Mathematics - Class 12 Practice Test 5

1. Find the integral of (ax²+b)².

-\(\frac{a^2 \,x^5}{5}-b^2 \,x+\frac{2abx^3}{3}+C\)

\(\frac{b^2 \,x^5}{5}+b^2 x+\frac{27x^3}{3}+C\)

\(\frac{a^2 \,x^5}{5}+b^2 \,x+\frac{2abx^3}{3}+C\)

\(\frac{a^2 \,x^5}{5}+x+\frac{2abx^3}{5}+C\)

2. Find the equation of the tangent of the tangent to the curve 2x²+3y²=3 at the point(3,4).

x-2y=11

-x+2y=11

x-2y=-11

x+2y=11

3. Integrate 3 sec²⁡x log⁡(tan⁡x) dx.

tan⁡x (log⁡(tan⁡x)-1)+C

-log⁡(tan⁡x) (tan⁡x-1)+C

log⁡(tan⁡x) (sec⁡x+1)+C

tan⁡x (log⁡sec⁡x +1)+C

4. What will be the approximate change in the surface area of a cube of side xm caused by increasing the side by 2%.

0.24x²

2.4x²

0.24x

0.4x²

5. What is the condition for a function f to be strictly decreasing if f be continuous and differentiable on (a,b)?

f’(x) ≤ 0 ∀ x₁, x₂ ∈ (a,b)

f’(x) < 0 ∀ x₁, x₂ ∈ (a,b)

f’(x) > 0 ∀ x₁, x₂ ∈ (a,b)

f’(x) = 0 ∀ x₁, x₂ ∈ (a,b)

6. Integrate \(3x^2 (cos⁡x^3+8)\).

-\(sin⁡x^3+8x^3+C\)

\(sin⁡x^3+8x^3+C\)

\(sin⁡x^3-8x^3+C\)

\(sin⁡x^3-x^3+C\)

7. Find the integral of \(\frac{e^{-x} (1-x)}{sin^2⁡(xe^{-x})}\).

-cot⁡xe^-x+C

cot⁡xe^-x+C

-cos^2⁡xe^-x+C

-cot⁡xe^x+C

8. Find \(\int \,3 \,cos⁡x+\frac{1}{x} dx\).

\(2 \,sin⁡x+\frac{1}{x^3}+C\)

\(3 \,sin⁡3x+\frac{1}{x}+C\)

\(3 \,sin⁡x-\frac{1}{x}+C\)

\(sin⁡x-\frac{1}{x^2}+C\)

9. Find the point at which the tangent to the curve y=\(\sqrt{4x^2+1}\)-2 has its slope2.

(\(\frac{1}{\sqrt{12}}\),3) and (\(\frac{1}{\sqrt{12}}\),-2)

(\(\frac{1}{\sqrt{12}}\),-1) and (\(\frac{1}{\sqrt{12}}\),-1)

(-\(\frac{1}{\sqrt{12}}\),3) and (-\(\frac{1}{\sqrt{12}}\),-1)

(\(\frac{1}{\sqrt{12}}\),2) and (-\(\frac{1}{\sqrt{12}}\),-2)

10. Integrate 2x sin⁡2x.

\(\frac{sin⁡2x}{2}\)+x cos⁡2x+C

\(\frac{sin⁡2x}{2}\)-cos⁡2x+C

\(\frac{cos⁡2x}{2}\)-x cos⁡2x+C

\(\frac{sin⁡2x}{2}\)-x cos⁡2x+C