{\sec ^{2} y}{\tan y} d y=\frac{-e^{x}}{1-e^{x}} d x$ Integrating both sides, we get: $\int \frac{\sec ^{2} y}{\tan y} d y=\int \frac{-e^{x}}{1-e^{x}} d x$ ...(1) Let $\tan y=u$. $\Rightarrow \frac{d}{d y}(\tan y)=\frac{d u}{d y}$ ...
Read More →The ratio between the radius of the base and the height of a cylinder is 2:3.
Question: The ratio between the radius of the base and the height of a cylinder is $2: 3$. Find the total surface area of the cylinder, if its volume is $1617 \mathrm{~cm}^{2}$. Solution: Let, r be the radius of the cylinder h be the height of the cylinder r/h=2/3 r = 2/3 * h....1 Volume of cylinder $=\pi r^{2}{ }^{*} \mathrm{~h}$ $1617=22 / 7 *(2 / 3 * h)^{2} * h$ $1617=22 / 7 *(2 / 3 * h)^{3}$ $\mathrm{h}^{3}=\frac{1617 * 7 * 3}{22 * 4}$ $h=\frac{3 * 7}{2}$ h = 10.5 cm from, eq 1 r =2/3 * 10.5...
Read More →The values of f(x)=2
Question: The values of $f(x)=2 \sin \sqrt{x^{2}+x+1}$ lie in the interval _________________ . Solution: $f(x)=2 \sin \sqrt{x^{2}+x+1}$ Since $-1 \leq \sin \theta \leq 1$ i. $\mathrm{e}-1 \leq \sin \sqrt{x^{2}+x+1} \leq 1$ i. $\mathrm{e}-2 \leq 2 \sin \sqrt{x^{2}+x+1} \leq 2$ i. e $2 \sin \sqrt{x^{2}+x+1} \in[-2,2]$ i. e. $2 \sin \sqrt{x^{2}+x+1}$ lies in interval $[-2,2]$...
Read More →In an equilateral triangle ABC, If AD ⊥ BC, then
Question: In an equilateral triangle ABC, If AD BC, then (a) $2 \mathrm{AB}^{2}=3 \mathrm{AD}^{2}$ (b) $4 \mathrm{AB}^{2}=3 \mathrm{AD}^{2}$ (c) $3 \mathrm{AB}^{2}=4 \mathrm{AD}^{2}$ (d) $3 \mathrm{AB}^{2}=2 \mathrm{AD}^{2}$ Solution: Given: In an equilateral $\triangle \mathrm{ABC}, \mathrm{AD} \perp \mathrm{BC}$ Applying Pythagoras theorem, In ΔABD, $\mathrm{AB}^{2}=\mathrm{AD}^{2}+\mathrm{BD}^{2}$ $\mathrm{AB}^{2}=\mathrm{AD}^{2}+\left(\frac{1}{2} \mathrm{BC}\right)^{2}\left(\right.$ SinceBD ...
Read More →The value of
Question: The value of $\frac{\cos 50^{\circ}}{\cos 130^{\circ}}$ is ______ . Solution: $\frac{\cos 50^{\circ}}{\cos 130^{\circ}}=\frac{\cos 50^{\circ}}{\cos \left(180^{\circ}-50^{\circ}\right)}$ $=\frac{\cos 50^{\circ}}{-\cos 50^{\circ}} \quad\left(\because \cos \left(180^{\circ}-\theta\right)=-\cos \theta\right)$ $=-1$ Hence $\frac{\cos 50^{\circ}}{\cos 130^{\circ}}=-1$...
Read More →In a ∆ABC, ∠A = 90°, AB = 5 cm and AC = 12 cm. If AD ⊥ BC, then AD =
Question: In a $\triangle \mathrm{ABC}, \angle \mathrm{A}=90^{\circ}, \mathrm{AB}=5 \mathrm{~cm}$ and $\mathrm{AC}=12 \mathrm{~cm}$. If $\mathrm{AD} \perp \mathrm{BC}$, then $\mathrm{AD}=$ (a)132cm(b)6013cm(c)1360cm(d)21513cm Solution: Given: In $\triangle \mathrm{ABC} \angle \mathrm{A}=90^{\circ}, \mathrm{AD} \perp \mathrm{BC}, \mathrm{AC}=12 \mathrm{~cm}$, and $\mathrm{AB}=5 \mathrm{~cm}$. To find: AD We know that the ratio of areas of two similar triangles is equal to the ratio of squares of ...
Read More →The value of
Question: The value of $\frac{\sin 70^{\circ}}{\sin 110^{\circ}}$ is ____________ . Solution: $\frac{\sin 70^{\circ}}{\sin 110^{\circ}}=\frac{\sin 70^{\circ}}{\sin \left(90^{\circ}+20^{\circ}\right)}$ $=\frac{\sin 70^{\circ}}{\cos 20^{\circ}} \quad\left(\right.$ Since $\left.\sin \left(90^{\circ}+\theta\right)=\cos \theta\right)$ $=\frac{\sin 70^{\circ}}{\cos \left(90^{\circ}-70^{\circ}\right)}$ $=\frac{\sin 70^{\circ}}{\sin 70^{\circ}}=1 \quad\left(\because \cos \left(90^{\circ}-\theta\right)=\...
Read More →Show that
Question: $\frac{d y}{d x}=\sin ^{-1} x$ Solution: The given differential equation is: $\frac{d y}{d x}=\sin ^{-1} x$ $\Rightarrow d y=\sin ^{-1} x d x$ Integrating both sides, we get: $\int d y=\int \sin ^{-1} x d x$ $\Rightarrow y=\int\left(\sin ^{-1} x \cdot 1\right) d x$ $\Rightarrow y=\sin ^{-1} x \cdot \int(1) d x-\int\left[\left(\frac{d}{d x}\left(\sin ^{-1} x\right) \cdot \int(1) d x\right)\right] d x$ $\Rightarrow y=\sin ^{-1} x \cdot x-\int\left(\frac{1}{\sqrt{1-x^{2}}} \cdot x\right) ...
Read More →If for real values of x,
Question: If for real values of $x, \cos \theta=x+\frac{1}{x}$, then (a)is an acute angle (b)is a right angle (c)is an obtuse angle (d) No value ofis possible Solution: Given for real value of $x, \cos \theta=x+\frac{1}{x}$ i. e $\cos \theta=\frac{x^{2}+1}{x}$ $\Rightarrow x \cos \theta=x^{2}+1$ $\Rightarrow x^{2}-x \cos \theta+1=0$ for $x \in \mathbb{R}$ i. e roots should be real i. e $b^{2}-4 a c \geq 0$ i. e $\cos ^{2} \theta-4(1)(1) \geq 0$ i. e $\cos ^{2} \theta \geq 4$ which is not possibl...
Read More →If for real values of x,
Question: If for real values of $x, \cos \theta=x+\frac{1}{x}$, then (a)is an acute angle (b)is a right angle (c)is an obtuse angle (d) No value ofis possible Solution: Given for real value of $x, \cos \theta=x+\frac{1}{x}$ i. e $\cos \theta=\frac{x^{2}+1}{x}$ $\Rightarrow x \cos \theta=x^{2}+1$ $\Rightarrow x^{2}-x \cos \theta+1=0$ for $x \in \mathbb{R}$ i. e roots should be real i. e $b^{2}-4 a c \geq 0$ i. e $\cos ^{2} \theta-4(1)(1) \geq 0$ i. e $\cos ^{2} \theta \geq 4$ which is not possibl...
Read More →ABCD is a trapezium such that BC || AD and AD = 4 cm.
Question: ABCD is a trapezium such that BC || AD and AD = 4 cm. If the diagonals AC and BD intersect at O such thatAOOC=DOOB=12, then BC = (a) 7 cm(b) 8 cm(c) 9 cm(d) 6 cm Solution: Given: ABCD is a trapezium in which BC||AD and AD = 4 cm The diagonals $\mathrm{AC}$ and $\mathrm{BD}$ intersect at $\mathrm{O}$ such that $\frac{\mathrm{AO}}{\mathrm{OC}}=\frac{\mathrm{DO}}{\mathrm{OB}}=\frac{1}{2}$ To find: DC In ΔAOD and ΔCOB OAD=OCBAlternateanglesODA=OBCAlternateanglesAOD=BOCVerticallyoppositeang...
Read More →If D, E, F are the mid-points of sides BC, CA and AB respectively of ∆ABC, then the ratio of the areas of triangles DEF and ABC is
Question: If D, E, F are the mid-points of sides BC, CA and AB respectively of ∆ABC, then the ratio of the areas of triangles DEF and ABC is (a) 1 : 4(b) 1 : 2(c) 2 : 3(d) 4 : 5 Solution: GIVEN: In ΔABC, D, E and F are the midpoints of BC, CA, and AB respectively. TO FIND: Ratio of the areas of ΔDEF and ΔABC Since it is given that D and, E are the midpoints of BC, and AC respectively. Therefore DE || AB, DE || FA(1) Again it is given that D and, F are the midpoints of BC, and, AB respectively. T...
Read More →The curved surface area of a cylinder is 1320 cm2
Question: The curved surface area of a cylinder is $1320 \mathrm{~cm}^{2}$ and its base had diameter $21 \mathrm{~cm}$. Find the height and volume of the cylinder. Solution: Let, r be the radius of the cylinder h be the height of the cylinder ⟹ 2r = 21cm ⟹ r=21/2 = 10.5 cm Given, Curved surface area (CSA) $=1320 \mathrm{~cm}^{2}$ ⟹ 2rh = 1320 ⟹ 2 * 22/7 * 10.5 * h = 1320 ⟹ h =1320/66 ⟹ h = 20 cm Volume of cylinder $=\pi r^{2} * \mathrm{~h}$ =22/7 * 10.5 * 10.5 * 20 = 22 * 1.5 * 10.5 * 20 $=6930 ...
Read More →Show that
Question: $x^{5} \frac{d y}{d x}=-y^{5}$ Solution: The given differential equation is: $x^{5} \frac{d y}{d x}=-y^{5}$ $\Rightarrow \frac{d y}{y^{5}}=-\frac{d x}{x^{5}}$ $\Rightarrow \frac{d x}{x^{5}}+\frac{d y}{y^{5}}=0$ Integrating both sides, we get: $\int \frac{d x}{x^{5}}+\int \frac{d y}{y^{5}}=k \quad$ (where $k$ is any constant) $\Rightarrow \int x^{-5} d x+\int y^{-5} d y=k$ $\Rightarrow \frac{x^{-4}}{-4}+\frac{y^{-4}}{-4}=k$ $\Rightarrow x^{-4}+y^{-4}=-4 k$ $\Rightarrow x^{-4}+y^{-4}=\ma...
Read More →Which of the following is incorrect?
Question: Which of the following is incorrect? (a) $\sin \theta=-\frac{1}{5}$ (b) $\cos \theta=1$ (c) $\sec \theta=\frac{1}{2}$ (d) $\tan \theta=20$ Solution: If $\sec \theta=\frac{1}{2}$ $\Rightarrow \cos \theta=2$ i. e $\theta=\cos ^{-1}(2)$ i.e No solution $\therefore \sec \theta=\frac{1}{2}$ is incorrect Hence, the correct answer is option C....
Read More →If ABC and DEF are similar triangles such that ∠A = 47° and ∠E = 83°, then ∠C =
Question: If $A B C$ and $D E F$ are similar triangles such that $\angle A=47^{\circ}$ and $\angle E=83^{\circ}$, then $\angle C=$ (a) $50^{\circ}$ (b) $60^{\circ}$ (c) $70^{\circ}$ (d) $80^{\circ}$ Solution: Given: If ΔABC and ΔDEF are similar triangles such that $\angle \mathrm{A}=47^{\circ}$ $\angle \mathrm{E}=83^{\circ}$ To find: Measure of angle C In similar ΔABC and ΔDEF, $\angle \mathrm{A}=\angle \mathrm{D}=47^{\circ}$ $\angle B=\angle E=83^{\circ}$ $\angle \mathrm{C}=\angle \mathrm{F}$ W...
Read More →Which of the following is incorrect?
Question: Which of the following is incorrect? (a) $\sin \theta=-\frac{1}{5}$ (b) $\cos \theta=1$ (c) $\sec \theta=\frac{1}{2}$ (d) $\tan \theta=20$ Solution: If $\sec \theta=\frac{1}{2}$ $\Rightarrow \cos \theta=2$ i. e $\theta=\cos ^{-1}(2)$ i.e No solution $\therefore \sec \theta=\frac{1}{2}$ is incorrect Hence, the correct answer is option C....
Read More →The ratio between the curved surface area and the total surface area of a right circular cylinder is 1: 2. Find
Question: The ratio between the curved surface area and the total surface area of a right circular cylinder is $1: 2$. Find the volume of the cylinder, if its total surface area is $616 \mathrm{~cm}^{2}$. Solution: Let, r be the radius of cylinder h be the radius of cylinder Total surface area (T.S.A) $=616 \mathrm{~cm}^{2}$ $\Rightarrow \frac{\text { curved sur face area }}{\text { total sur face area }}=\frac{1}{2}$ ⟹ CSA =1/2* TSA ⟹ CSA = 1/2 * 616 $\Rightarrow \mathrm{CSA}=308 \mathrm{~cm}^{...
Read More →If sin θ + cosec θ = 2,
Question: If sin+ cosec= 2, then sin2+ cosec2is equal to (a) 1 (b) 4 (c) 2 (d) none of these Solution: Givensin+ cosec= 2 ⇒ (sin+ cosec)2= 4 i.esin2+ cosec2+ 2 sin cosec = 4 i. e $\sin ^{2} \theta+\operatorname{cosec}^{2} \theta+2 \sin \theta \frac{1}{\sin \theta}=4$ i. e $\sin ^{2} \theta+\operatorname{cosec}^{2} \theta=2$ Hence, the correct answer is option C....
Read More →Show that
Question: $y \log y d x-x d y=0$ Solution: The given differential equation is: $y \log y d x-x d y=0$ $\Rightarrow y \log y d x=x d y$ $\Rightarrow \frac{d y}{y \log y}=\frac{d x}{x}$ Integrating both sides, we get: $\int \frac{d y}{y \log y}=\int \frac{d x}{x}$ ...(1) Let $\log y=t$. $\therefore \frac{d}{d y}(\log y)=\frac{d t}{d y}$ $\Rightarrow \frac{1}{y}=\frac{d t}{d y}$ $\Rightarrow \frac{1}{y} d y=d t$ Substituting this value in equation (1), we get: $\int \frac{d t}{t}=\int \frac{d x}{x}...
Read More →If sin θ and cos θ are the roots of the equation
Question: If sinand cosare the roots of the equationax2bx+c= 0, thena,bandcsatisfy the relation (a)a2+b2+ 2ac= 0 (b)a2b2+ 2ac= 0 (c)a2+c2+ 2ab= 0 (d)a2b2 2ac= 0 Solution: Given sinand cosare roots ofax2 bx + c = 0 Sum of roots is $\frac{b}{a}$ and product of root is $\frac{c}{a}$ i.e $\cos \theta+\sin \theta=\frac{b}{a}$ and $\cos \theta \sin \theta=\frac{c}{a}$ Since $\sin ^{2} \theta+\cos ^{2} \theta=1$ i. e $(\sin \theta+\cos \theta)^{2}=\sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \t...
Read More →In triangles ABC and DEF, ∠A = ∠E = 40°, AB : ED = AC : EF and ∠F = 65°, then ∠B =
Question: In triangles $\mathrm{ABC}$ and $\mathrm{DEF}, \angle \mathrm{A}=\angle \mathrm{E}=40^{\circ}, \mathrm{AB}: \mathrm{ED}=\mathrm{AC}: \mathrm{EF}$ and $\angle \mathrm{F}=65^{\circ}$, then $\angle \mathrm{B}=$ (a) $35^{\circ}$ (b) $65^{\circ}$ (c) $75^{\circ}$ (d) $85^{\circ}$ Solution: Given: In ΔABC and ΔDEF $\angle \mathrm{A}=\angle \mathrm{E}=40^{\circ}$ $\mathrm{AB}: \mathrm{ED}=\mathrm{AC}: \mathrm{EF}$ $\angle \mathrm{F}=65^{\circ}$ To find: Measure of angle B. In ΔABC and ΔDEF $\...
Read More →If tan
Question: If $\tan \theta=-\frac{4}{3}$, then $\sin \theta$ is equal to (a) $-\frac{4}{5}$ but not $\frac{4}{5}$ (b) $-\frac{4}{5}$ or $\frac{4}{5}$ (c) $\frac{4}{5}$ but not $-\frac{4}{5}$ (d) none of these Solution: Given $\tan \theta=-\frac{4}{3}$ Since tan is negative in 2nd or 4th quadrant ⇒ lies in 2nd or 4th quadrant Since AC2= AB2+ BC2= (4)2+ (3)2 AC2= 25 AC = 5 i.e. AC = 5 $\therefore \sin \theta=-\frac{4}{5}$ or $\frac{4}{5}$ Hence, the correct answer is option B....
Read More →Which of the following is correct?
Question: Which of the following is correct? (a) $\sin 1^{\circ}\sin 1$ (b) $\sin 1^{\circ}\sin 1$ (c) $\sin 1^{\circ}=\sin 1$ (d) $\sin 1^{\circ}=\frac{\pi}{180} \sin 1$ Solution: We know that, 1 radian is approximately 57. Also, the value of $\sin x$ is always increasing for $0 \leq x \leq 90^{\circ}$ ( or $\sin x$ is an increasing function for $0 \leq x \leq 90^{\circ}$ ). NOW, $1^{\circ}57^{\circ}$ Or $1^{\circ}1 \operatorname{radian}$ $\therefore \sin 1^{\circ}\sin 1$ Hence, the correct ans...
Read More →Which of the following is correct?
Question: Which of the following is correct? (a) $\sin 1^{\circ}\sin 1$ (b) $\sin 1^{\circ}\sin 1$ (c) $\sin 1^{\circ}=\sin 1$ (d) $\sin 1^{\circ}=\frac{\pi}{180} \sin 1$ Solution: We know that, 1 radian is approximately 57. Also, the value of $\sin x$ is always increasing for $0 \leq x \leq 90^{\circ}$ ( or $\sin x$ is an increasing function for $0 \leq x \leq 90^{\circ}$ ). NOW, $1^{\circ}57^{\circ}$ Or $1^{\circ}1 \operatorname{radian}$ $\therefore \sin 1^{\circ}\sin 1$ Hence, the correct ans...
Read More →