HSC Science (General) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Show that the combined equation of the pair of lines passing through the origin and each making an angle α with the line x + y = 0 is x² + 2(sec 2α)xy + y² = 0

Find the condition that the equation ay² + bxy + ex + dy = 0 may represent a pair of lines. 

If the lines given by ax² + 2hxy + by² = 0 form an equilateral triangle with the line lx + my = 1, show that (3a + b)(a + 3b) = 4h².

If the line x + 2 = 0 coincides with one of the lines represented by the equation x² + 2xy + 4y + k = 0, then prove that k = - 4. 

Prove that the combined of the pair of lines passing through the origin and perpendicular to the lines ax² + 2hxy + by² = 0 is bx² - 2hxy + ay² = 0.

If equation ax² - y² + 2y + c = 1 represents a pair of perpendicular lines, then find a and c.

Find k if the slope of one of the lines given by 3x² + 4xy + ky² = 0 is three times the other.

Find the joint equation of the line passing through the origin and having slopes 1 + `sqrt3` and 1 - `sqrt3`

Which of the following sentence is the statement in logic? Justify. Write down the truth value of the statement:

Please get me a glass of water.

Which of the following sentence is the statement in logic? Justify. Write down the truth value of the statement:

cos²θ − sin²θ = cos2θ for all θ∈R.

Which of the following sentence is the statement in logic? Justify. Write down the truth value of the statement:

If x is a whole number then x + 6 = 0.

`sqrt5` is an irrational but `3sqrt5` is a complex number.

Write the truth value of the following statement:

∀ n ∈ N, n² + n is even number while n² – n is an odd number.

Write the truth value of the following statement:

∃ n ∈ N such that n + 5 > 10.

Write the truth value of the following statement:

The square of any even number is odd or the cube of any odd number is odd.

Write the truth value of the following statement:

In ΔABC if all sides are equal then its all angles are equal.

Write the truth value of the following statement:

∀ n ∈ N, n + 6 > 8.

Find the vector equation of the line passing through the point having position vector `-2hat"i" + hat"j" + hat"k"  "and parallel to vector"  4hat"i" - hat"j" + 2hat"k"`.

Find the vector equation of the line passing through points having position vector `3hati + 4hatj - 7hatk and 6hati - hatj + hatk`.

Find the vector equation of line passing through the point having position vector `5hat"i" + 4hat"j" + 3hat"k"` and having direction ratios  –3, 4, 2.