List of practice Questions

Let

\(a→=α\^{i}+3\^{j}−\^{k}, \overrightarrow{b}=3\^{i}−β\^{j}+4\^{k} and \overrightarrow{c}=\^{i}+2\^{j}−2\^{k }\)

where α,β∈R, be three vectors. If the projection of

\(\overrightarrow{a} on \overrightarrow{c} is \frac{10}{3} and \overrightarrow{b}×\overrightarrow{c}=−6\^{i}+10\^{j}+7\^{k}, \)

then the value of α+β is equal to

Let the tangent to the circle C₁: x² + y² = 2 at the point M(–1, 1) intersect the circle C₂: (x – 3)² + (y – 2)² = 5, at two distinct points A and B. If the tangents to C₂ at the points A and B intersect at N, then the area of the triangle ANB is equal to

A monoatomic gas at pressure P and volume V is suddenly compressed to one eighth of its original volume. The final pressure at constant entropy will be :

Let \(X\) be a random variable having binomial distribution \(B(7, p)\). If \(P(X = 3) = 5P(X = 4)\), then the sum of the mean and the variance of \(X\) is:

The length of a given cylindrical wire is increased to double of its original length. The percentage increase in the resistance of the wire will be ______%.

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is _____.

The remainder on dividing 1 + 3 + 3² + 3³ + … + 3²⁰²¹ by 50 ____ is

The area (in sq. units) of the region enclosed between the parabola y² = 2x and the line x + y = 4 is _______.

Let a circle C : (x – h)² + (y – k)² = r², k > 0, touch the x-axis at (1, 0). If the line x + y = 0 intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of h + k + r is equal to ____.

The speed of light in media ‘A’ and ‘B’ are 2.0 × 10¹⁰ cm/s and 1.5 × 10¹⁰ cm/s respectively. A ray of light enters from the medium B to A at an incident angle ‘θ’. If the ray suffers total internal reflection, then

The number of values of a ∈ N such that the variance of 3, 7, 12, a, 43 – a is a natural number is :

From the base of a pole of height 20 meter, the angle of elevation of the top of a tower is 60°. The pole subtends an angle 30° at the top of the tower. Then the height of the tower is :

Let f and g be twice differentiable even functions on (–2, 2) such that
\(ƒ(\frac{1}{4})=0, ƒ(\frac{1}{2})=0, ƒ(1) =1\) and \(g(\frac{3}{4}) = 0 , g(1)=2\)
.Then, the minimum number of solutions of f(x)g′′(x) + f′(x)g′(x) = 0 in (–2, 2) is equal to_____.

A deuteron and a proton moving with equal kinetic energy enter into a uniform magnetic field at right angle to the field. If \(r_d\) and \(r_p\) are the radii of their circular paths respectively, then the ratio \(\frac{r_d}{r_p}\) will be \(\sqrt x:1\) where \(x\) is__________

Which of the following statement is a tautology?

If the sum of the coefficients of all the positive powers of x, in the Binomial expansion of \(\left(x^n+\frac{2}{x^5}\right)^7\) is 939, then the sum of all the possible integral values of n is ____________.

If \(y(x) = (x^x)^x, \quad x > 0\), then \(\frac{d^2x}{dy^2} + 20\) at \(x = 1\) is equal to __________.

The spin-only magnetic moment value of an octahedral complex among \(CoCl_3⋅4NH_3\), \(NiCl_2⋅6H_2O\) and \(PtCl_4⋅2HCl\), which upon reaction with excess of \(AgNO_3\) gives 2 moles of \(AgCl\) is _____ B.M. (Nearest Integer)

A dilute solution of sulphuric acid is electrolysed using a current of 0.10 A for 2 hours to produce hydrogen and oxygen gas. The total volume of gases produced at STP is ____ cm³. (Nearest integer)

[Given : Faraday constant F = 96500 C mol^–1 at STP, molar volume of an ideal gas is 22.7 L mol^–1]

The major product (P) of the given reaction is (where, Me is –CH₃)

If vertex of a parabola is (2, –1) and the equation of its directrix is 4x – 3y = 21, then the length of its latus rectum is :

Let \(\vec{a}\) be a vector which is perpendicular to the vector
\(3\hat{i}+\frac{1}{2}\hat{j}+2\hat{k}. \)If \(\vec{a}×(2\hat{i}+\hat{k})=2\hat{i}−13\hat{j}−4\hat{k}\)
, then the projection of the vector on the vector
\( 2\hat{i}+2\hat{j}+\hat{k}\) is:

The oxidation state of manganese in the product obtained in a reaction of potassium permanganate and hydrogen peroxide in basic medium is____.

How many of the following drugs is/are examples(s) of broad-spectrum antibiotics?
Ofloxacin, Penicillin G, Terpineol, Salvarsan.

Negation of the Boolean expression p⇔ (q⇒p) is