A rhombus-shaped sheet with perimeter 40 cm and one diagonal 12 cm,
Question: A rhombus-shaped sheet with perimeter 40 cm and one diagonal 12 cm, is painted on both sides at the rate of Rs 5 per cm2. Find the cost of painting. Solution: Let the sides of rhombus be of lengthxcm. Perimeter of rhombus = 4x⇒ 40 = 4x⇒x= 10 cmNow,In ∆ABC,The sides of the triangle are of length 10 cm, 10 cm and 12 cm. Semi-perimeter of the triangle is $s=\frac{10+10+12}{2}=\frac{32}{2}=16 \mathrm{~cm}$ By Heron's formula, Area of $\Delta A B C=\sqrt{s(s-a)(s-b)(s-c)}$ $=\sqrt{16(16-10)...
Read More →If the surface area of a cube is increasing at a rate
Question: If the surface area of a cube is increasing at a rate of $3.6$ $\mathrm{cm}^{2} / \mathrm{sec}$, retaining its shape; then the rate of change of its volume (in $\mathrm{cm}^{3} / \mathrm{sec}$.), when the length of a side of the cube is $10 \mathrm{~cm}$, is :(1) 18(2) 10(3) 20(4) 9Correct Option: , 4 Solution: Let the side of cube be $a$. $S=6 a^{2} \Rightarrow \frac{d S}{d t}=12 a \cdot \frac{d a}{d t} \Rightarrow 3.6=12 a \cdot \frac{d a}{d t}$ $\Rightarrow 12(10) \frac{d a}{d t}=3....
Read More →Suppose f(x) is a polynomial of degree four
Question: Suppose $f(x)$ is a polynomial of degree four, having critical points at $-1,0,1$. If $T=\{x \in \mathbf{R} \mid f(x)=f(0)\}$, then the sum of squares of all the elements of $T$ is :(1) 4(2) 6(3) 2(4) 8Correct Option: 1 Solution: $\because$ The critical points are $-1,0,1$ $\therefore f^{\prime}(x)=k \cdot x(x+1)(x-1)=k\left(x^{3}-x\right)$ $\Rightarrow f(x)=k\left(\frac{x^{4}}{4}-\frac{x^{2}}{2}\right)+C$\ $\Rightarrow f(0)=C$ $\because f(x)=f(0)$ $\Rightarrow k \frac{\left(x^{4}-2 x^...
Read More →The major product [B] in the following sequence of reactions is:
Question: The major product [B] in the following sequence of reactions is: Correct Option: , 2 Solution:...
Read More →A rectangular plot is given for constructing a house, having a measurement of 40 m
Question: A rectangular plot is given for constructing a house, having a measurement of 40 m long and 15 m in the front. According to the laws, a minimum of 3-m-wide space should be left in the front and back each and 2 m wide space on each of the other sides. Find the largest area where house can be constructed. Solution: LetABCDbe a rectangular plot is given for constructing a house, having a measurement of 40 m long and 15 m in the front. According to the laws, the length of the inner rectang...
Read More →The major product [B] in the following sequence of reactions is:
Question: The major product [B] in the following sequence of reactions is: Correct Option: , 2 Solution:...
Read More →Suppose f(x) is a polynomial of degree four
Question: Suppose $f(x)$ is a polynomial of degree four, having critical points at $-1,0,1$. If $T=\{x \in \mathbf{R} \mid f(x)=f(0)\}$, then the sum of squares of all the elements of $T$ is :(1) 4(2) 6(3) 2(4) 8Correct Option: 1 Solution: $\because$ The critical points are $-1,0,1$...
Read More →A field is in the shape of a trapezium having parallel sides 90 m and 30 m.
Question: A field is in the shape of a trapezium having parallel sides 90 m and 30 m. These sides meet the third side at right angles. The length of the fourth side is 100 m. If it costs Rs 5 to plough 1 m2of the field, find the total cost of ploughing the field. Solution: In the given figure,ABCDis a trapezium having parallel sides 90 m and 30 m. DrawDEperpendicular toAB, such thatDE=BC. In right angled ∆ADE, $A D^{2}=A E^{2}+E D^{2}$ (Pythagoras Theorem) $\Rightarrow 100^{2}=(90-30)^{2}+E D^{2...
Read More →Arrange the following compounds in increasing order of C - OH bond length:
Question: Arrange the following compounds in increasing order of $\mathrm{C}$ - OH bond length: methanol, phenol, $p$-ethoxyphenol methanol $p$-ethoxyphenol $$ phenolphenol $$ methanol $p$-ethoxyphenolphenol $p$-ethoxyphenol $$ methanolmethanol $$ phenol $p$-ethoxyphenolCorrect Option: Solution: Resonance is a deciding factor to determine the order of bond length in given compounds. Phenol exhibits least $\mathrm{C}-\mathrm{OH}$ bond length due to resonance whereas methanol will show maximum bon...
Read More →The major product of the following reaction is:
Question: The major product of the following reaction is: Correct Option: , 3 Solution:...
Read More →In the given figure, a ∆ABC has been given in which AB = 7.5 cm, AC = 6.5 cm and BC = 7 cm.
Question: In the given figure, a ∆ABChas been given in whichAB= 7.5 cm,AC= 6.5 cm andBC= 7 cm. On baseBC, a parallelogramDBCEof the same area as that of ∆ABCis constructed. Find the heightDLof the parallelogram. Solution: In ∆ABC,The sides of the triangle are of length 7.5 cm, 6.5 cm and 7 cm. Semi-perimeter of the triangle is $s=\frac{7.5+6.5+7}{2}=\frac{21}{2}=10.5 \mathrm{~cm}$ By Heron's formula, Area of $\Delta A B C=\sqrt{s(s-a)(s-b)(s-c)}$ $=\sqrt{10.5(10.5-7.5)(10.5-6.5)(10.5-7)}$ $=\sqr...
Read More →The function,
Question: The function, $f(x)=(3 x-7) x^{2 / 3}, x \in \mathbf{R}$, is increasing for all $x$ lying in :(1) $(-\infty, 0) \cup\left(\frac{14}{15}, \infty\right)$(2) $(-\infty, 0) \cup\left(\frac{3}{7}, \infty\right)$(3) $\left(-\infty, \frac{14}{15}\right)$(4) $\left(-\infty,-\frac{14}{15}\right) \cup(0, \infty)$Correct Option: 1, Solution: $f(x)=(3 x-7) \cdot x^{2 / 3}$ $f^{\prime}(x)=3 x^{2 / 3}+(3 x-7) \cdot \frac{2}{3} x^{-1 / 3}$ $=\frac{15 x-14}{3 x^{1 / 3}}$ For increasing function...
Read More →The area of a trapezium is 475 cm2 and its height is 19 cm.
Question: The area of a trapezium is $475 \mathrm{~cm}^{2}$ and its height is $19 \mathrm{~cm}$. Find the lengths of its two parallel sides if one side is $4 \mathrm{~cm}$ greater than the other. Solution: In the given figure,ABCDis a trapezium with parallel sidesABandCD. Let the length ofCDbex.Then, the length ofABbex+ 4. Area of trapezium $=\frac{1}{2} \times$ sum of parallel sides $\times$ height $\Rightarrow 475=\frac{1}{2} \times(x+x+4) \times 19$ $\Rightarrow 475 \times 2=19(2 x+4)$ $\Righ...
Read More →In the following reaction sequence, structures of A and B, respectively will be:
Question: In the following reaction sequence, structures of $\mathrm{A}$ and $\mathrm{B}$, respectively will be: Correct Option: , 3 Solution:...
Read More →In the given figure ABCD is a quadrilateral in which diagonal BD = 64 cm,
Question: In the given figureABCDis a quadrilateral in which diagonalBD= 64 cm,ALBDandCMBDsuch thatAL= 16.8 cm andCM= 13.2 cm. Calculate the area of quadrilateralABCD. Solution: Area of $A B C D=$ Area of $\triangle A B D+$ Area of $\triangle B D C$ $=\frac{1}{2} \times B D \times A L+\frac{1}{2} \times B D \times C M$ $=\frac{1}{2} \times B D(A L+C M)$ $=\frac{1}{2} \times 64(16.8+13.2)$ $=32 \times 30$ $=960 \mathrm{~cm}^{2}$...
Read More →The equation of the normal to the curve
Question: The equation of the normal to the curve $y=(1+x)^{2 y}+\cos ^{2}\left(\sin ^{-1} x\right)$ at $x=0$ is :(1) $y+4 x=2$(2) $y=4 x+2$(3) $x+4 y=8$(4) $2 y+x=4$Correct Option: , 3 Solution: $\because y=(1+x)^{2 y}+\cos ^{2}\left(\sin ^{-1} x\right)$ $y=e^{2 y \ln (1+x)}+\cos ^{2}\left(\cos ^{-1} \sqrt{1-x^{2}}\right)$ $=e^{2 y \ln (1+x)}+\left(1-x^{2}\right)$ $\frac{d y}{d x}=(1+x)^{2 y}\left[2 \ln (1+x) \frac{d y}{d x}+\frac{2 y}{1+x}\right]-2 x$ When $x=0$, then $y=2$ $\therefore \frac{d...
Read More →Find the area of a parallelogram ABCD in which AB = 14 cm,
Question: Find the area of a parallelogram $A B C D$ in which $A B=14 \mathrm{~cm}, B C=10 \mathrm{~cm}$ and $A C=16 \mathrm{~cm}$. [Given: $\sqrt{3}=1.73$ ] Solution: Let: $a=10 \mathrm{~cm}, b=16 \mathrm{~cm}$ and $c=14 \mathrm{~cm}$ $s=\frac{a+b+c}{2}=\frac{10+16+14}{2}=20 \mathrm{~cm}$ By Heron's formula, we have : Area of triangle $A B C=\sqrt{s(s-a)(s-b)(s-c)}$ $=\sqrt{20(20-10)(20-16)(20-14)}$ $=\sqrt{20 \times 10 \times 4 \times 6}$ $=\sqrt{10 \times 2 \times 10 \times 2 \times 2 \times ...
Read More →The correct order of stability for the following alkoxides is:
Question: The correct order of stability for the following alkoxides is: $(\mathrm{B})(\mathrm{A})(\mathrm{C})$$(\mathrm{C})(\mathrm{B})(\mathrm{A})$$(\mathrm{C})(\mathrm{A})(\mathrm{B})$$(\mathrm{B})(\mathrm{C})(\mathrm{A})$Correct Option: , 2 Solution: Electron withdrawing group like $\left(\mathrm{NO}_{2}\right)$ increase stability of alkoxide ion by dispersal of negative charge. In (B) and (C) structures negative charge is in conjugation with double bond and also stabilised by electron withd...
Read More →Then the function f :
Question: Let $f:(-1, \infty) \rightarrow \mathbf{R}$ be defined by $f(0)=1$ and $f(x)=\frac{1}{x} \log _{e}(1+x), x \neq 0 .$ Then the function $f$ :(1) decreases in $(-1,0)$ and increases in $(0, \infty)$.(2) increases in $(-1, \infty)$.(3) increases in $(-1,0)$ and decreases in $(0, \infty)$.(4) decreases in $(-1, \infty)$.Correct Option: , 4 Solution: $f^{\prime}(x)=\frac{\frac{x}{1+x}-\ln (1+x)}{x^{2}}$ $=\frac{x-(1+x) \ln (1+x)}{(1+x) x^{2}}0, \forall x \in(-1, \infty)-\{0\}$ [For $x \in(-...
Read More →1-methyl ethylene oxide when treated
Question: 1-methyl ethylene oxide when treated with an excess of HBr produces:Correct Option: , 2 Solution:...
Read More →Find the area of a parallelogram ABCD in which AB = 28 cm, BC = 26 cm and diagonal AC = 30 cm.
Question: Find the area of a parallelogramABCDin whichAB= 28 cm,BC= 26 cm and diagonalAC= 30 cm. Solution: Let: $a=26 \mathrm{~cm}, b=30 \mathrm{~cm}$ and $c=28 \mathrm{~cm}$ $s=\frac{a+b+c}{2}=\frac{26+30+28}{2}=42 \mathrm{~cm}$ By Heron's formula, we have : Area of triangle $A B C=\sqrt{s(s-a)(s-b)(s-c)}$ $=\sqrt{42(42-26)(42-30)(42-28)}$ $=\sqrt{42 \times 16 \times 12 \times 14}$ $=\sqrt{14 \times 3 \times 4 \times 4 \times 2 \times 2 \times 3 \times 14}$ $=14 \times 4 \times 2 \times 3$ $=33...
Read More →Let P(h, k) be a point on the curve
Question: Let $P(h, k)$ be a point on the curve $y=x^{2}+7 x+2$, nearest to the line, $y=3 x-3$. Then the equation of the normal to the curve at $P$ is :(1) $x+3 y+26=0$(2) $x+3 y-62=0$(3) $x-3 y-11=0$(4) $x-3 y+22=0$Correct Option: 1 Solution: The given curve is, $y=x^{2}+7 x+2$ $\Rightarrow \frac{d y}{d x}=2 x+7$ $\left(\frac{d y}{d x}\right)_{(h, k)}=2 h+7$ The tangent at $P(h, k)$ will be parallel to given line $2 h+7=3 \Rightarrow h=-2$ Point $P(h, k)$ lies on curve $k=(-2)^{2}-7 \times 2+2...
Read More →Find the area of the quadrilateral ABCD in which BCD is an equilateral triangle, each of whose sides is 26 cm,
Question: Find the area of the quadrilateral $A B C D$ in which $B C D$ is an equilateral triangle, each of whose sides is $26 \mathrm{~cm}, A D=24 \mathrm{~cm}$ and $\angle B A D=90^{\circ}$. Also, find the perimeter of the quadrilateral. (Given: $\sqrt{3}=1.73$.) Solution: We know that $\triangle B A D$ is a right-angled triangle. $\therefore A B=\sqrt{B D^{2}-A D^{2}}=\sqrt{26^{2}-24^{2}}=\sqrt{676-576}=\sqrt{100}=10 \mathrm{~cm}$ Now, Area of triangle $B A D=\frac{1}{2} \times$ Base $\times$...
Read More →A solution of phenol in chloroform when treated with aqueous
Question: A solution of phenol in chloroform when treated with aqueous $\mathrm{NaOH}$ gives compound $\mathrm{P}$ as a major product. The mass percentage of carbon in $\mathrm{P}$ is (to the nearest integer) (Atomic mass: $\mathrm{C}=12 ; \mathrm{H}=1 ; \mathrm{O}=16$ ) Solution:...
Read More →Find the perimeter and area of the quadrilateral ABCD in which AB = 21 cm,
Question: Find the perimeter and area of the quadrilateralABCDin whichAB= 21 cm,BAC= 90,AC= 20 cm,CD= 42 cm andAD= 34 cm. Solution: In right angled ∆ABC, $B C^{2}=A B^{2}+A C^{2}$ (Pythagoras Theorem) $\Rightarrow B C^{2}=21^{2}+20^{2}$ $\Rightarrow B C^{2}=441+400$ $\Rightarrow B C^{2}=841$ ⇒BC= 29 cm Area of $\triangle A B C=\frac{1}{2} \times A B \times A C$ $=\frac{1}{2} \times 21 \times 20$ $=210 \mathrm{~cm}^{2}$ ....(1) $\ln \Delta A C D$ The sides of the triangle are of length 20 cm, 34 ...
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