List of practice Questions

Let z₁, z₂ be the roots of the equation z² + a z + 12 = 0 and z₁, z₂ form an equilateral triangle with origin. Then, the value of |a| is ________.
Let f(x) and g(x) be two functions satisfying f(x²) + g(4-x) = 4x³ and g(4-x) + g(x) = 0, then the value of ∫$_{-4}^{4}$ f(x²) dx is __________.
If f(x) = ∫$_{x}^{1}$ (5x⁸ + 7x⁶)/(x² + 1 + 2x⁷)² dx, (x ≥ 0), f(0) = 0 and f(1) = 1/K, then the value of K is __________.
Let the plane ax + by + cz + d = 0 bisect the line joining the points (4, -3, 1) and (2, 3, -5) at the right angles. If a, b, c, d are integers, then the minimum value of (a² + b² + c² + d²) is __________.
The equation of the planes parallel to the plane x - 2y + 2z - 3 = 0 which are at unit distance from the point (1, 2, 3) is ax + by + cz + d = 0. If (b - d) = K(c - a), then the positive value of K is __________.
Match circles M, N, O, P and determine the shape formed by joining their centers in order.
The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is :
The equation of one of the straight lines which passes through the point (1, 3) and makes an angle tan⁻¹(√2) with the straight line, y + 1 = 3√2 x is :
The value of 3 + 1/(4 + 1/(3 + 1/(4 + 1/(3 + ⋯ ∞))) is equal to :
1/(3² - 1) + 1/(5² - 1) + 1/(7² - 1) + ⋯ + 1/(201)² - 1 is equal to :
If f(x) = { 1/|x| if |x| ≥ 1 ; a x² + b if |x| $< 1$ } is differentiable at every point of the domain, then the values of a and b are respectively :
The integral ∫ (2x - 1) cos√((2x - 1)² + 5) / √(4x² - 4x + 6) dx is equal to : (where c is a constant of integration)
If the equation a|z|² + āz + a z̄ + d = 0 represents a circle where a, d are real constants, then which of the following condition is correct ?
Let A + 2B = [1 2 0; 6 -3 3; -5 3 1] and 2A - B = [2 -1 5; 2 -1 6; 0 1 2]. If Tr(A) denotes the sum of all diagonal elements of the matrix A, then Tr(A) - Tr(B) has value equal to :
In order to prepare a buffer solution of pH 5.74, sodium acetate is added to acetic acid. If the concentration of acetic acid in the buffer is 1.0 M, the concentration of sodium acetate in the buffer is __________ M. (Round off to the Nearest Integer). [Given : pKa (acetic acid) = 4.74]
Match List - I with List - II : List - I (Class of Drug)
(a) Antacid
(b) Artificial Sweetener
(c) Antifertility
(d) Tranquilizers List - II (Example)
(i) Novestrol
(ii) Cimetidine
(iii) Valium
(iv) Alitame
Reagent, 1-naphthylamine and sulphanilic acid in acetic acid is used for the detection of :
Complete combustion of 3 g of ethane gives x × 10$^{22}$ molecules of water. The value of x is _________. (Round off to the Nearest Integer). [Use : $N_A$ = 6.023 × 10$^{23}$; Atomic masses in u : C : 12.0 ; O : 16.0 ; H : 1.0]
The statements that are TRUE : (A) methane leads to both global warming and photochemical smog (B) methane is generated from paddy fields (C) methane is a stronger global warming gas than CO₂ (D) methane is a part of reducing smog. Choose the most appropriate answer from the options given below :
The chemical that is added to reduce the melting point of the reaction mixture during the extraction of aluminium is :
Assertion A : During the boiling of water having temporary hardness, Mg(HCO₃)₂ is converted to MgCO₃.
Reason R : The solubility product of Mg(OH)₂ is greater than that of MgCO₃.
Match List - I with List - II : List - I
(a) Ca(OCl)₂
(b) CaSO₄ · 1/2 H₂O
(c) CaO
(d) CaCO₃ List - II
(i) Antacid
(ii) Cement
(iii) Bleach
(iv) Plaster of Paris
The number of ionisable hydrogens present in the product obtained from a reaction of phosphorus trichloride and phosphonic acid is :
Two separate wires A and B are stretched by 2 mm and 4 mm respectively, when they are subjected to a force of 2 N. Assume that both the wires are made up of same material and the radius of wire B is 4 times that of the radius of wire A. The length of the wires A and B are in the ratio of a : b. Then a/b can be expressed as 1/x where x is __________.
A bullet of mass 0.1 kg is fired on a wooden block to pierce through it, but it stops after moving a distance of 50 cm into it. If the velocity of bullet before hitting the wood is 10 m/s and it slows down with uniform deceleration, then the magnitude of effective retarding force on the bullet is 'x' N. The value of 'x' to the nearest integer is __________.