A triangle ABC is drawn to circumscribe a circle of radius 4 cm

Question: A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see fig.). Find the sides AB and AC. Solution: In fig. BD = 8 cm and DC = 6 cm Then we have $\mathrm{BE}=8 \mathrm{~cm} \quad(\because \mathrm{BE}=\mathrm{BD})$ and $\mathrm{CF}=6 \mathrm{~cm}(\because \mathrm{CF}=\mathrm{CD})$ Suppose $\mathrm{AE}=\mathrm{AF}=\mathrm{x} \mathrm{cm}$ In $\Delta \ma...

If X and Y are two sets such that X has 40 elements,

Question: If X and Y are two sets such that X has 40 elements, XY has 60 elements and XY has 10 elements, how many elements does Y have? Solution: It is given that: n(X) = 40,n(XY) = 60,n(XY) = 10 We know that: n(XY) =n(X) +n(Y) n(XY) 60 = 40 +n(Y) 10 n(Y) = 60 (40 10) = 30 Thus, the set Y has 30 elements....

What will be the pressure of the gaseous mixture when 0.5 L of H2 at 0.8 bar and

Question: What will be the pressure of the gaseous mixture when 0.5 L of H2at 0.8 bar and 2.0 L of dioxygen at 0.7 bar are introduced in a 1L vessel at 27C? Solution: Let the partial pressure of H2in the vessel be. Now, $\begin{array}{ll}p_{1}=0.8 \text { bar } p_{2}=p_{H_{2}}=? \\ V_{1}=0.5 \mathrm{~L} V_{2}=1 \mathrm{~L}\end{array}$ It is known that, $p_{1} V_{1}=p_{2} V_{2}$ $\Rightarrow p_{2}=\frac{p_{1} V_{1}}{V_{2}}$ $\Rightarrow p_{H_{2}}=\frac{0.8 \times 0.5}{1}$ $=0.4 \mathrm{bar}$ Now,...

If S and T are two sets such that S has 21 elements, T has 32 elements, and

Question: If S and T are two sets such that S has 21 elements, T has 32 elements, and ST has 11 elements, how many elements does ST have? Solution: It is given that: n(S) = 21,n(T) = 32,n(ST) = 11 We know that: n(ST) =n(S) +n(T) n(ST) n(ST) = 21 + 32 11 = 42 Thus, the set (ST) has 42 elements....

Explain the following processes:

Question: Explain the following processes: (a) Polarisation of the membrane of a nerve fibre (b) Depolarisation of the membrane of a nerve fibre (c) Conduction of a nerve impulse along a nerve fibre (d) Transmission of a nerve impulse across a chemical synapse Solution: (a)Polarisation of the membrane of a nerve fibre During resting condition, the concentration of K+ions is more inside the axoplasm while the concentration of Na+ions is more outside the axoplasm. As a result, the potassium ions m...

In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?

Question: In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English? Solution: Let H be the set of people who speak Hindi, and E be the set of people who speak English n(HE) = 400,n(H) = 250,n(E) = 200 n(HE) = ? We know that: n(HE) =n(H) +n(E) n(HE) 400 = 250 + 200 n(HE) ⇒400 = 450 n(HE) ⇒n(HE) = 450 400 n(HE) = 50 Thus, 50 people can speak both Hindi and English....

If X and Y are two sets such that X ∪Y has 18 elements, X has 8 elements and Y has 15 elements; how many elements does X ∩Y have?

Question: If X and Y are two sets such that XYhas 18 elements, X has 8 elements and Y has 15 elements; how many elements does XY have? Solution: It is given that: n(X Y) = ? We know that: $n(\mathrm{X} \cup \mathrm{Y})=n(\mathrm{X})+n(\mathrm{Y})-n(\mathrm{X} \cap \mathrm{Y})$ $\therefore 18=8+15-n(\mathrm{X} \cap \mathrm{Y})$ $\Rightarrow n(\mathrm{X} \cap \mathrm{Y})=23-18=5$ $\therefore n(\mathrm{X} \cap \mathrm{Y})=5$...

Prove that the parallelogram circumscribing a circle is a rhombus.

Question: Prove that the parallelogram circumscribing a circle is a rhombus. Solution: Let ABCD be a parallelogram such that its sides touch a circle with centre O. AP = AS [Tangents from an external point are equal] BP = BQ CR = CQ DR = DS Adding these equations $A P+B P+C R+D R=A S+D S+B Q+C Q$ $\mathrm{AB}+\mathrm{CD}=\mathrm{AD}+\mathrm{BC}$ 2AB = 2BC AB = BC AB = BC = CD = DA ABCD is a rhombus. Hence proved...

Compare the following:

Question: Compare the following: (a) Central neural system (CNS) and Peripheral neural system (PNS) (b) Resting potential and action potential (c) Choroid and retina Solution: (a)Central neural system (CNS) and Peripheral neural system (PNS) (b)Resting potential and action potential (c)Choroid and retina...

If X and Y are two sets such that n(X) = 17, n(Y) = 23 and n(X ∪ Y) = 38, find n(X ∩Y).

Question: If X and Y are two sets such thatn(X) = 17,n(Y) = 23 andn(XY) = 38, findn(XY). Solution: It is given that: n(X) = 17,n(Y) = 23,n(XY) = 38 n(XY) = ? We know that: $n(\mathrm{X} \cup \mathrm{Y})=n(\mathrm{X})+n(\mathrm{Y})-n(\mathrm{X} \cap \mathrm{Y})$ $\therefore 38=17+23-n(\mathrm{X} \cap \mathrm{Y})$ $\Rightarrow n(\mathrm{X} \cap \mathrm{Y})=40-38=2$ $\therefore n(\mathrm{X} \cap \mathrm{Y})=2$...

Prove that the angle between the two tangents drawn from an external point

Question: Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre. Solution: Let PA and PB be two tangents drawn from an external point P to a circle with centre O. Now, in right $\Delta \mathrm{OAP}$ and right $\Delta \mathrm{OBP}$, we have $\mathrm{PA}=\mathrm{PB}$ [Tangents to circle from an external point] $\mathrm{OA}=\mathrm{OB}$ [Radii of the same circle] ...

Fill in the blanks to make each of the following a true statement:

Question: Fill in the blanks to make each of the following a true statement: (i) (ii)Φ A = (iii) (iv) Solution: (i) (ii)Φ A = U A = A Φ A = A (iii)A A = Φ (iv)U A = Φ A = Φ U A = Φ...

What will be the pressure exerted by a mixture of 3.2 g of methane and 4.4 g

Question: What will be the pressure exerted by a mixture of 3.2 g of methane and 4.4 g of carbon dioxide contained in a 9 dm3flask at 27 C ? Solution: It is known that, $p=\frac{m}{M} \frac{\mathrm{RT}}{V}$ For methane (CH4), $p_{\mathrm{CH}_{4}}=\frac{3.2}{16} \times \frac{8.314 \times 300}{9 \times 10^{-3}}\left[\begin{array}{l}\text { Since } 9 \mathrm{dm}^{3}=9 \times 10^{-3} \mathrm{~m}^{3} \\ 27^{\circ} \mathrm{C}=300 \mathrm{~K}\end{array}\right]$ $=5.543 \times 10^{4} \mathrm{~Pa}$ For c...

Briefly describe the structure of the following:

Question: Briefly describe the structure of the following: (a) Brain (b) Eye (c) Ear Solution: (A)Brain:Brain is the main coordinating centre of the body. It is a part of nervous system that controls and monitors every organ of the body. It is well protected by cranial meninges that are made up of an outer layer called dura mater, a thin middle layer called arachnoid, and an inner layer called pia mater. It is divided into three regions forebrain, midbrain, and hindbrain. Forebrain:It is the mai...

Let $U$ be the set of all triangles in a plane. If $A$ is the set of all triangles with at least one angle different from $60^{circ}$, what is $A^{prime}$ ?

Question: Let $U$ be the set of all triangles in a plane. If $A$ is the set of all triangles with at least one angle different from $60^{\circ}$, what is $A^{\prime}$ ? Solution: $\mathrm{A}^{\prime}$ is the set of all equilateral triangles....

In fig., XY and X'Y' are two parallel tangents to a circle

Question: In fig., $\mathrm{XY}$ and $\mathrm{X}^{\prime} \mathrm{Y}^{\prime}$ are two parallel tangents to a circle with centre $\mathrm{O}$ and another tangent $\mathrm{AB}$ with point of contact $\mathrm{C}$ intersecting $\mathrm{XY}$ at $\mathrm{A}$ and $\mathrm{X}^{\prime} \mathrm{Y}^{\prime}$ at $\mathrm{B}$. Prove that $\angle \mathrm{AOB}=90^{\circ}$ Solution: In fig., Join OC and we have $\Delta \mathrm{s} \mathrm{AOP}$ and $\mathrm{AOC}$ for which AP = AC (Both tangents from A) OP = OC...

Draw appropriate Venn diagram for each of the following:

Question: Draw appropriate Venn diagram for each of the following: (i) $(A \cup B)^{\prime}$ (ii) $A^{\prime} \cap B^{\prime}$ (iii) $(\mathrm{A} \cap \mathrm{B})^{\prime}$ (iv) $\mathrm{A}^{\prime} \cup \mathrm{B}^{\prime}$ Solution: (i) $(\mathrm{A} \cup \mathrm{B})^{\prime}$ (ii) $A^{\prime} \cap B^{\prime}$ (iii) $(A \cap B)^{\prime}$ (iv) $\mathrm{A}^{\prime} \cup \mathrm{B}^{\prime}$...

Question: Briefly describe the structure of the following: (a) Brain (b) Eye (c) Ear Solution: (A)Brain:Brain is the main coordinating centre of the body. It is a part of nervous system that controls and monitors every organ of the body. It is well protected by cranial meninges that are made up of an outer layer called dura mater, a thin middle layer called arachnoid, and an inner layer called pia mater. It is divided into three regions forebrain, midbrain, and hindbrain. Forebrain:It is the mai...

The drain cleaner, Drainex contains small bits of aluminum which react with caustic soda to produce dihydrogen.

Question: The drain cleaner, Drainex contains small bits of aluminum which react with caustic soda to produce dihydrogen. What volume of dihydrogen at 20 C and one bar will be released when 0.15g of aluminum reacts? Solution: The reaction of aluminium with caustic soda can be represented as: $2 \mathrm{Al}+2 \mathrm{NaOH}+2 \mathrm{H}_{2} \mathrm{O} \longrightarrow 2 \mathrm{NaAlO}_{2}+3 \mathrm{H}_{2}$ $2 \times 27 \mathrm{~g}$ $3 \times 22400 \mathrm{~mL}$ At STP (273.15 K and 1 atm), 54 g (2 ...

A quadrilateral ABCD is drawn to circumscribe a circle (see fig.).

Question: A quadrilateral ABCD is drawn to circumscribe a circle (see fig.). Prove that AB + CD = AD + BC. Solution: In fig., we observe that $\mathrm{AP}=\mathrm{AS}$ ...(1) $\{\because \mathrm{AP}$ and $\mathrm{AS}$ are tangents to the circle drawn from the point $\mathrm{A}\}$ Similarly, $\quad B P=B Q \quad \ldots(2)$ $\mathrm{CR}=\mathrm{CQ} \quad \ldots(3)$ $\mathrm{DR}=\mathrm{DS} \quad \ldots(4)$ Adding (1), (2), (3), (4), we have $(\mathrm{AP}+\mathrm{BP})+(\mathrm{CR}+\mathrm{DR})=(\ma...

If U = {1, 2, 3, 4, 5,6,7,8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that

Question: If U = {1, 2, 3, 4, 5,6,7,8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that (i) $(\mathrm{A} \cup \mathrm{B})^{\prime}=\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}$ (ii) $(\mathrm{A} \cap \mathrm{B})^{\prime}=\mathrm{A}^{\prime} \cup \mathrm{B}^{\prime}$ Solution: U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {2, 4, 6, 8}, B = {2, 3, 5, 7} (i) $(\mathrm{A} \cup \mathrm{B})^{\prime}=\{2,3,4,5,6,7,8\}^{\prime}=\{1,9\}$ $\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}=\{1,3,5,7,9\} \cap(1,4,6,...

Two concentric circles are of radii 5 cm and 3 cm.

Question: Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle. Solution: In fig. the two concentric circles have their centre at O. The radius of the larger circle is 5 cm and that of the smaller circle is 3 cm. AB is a chord of the larger circle and it touches the smaller circle at P. Join $\mathrm{OA}, \mathrm{OB}$ and $\mathrm{OP}$. $\mathrm{OP}=3 \mathrm{~cm}$ and $\mathrm{OP} \perp \mathrm{AB}$, i.e., $\angle ...

Pressure of 1 g of an ideal gas A at 27 °C is found to be 2 bar.

Question: Pressure of 1 g of an ideal gas A at 27 C is found to be 2 bar. When 2 g of another ideal gas B is introduced in the same flask at same temperature the pressure becomes 3 bar. Find a relationship between their molecular masses. Solution: For ideal gas A, the ideal gas equation is given by, $p_{\mathrm{A}} V=n_{\mathrm{A}} \mathrm{R} T \ldots \ldots .(\mathrm{i})$ Where,pAandnArepresent the pressure and number of moles of gas A. For ideal gas B, the ideal gas equation is given by, $p_{B...

Taking the set of natural numbers as the universal set, write down the complements of the following sets:

Question: Taking the set of natural numbers as the universal set, write down the complements of the following sets: (i){x:xis an even natural number} (ii){x:xis an odd natural number} (iii){x:xis a positive multiple of 3} (iv){x:xis a prime number} (v){x:xis a natural number divisible by 3 and 5} (vi){x:xis a perfect square} (vii){x:xis perfect cube} (viii){x:x+ 5 = 8} (ix){x: 2x+ 5 = 9} (x){x:x 7} (xi){x:x N and 2x+ 1 10} Solution: U = N: Set of natural numbers (i){x:xis an even natural number}...

Question: Fill in the blank spaces: (a) All mammals (except a few) have __________ cervical vertebra. (b) The number of phalanges in each limb of human is __________ (c) Thin filament of myofibril contains 2 F actins and two other proteins namely __________ and __________. (d) In a muscle fibre Ca++is stored in __________ (e) __________ and __________ pairs of ribs are called floating ribs. (f) The human cranium is made of __________ bones. Solution: (a) All mammals (except a few) havecervical v...