#ssc 10th
- The sum of the roots of \(x^2 - 7x + 12 = 0\) is Answer: A. 7 Sum of roots = -b/a = --7/1 = 7 . 1
- If one root of \(x^2 + kx - 6 = 0\) is 2, then \(k = \) Answer: C. 1 Substitute x = 2 : 4 + 2k - 6 = 0 ⇒ 2k = 2 ⇒ k = 1 . 1
- The nature of the roots of \(x^2 + 4x + 4 = 0\) is Answer: A. real and equal Δ = 4² - 4(1)(4) = 0 , so the roots are real and equal (both are -2 ). 1
- The product of the roots of \(3x^2 + 7x - 6 = 0\) is Answer: B. -2 Product of roots = c/a = -6/3 = -2 . 1
- If the roots of \(x^2 - 6x + k = 0\) are real and equal, then \(k = \) Answer: C. 9 Real and equal roots need Δ = 0 : 36 - 4k = 0 ⇒ k = 9 . 1
- If \(\alpha\) and \(\beta\) are the roots of \(x^2 - 5x + 3 = 0\), then \(\alpha^2 + \beta^2 = \) Answer: D. 19 α + β = 5 , αβ = 3 . α² + β² = (α + β)² - 2αβ = 25 - 6 = 19 . 1
- The quadratic equation whose roots are 3 and \(-5\) is Answer: B. x² + 2x - 15 = 0 Sum = 3 + (-5) = -2 , product = -15 . The equation is x² - (sum)x + product = 0 , i.e. x² + 2x - 15 = 0 . 1
- The value of the determinant \(\begin{vmatrix} 5 & 3 \\ -7 & -4 \end{vmatrix}\) is Answer: D. 1 5×(-4) - 3×(-7) = -20 + 21 = 1 . 1
- For the equation \(4x + 5y = 19\), if \(x = 1\) then \(y = \) Answer: A. 3 4(1) + 5y = 19 ⇒ 5y = 15 ⇒ y = 3 . 1
- While solving simultaneous equations by Cramer's rule, if \(D_x = 24\) and \(x = -3\), then \(D = \) Answer: C. -8 x = D_x/D ⇒ D = D_x/x = 24/-3 = -8 . 1
- The solution of the simultaneous equations \(x + y = 5\) and \(x - y = 1\) is Answer: C. (3, 2) Adding, 2x = 6 , so x = 3 ; then y = 5 - 3 = 2 . 1
- Which of the following is a linear equation in two variables? Answer: D. 2x - 3y = 7 In 2x - 3y = 7 both variables have degree 1. The others contain x² , xy or y² , which have degree 2. 1
- For the equations \(3x + 2y = 8\) and \(2x - y = 3\), the value of \(D\) is Answer: C. -7 D = beginvmatrix 3 & 2 \\ 2 & -1 endvmatrix = 3(-1) - 2(2) = -3 - 4 = -7 . 1
- If \(2x + 3y = 12\) and \(3x + 2y = 13\), then \(x + y = \) Answer: B. 5 Adding the equations: 5x + 5y = 25 , so x + y = 5 . 1
- If \(\frac{1}{x} + \frac{1}{y} = 5\) and \(\frac{1}{x} - \frac{1}{y} = 1\), then \(x = \) Answer: A. 1/3 Adding: 2/x = 6 ⇒ 1/x = 3 ⇒ x = 1/3 . 1
- The probability of a certain (sure) event is Answer: C. 1 A certain event always happens, so its probability is 1. 1
- The simultaneous equations \(x + 2y = 4\) and \(2x + 4y = 12\) have Answer: C. no solution D = 1×4 - 2×2 = 0 . Dividing the second equation by 2 gives x + 2y = 6 , which contradicts x + 2y = 4 . The lines are... 1
- The sum of the present ages of a father and his son is 50 years. Five years ago, the father's age was 7 times the son's age. The father's present age is Answer: C. 40 years Let the ages be F and S . F + S = 50 and F - 5 = 7(S - 5) ⇒ F = 7S - 30 . So 8S - 30 = 50 ⇒ S = 10 , F = 40 . 1
- Two coins are tossed simultaneously. The number of sample points \(n(S)\) is Answer: A. 4 S = \HH, HT, TH, TT\ , so n(S) = 4 . 1
- A die is thrown. The probability of getting an even number is Answer: C. 1/2 Even numbers are 2, 4, 6: P = 3/6 = 1/2 . 1
- If \(x + y = 7\) and \(x - y = 3\), then \(x = \) Answer: B. 5 Adding: 2x = 10 ⇒ x = 5 . 1
- The value of \(\begin{vmatrix} 4 & 3 \\ 2 & 5 \end{vmatrix}\) is Answer: A. 14 4 × 5 - 3 × 2 = 20 - 6 = 14 . 1
- Which ordered pair satisfies the equation \(3x - 2y = 4\)? Answer: B. (2, 1) 3(2) - 2(1) = 6 - 2 = 4 . Check the others: (1,2) gives -1 , (0,2) gives -4 , (2,0) gives 6. 1
- \(ax^2 + bx + c = 0\) is a quadratic equation only when Answer: D. a ≠ 0 If a = 0 the x² term vanishes and the equation is no longer of degree 2. 1