Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $x=2 a t, y=a t^{2}$, where $a$ is a constant, then $\frac{d^{2} y}{d x^{2}}$ at $x=\frac{1}{2}$ is A. $1 / 2 a$ B. 1 C. $2 a$ D. none of these Solution: Given: $x=2 a t, y=a t^{2}$ $\frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=t$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}=\mathrm{t}$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\frac{\mathrm{d}}{\math...
Read More →If the diagonals of a rhombus get doubled,
Question: If the diagonals of a rhombus get doubled, then the area of the rhombus becomes __________ its original area. Solution: 4 times Explanation: Let p and q be the two diagonals of the rhombus We know that area of a rhombus = pq/2 If the diagonals are doubled, we will get A= (4p)(4q)/2 Take 4 outside, we will get A = 4(pq/2)...
Read More →The surface area of a cuboid formed
Question: The surface area of a cuboid formed by joining two cubes of side a face to face is __________. Solution: 10a2 Explanation: Let a be the side of two cubes. When the two cubes are joined face to face, the figure obtained should be a cuboid having the same breadth and height. As the combined cube has a length twice of the length of a cube. It means that l = 2a, b = a and h = a Hence, the total surface area of cuboid = 2(lb + bh + hl) = 2(2a a + a a + a 2a) Simplify the above expression, w...
Read More →If a, b, c are in GP, then show that log
Question: If $a, b, c$ are in GP, then show that $\log a^{n}, \log b^{n}, \log c^{n}$ are in AP. Solution: To prove: $\log a^{n}, \log b^{n}, \log c^{n}$ are in AP. Given: a, b, c are in GP Formula used: (i) log ab = log a + log b As a, b, c are in GP $\Rightarrow \mathrm{b}^{2}=\mathrm{ac}$ Taking power n on both sides $\Rightarrow \mathrm{b}^{2 \mathrm{n}}=(\mathrm{ac})^{\mathrm{n}}$ Taking log both side $\Rightarrow \log b^{2 n}=\log (a c)^{n}$ $\Rightarrow \log b^{2 n}=\log \left(a^{n} c^{n}...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $y=a \cos \left(\log _{e} x\right)+b \sin \left(\log _{e} x\right)$, then $x^{2} y_{2}+x y_{1}=$ A. 0 B. $y$ C. $-y$ D. none of these Solution: Given: $y=a \cos \left(\log _{e} x\right)+b \sin \left(\log _{e} x\right)$ $\frac{d y}{d x}=-a \sin \left(\log _{e} x\right) \frac{1}{x}+b \cos \left(\log _{e} x\right) \frac{1}{x}$ $x y_{1}=-a \sin \left(\log _{e} x\right)+b \cos \left(\log _{e} x\right)$ $\frac{d^{2} y}{d x^{2}}=-a \cos \left...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: Let $f(x)$ be a polynomial. Then, the second order derivative of $f\left(e^{x}\right)$ is A. $f^{\prime \prime}\left(e^{x}\right) e^{2 x}+f^{\prime}\left(e^{x}\right) e^{x}$ B. $f^{\prime \prime}\left(e^{x}\right) e^{x}+f^{\prime}\left(e^{x}\right)$ C. $f^{\prime \prime}\left(e^{x}\right) e^{2 x}+f^{\prime \prime}\left(e^{x}\right) e^{x}$ D. $f^{\prime \prime}\left(e^{x}\right)$ Solution: Given: $\frac{d}{d x}\left[\frac{d}{d x} f\left(e^...
Read More →If p, q, r are in AP, then prove that pth, qth and rth terms of any GP are in GP.
Question: If p, q, r are in AP, then prove that pth, qth and rth terms of any GP are in GP. Solution: To prove: $p^{\text {th }}, q^{\text {th }}$ and $r^{\text {th }}$ terms of any GP are in GP. Given: (i) p, q and r are in AP The formula used: (i) General term of GP, $T_{n}=a r^{n-1}$ As $p, q, r$ are in A.P. $\Rightarrow q-p=r-q=d=$ common difference $\ldots$ (i) Consider a G.P. with the first term as a and common difference $\mathrm{R}$ Then, the $p^{\text {th }}$ term will be $a r^{p-1}$ Th...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $y=\tan ^{-1}\left\{\frac{\log _{e}\left(e / x^{2}\right)}{\log _{e}\left(e x^{2}\right)}\right\}+\tan ^{-1}\left(\frac{3+2 \log _{e} x}{1-6 \log _{e} x}\right)$, then $\frac{d^{2} y}{d x^{2}}=$ A. 2 B. 1 C. 0 D. $-1$ Solution: Given: $y=\tan ^{-1}\left\{\frac{\log _{e}\left(\frac{e}{x^{2}}\right)}{\log _{e}\left(e x^{2}\right)}\right\}+\tan ^{-1}\left\{\frac{3+2 \log _{e} x}{1-6 \log _{e} x}\right\}$ $y=\tan ^{-1}\left\{\frac{\log _{e...
Read More →The number of bacteria in a certain culture doubles every hour.
Question: The number of bacteria in a certain culture doubles every hour. If there were 50 bacteria present in the culture originally, how many bacteria would be present at the end of (i) 2nd hour, (ii) 5th hour and (iii) nth hour? Solution: To find: The number of bacteria after (i) $2^{\text {nd }}$ hour (ii) $5^{\text {th }}$ hour (iii) nth hour Given: (i) Initially, there were 50 bacteria (ii) Rate 100% per hour The formula used: $A=P\left(1+\frac{r}{100}\right)^{t}$ (i) For $2^{\text {nd }}$...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $f(x)=\frac{\sin ^{-1} x}{\sqrt{\left(1-x^{2}\right)}}$ then $\left(1-x^{2}\right) f^{\prime}(x)-x f(x)=$ A. 1 B. $-1$ C. 0 D. none of these Solution: Given: $y=f(x)=\frac{\sin ^{-1} x}{\sqrt{\left(1-x^{2}\right)}}$ $\frac{d y}{d x}=\frac{1}{\left(\sqrt{\left.\left(1-x^{2}\right)\right)^{2}}\right.}\left\{\frac{1}{\sqrt{\left(1-x^{2}\right)}} \sqrt{\left(1-x^{2}\right)}-\sin ^{-1} x \frac{(-2 x)}{2 \sqrt{\left(1-x^{2}\right)}}\right\}$...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $y=a \sin m x+b \cos m x$, then $\frac{d^{2} y}{d x^{2}}$ is equal to A. $-m^{2} y$ B. $m^{2} y$ C. $-\mathrm{my}$ D. $m y$ Solution: Given: $y=a \sin m x+b \cos m x$ $\frac{d y}{d x}=m a \cos m x-m b \sin m x$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=-\mathrm{m}^{2} \mathrm{a} \sin \mathrm{m} \mathrm{x}-\mathrm{m}^{2} \mathrm{~b} \cos \mathrm{m} \mathrm{x}$ $=-m^{2}[a \sin m x+b \cos m x]$ $=-m^{2} y$...
Read More →A manufacturer reckons that the value of a machine which costs him
Question: A manufacturer reckons that the value of a machine which costs him 156250, will depreciate each year by 20%. Find the estimated value at the end of 5 years. Solution: To find: The amount after five years Given: (i) Principal 156250 (ii) Time 5 years (iii) Rate 20% per annum Formula used: $A=P\left(1-\frac{r}{100}\right)^{t}$ $\Rightarrow A=156250\left(1-\frac{20}{100}\right)^{5}$ $\Rightarrow A=156250\left(\frac{80}{100}\right)^{5}$ $\Rightarrow A=156250(0.8)^{5}$ $\Rightarrow A=156250...
Read More →A cube of side 5 cm is cut into 1 cm cubes.
Question: A cube of side 5 cm is cut into 1 cm cubes. The percentage increase in volume after such cutting is __________. Solution: No change Explanation: Volume of cube = 53= 125 Now, when the cube is cut into 1 cubic cm, we will get 125 small cubes Therefore, the volume of the big cube = volume of 125 cm with 1 cubic cm. It means that, there is no change in the volume....
Read More →A cube of side 4 cm is painted on
Question: A cube of side 4 cm is painted on all its sides. If it is sliced in 1 cu cm cubes, then number of such cubes that will have exactly two of their faces painted, is_______. Solution: 24 The volume of a cube of side 4 cm = 4 x 4 x 4 = 64cm3When it is sliced into 1cm3cubes, we will get 64 small cubes. In each side of the larger cube, the smaller cubes in the edges will have more than one face painted. The cubes which are situated at the corners of the big cube, have three faces painted. So...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $f(x)=(\cos x+i \sin x)(\cos 2 x+i \sin 2 x)(\cos 3 x+i \sin 3 x) \ldots(\cos n x+i \sin n x)$ and $f(1)=1$, then $f^{\prime \prime}(1)$ is equal to A. $\frac{\mathrm{n}(\mathrm{n}+1)}{2}$ B. $\left\{\frac{\mathrm{n}(\mathrm{n}+1)}{2}\right\}^{2}$ C. $-\left\{\frac{\mathrm{n}(\mathrm{n}+1)}{2}\right\}^{2}$ D. none of these Solution: Given: $f(x)=(\cos x+i \sin x)(\cos 2 x+i \sin 2 x)(\cos 3 x+i \sin 3 x) \ldots(\cos n x+i \sin n x)$ Si...
Read More →What will 5000 amount to in 10 years, compounded annually at 10% per
Question: What will 5000 amount to in 10 years, compounded annually at 10% per annum? [Given $\left.(1.1)^{10}=2.594\right]$ Solution: To find: The amount after ten years Given: (i) Principal 5000 (ii) Time 10 years (iii) Rate 10% per annum Formula used: $A=P\left(1+\frac{r}{100}\right)^{t}$ $\Rightarrow A=5000\left(1+\frac{10}{100}\right)^{10}$ $\Rightarrow A=5000\left(\frac{110}{100}\right)^{10}$ $\Rightarrow A=5000(1.1)^{10}$ $\Rightarrow A=5000 \times 2.594$ $\Rightarrow A=12970$ Ans) The am...
Read More →Ramesh has three containers.
Question: Ramesh has three containers. (a) Cylindrical container A having radius r and height h, (b) Cylindrical container B having radius 2r and height 1/2 h, and (c) Cuboidal container C having dimensions r r h The arrangement of the containers in the increasing order of their volumes is (a) A, B, C (b) B, C, A (c) C, A, B (d) cannot be arranged Solution: The correct answer is option(c) C, A, B Explanation: (i) If the cylinder have radius r and height h, then the volume will be r2h (ii) If the...
Read More →The surface areas of the six faces
Question: The surface areas of the six faces of a rectangular solid are 16, 16, 32, 32, 72 and 72 square centimetres. The volume of the solid, in cubic centimetres, is (a) 192 (b) 384 (c) 480 (d) 2592 Solution: The correct answer is option(a) 192 Explanation: It is given that, the solid has a rectangular faces, hence, lb=16 (1) bh = 32 .(2) lh = 72 (3) Multiply the equations (1), (2), (3), we will get (l)2(b)2(h)2= (16)(32)(72) = 36864 lbh = 192 Therefore, the volume of a solid is 192 cubic cent...
Read More →Two cubes have volumes in the ratio 1:64.
Question: Two cubes have volumes in the ratio 1:64. The ratio of the area of a face of first cube to that of the other is (a) 1:4 (b) 1:8 (c) 1:16 (d) 1:32 Solution: The correct answer is option(c) 1:16 Explanation: Let a and b be two cubes It is given that, a3/b3= 1/64 Then a/b = 1/4 Thus, the ratio of the areas are: (a/b)2= (1/4)2= 1/16...
Read More →Three years before the population of a village was 10000.
Question: Three years before the population of a village was 10000. If at the end of each year, 20% of the people migrated to a nearby town, what is its present population? Solution: To find: Present population of the village Given: (i) Three years back population - 10000 (ii) Time 3 years (iii) Rate 20% per annum Number of people migrated on the very first year is 20% of 10000 $\Rightarrow \frac{10000 \times 20}{100}=2000$ People left after migration in the very first year = 10000 2000 = 8000 N...
Read More →The ratio of radii of two cylinders is 1: 2
Question: The ratio of radii of two cylinders is 1: 2 and heights are in the ratio 2:3. The ratio of their volumes is (a) 1:6 (b) 1:9 (c) 1:3 (d) 2:9 Solution: The correct answer is option(a) 1:6 Explanation: Assume that r and R be the radii of the two cylinders and h and H be the height of the two cylinders It is given that r/R = and h/H = 2/3 We know that the volume of a cylinder = r2h Now, v/V = r2h / R2H v/ V = (r/R)2(h/H) v/V = (1/2)2(2/3) v/V = (1/4) (2/3) = 1/6 Therefore, the ratio of the...
Read More →A covered wooden box has the inner
Question: A covered wooden box has the inner measures as 115 cm, 75 cm and 35 cm and thickness of wood as 2.5 cm. The volume of the wood is (a) 85,000 cm3 (b) 80,000 cm3 (c) 82,125 cm3 (d) 84,000 cm3 Solution: The correct answer is option(c) 82,125 cm3 Explanation: The thickness of the wooden box is 2.5 cm Then the outer measure of the wooden box be 115+5, 75+5, 35+5 Thus, the outer volume be = (120)(80)(40) Outer volume = 384000 cm3 Given that, the inner volume = (115)(80)(40) Inner volume = 30...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $y=a+b x^{2}, a, b$ arbitrary constants, then A. $\frac{d^{2} y}{d x^{2}}=2 x y$ B. $x \frac{d^{2} y}{d x^{2}}=y_{1}$ C. $x \frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}+y=0$ D. $x \frac{d^{2} y}{d x^{2}}=2 x y$ Solution: Given: $y=a+b x^{2}$ $\frac{\mathrm{dy}}{\mathrm{dx}}=2 \mathrm{bx}$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=2 \mathrm{~b} \neq 2 \mathrm{xy}$ $x \frac{d^{2} y}{d x^{2}}=2 b x$ $=\frac{\mathrm{dy}}{\mathrm{dx}}$ ...
Read More →Three cubes of metal whose edges are 6 cm,
Question: Three cubes of metal whose edges are 6 cm, 8 cm and 10 cm respectively are melted to form a single cube. The edge of the new cube is (a) 12 cm (b) 24 cm (c) 18 cm (d) 20 cm Solution: The correct answer is option(a) 12 cm Explanation: Given that, the sum of the volume of the three metal cubes = 63+ 83+103 V = 216+ 512+ 1000 V = 1728 cm3 Let the side of the new cube be a Therefore, the volume of the new cube = sum of the volume of the three cubes a3= 1728 Hence, a = 12 cm...
Read More →A metal sheet 27 cm long,
Question: A metal sheet 27 cm long, 8 cm broad and 1 cm thick is melted into a cube. The side of the cube is (a) 6 cm (b) 8 cm (c) 12 cm (d) 24 cm Solution: The correct answer is option(a) 6 cm Explanation: Given that, the metal sheet dimension is 27 cm long, 8 cm broad and 1 cm thick. Thus, the volume of the sheet = (27)(8)(1) = 216 cm3 It is given that, the metal sheet is melted to make a cube Let the edge be a Hence, a3= 216 cm3 a = 6 cm...
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