In a 500 m race, the ratio of speeds of two participants \(A\) and \(B\) is \(4: 5\).
If \(A\) has a start of 180 m , then the distance by which \(A\) wins is :

For the linear programming problem,
Minimize (Z) =60x + 30y
subject to : \(2 x - y \geq-5 ; 3 x + y \geq 3 ; 2 x -3 y \leq 12; x , y \geq 0\)
the optimal value of \(z\) is

The angle between the line \(2 x=3 y=z\) and \(x\) - axis is :

The value of \(\int e^x\left[f(x)+f^{\prime}(x)\right] d x\) is :

A runs 5 times faster than B . If A gives B a start of 100 m , how far must the goal be so that A and B reach there at the same time.

Match List-I with List-II

List-I	List-II
(A) The degree of differential equation \(\frac{d^3 y}{d x^3}=e^{\frac{d y}{d x}}\)	(I) 2
(B) The order of differential equation \(\left(\frac{d y}{d x}\right)^2+\frac{d^3 y}{d x^3}=0\)	(II) 4
(C) The sum of order and degree of differential equation \(\frac{d}{d x}\left(\frac{d^2 y}{d x^2}\right)+\left(\frac{d y}{d x}\right)^5=x\)	(III) not defined
(D) The number of arbitrary constants in the general solution of a differential equation of order 2	(IV) 3

Choose the correct answer from the options given below :

Let \(A\) and \(B\) are square matrices of order 3 such that \(A+B=\left[\begin{array}{ccc}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3\end{array}\right]\). If \(A\) is a symmetric matrix, then the value of \(|B|\) is:

Read the passage carefully and answer the questions
Nearly all the transition elements display typical metallic properties such as high tensile strength, ductility, malleability, high thermal and electrical conductivity and metallic lustre. The transition metals (with the exception of Zn, Cd and Hg) are very hard and have low volatility. Their melting and boiling points are high. They have high enthalpies of atomization. lons of the same charge in a given series show progressive decrease in radius with increasing atomic number. However, the variation within a series is quite small. The filling of 4f before 5d orbital results in a regular decrease in atomic radii called Lanthanoid contraction which essentially compensates for the expected increase in atomic size with increasing atomic number. The net result of the lanthanoid contraction is that the second and the third d series exhibit similar radii (e.g., Zr 160 pm, Hf 159 pm) and have very similar physical and chemical properties. There is an increase in ionisation enthalpy along each series of the transition elements from left to right. The first ionisation enthalpy, in general, increases, but the magnitude of the increase in the second and third ionisation enthalpies for the successive elements, is much higher along a series. One of the notable features of a transition elements is the great variety of oxidation states these may show in their compounds.
Metallic radii of some transition elements are as follows:
Fe: 126 pm; Co: 125 pm; Ni: 125 pm; Cu: 128 pm
Which of these elements will have highest density?

If \(\vec{a}=3 \hat{i}-6 \hat{j}+\hat{k}\) and \(\vec{b}=2 \hat{i}-4 \hat{j}+\lambda \hat{k}\) are such that \(\vec{a} \| \vec{b}\), then \(3 \lambda+2=\)

In a diffraction pattern due to a narrow slit of width a, alternate dark and bright fringe pattern is obtained on a screen kept at distance D from the slit. Choose the correct option.
(A) The angular width of central maximum is given by 2D/a, linear width is given by 2D\lambda/a
(B) When the size of aperture, a is quite large, the diffraction effect is negligible,
(C) For light of smaller wavelength, angular width of central maximum increases, while linear width decreases,
(D) Angular width of central maximum does not decrease while linear width decreases on decreasing distance of screen from slit.
Choose the correct answer from the options given below:

A battery of emf 3 V and internal resistance r is connected in series with a resistor of \( 55 \text{ } \Omega \). An ammeter of resistance \( 2 \text{ } \Omega \) connected in circuit reads 50 mA. The value of r is

Match List - I with List - II.

List - I	List - II
(A) Range of \(y=\operatorname{cosec}^{-1} x\)	(I) \(\quad R-(-1,1)\)
(B) Domain of \(\sec ^{-1} x\)	(II) \(\quad(0, \pi)\)
(C) Domain of \(\sin ^{-1} x\)	(III) \(\quad[-1,1]\)
(D) Range of \(y=\cot ^{-1} x\)	(IV) \(\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]-\{0\}\)

Choose the correct answer from the options given below :

What is the order of hydrolysis of methyl acetate in acidic medium?