List of practice Questions

Let M and N be the number of points on the curve y⁵ – 9xy + 2x = 0, where the tangents to the curve are parallel to x-axis and y-axis, respectively. Then the value of M + N equals ________.

If z ≠ 0 be a complex number such that$|z-\frac{1}{z}|=2$, then the maximum value of |z| is

Let A and B be two $3 × 3$ matrices such that AB = I and |A| =$\frac{1}{8}$. Then |adj (B adj(2A))| is equal to

A bullet of mass $200 g$ having initial kinetic energy $90 J$ is shot inside a long swimming pool as shown in the figure. If it’s kinetic energy reduces to $40 J$ within $1 s$, the minimum length of the pool, the bullet has to travel so that it completely comes to rest is

A copper wire having length of 243m and diameter 4 mm was melted to form a sphere. Find the diameter of the sphere:

Suppose that the full employment level of output of an economy is Rs. 2200 million, expenditure determined level of output is Rs. 2163 million, and the marginal propensity to consume is 0.75. The deflationary gap equals Rs. _____ million (round off to 2 decimal places).

Let the hyperbola $H: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ pass through ($2\sqrt2,-2\sqrt2$ ). A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?

An ellipse
$E:\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
passes through the vertices of the hyperbola
$H:\frac{x^2}{49} - \frac{y^2}{64} = -1$
Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H, respectively. Let the product of the eccentricities of E and H be 1/2. If the length of the latus rectum of the ellipse E, then the value of 113l is equal to _____.

Let P₁ be a parabola with vertex (3, 2) and focus (4, 4) and P₂ be its mirror image with respect to the line x + 2y = 6. Then the directrix of P₂ is x + 2y = _______.

Given below are the two statements
Statement I: Glucose‐6‐phosphate can be formed from glucose, but not from glycogen.
Statement II: Glucose‐1‐phosphate may be hydrolyzed to yield free glucose in liver.
Choose the most appropriate answer from the options given below:

Let 3, 6, 9, 12, … upto 78 terms and 5, 9, 13, 17, … upto 59 terms be two series. Then, the sum of terms common to both the series is equal to _____.

For n ∈ N let $S_n$={z∈C:|z−3+2i|=$\frac{n}{4}$} and $T_n$={z ∈ C:|z−2+3i|=$\frac{1}{n}$}.Then the number of elements in the set

Consider a conical region of height h and base radius R with its vertex at the origin. Let the outward normal to its base be along the positive z-axis, as shown in the figure. A uniform magnetic field, $\overrightarrow{B}=B_0\hat{z}$ exists everywhere. Then the magnetic flux through the base ($\Phi_b$) and that through the curved surface of the cone ($\Phi_c$) are

20 mL of 0.1 M NH₄OH is mixed with 40 mL of 0.05 M HCl.The pH of the mixture is nearest to:(Given: K_b(NH₄OH) = 1 × 10^–5,log 2 = 0.30,log 3 = 0.48, log 5 = 0.69, log 7 = 0.84, log 11 =1.04)

Which of the following pair of molecules contain odd electron molecule and an expanded octet molecule?

The d spacing for the first-order X-ray ($\lambda$= 1.54 Å) diffraction event of metallic iron (fcc) at $2\theta= 20.2°$ is _______.Å. (round off to three decimal places)

Let n ≥ 5 be an integer. If 9ⁿ – 8n – 1 = 64α and 6ⁿ – 5n – 1 = 25β, then α – β is equal to

$\lim_{x\rightarrow \frac{\pi}{4}}$ 8$\sqrt2$−(cosx+sinx)$^{\frac{7}{\sqrt2}}$−$\sqrt2$sin2x is equal to

A particle experiences a variable force F=(4x$\hat i$+3y²$\hat j$) in a horizontal x–y plane. Assume distance in meters and force is Newton.If the particle moves from point (1, 2) to point (2, 3) in the x–y plane; then Kinetic Energy changes by

$2NOCl(g) ⇌ 2NO(g) + Cl_2(g)$
In an experiment, $2.0$ moles of $NOCl$ was placed in a one-litre flask and the concentration of $NO$ after equilibrium established, was found to be $0.4$ mol/ L. The equilibrium constant at $30°C$ is ____ $×10^{–4}$.

For the given operational amplifier circuit R₁ = 120 Ω, R₂ = 1.5 k and V_s = 0.6 V, then the output current I₀ is ____mA.

the given operational amplifier circuit R1=120Ω, R2=1.5k

Let A(α, -2), B(α, 6) and C($\frac{\alpha}{4}$, -2) be vertices of a ΔABC. If (5, $\frac{\alpha}{4}$) is the circumcentre of ΔABC, then which of the following is NOT correct about ΔABC?

Let the solution curve y = y(x) of the differential equation (4 + x²)dy – 2x(x² + 3y + 4)dx = 0 pass through the origin. Then y(2) is equal to _______.

Let f ∶ $\R^2 → \R$ be the function defined by f(x, y) = x² - 12y. If M and m be the maximum value and the minimum value, respectively, of the function f on the circle x² + y² = 49, then |M| + |m| equals __________

If
$\frac{1}{2\times 3 \times 4} + \frac{1}{3\times 4 \times 5 } + \frac{1}{4 \times 5 \times 6 }+ \dots + \frac{1}{100 \times 101 \times 102} = \frac{k}{101}$
then 34 k is equal to ____________.