Category Archives: School Learning

Evaluate the following integrals: Question 21. ∫(log⁡x)2 x dx Solution: Given that, I = ∫(log⁡x)2 x dx Using integration by parts,        ∫u… Read More
Evaluate the following integrals: Question 1. ∫x cos⁡xdx Solution: Given that, I = ∫x cos⁡xdx Using integration by parts,        ∫u v dx… Read More
Question 1. Rationalise the denominator of each of the following (i-vii): (i) (ii) (iii) (iv) (v) (vi) (vii) Solution: (i) We know that rationalisation factor… Read More
Question 23. Prove that the points (a, 0), (0, b) and (1, 1) are collinear if, 1/a + 1/b = 1. Solution: Let us assume… Read More
Question 1. If a line makes angles of 90°, 60°, and 30° with the positive direction of x, y, and z-axis respectively. Find its direction… Read More
Question 1. Find the vector equation of a plane which is at a distance of 3 units from the origin and hasas the unit vector… Read More
Question 12. Prove that the points (a, b), (a1, b1) and (a – a1, b – b1) are collinear if ab1 = a1b. Solution: As… Read More
Question 6. In each of the following determine rational numbers a and b: (i) (ii) (iii) (iv) (v) (vi) Solution: (i) We know that rationalisation… Read More
Question 1. Find the area of a triangle whose vertices are: (i) (6, 3), (-3, 5) and (4, -2) Solution: Let us assume ABC is… Read More
Question 1. The vertices of the triangle are A(5, 4, 6), B(1, -1, 3), and C(4, 3, 2). The internal bisector of angle A meets… Read More
Question 1. If f is defined by f(x) = x2, find f'(2). Solution: Hence, f'(2) = 4. Question 2.  If f is defined by f(x)… Read More
Question 1. Using binomial theorem, write down the expressions of the following: (i) (2x + 3y)5 Solution: Using binomial theorem, we have, (2x + 3y)5… Read More
Question 21. Differentiate (2x2 – 3) sin x with respect to x. Solution: We have, => y = (2x2 – 3) sin x On differentiating… Read More
Question 11. Differentiate (x sin x + cos x) (x cos x − sin x) with respect to x. Solution: We have, => y =… Read More
Question 1. Differentiate x3 sin x with respect to x. Solution: We have,  => y = x3 sin x On differentiating both sides with respect… Read More