Rolle's theorem holds for the function \(x^3+b x^2+c x, 1 \leq x \leq 2\) at the point \(\frac{4}{3}\), the respective values of \(b\) and \(c\) are :

Ramesh plans to save some amount required after 10 years for higher studies of his son.
He expects the cost of these studies to be Rs. 1,00,000.
How much should be saved at the beginning of each year to accumulate this amount at the end of 10 years, if the interest rate is 12% compounded annually? \(\left(\right.\)Given \(\left.(1.12)^{11}=3.477\right)\)

The area (in sq.units) of the region bounded by the curves \(y = x\) and y = \(x^3\) is:

Match List-I with List-II

List-I	List-II
(A) The value of \(\int_0^4|x| d x\) is	(I) 3
(B) The value of \(\int_{-2}^2|x| d x\) is	(II) -1
(C) The value of \(\int_0^3[x] d x\) is	(III) 8
(D) The value of \(\int_{-1}^1[x] d x\) is	(IV) 4

Choose the correct answer from the options given below :

If \(y=7 \sqrt{x}+3 e^x\), then \(\frac{d^2 y}{d x^2}\) is :

Which of the following is the correct equation to find the area of the shaded region, with reference to the figure given below ?

Match List - I with List - II.

LIST - I	LIST - II
(A) \(x^x\) has a stationary point at \(x\) equal to	(I) \(e\)
(B) For \(x>0\), minimum value of \(x^x\)	(II) \(1 / e\)
(C) The greatest value of \(\left(\frac{1}{x}\right)^x\)	(III) \(e^{1 / e}\)
(D) The stationary point of \(\frac{\log x}{x}\) for \(x\), where \(x>0\), is	(IV) \(e^{-1 / e}\)

Choose the correct answer from the otions given below:

Read the passage carefully and answer the Questions:
According to Valence Bond theory, the metal atom or ion under the influence of ligands can use its \((n-1) d, n s, n p\) or \(n s, n p, n d\) orbitals for hybridisation to yield a set of equivalent orbitals of definite geometry such as octahedral, tetrahedral, square planar and so on. These hybridised orbitals are allowed to overlap with ligand orbitals that can donate electron pairs for bonding.
While the VB theory, to a larger extent, explains the formation, structures and magnetic behaviour of coordination compounds, it suffers from the many shortcomings. The Crystal field theory (CFT) is an electrostatic model which considers the metal-ligand bond to be ionic arising purely from electrostatic interactions between the metal ion and the ligand. The five \(d\) orbitals in an isolated gaseous metal atom/ion have same energy, i.e., they are degenerate. This degeneracy is maintained if a spherically symmetrical field of negative charges surrounds the metal atom/ion. However, when this negative field is due to ligands in a complex, it becomes asymmetrical and the degeneracy of the \(d\) orbitals is lifted. It results in splitting of the \(d\) orbitals. The pattern of splitting depends upon the nature of the crystal field.
The spin only magnetic moment of \(\left[ MnBr _4\right]^{2-}\) is 5.9 BM. Predict the geometry of the complex ion?

Read the passage carefully and answer the Questions:
According to Valence Bond theory, the metal atom or ion under the influence of ligands can use its ( \(n-1\) )d, \(n s, n p\) or \(n s, n p, n d\) orbitals for hybridisation to yield a set of equivalent orbitals of definite geometry such as octahedral, tetrahedral, square planar and so on. These hybridised orbitals are allowed to overlap with ligand orbitals that can donate electron pairs for bonding. While the VB theory, to a larger extent, explains the formation, structures and magnetic behaviour of coordination compounds, it suffers from the many shortcomings. The Crystal field theory (CFT) is an electrostatic model which considers the metal-ligand bond to be ionic arising purely from electrostatic interactions between the metal ion and the ligand. The five \(d\) orbitals in an isolated gaseous metal atom/ion have same energy, i.e., they are degenerate. This degeneracy is maintained if a spherically symmetrical field of negative charges surrounds the metal atom/ion. However, when this negative field is due to ligands in a complex, it becomes asymmetrical and the degeneracy of the \(d\) orbitals is lifted. It results in splitting of the \(d\) orbitals. The pattern of splitting depends upon the nature of the crystal field.
Select the incorrect statement about the Crystal field splitting in octahedral complexes.

If a random variable \(X\) has the following probability distribution :

X	0	1	2	3	4
P(X)	k	2k	3k	\(k^2\)	\(6 k^2\)

then Match List-I with List-II

List-I	List-II
(A) k	(I) 3/7
(B) \(P(x<2)\)	(II) \(6 / 49\)
(C) \(P(X>3)\)	(III) \(1 / 7\)
(D) \(P(2 \leq X \leq 3)\)	(IV) 22/49

Choose the correct answer from the options given below :

In an ecosystem

Which of the following statements are true with reference to homology or homologous organs?
(A) Homology indicates common ancestry.
(B) Whale and Cheetah share similarities in the pattern of the bones of the forelimbs.
(C) Vertebrate heart is an example of homologous organs.
(D) Thorn of Bougainvillea and tendrils of cucurbita represent homology.
Choose the correct answer from the options given below:

Read the following paragraph carefully and answer the given questions.
Certain types of leukocytes (WBC) of our body like polymorpho-nuclear leukocytes (PMNL-neutrophils) and monocytes and natural killer (type of lymphocytes) in the blood as well as macrophages in tissues can phagocytose and destroy microbes. For determining the cause of allergy, the patient is exposed to or injected with very small doses of possible allergens, and the reactions studied. The use of drugs like anti-histamine, adrenalin and steroids quickly reduce the symptoms of allergy. Heroin, commonly called smack is chemically diacetylmorphine which is a white, odourless, bitter crystalline compound. This is obtained by acetylation of morphine. Techniques like radiography (use of X-rays), CT (computed tomography) and MRI (magnetic resonance imaging) are very useful to detect cancers of the internal organs. Different species of Plasmodium (P. vivax, P. malaria and P. falciparum) are responsible for different types of malaria. Of these, malignant malaria caused by Plasmodium falciparum is the most serious one and can even be fatal.
Malignant malaria is caused by -