Mathematics and Pedagogy MCQ — 140 Practice Questions with Answers

Explanation

Let the cylinder radius be R; its height is also R, so its volume is pi R^3. If each ball has radius r, the total volume of 48 balls is 48 x (4/3)pi r^3 = 64pi r^3. Equating volumes gives R^3 = 64r^3, so r:R = 1:4; option D is correct.

Explanation

From the first two equations, y=5/(3−2k) and x=5(k−3)/(3−2k). Substituting these in kx−12y=14 gives 5k^2+13k−102=0, i.e. (k+6)(5k−17)=0. Thus k=−6 or 17/5, so C is correct.

Explanation

Using the double-angle identity, sin15° cos15° = 1/2 sin30° = 1/4, which is rational. The individual values sin15° and cos15° contain surds, and sin15° cos75° equals sin²15°, which is also irrational. Therefore option C is correct.

Explanation

For y = e^(tan^-1x), dy/dx = y/(1 + x^2). Differentiating again gives y'' = ((1 - 2x)y)/(1 + x^2)^2, so (1 + x^2)y'' = (1 - 2x)y/(1 + x^2) = (1 - 2x)dy/dx. Therefore option B is correct.

Explanation

The line through B(-2, 0) and C(6, 1) is x - 8y + 2 = 0. For A(0, 3), this expression is -22, so P(2x, x) must also give a negative value to lie on the same side. Substituting P gives 2x - 8x + 2 < 0, so x > 1/3; option B is correct.