If f(x)=
Question: If $f(x)=64 x^{3}+\frac{1}{x^{3}}$ and $\alpha, \beta$ are the roots of $4 x+\frac{1}{x}=3$. Then, (a)f() =f() = 9 (b)f() =f() = 63 (c)f()f() (d) none of these Solution: (a) $f(\alpha)=f(\beta)=-9$ Given: $f(x)=64 x^{3}+\frac{1}{x^{3}}$ $\Rightarrow f(x)=\left(4 x+\frac{1}{x}\right)\left(16 x^{2}+\frac{1}{x^{2}}-4\right)$ $\Rightarrow f(x)=\left(4 x+\frac{1}{x}\right)\left(\left(4 x+\frac{1}{x}\right)^{2}-12\right)$ $\Rightarrow f(\alpha)=\left(4 \alpha+\frac{1}{\alpha}\right)\left(\le...
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Question: $\int_{0}^{2} \frac{6 x+3}{x^{2}+4} d x$ Solution: Let $I=\int_{0}^{2} \frac{6 x+3}{x^{2}+4} d x$ $\int \frac{6 x+3}{x^{2}+4} d x=3 \int \frac{2 x+1}{x^{2}+4} d x$ $=3 \int \frac{2 x}{x^{2}+4} d x+3 \int \frac{1}{x^{2}+4} d x$ $=3 \log \left(x^{2}+4\right)+\frac{3}{2} \tan ^{-1} \frac{x}{2}=\mathrm{F}(x)$ By second fundamental theorem of calculus, we obtain $I=\mathrm{F}(2)-\mathrm{F}(0)$ $=\left\{3 \log \left(2^{2}+4\right)+\frac{3}{2} \tan ^{-1}\left(\frac{2}{2}\right)\right\}-\left\...
Read More →f is a real valued function given by f(x)
Question: $f$ is a real valued function given by $f(x)=27 x^{3}+\frac{1}{x^{3}}$ and $\alpha, \beta$ are roots of $3 x+\frac{1}{x}=12$. Then, (a)f() f() (b)f() = 10 (c)f() = 10 (d) None of these Solution: (d) None of these Given: $f(x)=27 x^{3}+\frac{1}{x^{3}}$ $\Rightarrow f(x)=\left(3 x+\frac{1}{x}\right)\left(9 x^{2}+\frac{1}{x^{2}}-3\right)$ $\Rightarrow f(x)=\left(3 x+\frac{1}{x}\right)\left(\left(3 x+\frac{1}{x}\right)^{2}-9\right)$ $\Rightarrow f(\alpha)=\left(3 \alpha+\frac{1}{\alpha}\ri...
Read More →f is a real valued function given by f(x)
Question: $f$ is a real valued function given by $f(x)=27 x^{3}+\frac{1}{x^{3}}$ and $\alpha, \beta$ are roots of $3 x+\frac{1}{x}=12$. Then, (a)f() f() (b)f() = 10 (c)f() = 10 (d) None of these Solution: (d) None of these Given: $f(x)=27 x^{3}+\frac{1}{x^{3}}$ $\Rightarrow f(x)=\left(3 x+\frac{1}{x}\right)\left(9 x^{2}+\frac{1}{x^{2}}-3\right)$ $\Rightarrow f(x)=\left(3 x+\frac{1}{x}\right)\left(\left(3 x+\frac{1}{x}\right)^{2}-9\right)$ $\Rightarrow f(\alpha)=\left(3 \alpha+\frac{1}{\alpha}\ri...
Read More →On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeeper gains Rs 2000.
Question: On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeeper gains Rs 2000. But if he sells the T.V. at 10% gain the fridge at 5% loss. He gains Rs 1500 on the transaction. Find the actual prices of T.V. and fridge. Solution: Given: (i) On selling of a T.V. at 5% gain and a fridge at 10% gain, shopkeeper gain Rs.2000. (ii) Selling T.V. at 10% gain and fridge at 5% loss. He gains Rs. 1500. To find: Actual price of T.V. and fridge. Let the S.P. of T.V = Rs. $x$; Let the S.P. of fr...
Read More →A triangle and a parallelogram have the same base and the same area.
Question: A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 13 cm, 14 cm and 15 cm and the parallelogram stands on the base 14 cm, find the height of a parallelogram. Solution: The sides of the triangle DCE are DC = 15 cm, CE = 13 cm, ED = 14 cm Let the h be the height of parallelogram ABCD Now, for the area of triangle DCE Perimeter = DC + CE + ED 2s= 15 cm + 13 cm + 14 cm s = 21 cm By using Heron's Formula, Area of the triangle $\mathrm{AOB}=...
Read More →If
Question: If $e^{f(x)}=\frac{10+x}{10-x}, x \in(-10,10)$ and $f(x)=k f\left(\frac{200 x}{100+x^{2}}\right)$, then $k=$ (a) 0.5 (b) 0.6 (c) 0.7 (d) 0.8 Solution: (a) 0.5 $e^{f(x)}=\frac{10+x}{10-x}$ $\Rightarrow f(x)=\log _{e}\left(\frac{10+x}{10-x}\right) \ldots(1)$ $f(x)=k f\left(\frac{200 x}{100+x^{2}}\right)$ $\Rightarrow \log _{e}\left(\frac{10+x}{10-x}\right)=k \log _{e}\left(\frac{10+\frac{200 x}{100+x^{2}}}{10-\frac{200 x}{100+x^{2}}}\right) \quad\{$ from $(1)\}$ $\Rightarrow \log _{e}\le...
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Question: $\int_{0}^{\pi}\left(\sin ^{2} \frac{x}{2}-\cos ^{2} \frac{x}{2}\right) d x$ Solution: Let $I=\int_{0}^{\pi}\left(\sin ^{2} \frac{x}{2}-\cos ^{2} \frac{x}{2}\right) d x$ $=-\int_{0}^{\pi}\left(\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}\right) d x$ $=-\int_{0}^{\pi} \cos x d x$ $\int \cos x d x=\sin x=\mathrm{F}(x)$ By second fundamental theorem of calculus, we obtain $I=\mathrm{F}(\pi)-\mathrm{F}(0)$ $=\sin \pi-\sin 0$ $=0$...
Read More →If
Question: If $e^{f(x)}=\frac{10+x}{10-x}, x \in(-10,10)$ and $f(x)=k f\left(\frac{200 x}{100+x^{2}}\right)$, then $k=$ (a) 0.5 (b) 0.6 (c) 0.7 (d) 0.8 Solution: (a) 0.5 $e^{f(x)}=\frac{10+x}{10-x}$ $\Rightarrow f(x)=\log _{e}\left(\frac{10+x}{10-x}\right) \ldots(1)$ $f(x)=k f\left(\frac{200 x}{100+x^{2}}\right)$ $\Rightarrow \log _{e}\left(\frac{10+x}{10-x}\right)=k \log _{e}\left(\frac{10+\frac{200 x}{100+x^{2}}}{10-\frac{200 x}{100+x^{2}}}\right) \quad\{$ from $(1)\}$ $\Rightarrow \log _{e}\le...
Read More →A hand fan is made by sticking 10 equal size triangular strips of two different types of paper as shown in the figure.
Question: A hand fan is made by sticking 10 equal size triangular strips of two different types of paper as shown in the figure. The dimensions of equal strips are 25 cm, 25 cm and 14 cm. Find the area of each type of paper needed to make the hand fan. Solution: Given that, The sides of AOB AO = 25 cm OB = 25 cm BA = 14 cm Area of each strip = Area of triangle AOB Now, for the area of triangle AOB Perimeter = AO + OB + BA 2s = 25 cm +25 cm + 14 cm s = 32 cm By using Heron's Formula, Area of the ...
Read More →If f : [−2, 2]
Question: If $f:[-2,2] \rightarrow \mathrm{R}$ is defined by $f(x)=\left\{\begin{aligned}-1, \text { for }-2 \leq x \leq 0 \\ x-1, \text { for } 0 \leq x \leq 2 \end{aligned}\right.$, then $\{x \in[-2,2]: x \leq 0$ and $f(|x|)=x\}=$ (a) {1} (b) {0} (c) $\left\{-\frac{1}{2}\right\}$ (d) $\phi$ Solution: (c) $\left\{-\frac{1}{2}\right\}$ Given: $f(x)=\left\{\begin{aligned}-1, \text { for }-2 \leq x \leq 0 \\ x-1, \text { for } 0 \leq x \leq 2 \end{aligned}\right.$ We know, $|x| \geq 0$ $\Rightarro...
Read More →Find the area of the blades of the magnetic compass shown in figure given below:
Question: Find the area of the blades of the magnetic compass shown in figure given below: Solution: Area of the blades of magnetic compass = Area of triangle ADB + Area of triangle CDB Now, for the area of triangle ADB Perimeter = 2s = AD + DB + BA 2s = 5 cm + 1 cm + 5 cm s = 5.5 cm By using Heron's Formula, Area of the triangle DEF $=\sqrt{s \times(s-a) \times(s-b) \times(s-c)}$ $=\sqrt{5.5 \times(0.5) \times(4.5) \times(0.5)}$ $=2.49 \mathrm{~cm}^{2}$ Also, area of triangle ADB = Area of tria...
Read More →If f : [−2, 2]
Question: If $f:[-2,2] \rightarrow \mathrm{R}$ is defined by $f(x)=\left\{\begin{aligned}-1, \text { for }-2 \leq x \leq 0 \\ x-1, \text { for } 0 \leq x \leq 2 \end{aligned}\right.$, then $\{x \in[-2,2]: x \leq 0$ and $f(|x|)=x\}=$ (a) {1} (b) {0} (c) $\left\{-\frac{1}{2}\right\}$ (d) $\phi$ Solution: (c) $\left\{-\frac{1}{2}\right\}$ Given: $f(x)=\left\{\begin{aligned}-1, \text { for }-2 \leq x \leq 0 \\ x-1, \text { for } 0 \leq x \leq 2 \end{aligned}\right.$ We know, $|x| \geq 0$ $\Rightarro...
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Question: $\int_{0}^{\frac{\pi}{4}}\left(2 \sec ^{2} x+x^{3}+2\right) d x$ Solution: Let $I=\int_{0}^{\frac{\pi}{4}}\left(2 \sec ^{2} x+x^{3}+2\right) d x$ $\int\left(2 \sec ^{2} x+x^{3}+2\right) d x=2 \tan x+\frac{x^{4}}{4}+2 x=\mathrm{F}(x)$ By second fundamental theorem of calculus, we obtain $I=\mathrm{F}\left(\frac{\pi}{4}\right)-\mathrm{F}(0)$ $=\left\{\left(2 \tan \frac{\pi}{4}+\frac{1}{4}\left(\frac{\pi}{4}\right)^{4}+2\left(\frac{\pi}{4}\right)\right)-(2 \tan 0+0+0)\right\}$ $=2 \tan \f...
Read More →A and B each have a certain number of mangoes. A says to B, "if you give 30 of your mangoes,
Question: A and B each have a certain number of mangoes. A says to B, "if you give 30 of your mangoes, I will have twice as many as left with you." B replies, "if you give me 10, I will have thrice as many as left with you." How many mangoes does each have? Solution: To find: (1) Total mangoes of A. (2) Total mangoes of B. Suppose A hasxmangoes and B hasymangoes, According to the given conditions, $x+30=2(y-30)$ $x+30=2 y-60$ $x-2 y+30+60=0$ $x-2 y+90=0$....(1) $y+10=3(x-10)$ $y+0=3 x-30$ $y-3 x...
Read More →The adjacent sides of a parallelogram ABCD measure 34 cm and 20 cm, and the diagonal
Question: The adjacent sides of a parallelogram ABCD measure 34 cm and 20 cm, and the diagonal AC measures 42 cm. Find the area of parallelogram. Solution: The adjacent sides of a parallelogram ABCD measures 34 cm and 20 cm, and the diagonal AC measures 42 cm. Area of the parallelogram = Area of triangle ADC + Area of triangle ABC Note: Diagonal of a parallelogram divides into two congruent triangles Therefore, Area of the parallelogram = 2(Area of triangle ABC) Now, for area of triangle ABC Per...
Read More →If f : R → R and g :
Question: Iff: R R andg: R R are defined byf(x) = 2x+ 3 andg(x) =x2+ 7, then the values ofxsuch that g(f(x)) = 8 are (a) 1, 2 (b) 1, 2 (c) 1, 2 (d) 1, 2 Solution: (c) 1, 2 f(x) = 2x+ 3 andg(x) =x2+ 7 $g(f(x))=8$ $\Rightarrow(f(x))^{2}+7=8$ $\Rightarrow(2 x+3)^{2}+7=8$ $\Rightarrow x^{2}+3 x+2=0$ $\Rightarrow(x+2)(x+1)=0$ $\Rightarrow x=1 \mid=1,-2$...
Read More →If f : R → R and g :
Question: Iff: R R andg: R R are defined byf(x) = 2x+ 3 andg(x) =x2+ 7, then the values ofxsuch that g(f(x)) = 8 are (a) 1, 2 (b) 1, 2 (c) 1, 2 (d) 1, 2 Solution: (c) 1, 2 f(x) = 2x+ 3 andg(x) =x2+ 7 $g(f(x))=8$ $\Rightarrow(f(x))^{2}+7=8$ $\Rightarrow(2 x+3)^{2}+7=8$ $\Rightarrow x^{2}+3 x+2=0$ $\Rightarrow(x+2)(x+1)=0$ $\Rightarrow x=1 \mid=1,-2$...
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Question: $\int_{0}^{1} \frac{5 x^{2}}{x^{2}+4 x+3}$ Solution: Let $I=\int_{1}^{2} \frac{5 x^{2}}{x^{2}+4 x+3} d x$ Dividing $5 x^{2}$ by $x^{2}+4 x+3$, we obtain $I=\int_{1}^{2}\left\{5-\frac{20 x+15}{x^{2}+4 x+3}\right\} d x$ $=\int_{1}^{2} 5 d x-\int_{1}^{2} \frac{20 x+15}{x^{2}+4 x+3} d x$ $=[5 x]_{1}^{2}-\int_{1}^{2} \frac{20 x+15}{x^{2}+4 x+3} d x$ $I=5-I_{1}$, where $I=\int_{1}^{2} \frac{20 x+15}{x^{2}+4 x+3} d x$ ....(1) Consider $I_{1}=\int_{1}^{2} \frac{20 x+15}{x^{2}+4 x+8} d x$ Let $...
Read More →Let A = {x ∈ R : x ≠ 0, −4 ≤ x ≤ 4} and f :
Question: Let $\mathrm{A}=\{x \in \mathrm{R}: x \neq 0,-4 \leq x \leq 4\}$ and $f: \mathrm{A} \in \mathrm{R}$ be defined by $f(x)=\frac{|x|}{x}$ for $x \in \mathrm{A}$. Then th (is (a) [1, 1] (b) [x: 0 x 4] (c) {1} (d) {x: 4 x 0} Solution: As, $|x|=\left\{\begin{array}{l}x, x \geq 0 \\ -x0\end{array}\right.$ So, $f(x)=\frac{x}{|x|}$ When $x0$ i. e. $x \in[-4,0)$ $f(x)=\frac{x}{-x}=-1$ and when $x0$ i. e. $x \in(0,4]$ $f(x)=\frac{x}{x}=1$ So, range $(f)=\{-1,1\}$ Disclaimer: The question in the b...
Read More →5 books and 7 pens together cost Rs 79 whereas 7 books and 5 pens together cost Rs 77.
Question: 5 books and 7 pens together cost Rs 79 whereas 7 books and 5 pens together cost Rs 77. Find the total cost of 1 book and 2 pens. Solution: Given: (i) Cost of 5 books and 7 pens = Rs. 79. (ii) Cost of 7 books and 5 pens = Rs. 77. To find: Cost of 1 book and 2 pens. Suppose the cost of 1 book = Rsx. and the cost of 1 pen = Rsy. According to the given conditions, we have 5x+ 7y= 79 5x+ 7y 79 = 0 (1) 7x+ 5y= 77, 5x+ 7y 77 = 0 (2) Thus we get the following system of linear equation, $5 x+7 ...
Read More →Find the perimeter and the area of the quadrilateral
Question: Find the perimeter and the area of the quadrilateral ABCD in which AB = 17 cm, AD = 9 cm, CD = 12 cm, AC = 15 cm and angle ACB = 90. Solution: Given are the sides of the quadrilateral ABCD in which AB = 17 cm, AD = 9 cm, CD = 12 cm, AC = 15 cm and an angle ACB =90 By using Pythagoras theorem $B C^{2}=A B^{2}-A C^{2}$ $B C^{2}=17^{2}-15^{2}$ BC = 8 cm Now, area of triangle $A B C=1 / 2 \times A C \times B C$ Area of triangle $\mathrm{ABC}=1 / 2 \times 8 \times 15$ Area of triangle $A B ...
Read More →The function f :
Question: The functionf: R R is defined byf(x) = cos2x+ sin4x. Then,f(R) = (a) [3/4, 1) (b) (3/4, 1] (c) [3/4, 1] (d) (3/4, 1) Solution: (c) (3/4, 1) Given: f(x) = cos2x+ sin4x $\Rightarrow f(x)=1-\sin ^{2} x+\sin ^{4} x$ $\Rightarrow f(x)=\left(\sin ^{2} x-\frac{1}{2}\right)^{2}+\frac{3}{4}$ The minimum value of $f(x)$ is $\frac{3}{4}$. Also, $\sin ^{2} x \leq 1$ $\Rightarrow \sin ^{2} x-\frac{1}{2} \leq \frac{1}{2}$ $\Rightarrow\left(\sin ^{2} x-\frac{1}{2}\right)^{2} \leq \frac{1}{4}$ $\Right...
Read More →The function f :
Question: The functionf: R R is defined byf(x) = cos2x+ sin4x. Then,f(R) = (a) [3/4, 1) (b) (3/4, 1] (c) [3/4, 1] (d) (3/4, 1) Solution: (c) (3/4, 1) Given: f(x) = cos2x+ sin4x $\Rightarrow f(x)=1-\sin ^{2} x+\sin ^{4} x$ $\Rightarrow f(x)=\left(\sin ^{2} x-\frac{1}{2}\right)^{2}+\frac{3}{4}$ The minimum value of $f(x)$ is $\frac{3}{4}$. Also, $\sin ^{2} x \leq 1$ $\Rightarrow \sin ^{2} x-\frac{1}{2} \leq \frac{1}{2}$ $\Rightarrow\left(\sin ^{2} x-\frac{1}{2}\right)^{2} \leq \frac{1}{4}$ $\Right...
Read More →3 bags and 4 pens together cost Rs 257 whereas 4 bags and 3 pens together cost Rs 324.
Question: 3 bags and 4 pens together cost Rs 257 whereas 4 bags and 3 pens together cost Rs 324. Find the total cost of 1 bag and 10 pens. Solution: Given: (i) Cost of 3 bags and 4 pens = Rs. 257. (ii) Cost of 4 bags and 3 pens = Rs. 324. To Find: Cost of 1 bag and 10 pens. Suppose, the cost of 1 bag = Rs.x. and the cost 1 pen = Rs.y. According to the given conditions, we have 3x+ 4y= 257, 3x+ 4y 257 = 0 (1) 4x+ 3y= 324 4x+3y 324 = 0 (2) Solving equation 1 and 2 by cross multiplication $\frac{x}...
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