The value of a machine costing 80000 depreciates at the rate
Question: The value of a machine costing 80000 depreciates at the rate of 15% per annum. What will be the worth of this machine after 3 days? Solution: To find: The amount after three days Given: (i) Principal - 80000 (ii) Time $-3$ days (iii) Rate - 15\% per annum Deduction = P R T $=80000 \times \frac{15}{100} \times \frac{3}{365}$ = 98.63 The final amount after deduction = 80000 98.63 = 79901.37 The value of the machine after 3 days is Rs. 79901.37...
Read More →Area of a quadrilateral ABCD is
Question: Area of a quadrilateral ABCD is 20 cm2and perpendiculars on BD from opposite vertices are 1 cm and 1.5 cm. The length of BD is (a) 4 cm (b) 15 cm (c) 16 cm (d) 18 cm Solution: The correct answer is option(c) 16 cm Explanation: Given that, the area of a quadrilateral = 20 cm2 We know that, the area of a quadrilateral = (1/2) (diagonal) (sum of the altitudes) 20 = (1/2) (1+1.5) BD 20 = (1/2) (2.5) BD 202 = 2.5 BD 40 = 2.5 BD BD = 16 cm...
Read More →The perimeter of a trapezium is 52 cm
Question: The perimeter of a trapezium is 52 cmand its each non-parallel side is equal to 10 cm with its height 8 cm. Its area is (a) 124 cm2 (b) 118 cm2 (c) 128 cm2 (d) 112 cm2 Solution: Explanation: Given: The perimeter of a trapezium = 52 cm The sum of its parallel sides = 52 (10+10) = 32 cm We know that, the area of a trapezium = (1/2) (a+b) h A = (1/2) (32) (8) A = 128 cm2...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $x=t^{2}, y=t^{3}$, then $\frac{d^{2} y}{d x^{2}}=$ A. $\frac{3}{2}$ B. $\frac{3}{4 t}$ C. $\frac{3}{2 t}$ D. $\frac{3 t}{2}$ Solution: Given: $x=t^{2} ; y=t^{3}$ $\frac{\mathrm{dy}}{\mathrm{dt}}=3 \mathrm{t}^{2} ; \frac{\mathrm{dx}}{\mathrm{dt}}=2 \mathrm{t}$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}=\frac{3 \mathrm{t}}{2}$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}...
Read More →The area of a parallelogram is
Question: The area of a parallelogram is 60 cm2and one of its altitude is 5 cm. The length of its corresponding side is (a) 12 cm (b) 6 cm (c) 4 cm (d) 2 cm Solution: The correct answer is option(a) 12 cm Explanation: The area of a parallelogram = base x altitude b. h = A b (5) = 60 b = 60/5 b = 12cm...
Read More →The volume of a cube
Question: The volume of a cube whose edge is 3x is (a) 27x3 (b) 9x3 (c) 6x3 (d) 3x3 Solution: The correct answer is option(a) 27x3 Explanation: The volume of a cube is (side)3 V = (3x)3 V = 27x3...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: $\frac{d^{20}}{d x^{20}}(2 \cos x \cos 3 x)=$ A. $2^{20}\left(\cos 2 x-2^{20} \cos 4 x\right)$ B. $2^{20}\left(\cos 2 x+2^{20} \cos 4 x\right)$ c. $2^{20}\left(\sin 2 x-2^{20} \sin 4 x\right)$ D. $2^{20}\left(\sin 2 x-2^{20} \sin 4 x\right)$ Solution: Given: Let $y=2 \cos x \cos 3 x$ $2 \cos A \cos B=\cos \left(\frac{A+B}{2}\right)+\cos \left(\frac{A-B}{2}\right)$ So $y=\cos 2 x+\cos 4 x$ $\frac{d y}{d x}=-2 \sin 2 x-4 \sin 4 x$ $=(-2)^{1...
Read More →The volume of a cylinder
Question: The volume of a cylinder whose radius r is equal to its height is (a) 1/4 r3 (b) r3/32 (c) r3 (d) r3/8 Solution: The correct answer is option(c) r3 Explanation: The volume of cylinder = r2h Given that r = h Then, the volume of cylinder = r2(r) V = r3...
Read More →How many small cubes with edge of
Question: How many small cubes with edge of 20 cm each can be just accommodated in a cubical box of 2 m edge? (a) 10 (b) 100 (c) 1000 (d) 10000 Solution: The correct answer is option(c) 1000 Explanation: We know that, the volume of cube is (side)3 Therefore, the volume of each small cube is (20)3 = 8000 cm3 When it is converted into m3, we get V = 0.008 m3 It is given that, the volume of the cuboidal box is 23= 8 m3 Now, the number of small cubes that can be accommodated in the cuboidal box is =...
Read More →If the radius of a cylinder is tripled
Question: If the radius of a cylinder is tripled but its curved surface area is unchanged, then its height will be (a) tripled (b) constant (c) one sixth (d) one third Solution: The correct answer is option(d) one third Explanation: We know that the curved surface area of a cylinder is 2rh, when the radius is r and height is h. Let H be the new height. When the radius of a cylinder is tripled, then the CSA of a cylinder becomes, CSA = 2 (3r) H CSA = 6r. H Now, compare the CSA of the cylinder to ...
Read More →The volume of a cube is
Question: The volume of a cube is 64 cm3. Its surface area is (a) 16 cm2 (b) 64 cm2 (c) 96 cm2 (d) 128 cm2 Solution: The correct answer is option(c) 96 cm2 Explanation: Let a be the side of the cube Given that, the volume of cube is 64 cm3 It means that a3= 64 cm3 Hence, a = 4 cm Therefore, the surface area of a cube = 6 x 42= 6 x 16 = 96...
Read More →The dimensions of a godown are 40 m, 25 m and 10 m.
Question: The dimensions of a godown are 40 m, 25 m and 10 m. If it is filled with cuboidal boxes each of dimensions 2 m 1.25 m 1 m, then the number of boxes will be (a) 1800 (b) 2000 (c) 4000 (d) 8000 Solution: The correct answer is option(c) 4000 Explanation: Given that, the dimensions of the godown are 40 m, 25 m and 10 m Volume = 40 m 25 m 10 m = 10000 m3 Given that, volume of each cuboidal box is 2 m 1.25 m 1 m = 2.5 m3 Hence, the total number of boxes to be filled in the godown is = 10000/...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $y=a x^{n+1}+b x^{-n}$, then $x^{2} \frac{d^{2} y}{d x^{2}}=$ A. $n(n-1) y$ B. $n(n+1) y$ C. ny D. $n^{2} y$ Solution: Given: $y=a x^{n+1}+b x^{-n}$ $\frac{\mathrm{dy}}{\mathrm{dx}}=(\mathrm{n}+1) \mathrm{ax}^{\mathrm{n}}+(-\mathrm{n}) \mathrm{bx}^{-\mathrm{n}-1}$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\mathrm{n}(\mathrm{n}+1) \mathrm{ax}^{\mathrm{n}-1}+(-\mathrm{n})(-\mathrm{n}-1) \mathrm{bx}^{-\mathrm{n}-2}$ $x^{2} \frac{...
Read More →A regular hexagon is inscribed
Question: A regular hexagon is inscribed in a circle of radius r. The perimeter of the regular hexagon is (a) 3r (b) 6r (c) 9r (d) 12r Solution: The correct answer is option(b) 6r Explanation: We know that a hexagon contains six equilateral triangles, where one of the vertices of each equilateral triangles meet at the centre of the hexagon. The radius of the smallest which is inscribing the hexagon is equal to the sides of the equilateral triangle. Therefore, the perimeter of a regular hexagon i...
Read More →The surface area of the three coterminous
Question: The surface area of the three coterminous faces of a cuboid are 6, 15 and 10 cm2respectively. The volume of the cuboid is (a) 30 cm3 (b) 40 cm3 (c) 20 cm3 (d) 35 cm3 Solution: The correct answer is option(a) 30 cm3 Explanation: It is given that, the coterminous faces of a cuboid is given as: l b = 6 l h = 15 b h = 10 The formula for volume of a cuboid is l b h l2 b2 h2= 6 15 10 (lbh) = (900) = 30...
Read More →If the height of a cylinder becomes 1/4 of
Question: If the height of a cylinder becomes 1/4 of the original height and the radius is doubled, then which of the following will be true? (a) Total surface area of the cylinder will be doubled. (b) Total surface area of the cylinder will remain unchanged. (c) Total surface of the cylinder will be halved. (d) None of the above. Solution: The correct answer is option(d) None of the above. Explanation: We know that, the total surface area of a cylinder is 2 r(h + r), when the radius is r and he...
Read More →What will 15625 amount to in 3 years after its deposit in a bank which pays
Question: What will 15625 amount to in 3 years after its deposit in a bank which pays annual interest at the rate of $8 \%$ per annum, compounded annually? Solution: To find: The amount after three years Given: (i) Principal - 15625 (ii) Time $-3$ years (iii) Rate $-8 \%$ per annum Formula used: $A=P\left(1+\frac{r}{100}\right)^{t}$ $A=15625\left(1+\frac{8}{100}\right)^{3}$ $A=15625\left(\frac{108}{100}\right)^{3}$ A = 19683 Ans) 19683...
Read More →If the height of a cylinder becomes1/4 of
Question: If the height of a cylinder becomes1/4 of the original height and the radius is doubled, then which of the following will be true? (a) Curved surface area of the cylinder will be doubled. (b) Curved surface area of the cylinder will remain unchanged. (c) Curved surface area of the cylinder will be halved. (d) Curved surface area will be 1/4 of the original curved surface. Solution: The correct answer is option(c) Curved surface area of the cylinder will be halved. Explanation: We know ...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $x=a t^{2}, y=2 a t$, then $\frac{d^{2} y}{d x^{2}}=$ A. $-\frac{1}{t^{2}}$ B. $\frac{1}{2 \mathrm{at}^{3}}$ C. $-\frac{1}{t^{3}}$ D. $-\frac{1}{2 \mathrm{at}^{3}}$ Solution: Given: $y=2 a t, x=a t^{2}$ $\frac{\mathrm{dx}}{\mathrm{dt}}=2 \mathrm{at} ; \frac{\mathrm{dy}}{\mathrm{dt}}=2 \mathrm{a}$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}$ $=\frac{1}{t}$ $\frac{\mathrm{d}^{2...
Read More →If the height of a cylinder becomes 1/4 of
Question: If the height of a cylinder becomes 1/4 of the original height and the radius is doubled, then which of the following will be true? (a) Volume of the cylinder will be doubled. (b) Volume of the cylinder will remain unchanged. (c) Volume of the cylinder will be halved. (d) Volume of the cylinder will be1/4 of the original volume Solution: The correct answer is option(b) Volume of the cylinder will remain unchanged. Explanation: We know that, the volume of a cylinder is r2 h We know that...
Read More →What is the area of the largest triangle
Question: What is the area of the largest triangle that can be fitted into a rectangle of length l units and width w units? (a) lw /2 (b) lw /3 (c) lw/6 (d) lw/4 Solution: The correct answer is option(a)lw /2 Explanation: We know that, the area of a triangle is (1/2) x base x height Let ABCD be a triangle with length l and width w. Here, we have to construct a triangle of maximum area inside the rectangle in all possible ways. Now, the maximum base length is l Maximum height is w. Therefore, the...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $x=a \cos n t-b \sin n t$, then $\frac{d^{2} x}{d t^{2}}$ is A. $n^{2} x$ B. $-n^{2} x$ C. $-n x$ D. $n x$ Solution: Given: $x=a \cos n t-b \sin n t$ $\frac{\mathrm{dx}}{\mathrm{dt}}=-\mathrm{an} \sin \mathrm{nt}-\mathrm{bn} \cos \mathrm{nt}$ $\frac{d^{2} x}{d t^{2}}=-a n^{2} \cos n t+b n^{2} \sin n t$ $=-n^{2}(a \cos n t-b \sin n t)$ $=-n^{2} x$...
Read More →A circle of maximum possible size is cut from
Question: A circle of maximum possible size is cut from a square sheet of board. Subsequently, a square of maximum possible size is cut from the resultant circle. What will be the area of the final square? (a) 3/4 of the original square. (b) 1/2 of the original square. c) 1/4 of the original square. (d) 2/3 of the original square. Solution: The correct answer is option(b) 1/2 of the original square Explanation: Let a be the side of the square sheet Thus, the area of the bigger square sheet = a2....
Read More →Show that the ratio of the sum of first n terms of a GP to the sum of the
Question: Show that the ratio of the sum of first n terms of a GP to the sum of the terms from $(n+1)$ th to $(2 n)$ th term is $\frac{1}{r^{n}}$. Solution: Sum of a G.P. series is represented by the formula $S_{n}=a \frac{r^{n}-1}{r-1}$ when r1. Sn represents the sum of the G.P. series upto nth terms, a represents the first term, r represents the common ratio and n represents the number of terms. Thus, the sum of the first n terms of the G.P. series is, $\mathrm{S}_{\mathrm{n}}=\mathrm{a} \frac...
Read More →A cube of side 4 cm is cut into 1 cm cubes.
Question: A cube of side 4 cm is cut into 1 cm cubes. What is the ratio of the surface areas of the original cubes and cut-out cubes? (a) 1 : 2 (b) 1 : 3 (c) 1 : 4 (d) 1 : 6 Solution: The correct answer is option(c) 1: 4 Explanation: Given: The cube side is 4cm The side of cube 4cm is cut into small cubes, in which each of 1 cm Therefore, the total number of cubes = 4 x 16 = 64 cubes Thus, the number of cut-out cubes = 64/1 Now, the surface area of the cut-out cubes = c x 1 cm2 The surface area ...
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