The unit vector in the xy-plane, making an angle of \(60^{\circ}\) with the positive direction of x-axis is:

The tangent to the curve \(y=x^3-11 x+5\) at \(x=2\) is parallel to :
(A) \( y=x+1\)
(B) \(y=-x+2\)
(C) \(2 y=x+3\)
(D) \(2 y=2 x+4\)
Choose the correct answer from the options given below :

If \(A\) is a square matrix of order 3 and \(|A|=-2\), then the value of \(\left|-A^{-1}\right|\) is :

If \(A\) is a square matrix of order \(3 \times 3\) and \(|A|=4\). The value of \(\mid(\operatorname{adj} A)\). \(A \mid\) is:

The area enclosed by the curves \(x y=4, y=x, x=e^2\) and the \(x\)-axis, is :

If \(\vec{a}, \vec{b}\) and \(\sqrt{3} \vec{a}+\vec{b}\) are unit vectors, then the angle between \(\vec{a}\) and \(\vec{b}\) is:

Consider the linear programming problem (LPP):
Minimize \(Z=x+y\)
subject to
\(x+2 y \leq 4,\)
\(3 x+y \geq 3\),
\(4 x+3 y \geq 6\),
\(x, y \geq 0\).
Which of the following is correct for the above linear programming problem (LPP):
(A) The LPP has a bounded feasible region.
(B) The LPP has a unique optimal solution.
(C) The optimal value of the LPP exists at the point (3/2,0)
(D) The corner points of the feasible region are \((3 / 2,0),(3 / 5,6 / 5),(2 / 5,6 / 5)\) and \((4,0)\)
Choose the correct answer from the options given below:

Identify the statements which hold true with respect to origin of life.
(A) Louis Pasteur demonstrated that life comes only from pre-existing life.
(B) Big Bang theory attempts to explain the origin of universe.
(C) Oparin and Haldane proposed first form of life came from pre-existing non-living organic molecules.
(D) Miller's experiment was unable to prove chemical evolution.

Which one of the following is opposite part of micropylar end of ovule.

Read the given passage carefully and answer the questions
Antibiotics have greatly improved our capacity to treat deadly diseases such as plague, whooping cough (kali khansi), diphtheria (gal ghotu) and leprosy (kusht rog), which used to kill millions all over the globe.
Streptokinase produced by the bacterium Streptococcus and modified by genetic engineering is used as a 'clot buster' for removing clots from the blood vessels of patients who have undergone myocardial infarction leading to heart attack. Microbes are diverse-protozoa, bacteria, fungi and microscopic animal and plant viruses, viroids and also prions that are proteinacious infectious agents. The puffed-up appearance of dough is due to the production of CO₂ gas. Similarly the dough, which is used for making bread, is fermented using baker's yeast (Saccharomyces cerevisiae). Primary sewage treatment basically involve physical removal of particles - large and small from the sewage through filtration and sedimentation. These are removed in stages; initially, floating debris is removed by sequential filtration. Then the grit (soil and small pebbles) are removed by sedimentation. All solids that settle form the primary sludge, and the supernatant forms the effluent. The effluent from the primary settling tank is taken for secondary treatment. The primary effluent is passed into large aeration tanks, where it is constantly agitated mechanically and air is pumped into it. This allows vigorous growth of useful aerobic microbes into flocs. While growing, these microbes consume the major part of the organic matter in the effluent.
Genetically modified Streptokinase is used as a 'clot buster' for removing clots from the blood vessels is produced by :

Rohit has been given two converging lenses of focal lengths 1.25 cm and 5 cm, respectively to design a compound microscope. If it is desired to have a magnification of 30, what would be the separation between the objective and the eyepiece when the final image is formed at the least distance of distinct vision?

By increasing the temperature, resistivity of a conductor and a semiconductor

If red light is replaced by blue light in single slit diffraction then