The Electric flux through the surface
Question: The Electric flux through the surface (a)in Fig.1.3 (iv) is the largest. (b)in Fig. 1.3 (iii) is the least. (c)in Fig. 1.3 (ii) is same as Fig. 1.3 (iii) but is smaller than Fig. 1.3 (iv) (d)is the same for all the figures. Solution: The flux of electric field through any closed surface S is 1/0times the total charge enclosed by S (Irrespective of shape and size of the surface). All the figures given in the questions have the same charge i.e., +q. So, the electric flux through the surf...
Read More →A point positive charge is brought near an isolated
Question: A point positive charge is brought near an isolated conducting sphere (figure). The electric field is best given by (a)Fig (i) (b)Fig (ii) (c)Fig (iii) (d)Fig (iv) Solution: Electric field originates from positive charge. It is perpendicular to the surface of the conductor. Also, inside the conducting sphere, it will be zero....
Read More →In figure two positive charges q2 and q3 fixed along the y-axis,
Question: In figure two positive chargesq2andq3fixed along they-axis, exert a net electric force in the +x-direction on a chargeq1, fixed along thex-axis. If a positive charge Q is added at (x, 0), the force onq1, (a)shall increase along the positive x-axis (b)shall decrease along the positive x-axis (c)shall point along the negative x-axis (d)shall increase but the direction changes because of the intersection of Q withq2andq3 Solution: Net force on chargeq1, by other chargesq2andq3is along the...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{2 x-1}{(x-1)^{2}} d x$ Solution: Let $I=\int \frac{2 x-1}{(x-1)^{2}} d x$ Substituting $x-1=t \Rightarrow d x=d t$ $\Rightarrow I=\int \frac{2(t+1)-1}{t^{2}} d t$ $\Rightarrow I=\int \frac{2 t+1}{t^{2}} d t$ $\Rightarrow I=\int\left(\frac{2}{t}+\frac{1}{t^{2}}\right) d t$ $\Rightarrow I=2 \log |t|+\frac{1}{t}+c$ $\Rightarrow I=2 \log |x-1|+\frac{1}{x-1}+c$ Therefore, $\int \frac{2 x-1}{(x-1)^{2}} d x=2 \log |x-1|+\frac{1}{x-1}+c$...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\mathrm{x}^{2}}{\sqrt{3 \mathrm{x}+4}} \mathrm{dx}$ Solution: Let $\mathrm{I}=\int \frac{\mathrm{x}^{2}}{\sqrt{3 \mathrm{x}+4}} \mathrm{dx}$ Substituting $3 x+4=t \Rightarrow 3 d x=d t$ $\Rightarrow I=\int \frac{\left(\frac{t-4}{3}\right)^{2}}{3 \sqrt{t}} d t$ $\Rightarrow \mathrm{I}=\frac{1}{27} \int \frac{\mathrm{t}^{2}+16-8 \mathrm{t}}{\sqrt{\mathrm{t}}} \mathrm{dt}$ $\Rightarrow \mathrm{I}=\frac{1}{27} \int\left(\mathrm{t}^{\frac{3}{2}}...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\mathrm{x}^{2}}{\sqrt{\mathrm{x}-1}} \mathrm{dx}$ Solution: Let $\mathrm{I}=\int \frac{\mathrm{x}^{2}}{\sqrt{\mathrm{x}-1}} \mathrm{dx}$ Substituting $x-1=t \Rightarrow d x=d t$ $\Rightarrow I=\int \frac{(t+1)^{2}}{\sqrt{t}} d t$ $\Rightarrow I=\int \frac{t^{2}+2 t+1}{\sqrt{t}} d t$ $\Rightarrow I=\int\left(t^{\frac{3}{2}}+2 t^{\frac{1}{2}}+t^{-\frac{1}{2}}\right) d t$ $\Rightarrow I=\frac{\left(6 t^{\frac{5}{2}}+30 t^{\frac{1}{2}}+20 t^{\f...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \mathrm{x}^{2} \sqrt{\mathrm{x}+2} \mathrm{dx}$ Solution: Let $\mathrm{I}=\int \mathrm{x}^{2} \sqrt{\mathrm{x}+2} \mathrm{dx}$ Substituting, $x+2=t \Rightarrow d x=d t$, $I=\int(t-2)^{2} \sqrt{t} d t$ $\Rightarrow I=\int\left(t^{2}-4 t+4\right) \sqrt{t} d t$ $\Rightarrow I=\int\left(t^{\frac{5}{2}}-4 t^{\frac{3}{2}}+4 t^{\frac{1}{2}}\right) d t$ $\Rightarrow I=\frac{2}{7} t^{\frac{7}{2}}-\frac{8}{5} t \frac{5}{2}+\frac{8}{2} t \frac{3}{2}+c$ $\Ri...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\sin ^{5} x}{\cos ^{4} x} d x$ Solution: $\sin ^{5} x=\sin ^{4} x \cdot \sin x$ Assume $\cos x=t$ $d(\cos x)=d t$ $-\sin x \cdot d x=d t$ $\Rightarrow \mathrm{dx}=\frac{-\mathrm{dt}}{\sin \mathrm{x}}$ Substitute $t$ and $d t$ we get $\Rightarrow \int \frac{\sin ^{4} x \cdot \sin x}{\cos ^{4} x} \times \frac{-d t}{\sin x}$ $\Rightarrow \int \frac{-d t\left(1-\cos ^{2} x\right)^{2}}{\cos ^{4} x}$ $\Rightarrow \int \frac{-d t\left(1-t^{2}\righ...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{x^{2}\left(x^{4}+1\right)^{3 / 4}} d x$ Solution: $I=\int \frac{1}{x^{2}\left(x^{4}+1\right)^{\frac{3}{4}}} d x$ $\Rightarrow \int \frac{1}{x^{5}\left(1+\frac{1}{x^{4}}\right)^{\frac{3}{4}}} d x$ Let $1+\frac{1}{x^{4}}=t$ $\Rightarrow-\frac{4}{x^{5}} d x=d t$ $\Rightarrow \frac{1}{x^{5}} d x=\frac{-d t}{4}$ $I=\frac{-1}{4} \int \frac{1}{t^{\frac{3}{4}}} d t$ $\Rightarrow \frac{-1}{4}\left(\frac{\mathrm{t}^{\frac{1}{4}}}{\frac{1}{4}}\righ...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{\sqrt{x}+x} d x$ Solution: $x=t^{2}$ $d(x)=2 t \cdot d t$ $d x=2 t \cdot d t$ Substituting $t$ and dt we get $\Rightarrow \int \frac{2 t \cdot d t}{t^{2}+t}$ $\Rightarrow 2 \int \frac{t \cdot d t}{t^{2}+t}$ $\Rightarrow 2 \int \frac{1}{1+t} d t$ $\Rightarrow 2(\ln |1+t|)$ But $t=\sqrt{x}$ $\Rightarrow 2(\ln |1+\sqrt{x}|)+c$...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int 4 x^{3} \sqrt{5-x^{2}} d x$ Solution: Assume $5-x^{2}=t^{2}$ $d\left(5-x^{2}\right)=d\left(t^{2}\right)$ $-2 x \cdot d x=2 t \cdot d t$ $\Rightarrow x d x=-t . d x$ $\Rightarrow \mathrm{dx}=\frac{-\mathrm{t}}{\mathrm{x}} \mathrm{dt}$ Substituting $t$ and $d t$ $\Rightarrow \int 4 x^{3} \sqrt{t^{2}} \frac{-t}{x} d t$ $\Rightarrow 4 \int x^{2} t^{2}$ $\Rightarrow x^{2}=5-t^{2}$ $\Rightarrow 4 \int\left(5-t^{2}\right) t^{2} \cdot d t$ $\Rightarrow 2...
Read More →Hypothalamus is a super master endocrine
Question: Hypothalamus is a super master endocrine gland. Elaborate. Solution: Hypothalamus is a super master endocrine gland as it secretes hormones that regulate the synthesis and secretion of the pituitary gland which is the master gland. It is connected to the anterior lobe of the pituitary gland by hypo physical portal veins. The hormones which are secreted by hypothalamus are: 1. Adrenocorticoicotropic releasing hormone (ARH) or corticotrophin 2. Growth hormone-releasing hormone (GHRH) 3. ...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x^{5}}{\sqrt{1+x^{3}}} d x$ Solution: Assume $x^{3}+1=t^{2}$ $d\left(x^{3}+1\right)=d\left(t^{2}\right)$ $3 x^{2} \cdot d x=2 t \cdot d t$ $\Rightarrow \mathrm{dx}=\frac{2 \mathrm{t}}{3 \mathrm{x}^{2}} \mathrm{dt}$ $x^{3}+1=t^{2}$ $\Rightarrow \mathrm{dx}=\frac{2 t}{3 \mathrm{x}^{2}} \mathrm{dt}$ Substituting $\mathrm{t}$ and $\mathrm{dt}$ $\Rightarrow \int \frac{x^{5}}{\sqrt{t^{2}}} \cdot \frac{2 t}{3 x^{2}} d t$ $\Rightarrow \int \frac{x^...
Read More →Illustrate the differences between
Question: Illustrate the differences between the mechanism of action of a protein and a steroid hormone. Solution: Protein hormone: Protein hormones are insulin, glucagon, prolactin etc. They have to bind their receptors which are present on the plasma membrane. They form their hormone receptor complex on the membrane. They are water-soluble hormone so they cant diffuse to lipid layer plasma membrane and bind to a receptor on membrane. Steroid hormone: Steroid hormone is an androgen, estrogen et...
Read More →Calcium plays a very important role in the formation of bones.
Question: Calcium plays a very important role in the formation of bones. Write on the role of endocrine glands and hormones responsible for maintaining Calcium homeostasis. Solution: Thyroid gland and the parathyroid gland that is an antagonist with each other are responsible for maintaining calcium homeostasis. Calcitonin is responsible for the high calcium level in the blood. It also decreases plasma calcium concentration by decreasing mobilization of calcium from bones. Parathyroid hormone in...
Read More →A sample of urine was diagnosed to contain a high content
Question: A sample of urine was diagnosed to contain a high content of glucose and ketone bodies. Based on this observation, answer the following: a. Which endocrine gland and hormone is related to this condition? b. Name the cells on which this hormone acts. c. What is the condition called and how can it be rectified? Solution: a. The pancreas is the gland and hormone is Insulin which is responsible for this condition. b. Hepatocytes and adipocytes cells are the main cells on which insulin act....
Read More →A milkman is very upset one morning as his cow refuses to give any milk.
Question: A milkman is very upset one morning as his cow refuses to give any milk. The milkmans wife gets the calf from the shed. On fondling by the calf, the cow gave sufficient milk. Describe the role of endocrine gland and pathway associated with this response? Solution: Oxytocin hormone is the hormone which stimulates the release of milk from the mammary gland by causing surrounding cells to contract. When the infant starts feeding breast stimulates the posterior pituitary to produce the oxy...
Read More →Mention the difference between hypothyroidism
Question: Mention the difference between hypothyroidism and hyperthyroidism. Solution: Hypothyroidism is when the thyroid hormone secretes less amount causes this disease. The symptoms include weight gain, loss of appetite, dry and coarse skin. Hyperthyroidism is due to the high amount of secretion which causes the excessive amount of thyroid hormone and patient leads to hyperthyroidism. The symptoms include weight loss, increased appetite and moist skin....
Read More →What are the effects of hypothyroidism (observed during pregnancy)
Question: What are the effects of hypothyroidism (observed during pregnancy) on the development and maturation of a growing baby? Solution: Hypothyroidism during pregnancy causes defective development and stunted growth (cretinism). Mental retardation and low intelligence quotient....
Read More →Old people have a weak immune system.
Question: Old people have a weak immune system. What could be the reason? Solution: Thymosin is the important hormone that plays an important role in the immune system but with age, thymus gland starts to degenerate and thus old people have a weak immune system....
Read More →Inflammatory responses can be controlled
Question: Inflammatory responses can be controlled by a certain steroid. Name the steroid, its source and also its other important functions. Solution: Cortisol is a steroid hormone which is secreted by the adrenal cortex that produces anti-inflammatory responses and suppresses the immune response....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{(x+1)\left(x^{2}+2 x+2\right)} d x$ Solution: We can write $x^{2}+2 x+1+1=(x+1)^{2}+1$ $\Rightarrow \frac{1 \cdot d x}{(x+1)(x+1)^{2}+1}$ Assume $x+1=$ tant $\Rightarrow d x=\sec ^{2} t \cdot d x$ $\Rightarrow \int \frac{\sec ^{2} t d t}{\tan t \cdot \tan ^{2} t+1}$ $\Rightarrow \tan ^{2} t+1=\sec ^{2} t$ $\Rightarrow \int \frac{d t}{\tan t}$ $\Rightarrow \frac{\cos t}{\sin t} d t$ $\Rightarrow \log |\sin t|+c$ $\Rightarrow \sin t=\frac{...
Read More →George comes on a vacation to India from the US.
Question: George comes on a vacation to India from the US. The long journey disturbs his biological system and he suffers from jet lag. What is the cause of his discomfort? Solution: Melatonin is the hormone which is responsible for this biological system discomfort. Jet lag is a temporary sleep disorder....
Read More →On an educational trip to Uttaranchal,
Question: On an educational trip to Uttaranchal, Ketki and her friends observe that many local people were having swollen necks. Please help Ketki and her friends to find out the solutions to the following questions. a. Which probable disease are these people suffering from? b. How is it caused? c. What effect does this condition have on pregnancy? Solution: a. Goitre. This is due to the overstimulation of thyroid gland which causes an enlargement in the gland. b. It is caused due to the deficie...
Read More →What is the role of the second messenger
Question: What is the role of the second messenger in hormone action? Solution: Secondary messenger like (cAMP) act as intracellular hormonal mediator which deliver the information inside the target cell. Secondary messenger is cAMP, cGMP, diacylglycerol, inositol triphosphate, phosphoinositides, calcium etc....
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